Global existence and blow-up of solutions for mixed local and nonlocal hyperbolic equations

Abstract

In this paper, we consider the following mixed local and nonlocal hyperbolic equation: $\begin{aligned} {\begin{cases} u_{t t} - Δ u + μ {(- Δ)}^{s} u = | u |^{p - 2} u, & in Ω \times R^{+}, \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & in Ω, \\ u (x, t) = 0, & in (R^{N} ∖ Ω) \times R_{0}^{+}, \end{cases} \end{aligned}$ where $s \in (0, 1)$ , $N > 2$ , $p \in (2, 2_{s}^{*}]$ , μ is a nonnegative real parameter, $Ω \subset R^{N}$ is a bounded domain with Lipschitz boundary $\partial Ω$ , Δ is the Laplace operator, $(- Δ)^{s}$ is the fractional Laplace operator. By combining the Galerkin approach with the modified potential well method, we obtain the global existence, vacuum isolating, and blow-up of solutions for the aforementioned problem, provided certain assumptions are fulfilled. Specifically, we study the existence of global solutions for the above problem in the cases of subcritical and critical initial energy levels, as well as the finite time blow-up of solutions. Then, we investigate the blow-up of solutions for the above problem in the case of supercritical initial energy level, as well as upper and lower bounds of blow-up time of solutions.

Keywords

Mixed local and nonlocal problem hyperbolic equation global existence blow-up modified potential well

1. .Introduction

This paper investigates some qualitative results related to properties such as global existence and blow-up in finite time for the following mixed local and nonlocal hyperbolic problem:

\begin{aligned} {\begin{cases} u_{t t} - Δ u + μ {(- Δ)}^{s} u = f (u), & in Ω \times R^{+}, \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & in Ω, \\ u (x, t) = 0, & in (R^{N} ∖ Ω) \times R_{0}^{+}, \end{cases} \end{aligned}

(1.1)

where

u_{t t} = \partial^{2} u / \partial t^{2}

s \in (0, 1)

N > 2

, μ is a nonnegative real parameter,

Ω \subset R^{N}

is a bounded domain with Lipschitz boundary

\partial Ω

\begin{aligned} (- Δ)^{s} u = 2 P . V . \int_{R^{N}} \frac{u (x) - u (y)}{| x - y |^{N + 2 s}} d y, \end{aligned}

where

P . V

. denotes the Cauchy principal value. Namely,

\begin{aligned} P . V . \int_{R^{N}} \frac{u (x) - u (y)}{| x - y |^{N + 2 s}} d y = lim_{ε \to 0^{+}} \int_{y \in R^{N} : | y - x | ⩾ ε} \frac{u (x) - u (y)}{| x - y |^{N + 2 s}} d y . \end{aligned}

Here we call the $- Δ + (- Δ)^{s}$ is a mixed operator. Indeed mixed operators naturally arise in the applied sciences, to investigate the effects of local and nonlocal variations on many physical phenomena. They describe the stochastic processes that involve Lévy flight and the classical random walk at the same time. The key to the situation is the coexistence of a subordinate fractional operator and a dominating local one. In a recent spate of fascinating papers, Biagi, Dipierro, Valdinoci, and Vecchi [2,3,8] started a methodical investigation into mixed operator problems, proving various results about the regularity and qualitative behavior for solutions, maximum principles, and related variational principles.

Next, we present some known results on mixed operators. The simplest model case being

\begin{aligned} - Δ u + (- Δ)^{s} u = 0, for s \in (0, 1) . \end{aligned}

(1.2)

In [4], authors established a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for non-negative functions which are harmonic with respect to

Δ + b Δ^{α / 2}

C^{1, 1}

open sets. For nonnegative functions in the Lipschitz open set that are harmonic with respect to

Δ + b Δ^{α / 2}

, the authors employed a combination of probabilistic and analytic techniques to establish a consistent Carleson type estimate. For quantitative and qualitative properties of solutions to (1.2) such as the maximum principle, interior Sobolev regularity, and symmetry results, see references [2,3,8]. Dipierro et al. proposed the following mathematical model to consider scenarios that are more favorable for species survival (see [8]).

\begin{aligned} {\begin{cases} - α Δ u + β {(- Δ)}^{s} u = (m - μ u) u + τ J ⋆ u, & in Ω \\ with (α, β) - Neumann condition . \end{cases} \end{aligned}

The authors proved the existence of a minimal solution for this problem and conducted a comprehensive discussion on the possibility of obtaining nontrivial solutions, which correspond to the survival of the population. Maione et al. in [19] demonstrated the existence of weak solutions for semilinear elliptic marginals problems driven by mixed local and nonlocal operators. They also concluded that such operators are allowed to be nonpositive definite. Additionally, some authors has discovered the existence and the multiplicity of weak solutions to the following issue:

\begin{aligned} {\begin{cases} - Δ u + {(- Δ)}^{s} u = u^{- γ} + λ (\int_{Ω} \frac{| u |^{2_{μ}^{*}} (y)}{| x - y |^{μ}}) {| u |}^{2_{μ}^{*} - 2} u, & u > 0 in Ω, \\ u = 0 & in R^{N} ∖ Ω, \end{cases} \end{aligned}

where

γ > 0

0 < μ < N

and

| u |^{2_{μ}^{*}} = (2 N - μ) / (N - 2)

(see [1]).

For the nonlinear scenario of (1.2)

\begin{aligned} - Δ_{p} u + {(- Δ)}_{p}^{s} u = 0, for s \in (0, 1) . \end{aligned}

(1.3)

In [10], Garain and Kinnunen discussed the regularity theory, covering aspects such as semicontinuity, Harnack inequality, local boundedness, and Hölder continuity of weak solutions. The study in [11] established the existence, uniqueness, Hölder regularity, and local bounds of weak solutions for problem (1.3) with the nonhomogeneous terms. In particular, De Filippis and Mingione [6] first proposed the functionals with nonstandard growth

\begin{aligned} u \mapsto \int_{Ω} ({| D u |}^{q} - f u) d x + \iint_{R^{2 N}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + s p}} d x d y, \end{aligned}

(1.4)

where

q > s p

. The local Hölder continuity of the gradient of minimizers and maximal regularity of solutions to mixed problems in nonlinear, potentially degenerate circumstances, as in (1.4), were demonstrated by the authors.

For a parabolic equation of type (1.3), Shang and Zhang in [25,26] showed the local boundedness and Hölder continuity of weak solutions for such equations by combining a De Giorgi–Nash–Moser iteration with an appropriate Caccioppoli-type inequality and Logarithmic lemma. Additionally, with the aid of the energy estimates, De Giorgi lemma, and measure theoretical argument, an inherent Harnack inequality of non-negative weak solutions to such a problem was established under the condition that $2 < p < \infty$ . For the following case

\begin{aligned} \partial_{t} u (x, t) - Δ_{q} u (x, t) + (- Δ)_{p}^{s} u = 0, for s \in (0, 1) . \end{aligned}

Shang and Zhang [16] proved local boundedness, lower semicontinuity of weak supersolutions, and pointwise behavior of mixed local and nonlocal double phase parabolic equation at

q \neq p

by energy estimates and De Giorgi–Nash–Moser iteration. In [9], Dipierno proposed a new model to describe the dispersal of populations of organisms living in ecological niches and subject to both local and nonlocal dispersal. In some sense, the model refines the development of a mixed system of operators for parabolic equations. After that, Zhao and Zhang [33] studied equations like this:

\begin{aligned} {\begin{cases} u_{t} - Δ u + {(- Δ)}^{s} u = | u |^{p - 2} u & in Ω \times R^{+}, \\ u (x, 0) = u_{0} (x) & in Ω, \\ u (x, t) = 0 & in R^{N} ∖ Ω \times R_{0}^{+} . \end{cases} \end{aligned}

They obtained the global existence and blow-up of weak solutions at three initial energy levels

G (u) < d

G (u) = d

and

G (u) > d

by using modified potential well method and the Galerkin method.

In recent years, there have been many studies on mixed local and nonlocal problems. But there are few results concerning the global existence and blow-up in finite time of solutions to problem like (1.1) involving mixed operators. Of course, either local or non-local problems has been explored extensively, in particular, some properties such as local boundedness of solutions, Hölder continuity and regularity, have been investigated. While few authors studied mixed operator problems involving the dynamic behaviors. As far as we know, some important properties about the global existence and blow-up of solutions to local problems have been richly studied by many authors, see for instance [12–14,22,28]. For the study of nonlocal equations, we refer the readers to [8,20,21,23,24,29]. There are also some interesting models studying the global existence and long time blow-up behaviour of solutions, such as an anomalous pseudo-parabolic Kirchhoff-type dynamical model (see [5]), a class of fourth-order strongly damped nonlinear wave equations (see [30]), and so on. In recent years, the study of the existence and nonexistence of solutions for wave equations with nonlocal dissipative effects has become more extensive, and a wealth of results have been given by many authors. For example, the initial boundary value issue wave equation with nonlocal damping and anti-damping was studied by Zhao et al. [31,32] under suitable conditions. To our best knowledge, there is no result concerning the existence of global weak solutions to problem (1.1) and blow-up in finite time.

In this paper, we put $F (u) = \int_{0}^{u} f (τ) d τ$ and suppose that $f (u)$ satisfies the assumption $(Z)$ : (Z)

$(Z_{1})$ $f (u) \in C^{1} (R), f (0) = f^{'} (0) = 0, f ⧸ \equiv 0$ in a small neighbourhood of the origin;

$(Z_{2})$ $f (u)$ is monotone in $R$ , convex when $u > 0$ and concave when $u < 0$ ;

$(Z_{3})$ $f (u) u - u^{2} f^{'} (u) ⩽ 0$ , where $2 < p ⩽ 2_{s}^{*} = \frac{2 N}{N - 2 s}$ ;

$(Z_{4})$ $p F (u) ⩽ u f (u)$ and $| u f (u) | ⩽ γ | F (u) |$ , where $p ⩽ γ ⩽ 2_{s}^{*}$ .

A basic example is provided as

f (u) = | u |^{p - 2} u

Inspired by the above work, together with the modified potential well method developed by Liu and Zhao in [18], we attempt to study the global existence, nonexistence and the vacuum isolating of solutions for problem (1.1) by combining the Galerkin method with potential well theory. The potential well method was first introduced by Sattinger [27] in 1968 and has since been widely used to prove the global existence of solutions to evolution equations. Undoubtedly, due to the local and nonlocal properties of the mixed operators and the existence of the parameter μ, we have encountered more difficulties in the exploration of problem (1.1). Therefore, in order to overcome the difficulties inherent in these new features, we have to perform more precise and skillful calculations and derivations.

The structure of this article is as follows. In Section 2, we begin with spaces knowledge related to mixed operators and foundational lemmas that will be utilized throughout the work. In Section 3, we show the existence of global solutions in the subcritical initial energy level, and the vacuum isolation and finite time blow-up properties of solutions. In Section 4, we study the global existence of solutions and blow-up in the critical initial energy level. Section 5 investigates the blow-up property of solutions in the supercritical initial energy level. In Section 6, we specialize in the upper and lower bounds of blow-up time of solutions for the special nonlinear source term $f (u) = | u |^{p - 2} u$ in the supercritical initial energy level.

2. .Preliminary lemmas

First, we go over some essential definitions and characteristics of the mixed operator spaces in this section, for more information, see also [8,33]. Moreover, we will give some preliminary lemmas which will be used in the rest of the paper.

Next, for convenience sake, we denote $Q = R^{2 N} ∖ O$ , where $O = C (Ω) \times C (Ω) \subset R^{2 N}$ , and $C (Ω) = R^{N} ∖ Ω$ . We note that a common Lebesgue space is $L^{q} (Ω)$ , with ${‖ u ‖}_{q} = {(\int_{Ω} {| u |}^{q} d x)}^{\frac{1}{q}}$ for $q ⩾ 2$ , and $(u, v) = \int_{Ω} u v d x$ when $q = 2$ .

Let $s \in (0, 1)$ , if $u : R^{N} \to R$ is a measurable function, we recall the space U introduced in [20] and defined as

\begin{aligned} U := {u : R^{N} \to R such that u \in L^{2} (Ω) and \iint_{Q} \frac{| u (x) - u (y) |^{2}}{| x - y |^{N + 2 s}} d x d y < + \infty} . \end{aligned}

We set

\begin{aligned} [u]_{s} = {(\iint_{R^{2 N}} \frac{| u (x) - u (y) |^{2}}{| x - y |^{N + 2 s}} d x d y)}^{\frac{1}{2}}, \end{aligned}

and we refer to

[u]_{s}

as the Gagliardo seminorm of u (of order s).

Next, for the convenience of discussion, we will define the closed linear subspace $U_{0}$ of U

\begin{aligned} U_{0} = {u \in U : u (x) = 0 a . e . in R^{N} ∖ Ω} . \end{aligned}

The space

U_{0}

is equipped with the norm

\begin{aligned} ‖ u ‖_{U_{0}} = [u]_{s, Q} = {(\iint_{Q} \frac{| u (x) - u (y) |^{2}}{| x - y |^{N + 2 s}} d x d y)}^{\frac{1}{2}} . \end{aligned}

According to [23], it is easy to see that

‖ u ‖_{U_{0}}

is an equivalent norm of

U_{0}

. Furthermore, we define the mixed operator space

\begin{aligned} X_{μ}^{s} (Ω) = {\begin{cases} H^{1} (Ω), & μ = 0, \\ X_{0}^{s} (Ω), & μ = 1, \\ H^{1} (Ω) \cap U_{0}, & 0 < μ < 1 and μ > 1, \end{cases} \end{aligned}

where

\begin{aligned} X_{0}^{s} (Ω) & := {\bar{C_{0}^{\infty} (Ω)}}^{‖ u ‖_{H^{1} (R^{N})}} = {u \in H^{1} (R^{N}) : u |_{Ω} \in H_{0}^{1} (Ω), [u]_{s, Q} < \infty and u \equiv 0 a . e . in R^{N} ∖ Ω}, \end{aligned}

with the inner product

\begin{aligned} ⟨ u, v ⟩_{X_{0}^{s}} := (- Δ u, v) + ⟨ u, v ⟩_{U_{0}} = \int_{Ω} \nabla u \cdot \nabla v d x + \iint_{Q} \frac{[u (x) - u (y)] [v (x) - v (y)]}{| x - y |^{N + 2 s}} d x d y, \end{aligned}

and the corresponding norm

\begin{aligned} ‖ u ‖_{X_{0}^{s} (Ω)} = {(‖ \nabla u ‖_{2}^{2} + ‖ u ‖_{U_{0}}^{2})}^{\frac{1}{2}} . \end{aligned}

Thus, we can give a definition of norm of

X_{μ}^{s} (Ω)

\begin{aligned} ‖ u ‖_{X_{μ}^{s} (Ω)} = {\begin{cases} ‖ \nabla u ‖_{2}, & μ = 0, \\ {(‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2})}^{\frac{1}{2}}, & μ > 0. \end{cases} \end{aligned}

Next, we shall now introduce the primary symbols used throughout this paper. If not stated otherwise, we consistently assume that $f = f (u)$ satisfies assumption $(Z)$ . For $0 < σ < \bar{σ}$ , where the definition of $\bar{σ}$ refers to Lemma 2.9, we give the definition of energy functional $G (u)$ and Nehari functional $H (u)$ :

\begin{aligned} G (u) = \frac{1}{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} ‖ u ‖_{U_{0}}^{2} - \int_{Ω} F (u) d x, \\ H (u) = ‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2} - \int_{Ω} u f (u) d x, \\ H_{σ} (u) = σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2} - \int_{Ω} u f (u) d x, \\ M = {u \in X_{μ}^{s} (Ω) ∖ {0} : H (u) = 0}, \\ M_{σ} = {u \in X_{μ}^{s} (Ω) ∖ {0} : H_{σ} (u) = 0}, \end{aligned}

(2.1)

the potential well and the family of potential wells in the stationary setting as follows:

\begin{aligned} W = {u \in X_{μ}^{s} (Ω) | H (u) > 0, G (u) < d} \cup {0}, \\ W_{σ} = {u \in X_{μ}^{s} (Ω) : H_{σ} (u) > 0, G (u) < d (σ)} \cup {0}, \\ {\bar{W}}_{σ} = W_{σ} \cup \partial W_{σ} = {u \in X_{μ}^{s} (Ω) : H_{σ} (u) ⩾ 0, G (u) ⩽ d (σ)}, \\ V = {u \in X_{μ}^{s} (Ω) : H (u) < 0, G (u) < d}, \\ V_{σ} = {u \in X_{μ}^{s} (Ω) : H_{σ} (u) < 0, G (u) < d (σ)}, \\ {\bar{V}}_{σ} = V_{σ} \cup \partial V_{σ} = {u \in X_{μ}^{s} (Ω) : H_{σ} (u) ⩽ 0, G (u) ⩽ d (σ)}, \\ G_{σ} = {u \in X_{μ}^{s} (Ω) : ‖ \nabla u ‖_{2} < r_{1} (σ), ‖ u ‖_{U_{0}} < r_{2} (σ)}, \\ {\bar{G}}_{σ} = G_{σ} \cup \partial G_{σ} = {u \in X_{μ}^{s} (Ω) : ‖ \nabla u ‖_{2} ⩽ r_{1} (σ), ‖ u ‖_{U_{0}} ⩽ r_{2} (σ)} \cup {0}, \\ G_{σ}^{c} = {u \in X_{μ}^{s} (Ω) : ‖ \nabla u ‖_{2} ⩾ r_{1} (σ), ‖ u ‖_{U_{0}} ⩾ r_{2} (σ)}, \end{aligned}

(2.2)

where

\begin{aligned} d = inf_{u \in M} G (u), \\ d (σ) = inf_{u \in M_{σ}} G (u) . \end{aligned}

(2.3)

Next we will give the definition of global weak solutions (1.1) and the energy identity equation.

Definition 2.1 Maximal existence time T

Assume that $u (x, t)$ is a weak solution of problem (1.1).

(1)
If $u (t)$ exists at all time, then maximal existence time $T = + \infty$ .
(2)
If $u (t)$ exists for $0 ⩽ t < t_{0}$ and $0 < t_{0} < \infty$ , and does not exist at $t = t_{0}$ , then maximal existence time $T = t_{0}$ .

Definition 2.2.
A function $u = u (x, t)$ as a weak solution of problem (1.1), if
$\begin{aligned} \begin{array}{ll} u \in L^{\infty} (0, T; X_{μ}^{s} (Ω)), & u_{t} \in L^{\infty} (0, T; L^{2} (Ω)), \\ u (x, 0) = u_{0} in X_{μ}^{s} (Ω), & u_{t} (x, 0) = u_{1} in L^{2} (Ω), \end{array} \\ (u_{t}, ϕ) + \int_{0}^{t} (\nabla u, \nabla ϕ) d τ + μ \int_{0}^{t} ⟨ u, ϕ ⟩_{U_{0}} d τ = \int_{0}^{t} (f (u)), ϕ) d τ + (u_{1}, ϕ), \end{aligned}$
for any $ϕ \in L^{1} (0, T; X_{μ}^{s} (Ω))$ and $t \in R_{0}^{+}$ .

For every $t \in R$ , the energy functional along a solution u of (1.1) is defined by
$\begin{aligned} E (t) = \frac{1}{2} ‖ u_{t} ‖_{2}^{2} + \frac{1}{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} ‖ u ‖_{U_{0}}^{2} - \int_{Ω} F (u) d x = \frac{1}{2} ‖ u_{t} ‖_{2}^{2} + G (u), \end{aligned}$
by virtue of (2.1).
Lemma 2.1.
Assume that $u : Ω \times [0, T) \to R^{N}$ is a weak solution of problem (1.1), with $(u_{0}, u_{1}) \in X_{μ}^{s} (Ω) \times L^{2} (Ω)$ . Then the total energy function $E (t)$ of problem (1.1) is invariant with respect to t, i.e. $E (t) = E (0)$ .
Proof.
Multiplying $u_{t}$ identically on both sides of problem (1.1) and integrating over Ω yield
$\begin{aligned} \int_{Ω} u_{t t} \cdot u_{t} d x + \int_{Ω} \nabla u \cdot \nabla u_{t} d x + μ \iint_{Q} \frac{[u (x) - u (y)] [u_{t} (x) - u_{t} (y)]}{| x - y |^{N + 2 s}} d x d y = \int_{Ω} f (u) \cdot u_{t} d x . \end{aligned}$
Organizing the above equation, we get
$\begin{aligned} \frac{d}{d t} (\frac{1}{2} ‖ u_{t} ‖_{2}^{2} + \frac{1}{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} ‖ u ‖_{U_{0}}^{2} - \int_{Ω} F (u) d x) = 0, \end{aligned}$
integrating this equation from 0 to t, it can be written as
$\begin{aligned} \frac{1}{2} ‖ u_{t} ‖_{2}^{2} + \frac{1}{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} ‖ u ‖_{U_{0}}^{2} - \int_{Ω} F (u) d x = \frac{1}{2} {‖ u_{t} (0) ‖}_{2}^{2} + \frac{1}{2} ‖ \nabla u_{0} ‖_{2}^{2} + \frac{μ}{2} ‖ u_{0} ‖_{U_{0}}^{2} - \int_{Ω} F (u_{0}) d x, \end{aligned}$
i.e. $E (t) = E (0)$ .

From now on, instead of going into details, we will state the main citations used in this paper.
Lemma 2.2.
Let $2 < p ⩽ γ ⩽ 2_{s}^{}$ , then we can get the following statements:* (1)
$‖ u ‖_{γ} ⩽ C_{} ‖ u ‖_{U_{0}}$ , where* $C_{} > 0$ is the best embedding constant of the embedding* $U_{0} ↪ L^{γ} (Ω)$ . Particularly, $C_{1} ‖ u ‖_{2}^{2} ⩽ ‖ u ‖_{U_{0}}^{2}$ , where $(C_{1})^{- \frac{1}{2}} > 0$ is the best embedding constant of the embedding $U_{0} ↪ L^{2} (Ω)$ .
(2)
$‖ u ‖_{γ} ⩽ C^{} ‖ \nabla u ‖_{2}$ , where* $C^{} > 0$ is the best embedding constant of the embedding* $H^{1} (Ω) ↪ L^{γ} (Ω)$ .

Proof.
(1) We can obtain this directly from Theorem 6.5 in [7] and Theorem 2.3 in [20].

(2) According to the definition of $2^{}$ and $2_{s}^{}$ , we can write the embedding relation of $L^{2} (Ω)$ , $L^{2^{}} (Ω)$ and $L^{2_{s}^{}} (Ω)$ as $L^{2^{}} (Ω) ↪ L^{2_{s}^{}} (Ω) ↪ L^{2} (Ω)$ . Thus, $H_{1} (Ω) ↪ L^{2^{}} (Ω)$ implies $H^{1} (Ω) ↪ L^{2_{s}^{}} (Ω) ↪ L^{γ} (Ω)$ . The conclusion is hence proven.
Lemma 2.3.
The following accounts hold: (1)
For all $u \in R$ and some $A > 0$ , there holds $| F (u) | ⩽ A | u |^{γ}$ .
(2)
For all $u \in R$ and some $B > 0$ , there holds $F (u) ⩾ B | u |^{p} - K$ , $where K > 0 i s a constant$ .
(3)
$u (u f^{'} (u) - f (u)) ⩾ 0$ , and the equality sign is valid only when $u = 0$ .

Lemma 2.4.
The following accounts hold: (1)
For all $u \in R$ and for some $A > 0$ , there will be $| f (u) | ⩽ γ A | u |^{γ - 1}$ and $| u f (u) | ⩽ γ A | u |^{γ}$ .
(2)
If $u ⩾ 0$ and in the case of hypothesis $(Z_{2})$ , for some $B > 0$ , there will be
$\begin{aligned} u f (u) ⩾ p B | u |^{p} - p K, where K > 0 i s a constant . \end{aligned}$

The aforementioned results were demonstrated in Lemma 2.3 and Corollary 2.2 in [18].
Lemma 2.5.
For $u \in X_{μ}^{s} (Ω)$ , with $u \neq 0$ , let $φ (θ) = \frac{1}{θ} \int_{Ω} u f (θ u) d x$ for every $θ \in (0, + \infty)$ . (1)
φ is strictly increasing;
(2)
$lim_{θ \to 0^{+}} φ (θ) = 0$ , $lim_{θ \to \infty} φ (θ) = + \infty$ .

Proof.
(1)
$\begin{aligned} φ^{'} (θ) = \frac{1}{θ^{3}} [\int_{Ω} f^{'} (θ u) (θ u)^{2} d x - \int_{Ω} f (θ u) (θ u) d x] > 0. \end{aligned}$
Then, we can demonstrate (1) by $(Z_{3})$ .

(2) Lemma 2.4 gives
$\begin{aligned} | φ (θ) | = | \frac{1}{θ^{2}} \int_{Ω} θ u f (θ u) d x | ⩽ \frac{1}{θ^{2}} \int_{Ω} | θ u f (θ u) | d x ⩽ \frac{1}{θ^{2}} \int_{Ω} | θ u γ A | θ u |^{γ - 1} | d x = γ A θ^{γ - 2} ‖ u ‖_{γ}^{γ} . \end{aligned}$
Hence $lim_{θ \to 0^{+}} φ (θ) = 0$ .

On the other hand, in the case of hypothesis $(Z_{2})$ , we get $u f (u) > 0$ . Lemma 2.4 yields
$\begin{aligned} φ (θ) = \frac{1}{θ^{2}} \int_{Ω} θ u f (θ u) d x ⩾ \frac{1}{θ^{2}} \int_{Ω} p B {| θ u |}^{p} d x = p B θ^{p - 2} ‖ u ‖_{p}^{p}, \end{aligned}$
it immediately follows that $lim_{θ \to + \infty} φ (θ) = + \infty$ . From this we can draw the desired conclusions.
Lemma 2.6.
Let $u \in X_{μ}^{s} (Ω)$ , $u \neq 0$ , and let $θ ⩾ 0$ be a real parameter. (1)
$lim_{θ \to 0} G (θ u) = 0$ , $lim_{θ \to + \infty} G (θ u) = - \infty$ .
(2)
There exists a unique threshold $θ^{} = θ^{} (u) > 0$ , such that
$\begin{aligned} {\frac{\partial G (θ u)}{\partial θ} |}_{θ = θ^{}} = 0 a n d {\frac{\partial G^{2} (θ u)}{\partial θ^{2}} |}_{θ = θ^{}} < 0. \end{aligned}$

(3)
$G (θ u)$ is nondecreasing in $[0, θ^{})$ , nonincreasing in* $(θ^{}, \infty)$ , and takes the maximum value at* $θ = θ^{}$ .
(4)
$H (θ u) > 0$ in $(0, θ^{})$ , $H (θ u) < 0$ in $(θ^{}, \infty)$ and* $H (θ^{} u) = 0$ .

Proof.
(1) Lemma 2.3 and (2.1) yield that
$\begin{aligned} G (θ u) = \frac{1}{2} θ^{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} θ^{2} ‖ u ‖_{U_{0}}^{2} - \int_{Ω} F (θ u) d x ⩾ \frac{1}{2} θ^{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} θ^{2} ‖ u ‖_{U_{0}}^{2} - A θ^{γ} ‖ u ‖_{γ}^{γ}, \end{aligned}$
and
$\begin{aligned} G (θ u) = \frac{1}{2} θ^{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} θ^{2} ‖ u ‖_{U_{0}}^{2} - \int_{Ω} F (θ u) d x ⩽ \frac{1}{2} θ^{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} θ^{2} ‖ u ‖_{U_{0}}^{2} . \end{aligned}$
It is clear to see that
$\begin{aligned} lim_{θ \to 0} (\frac{1}{2} θ^{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} θ^{2} ‖ u ‖_{U_{0}}^{2} - A θ^{γ} ‖ u ‖_{γ}^{γ}) = 0, \\ lim_{θ \to 0} (\frac{1}{2} θ^{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} θ^{2} ‖ u ‖_{U_{0}}^{2}) = 0, \end{aligned}$
and since $2 < p ⩽ γ$ we can get
$\begin{aligned} lim_{θ \to + \infty} (\frac{1}{2} θ^{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} θ^{2} ‖ u ‖_{U_{0}}^{2} - A θ^{γ} ‖ u ‖_{γ}^{γ}) = - \infty, \\ lim_{θ \to + \infty} (\frac{1}{2} θ^{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} θ^{2} ‖ u ‖_{U_{0}}^{2}) = - \infty . \end{aligned}$
Therefore, $lim_{θ \to 0} G (θ u) = 0$ , $lim_{θ \to + \infty} G (θ u) = - \infty$ .

(2) By the definition of $φ (θ)$ in Lemma 2.5, we can know
$\begin{aligned} \frac{\partial G (θ u)}{\partial θ} & = θ ‖ \nabla u ‖_{2}^{2} + θ μ ‖ u ‖_{U_{0}}^{2} - \int_{Ω} f (θ u) u d x \\ = θ (‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2} - \frac{1}{θ} \int_{Ω} f (θ u) u d x) \\ = θ (‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2} - φ (θ)) . \end{aligned}$
The strictly increasing nature of $φ (θ)$ for $θ \in (0, + \infty)$ implies that there exists a unique $θ^{} = θ^{} (u)$ such that $\frac{\partial G (θ u)}{\partial θ} = 0$ .
$\begin{aligned} \frac{\partial G^{2} (θ u)}{\partial θ^{2}} & = ‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2} - \int_{Ω} \frac{\partial f (θ u)}{\partial θ} u^{2} d x \\ = ‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2} - \frac{1}{θ^{2}} \int_{Ω} θ^{2} u^{2} \frac{\partial f (θ u)}{\partial θ} d x \\ = \frac{1}{θ^{2}} [θ^{2} ‖ \nabla u ‖_{2}^{2} + θ^{2} μ ‖ u ‖_{U_{0}}^{2} - \int_{Ω} θ^{2} u^{2} \frac{\partial f (θ u)}{\partial θ} d x] . \end{aligned}$
Now from ${\frac{\partial G (θ u)}{\partial θ} |}_{θ = θ^{}} = 0$ and $(Z_{3})$ , we can write
$\begin{aligned} {\frac{\partial G^{2} (θ u)}{\partial θ^{2}} |}_{θ = θ^{}} & = \frac{1}{{(θ^{})}^{2}} [{(θ^{})}^{2} ‖ \nabla u ‖_{2}^{2} + {(θ^{})}^{2} μ ‖ u ‖_{U_{0}}^{2} - \int_{Ω} (θ^{})^{2} u^{2} {\frac{\partial f (θ u)}{\partial θ} |}_{θ = θ^{}} d x] \\ = \frac{1}{{(θ^{})}^{2}} [\int_{Ω} f (θ^{} u) θ^{} u d x - \int_{Ω} (θ^{})^{2} u^{2} {\frac{\partial f (θ u)}{\partial θ} |}_{θ = θ^{}} d x] \\ < 0. \end{aligned}$
Thus, we prove (3).

(4) By the definition of $H (u)$ , one has
$\begin{aligned} H (θ u) & = θ^{2} [‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2} - \frac{1}{θ} \int_{Ω} u f (θ u) d x] \\ = θ^{2} [‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2} - φ (θ)] \\ = θ \frac{\partial G (θ u)}{\partial θ} . \end{aligned}$
Therefore, $H (θ u) > 0$ for $0 < θ < θ^{}$ , $H (θ u) < 0$ for $θ^{} < θ < \infty$ and $H (θ^{} u) = 0$ .
Lemma 2.7.
Let $u_{0} \in X_{μ}^{s} (Ω)$ , $r_{1} (σ)$ and $r_{2} (σ)$ are defined as follows:
$\begin{aligned} {\begin{cases} o n l y r_{1} (σ) = {(\frac{σ}{γ A {(C^{})}^{γ}})}^{\frac{1}{γ - 2}}, & μ = 0, \\ r_{1} (σ) = {(\frac{σ}{γ A {(C^{})}^{γ}})}^{\frac{1}{γ - 2}}, r_{2} (σ) = {(\frac{μ σ}{γ A {(C_{})}^{γ}})}^{\frac{1}{γ - 2}}, & 0 < μ < 1, \\ r_{1} (σ) = {(\frac{σ}{γ A {(C^{})}^{γ}})}^{\frac{1}{γ - 2}}, r_{2} (σ) = {(\frac{σ}{γ A {(C_{})}^{γ}})}^{\frac{1}{γ - 2}}, & μ ⩾ 1, \end{cases} \end{aligned}$
(2.4)
where the definition of* $C^{}$ and* $C_{}$ are referred to Lemma* 2.2, then (1)
If $μ = 0$ and $0 < ‖ \nabla u ‖_{2} < r_{1} (σ)$ , it holds that $H_{σ} (u) > 0$ . If $μ > 0$ , $0 < ‖ \nabla u ‖_{2} < r_{1} (σ)$ and $0 < ‖ u ‖_{U_{0}} < r_{2} (σ)$ , it holds that $H_{σ} (u) > 0$ . Particularly, if $μ = 0$ and $0 < ‖ \nabla u ‖_{2} < r_{1} (1)$ , it holds that $H (u) > 0$ . If $μ > 0$ , $0 < ‖ \nabla u ‖_{2} < r_{1} (1)$ and $0 < ‖ u ‖_{U_{0}} < r_{2} (1)$ , it holds that $H (u) > 0$ .
(2)
If $μ = 0$ and $H_{σ} (u) < 0$ , it holds that $‖ \nabla u ‖_{2} > r_{1} (σ)$ . If $μ > 0$ and $H_{σ} (u) < 0$ , it holds that $‖ \nabla u ‖_{2} > r_{1} (σ)$ and $‖ u ‖_{U_{0}} > r_{2} (σ)$ . Particularly, if $μ = 0$ and $H (u) < 0$ , it holds that $‖ \nabla u ‖_{2} > r_{1} (1)$ . If $μ > 0$ and $H (u) < 0$ , it holds that $‖ \nabla u ‖_{2} > r_{1} (1)$ and $‖ u ‖_{U_{0}} > r_{2} (1)$ .
(3)
If $μ = 0$ and $H_{σ} (u) = 0$ , it holds that $‖ \nabla u ‖_{2} ⩾ r_{1} (σ)$ or $‖ \nabla u ‖_{2} = 0$ . If $μ > 0$ and $H_{σ} (u) = 0$ , it holds that $‖ \nabla u ‖_{2} ⩾ r_{1} (σ)$ and $‖ u ‖_{U_{0}} = 0$ or $‖ \nabla u ‖_{2} = 0$ and $‖ u ‖_{U_{0}} ⩾ r_{2} (σ)$ or $‖ \nabla u ‖_{2} ⩾ r_{1} (σ)$ and $‖ u ‖_{U_{0}} ⩾ r_{2} (σ)$ or $‖ \nabla u ‖_{2} = 0 (‖ u ‖_{U_{0}} < r_{2} (σ))$ and $‖ u ‖_{U_{0}} = 0$ . Particularly, if $μ = 0$ and $H (u) = 0$ , it holds that $‖ \nabla u ‖_{2} ⩾ r_{1} (1)$ or $‖ \nabla u ‖_{2} = 0$ . If $μ > 0$ and $H (u) = 0$ , it holds that $‖ \nabla u ‖_{2} ⩾ r_{1} (1)$ and $‖ u ‖_{U_{0}} = 0$ or $‖ \nabla u ‖_{2} = 0$ and $‖ u ‖_{U_{0}} ⩾ r_{2} (1)$ or $‖ \nabla u ‖_{2} ⩾ r_{1} (1)$ and $‖ u ‖_{U_{0}} ⩾ r_{2} (1)$ or $‖ \nabla u ‖_{2} = 0 (‖ u ‖_{U_{0}} < r_{2} (1))$ and $‖ u ‖_{U_{0}} = 0$ .

Proof.
(1) If $μ = 0$ and $0 < ‖ \nabla u ‖_{2} < r_{1} (σ)$ , combining with Lemma 2.2 and Lemma 2.4 we will derive
$\begin{aligned} H_{σ} (u) & = σ ‖ \nabla u ‖_{2}^{2} - \int_{Ω} u f (u) d x \\ ⩾ σ ‖ \nabla u ‖_{2}^{2} - γ A ‖ u ‖_{γ}^{γ} \\ ⩾ σ ‖ \nabla u ‖_{2}^{2} - γ A (C^{})^{γ} ‖ \nabla u ‖_{2}^{γ - 2} ‖ \nabla u ‖_{2}^{2} \\ > σ ‖ \nabla u ‖_{2}^{2} - γ A (C^{})^{γ} r_{1}^{γ - 2} (σ) ‖ \nabla u ‖_{2}^{2} \\ = 0. \end{aligned}$
(2.5)

If $μ > 0$ , $0 < ‖ \nabla u ‖_{2} < r_{1} (σ)$ and $0 < ‖ u ‖_{U_{0}} < r_{2} (σ)$ , similarly to $μ = 0$ we can get
$\begin{aligned} H_{σ} (u) & = σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2} - \int_{Ω} u f (u) d x \\ ⩾ σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2} - γ A ‖ u ‖_{γ}^{γ} \\ ⩾ σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2} - γ A (C^{})^{γ} ‖ \nabla u ‖_{2}^{γ - 2} ‖ \nabla u ‖_{2}^{2} - γ A (C_{})^{γ} ‖ u ‖_{U_{0}}^{γ - 2} ‖ u ‖_{U_{0}}^{2} \\ > σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2} - γ A (C^{})^{γ} r_{1}^{γ - 2} (σ) ‖ \nabla u ‖_{2}^{2} - γ A (C_{})^{γ} r_{2}^{γ - 2} (σ) ‖ u ‖_{U_{0}}^{2} \\ = 0. \end{aligned}$
(2.6)
Particularly, if $σ = 1$ , the conclusion is clearly valid.

(2) If $H_{σ} (u) < 0$ holds, combining with Lemma 2.2 and Lemma 2.4, one has
$\begin{aligned} {\begin{cases} σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2} < \int_{Ω} u f (u) d x ⩽ γ A ‖ u ‖_{γ}^{γ} ⩽ γ A (C^{})^{γ} ‖ \nabla u ‖_{2}^{γ - 2} ‖ \nabla u ‖_{2}^{2}, \\ σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2} < \int_{Ω} u f (u) d x ⩽ γ A ‖ u ‖_{γ}^{γ} ⩽ γ A (C_{})^{γ} ‖ u ‖_{U_{0}}^{γ - 2} ‖ u ‖_{U_{0}}^{2} . \end{cases} \end{aligned}$
(2.7)
Furthermore, if $μ = 0$ , we can turn (2.7) into
$\begin{aligned} σ ‖ \nabla u ‖_{2}^{2} < γ A (C^{})^{γ} ‖ \nabla u ‖_{2}^{γ - 2} ‖ \nabla u ‖_{2}^{2} . \end{aligned}$
If $0 < μ < 1$ , we note
$\begin{aligned} {\begin{cases} σ ‖ \nabla u ‖_{2}^{2} < γ A (C^{})^{γ} ‖ \nabla u ‖_{2}^{γ - 2} ‖ \nabla u ‖_{2}^{2}, \\ σ μ ‖ u ‖_{U_{0}}^{2} < γ A (C_{})^{γ} ‖ u ‖_{U_{0}}^{γ - 2} ‖ u ‖_{U_{0}}^{2} . \end{cases} \end{aligned}$
If $μ ⩾ 1$ , we obtain
$\begin{aligned} {\begin{cases} σ ‖ \nabla u ‖_{2}^{2} < γ A (C^{})^{γ} ‖ \nabla u ‖_{2}^{γ - 2} ‖ \nabla u ‖_{2}^{2}, \\ σ μ ‖ u ‖_{U_{0}}^{2} < γ A (C_{})^{γ} μ ‖ u ‖_{U_{0}}^{γ - 2} ‖ u ‖_{U_{0}}^{2} . \end{cases} \end{aligned}$
Hence, the second claim is proved. Particularly, if $σ = 1$ , the conclusion is clearly valid.

(3) If $H_{σ} (u) = 0$ and $0 < μ < 1$ , combining with Lemma 2.2 and Lemma 2.4, we have
$\begin{aligned} 0 & = σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2} - \int_{Ω} u f (u) d x \\ ⩾ σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2} - γ A ‖ u ‖_{γ}^{γ} \\ ⩾ σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2} - γ A (C^{})^{γ} ‖ \nabla u ‖_{2}^{γ - 2} ‖ \nabla u ‖_{2}^{2} - γ A (C_{})^{γ} ‖ u ‖_{U_{0}}^{γ - 2} ‖ u ‖_{U_{0}}^{2} \\ = ‖ \nabla u ‖_{2}^{2} (σ - γ A (C^{})^{γ} ‖ \nabla u ‖_{2}^{γ - 2}) + μ ‖ u ‖_{U_{0}}^{2} (σ - \frac{1}{μ} γ A {(C_{})}^{γ} ‖ u ‖_{U_{0}}^{γ - 2}) . \end{aligned}$
Clearly this also holds when $μ = 0$ .
$\begin{aligned} 0 & = σ ‖ \nabla u ‖_{2}^{2} - \int_{Ω} u f (u) d x \\ ⩾ σ ‖ \nabla u ‖_{2}^{2} - γ A ‖ u ‖_{γ}^{γ} \\ ⩾ σ ‖ \nabla u ‖_{2}^{2} - γ A (C^{})^{γ} ‖ \nabla u ‖_{2}^{γ - 2} ‖ \nabla u ‖_{2}^{2} \\ = ‖ \nabla u ‖_{2}^{2} (σ - γ A (C^{})^{γ} ‖ \nabla u ‖_{2}^{γ - 2}) . \end{aligned}$
If $H_{σ} (u) = 0$ and $μ ⩾ 1$ , we obtain
$\begin{aligned} 0 & = σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2} - \int_{Ω} u f (u) d x \\ ⩾ σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2} - γ A ‖ u ‖_{γ}^{γ} \\ ⩾ σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2} - γ A (C^{})^{γ} ‖ \nabla u ‖_{2}^{γ - 2} ‖ \nabla u ‖_{2}^{2} - γ A (C_{})^{γ} μ ‖ u ‖_{U_{0}}^{γ - 2} ‖ u ‖_{U_{0}}^{2} \\ = ‖ \nabla u ‖_{2}^{2} (σ - γ A (C^{})^{γ} ‖ \nabla u ‖_{2}^{γ - 2}) + μ ‖ u ‖_{U_{0}}^{2} (σ - γ A (C_{})^{γ} ‖ u ‖_{U_{0}}^{γ - 2}) . \end{aligned}$

From this, we can conclude that (i) if $‖ \nabla u ‖_{2} = 0$ , we deduce $‖ \nabla u ‖_{2} ⩾ r_{1} (σ)$ ; (ii) if $‖ u ‖_{U_{0}} = 0$ , we deduce $‖ \nabla u ‖_{2} ⩾ r_{1} (σ)$ ; (iii) if $‖ \nabla u ‖_{2} ⩾ r_{1} (σ)$ , we deduce $‖ u ‖_{U_{0}} ⩾ r_{2} (σ)$ ; (iv) if $‖ \nabla u ‖_{2} = 0 (‖ u ‖_{U_{0}} < r_{2} (σ))$ , we deduce $‖ u ‖_{U_{0}} = 0$ . It also holds when $σ = 1$ . Then we can draw the desired conclusion.
Lemma 2.8.
For* $0 < σ < \bar{σ}$ , the $d (σ)$ can indeed be represented as $d (σ) = inf {G (u) | u \in M_{σ}}$ .
Proof.
For all $u \in X_{μ}^{s} (Ω)$ by the definition of $G (u)$ , $H_{σ} (u)$ and $(Z_{4})$ we have
$\begin{aligned} G (u) & = \frac{1}{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} ‖ u ‖_{U_{0}}^{2} - \int_{Ω} F (u) d x \\ ⩾ \frac{1}{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} ‖ u ‖_{U_{0}}^{2} - \frac{1}{p} \int_{Ω} u f (u) d x \\ = (\frac{1}{2} - \frac{σ}{p}) (‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2}) + \frac{1}{p} H_{σ} (u) . \end{aligned}$
(2.8)
For $u \in X_{μ}^{s} (Ω)$ , with $‖ \nabla u ‖_{2} \neq 0$ , $‖ u ‖_{U_{0}} \neq 0$ and $H_{σ} (u) = 0$ . Analogue to (2.7), one has
$\begin{aligned} {\begin{cases} σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2} ⩽ γ A (C^{})^{γ} ‖ \nabla u ‖_{2}^{γ - 2} ‖ \nabla u ‖_{2}^{2}, \\ σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2} ⩽ γ A (C_{})^{γ} ‖ u ‖_{U_{0}}^{γ - 2} ‖ u ‖_{U_{0}}^{2}, \end{cases} \end{aligned}$
which together with Lemma 2.7 yields that if $μ = 0$ , $‖ \nabla u ‖_{2} ⩾ {(\frac{σ}{γ A (C^{})^{γ}})}^{\frac{1}{γ - 2}} = r_{1} (σ)$ , then
$\begin{aligned} G (u) ⩾ (\frac{1}{2} - \frac{σ}{p}) r_{1}^{2} (σ) . \end{aligned}$
If $0 < μ < 1$ , we note
$\begin{aligned} {\begin{cases} ‖ \nabla u ‖_{2} > {(\frac{σ}{γ A {(C^{})}^{γ}})}^{\frac{1}{γ - 2}} = r_{1} (σ), \\ ‖ u ‖_{U_{0}} > {(\frac{μ σ}{γ A {(C_{})}^{γ}})}^{\frac{1}{γ - 2}} = r_{2} (σ) . \end{cases} \end{aligned}$
If $μ ⩾ 1$ ,
$\begin{aligned} {\begin{cases} ‖ \nabla u ‖_{2} > {(\frac{σ}{γ A {(C^{})}^{γ}})}^{\frac{1}{γ - 2}} = r_{1} (σ), \\ ‖ u ‖_{U_{0}} > {(\frac{σ}{γ A {(C_{})}^{γ}})}^{\frac{1}{γ - 2}} = r_{2} (σ) . \end{cases} \end{aligned}$
Hence, (2.8) gives
$\begin{aligned} G (u) > (\frac{1}{2} - \frac{σ}{p}) (r_{1}^{2} (σ) + μ r_{2}^{2} (σ)) . \end{aligned}$
This completes the proof.
Lemma 2.9.
Some properties of the function* $d (σ) : R^{+} \to R$ are given below. (1)
$d (σ) ⩾ a (σ) [r_{1}^{2} (σ) + μ r_{2}^{2} (σ)]$ , where $a (σ) = \frac{1}{2} - \frac{σ}{p}$ for $0 < σ < \frac{p}{2}$ .
(2)
$lim_{σ \to 0^{+}} d (σ) = 0$ and there exists a unique $\bar{σ} ⩾ \frac{p}{2}$ such that $d (\bar{σ}) = 0$ . It can also be said that $d (σ) > 0$ for $0 < σ < \bar{σ}$ .
(3)
$d (σ)$ is strictly monotonically increasing on the interval $(0, 1)$ , strictly monotonically decreasing on the interval $(1, \bar{σ})$ and sets $d (1) = d$ as the maximum value at $σ = 1$ .

Proof.
(1) For $u \in M_{σ}$ , then $‖ \nabla u ‖_{2} ⩾ r_{1} (σ)$ and $‖ u ‖_{U_{0}} ⩾ r_{2} (σ)$ by Lemma 2.7. Hence, (2.8) gives
$\begin{aligned} G (u) ⩾ (\frac{1}{2} - \frac{σ}{p}) (‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2}) + \frac{1}{p} H_{σ} (u) ⩾ a (σ) [r_{1}^{2} (σ) + μ r_{2}^{2} (σ)] . \end{aligned}$
Then by the definition of $d (σ)$ , we can draw the desired conclusion.

(2) Lemma 2.5 states that we can establish an unique positive number $θ = θ (σ)$ as follows: for all $u \in X_{μ}^{s} (Ω)$ where $u \neq 0$ and for $σ > 0$ , we can examine the solution of the following equation with respect to θ,
$\begin{aligned} σ ‖ θ \nabla u ‖_{2}^{2} + σ μ ‖ θ u ‖_{U_{0}}^{2} = \int_{Ω} θ u f (θ u) d x = θ^{2} φ (θ), \end{aligned}$
which implies $σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2} = φ (θ)$ , hence $θ (σ) = φ^{- 1} (σ ‖ \nabla u ‖_{2}^{2} + σ μ ‖ u ‖_{U_{0}}^{2})$ and $H_{σ} (θ u) = 0$ by the definition of $H_{σ} (u)$ .

Furthermore, we yield that
$\begin{aligned} lim_{σ \to 0^{+}} θ (σ) = 0, lim_{σ \to + \infty} θ (σ) = + \infty . \end{aligned}$
Hence, by Lemma 2.6 we have
$\begin{aligned} lim_{σ \to 0^{+}} G (θ u) = lim_{θ \to 0^{+}} G (θ u) = 0, lim_{σ \to 0^{+}} d (σ) = 0, \\ lim_{σ \to + \infty} G (θ u) = lim_{θ \to + \infty} G (θ u) = - \infty, lim_{σ \to + \infty} d (σ) = - \infty . \end{aligned}$
Thus, through the continuity of $d (σ)$ and (1) of this lemma, we can obtain that there exists $\bar{σ} ⩾ p / 2$ such that $d (\bar{σ}) = 0$ and $d (σ) > 0$ for $0 < σ < \bar{σ}$ .

(3) For any $σ^{'}$ , $σ^{″}$ , we assert that $d (σ^{'}) < d (σ^{″})$ , either $0 < σ^{'} < σ^{″} < 1$ or $1 < σ^{″} < σ^{'} < \bar{σ}$ . It is enough to prove that for any $u \in M_{σ^{″}}$ , there exists $v \in M_{σ^{'}}$ and a number $ϑ = ϑ (σ^{'}, σ^{″})$ such that $G (v) < G (u) - ϑ (σ^{'}, σ^{″})$ . Indeed, for such $u \in X_{μ}^{s} (Ω)$ and $σ > 0$ , the definition of $θ (σ)$ in (2) yields $θ (σ) u \in M_{σ}$ , then $H_{σ} (θ (σ) u) = 0$ . Put $h (θ) = G (θ u)$ . Hence
$\begin{aligned} \frac{d}{d θ} h (θ) & = θ ‖ \nabla u ‖_{2}^{2} + θ μ ‖ u ‖_{U_{0}}^{2} - \int_{Ω} u f (θ u) d x = \frac{1}{θ} [‖ θ \nabla u ‖_{2}^{2} + μ ‖ θ u ‖_{U_{0}}^{2} - \int_{Ω} θ u f (θ u) d x] \\ = \frac{1}{θ} [(1 - σ) (‖ θ \nabla u ‖_{2}^{2} + μ ‖ θ u ‖_{U_{0}}^{2}) + H_{σ} (θ u)] = θ (1 - σ) (‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2}) . \end{aligned}$

Next, we discuss this issue from two cases. Let $v = θ (σ^{'}) u$ , so that $v \in M_{σ^{'}}$ .

(i) If $0 < σ^{'} < σ^{″} < 1$ , Lemma 2.7 and the mean value theorem provide
$\begin{aligned} G (u) - G (v) & = h (1) - h (θ (σ^{'})) \\ = θ (ξ) (1 - ξ) (‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2}) (1 - θ (σ^{'})) \\ > (1 - σ^{″}) θ (σ^{'}) [r_{1}^{2} (σ^{″}) + μ r_{2}^{2} (σ^{″})] (1 - θ (σ^{'})) \\ \equiv ϑ (σ^{'}, σ^{″}), \end{aligned}$
where $0 < σ^{'} < ξ < σ^{″} < 1$ .

(ii) If $1 < σ^{″} < σ^{'} < \bar{σ}$ , we obtain
$\begin{aligned} G (u) - G (v) = h (1) - h (θ (σ^{'})) \\ > (σ^{″} - 1) θ (σ^{″}) [r_{1}^{2} (σ^{″}) + μ r_{2}^{2} (σ^{″})] (θ (σ^{'}) - 1) \\ \equiv ϑ (σ^{'}, σ^{″}), \end{aligned}$
where $1 < σ^{″} < ξ < σ^{'} < \bar{σ}$ . This completes the proof.
Lemma 2.10.
Let $0 < σ < p / 2$ . Then
$\begin{aligned} G_{r^{'} (σ)} \subset W_{σ} \subset G_{r^{″} (σ)} a n d V_{σ} \subset G_{σ}^{c}, \end{aligned}$
where
$\begin{aligned} G_{r^{'} (σ)} = {u \in X_{μ}^{s} (Ω) : ‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2} < min {r_{1}^{2} (σ) + μ r_{2}^{2} (σ), 2 d (σ)}}, \end{aligned}$
and
$\begin{aligned} G_{r^{″} (σ)} = {u \in X_{μ}^{s} (Ω) : ‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2} < \frac{d (σ)}{a (σ)}} . \end{aligned}$

Proof.
For $0 < σ < p / 2$ and $u \in X_{μ}^{s} (Ω)$ . On the one hand, if $‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2} < r_{1}^{2} (σ) + μ r_{2}^{2} (σ)$ , Lemma 2.7(1) yields either $‖ \nabla u ‖_{2} = 0$ and $‖ u ‖_{U_{0}} = 0$ or $H_{σ} (u) > 0$ . On the other hand, from $G (u) ⩽ \frac{1}{2} (‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2})$ and $‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2} < 2 d (σ)$ , we can obtain $G (u) < d (σ)$ . Therefore, $G_{r^{'} (σ)} \subset W_{σ}$ .

Using the definition of $a (σ)$ in Lemma 2.8(1) and (2.8), we obtain
$\begin{aligned} a (σ) (‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2}) < (\frac{1}{2} - \frac{σ}{p}) (‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2}) + \frac{1}{p} H_{σ} (u) ⩽ G (u) < d (σ), \end{aligned}$
which implies $‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2} < \frac{d (σ)}{a (σ)}$ , i.e. $u \in G_{r^{″} (σ)}$ . Hence $W_{σ} \subset G_{r^{″} (σ)}$ .

Lemma 2.7(2) suggests right away that $V_{σ} \subset G_{σ}^{c}$ . This proves the conclusion.
Lemma 2.11.
The following accounts hold: (1)
If $0 < σ^{'} < σ^{″} < 1$ , then $W_{σ^{'}} \subset W_{σ^{″}}$ .
(2)
If $1 ⩽ σ^{″} < σ^{'} < \bar{σ}$ , then $V_{σ^{'}} \subset V_{σ^{″}}$ .

The proof of Lemma 2.11 follows directly from the definitions of $W_{σ}$ , $V_{σ}$ and Lemma 2.9.
Lemma 2.12.
Suppose that $0 < G (u) < d$ , for any $u \in X_{μ}^{s} (Ω)$ . Let $σ_{1}$ and $σ_{2}$ , with $0 < σ_{1} < σ_{2}$ , represent the two roots of $d (σ) = G (u)$ . Thus, the sign of $H_{σ} (u)$ remains unchanged in $(σ_{1}, σ_{2})$ .

For the proof of this lemma we refer to a similar version, see Lemma 3.13 in [20].
3. .The subcritical energy level

The question of the existence of global solutions, vacuum isolating and blow-up in finite time to problem (1.1) at the subcritical energy level concerns us. This section assumes that the function f satisfies assumption $(Z)$ .

Theorem 3.1.
Let $u_{0} \in X_{μ}^{s} (Ω)$ and $u_{1} \in L^{2} (Ω)$ , and suppose that $σ_{1}$ and $σ_{2}$ , with $0 < σ_{1} < σ_{2}$ , are the two roots of the equation $d (σ) = e$ , where $0 < e < d = d (1)$ . Then, the subsequent properties are true: (1)
Provided that either $H (u_{0}) > 0$ or $‖ \nabla u_{0} ‖_{2} = 0$ and $‖ u_{0} ‖_{U_{0}} = 0$ , all of the (1.1) solutions $u (x, t)$ , with $E (0) = e$ , are in $W_{σ}$ for $σ \in (σ_{1}, σ_{2})$ and $t \in [0, T)$ .
(2)
Provided that $H (u_{0}) < 0$ , all of the (1.1) solutions $u (x, t)$ , with $E (0) = e$ , are in $V_{σ}$ for $σ \in (σ_{1}, σ_{2})$ and $t \in [0, T)$ .

Proof.
$(1)$ Put $u = u (x, t)$ be a solution of (1.1), and satisfying $E (0) = e$ , either $H (u_{0}) > 0$ or $‖ \nabla u_{0} ‖_{2} = 0$ and $‖ u_{0} ‖_{U_{0}} = 0$ . Suppose that T is the maximal existence time of the solution. If $‖ \nabla u_{0} ‖_{2} = 0$ and $‖ u_{0} ‖_{U_{0}} = 0$ , thus $u_{0} \in W_{σ}$ for $0 < σ < \bar{σ}$ . If $H (u_{0}) > 0$ , according to
$\begin{aligned} E (0) = \frac{1}{2} {‖ u_{t} (0) ‖}_{2}^{2} + G (u_{0}) = d (σ_{1}) = d (σ_{2}) < d (σ), for σ_{1} < σ < σ_{2}, \end{aligned}$
(3.1)
it shows $G (u_{0}) < d (σ)$ . And Lemma 2.12 gives $H_{σ} (u_{0}) > 0$ , that is, for $σ \in (σ_{1}, σ_{2})$ , $u_{0} \in W_{σ}$ .

According to the continuity, we proof $u \in W_{σ}$ for $σ_{1} < σ < σ_{2}$ and $0 < t < T$ . By contradiction, for any $σ \in (σ_{1}, σ_{2})$ there exists $t_{0} \in (0, T)$ such that $u (t_{0}) \in \partial W_{σ}$ , i.e. either $H_{σ} (u (t_{0})) = 0$ , $‖ \nabla u (t_{0}) ‖_{2} \neq 0$ and $‖ u (t_{0}) ‖_{U_{0}} \neq 0$ or $G (u (t_{0})) = d (σ)$ . Through the identity of energy
$\begin{aligned} \frac{1}{2} ‖ u_{t} ‖_{2}^{2} + G (u) = E (t) = E (0) < d (σ), for σ_{1} < σ < σ_{2} and 0 < t < T, \end{aligned}$
(3.2)
we get $G (u (t_{0})) \neq d (σ)$ . On the other hand, if $H_{σ} (u (t_{0})) = 0$ , $‖ \nabla u (t_{0}) ‖_{2} \neq 0$ and $‖ u (t_{0}) ‖_{U_{0}} \neq 0$ , we deduce $G (u (t_{0})) ⩾ d (σ)$ by the definition of $d (σ)$ , which is in opposition to (3.2). And this proves the claim.

$(2)$ The proof is similar to (1).
Theorem 3.2.
Let $u_{0} \in X_{μ}^{s} (Ω)$ , $u_{1} \in L^{2} (Ω)$ . If $0 < E (0) < d$ , either $H (u_{0}) > 0$ or $‖ \nabla u_{0} ‖_{2} = 0$ and $‖ u_{0} ‖_{U_{0}} = 0$ , then there is a global solution $u = u (x, t)$ to problem (1.1) for any $t \in [0, T)$ , with $u \in L^{\infty} (0, \infty; X_{μ}^{s} (Ω))$ , $u_{t} \in L^{\infty} (0, \infty; L^{2} (Ω))$ and $u \in W$ .
Proof.
Choosing an orthonormal basis sequence ${w_{j}}_{j = 1}^{\infty} \subset C_{0}^{\infty} (Ω)$ in $X_{μ}^{s} (Ω)$ . By the method in [33], we can construct an approximate solution of problem (1.1):
$\begin{aligned} u_{n} (x, t) = \overset{n}{\sum_{j = 1}} h_{j n} (t) w_{j} (x), n = 1, 2, \dots, \end{aligned}$
that satisfies:
$\begin{aligned} (u_{n t t}, w_{s}) + (\nabla u_{n}, \nabla w_{s}) + μ {⟨ u_{n}, w_{s} ⟩}_{U_{0}} = (f (u_{n}), w_{s}), s = 1, 2, \dots n, t ⩾ 0, \end{aligned}$
(3.3)

$\begin{aligned} u_{n} (x, 0) = u_{0 n} = \sum_{j = 1}^{n} a_{j n} w_{j} (x) \to u_{0} in X_{μ}^{s} (Ω) as n \to \infty, u_{0 n} \in W, \end{aligned}$
(3.4)

$\begin{aligned} u_{n t} (x, 0) = u_{1 n} = \sum_{j = 1}^{n} b_{j n} w_{j} (x) \to u_{1} in L^{2} (Ω) as n \to \infty, \end{aligned}$
(3.5)
where $a_{j n} = (u_{n} (0), w_{j})$ and $b_{j n} = (u_{n t} (0), w_{j})$ . Multiplying (3.3) by $h_{i n}^{^{'}} (t)$ and summing for i, we get
$\begin{aligned} \int_{Ω} u_{n t t} u_{n t} d x + \int_{Ω} \nabla u_{n} \cdot \nabla u_{n t} d x + μ \iint_{Q} \frac{(u_{n} (x, t) - u_{n} (y, t)) (u_{n t} (x, t) - u_{n t} (y, t))}{| x - y |^{N + 2 s}} d x d y = \int_{Ω} f (u_{n}) u_{n t} d x, \end{aligned}$
(3.6)
i.e.
$\begin{aligned} \frac{1}{2} \frac{d}{d t} \int_{Ω} u_{n t}^{2} d x + \frac{1}{2} \frac{d}{d t} \int_{Ω} {| \nabla u_{n} |}^{2} d x + \frac{μ}{2} \frac{d}{d t} \int \int_{Ω} \frac{| u_{n} (x, t) - u_{n} (y, t) |^{2}}{| x - y |^{N + 2 s}} d x d y = \int_{Ω} f (u_{n}) u_{n t} d x . \end{aligned}$
(3.7)
Integrating (3.7) by $0 \to t$ with regard to τ, for $t ⩾ 0$ , we have
$\begin{aligned} \frac{1}{2} \int_{0}^{t} \frac{d}{d t} (‖ u_{n τ} ‖_{2}^{2} d x) d τ + \frac{1}{2} \int_{0}^{t} \frac{d}{d t} ‖ \nabla u_{n} ‖_{2}^{2} d τ + \frac{μ}{2} \int_{0}^{t} \frac{d}{d t} ‖ u_{n} ‖_{U_{0}}^{2} d τ - \int_{0}^{t} \int_{Ω} f (u_{n}) u_{n τ} d x d τ = 0. \end{aligned}$
The above equation can also be written as
$\begin{aligned} \frac{1}{2} ‖ u_{n t} ‖_{2}^{2} - \frac{1}{2} ‖ u_{1 n} ‖_{2}^{2} + \frac{1}{2} ‖ \nabla u_{n} ‖_{2}^{2} - \frac{1}{2} ‖ \nabla u_{0 n} ‖_{2}^{2} + \frac{μ}{2} ‖ u_{n} ‖_{U_{0}}^{2} - \frac{μ}{2} ‖ u_{0 n} ‖_{U_{0}}^{2} - \int_{Ω} F (u_{n}) d x + \int_{Ω} F (u_{0 n}) d x = 0, \end{aligned}$
and from the energy equation, we can obtain
$\begin{aligned} \frac{1}{2} ‖ u_{n t} ‖_{2}^{2} + G (u_{n}) = \frac{1}{2} ‖ u_{1 n} ‖_{2}^{2} + \frac{1}{2} ‖ \nabla u_{0 n} ‖_{2}^{2} + \frac{μ}{2} ‖ u_{0 n} ‖_{U_{0}}^{2} - \int_{Ω} F (u_{0 n}) d x = E_{n} (0) < d . \end{aligned}$
(3.8)

In the following we prove that $u_{n} (t) \in W$ for any $0 < t < \infty$ and sufficiently large n. We prove the conclusion by contradiction. Suppose that $t^{0}$ is the first time such that $u_{n} (t)$ is not contained in W for $t \in (0, t^{0})$ and $u_{n} (t^{0}) \in \partial W$ . So we can say $G (u_{n} (t^{0})) = d$ or $H (u_{n} (t^{0})) = 0$ , $‖ \nabla u_{n} (t^{0}) ‖_{2} \neq 0$ and $‖ u_{n} (t^{0}) ‖_{U_{0}} \neq 0$ . Indeed, $u_{0 n} \in W$ and the definition of d gives $G (u_{n} (t^{0})) ⩾ d$ , which contradicts (3.8). Hence, $u_{n} (t) \in W$ for any $0 < t < \infty$ .

According to the definition of $G (u)$ and condition $(Z_{4})$ , we can obtain the following relation
$\begin{aligned} G (u_{n}) & = \frac{1}{2} ‖ \nabla u_{n} ‖_{2}^{2} + \frac{μ}{2} ‖ u_{n} ‖_{U_{0}}^{2} - \int_{Ω} F (u_{n}) d x \\ ⩾ (\frac{1}{2} - \frac{1}{p}) (‖ \nabla u_{n} ‖_{2}^{2} + μ ‖ u_{n} ‖_{U_{0}}^{2}) + \frac{1}{p} H (u_{n}) \\ > \frac{p - 2}{2 p} (‖ \nabla u_{n} ‖_{2}^{2} + μ ‖ u_{n} ‖_{U_{0}}^{2}) . \end{aligned}$
Combining with (3.8), we get
$\begin{aligned} \frac{1}{2} ‖ u_{n t} ‖_{2}^{2} + \frac{p - 2}{2 p} (‖ \nabla u_{n} ‖_{2}^{2} + μ ‖ u_{n} ‖_{U_{0}}^{2}) < d, for 0 ⩽ t < \infty . \end{aligned}$
(3.9)
Therefore, $‖ \nabla u_{n} ‖_{2}^{2} < \frac{2 p}{p - 2} d$ , $‖ u_{n} ‖_{U_{0}}^{2} < \frac{2 p}{μ (p - 2)} d$ (If $μ = 0$ , only $‖ \nabla u_{n} ‖_{2}^{2} < \frac{2 p}{p - 2} d$ .) and $‖ u_{n t} ‖_{2}^{2} < 2 d$ . Furthermore, Lemmas 2.2 and 2.4 imply that
$\begin{aligned} {‖ f (u_{n}) ‖}_{q}^{q} & ⩽ (γ A)^{q} ‖ u_{n} ‖_{γ}^{γ} \\ ⩽ (γ A)^{q} ((C^{})^{γ} ‖ \nabla u_{n} ‖_{2}^{γ} + (C_{})^{γ} ‖ u_{n} ‖_{U_{0}}^{γ}) \\ ⩽ (γ A)^{q} (C^{})^{γ} {(\frac{2 p d}{p - 2})}^{\frac{γ}{2}} + (γ A)^{q} (C_{})^{γ} {(\frac{2 p d}{μ (p - 2)})}^{\frac{γ}{2}}, \end{aligned}$
then we note
$\begin{aligned} {‖ f (u_{n}) ‖}_{q}^{q} ⩽ {\begin{cases} {(γ A)}^{q} {(C^{})}^{γ} {(\frac{2 p d}{p - 2})}^{\frac{γ}{2}}, & μ = 0, \\ 2 {(γ A)}^{q} Λ^{γ} {(\frac{2 p d}{μ (p - 2)})}^{\frac{γ}{2}}, & 0 < μ < 1, \\ 2 {(γ A)}^{q} Λ^{γ} {(\frac{2 p d}{p - 2})}^{\frac{γ}{2}}, & μ ⩾ 1, \end{cases} \end{aligned}$
(3.10)
where $Λ = max {C^{}, C_{}}$ , $q = γ / (γ - 1)$ and $t \in [0, \infty)$ .

As $n \to \infty$ , there exists u, ι and a subsequence of $(u_{n})_{n}$ (still denoted by $(u_{n})_{n}$ ), such that
$\begin{aligned} u_{n} \overset{}{⇀} u in L^{\infty} (0, \infty; X_{μ}^{s} (Ω)), \\ u_{n t} \overset{}{⇀} u_{t} in L^{\infty} (0, \infty; L^{2} (Ω)), \\ f (u_{n}) \overset{}{⇀} ι in L^{\infty} (0, \infty; L^{q} (Ω)), \end{aligned}$
(3.11)
and
$\begin{aligned} u_{n} \to u a . e . in Ω \times R_{0}^{+}, \\ f (u_{n}) \to f (u) a . e . in Ω \times R_{0}^{+} . \end{aligned}$
(3.12)
Furthermore, Lemma 1.3 in [17] yields that $ι = f (u)$ . As we integrate (3.3) by $0 \to t$ with regard to τ, we have
$\begin{aligned} (u_{n t}, w_{s}) + \int_{0}^{t} (\nabla u_{n}, \nabla w_{s}) d τ + μ \int_{0}^{t} ⟨ u_{n}, w_{s} ⟩_{U_{0}} d τ = \int_{0}^{t} (f (u_{n}), w_{s}) d τ + (u_{n t} (0), w_{s}), \end{aligned}$
(3.13)
for $t \in R_{0}^{+}$ , $s = 1, 2, \dots, n$ and n sufficiently large.

In (3.13), we can fix s and let $n \to \infty$ , for all $t \in R_{0}^{+}$ (3.11) and (3.12) give
$\begin{aligned} (u_{t}, w_{s}) + \int_{0}^{t} (\nabla u, \nabla w_{s}) d τ + μ \int_{0}^{t} ⟨ u, w_{s} ⟩_{U_{0}} d τ = \int_{0}^{t} (f (u), w_{s}) d τ + (u_{1}, w_{s}) . \end{aligned}$
Letting $w_{s} = v$ , it is easy to get
$\begin{aligned} (u_{t}, v) + \int_{0}^{t} (\nabla u, \nabla v) d τ + μ \int_{0}^{t} ⟨ u, v ⟩_{U_{0}} d τ = \int_{0}^{t} (f (u), v) d τ + (u_{1}, v), for v \in X_{μ}^{s} (Ω) . \end{aligned}$
For any $ϕ (x, t) \in L^{1} (0, \infty; X_{μ}^{s} (Ω))$ and $t \in [0, \infty)$ , we set $v = ϕ (x, t)$ . Using Definition 2.2 as a guide, we integrate with regard to t and conclude that $u (x, t)$ is a global solution of (1.1) since (3.4) implies that $u (x, 0) = u_{0}$ in $X_{μ}^{s} (Ω)$ . The proof is completed.
Corollary 3.3.
Let $u_{0} \in X_{μ}^{s} (Ω)$ and $u_{1} \in L^{2} (Ω)$ . If $d (σ) = E (0)$ , with $σ_{1} < σ < σ_{2}$ , and $0 < E (0) < d$ and $H_{σ_{2}} (u_{0}) > 0$ or $‖ \nabla u_{0} ‖_{2} = 0$ and $‖ u_{0} ‖_{U_{0}} = 0$ , where $σ_{1}$ , $σ_{2}$ are the two roots of equation $d (σ) = E (0)$ , then the problem (1.1) has a global weak solution $u = u (x, t)$ satisfying $u \in L^{\infty} (0, \infty; X_{μ}^{s} (Ω))$ , $u_{t} \in L^{\infty} (0, \infty; L^{2} (Ω))$ and $u \in W_{σ}$ for $0 ⩽ t < \infty$ , $σ_{1} < σ < σ_{2}$ .
Corollary 3.4.
Under the assumptions and symbols of Theorem 3.1, the conclusion of Theorem 3.1remains true, if $0 < E (0) ⩽ e$ is used in place of the assumption $E (0) = e$ .
Corollary 3.5.
Under the assumptions and symbols of Theorem 3.1, for each solution $u = u (x, t)$ of (1.1), with $0 < E (0) ⩽ e$ , (1)
Provided that $H (u_{0}) > 0$ , it holds that $u (t) \in {\bar{W}}_{σ_{1}}$ for $t \in [0, T)$ .
(2)
Provided that $H (u_{0}) < 0$ , it holds that $u (t) \in {\bar{V}}_{σ_{2}}$ for $t \in [0, T)$ .

Proof.
(3.1) gives
$\begin{aligned} G (u) ⩽ E (0) ⩽ e = d (σ_{1}) (or G (u) ⩽ E (0) ⩽ e = d (σ_{2})) for t \in [0, T) . \end{aligned}$
For any $t \in [0, T)$ , if $H_{σ} (u) > 0$ , the invariant sign of $H_{σ} (u)$ gives $H_{σ_{1}} (u) ⩾ 0$ . Hence we can obtain the conclusion of (1). Similarly, if $H_{σ} (u) < 0$ , the invariant sign of $H_{σ} (u)$ gives $H_{σ_{1}} (u) ⩽ 0$ . It is shows (2). This completes the proof.

The vacuum isolation property of problem (1.1) is discussed next. Firstly, Lemma 2.7 and Corollary 3.5 together yield Theorem 3.6.
Theorem 3.6.
Under the assumptions and symbols of Theorem 3.1, when the initial energy $E (0)$ of each solution $u = u (x, t)$ of (1.1) satisfies $0 < E (0) ⩽ e$ , it follows that (1)
If $u_{0} \in G_{1}$ , for any $t \in [0, T)$ , it holds that $u \in {\bar{G}}_{σ_{1}}$ .
(2)
If $u_{0} \in G_{1}^{c}$ , for any $t \in [0, T)$ , it holds that $u \in G_{σ_{2}}^{c}$ .

The result of Theorem 3.6 implies that for the set of all solutions for problem (1.1), when $0 < E (0) ⩽ e$ , there exists a vacuum region $R_{e}$ . That corresponds to it is
$\begin{aligned} R_{e} = {u \in X_{μ}^{s} (Ω) : r_{1} (σ_{1}) < ‖ \nabla u ‖_{2} < r_{1} (σ_{2}) and r_{2} (σ_{1}) < ‖ u ‖_{U_{0}} < r_{2} (σ_{2})}, \end{aligned}$
where $σ \mapsto r_{1} (σ)$ and $σ \mapsto r_{2} (σ)$ are defined in Lemma 2.7.

The vacuum region $R_{e}$ increases with decreasing of e. As limiting case of the vacuum region we denote as $R_{1}$ , where
$\begin{aligned} R_{1} = {u \in X_{μ}^{s} (Ω) : 0 < ‖ \nabla u ‖_{2} < r_{1} (\bar{σ}) and 0 < ‖ u ‖_{U_{0}} < r_{2} (\bar{σ})} . \end{aligned}$

In fact, we can conclude the following.
Theorem 3.7.
Under the assumptions and symbols of Theorem 3.1, all the nontrivial solutions of problem (1.1), with $E (0) = 0$ , belong to
$\begin{aligned} G_{1}^{c} = {u \in X_{μ}^{s} (Ω) : ‖ \nabla u ‖_{2} ⩾ r_{1} (1) and ‖ u ‖_{U_{0}} ⩾ r_{2} (1)} . \end{aligned}$

Proof.
With $E (0) = 0$ in mind, let $u = u (x, t)$ be any solution of (1.1), and let T be as defined in Definition 2.1. From the energy identity
$\begin{aligned} \frac{1}{2} ‖ u_{t} ‖^{2} + G (u) = E (0) = 0, for t \in [0, T), \end{aligned}$
we can yield $G (u) ⩽ 0$ for any $t \in [0, T)$ . And by $(Z_{4})$ , we have
$\begin{aligned} \frac{1}{γ} (‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2}) & ⩽ \frac{1}{2} (‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2}) ⩽ \int_{Ω} F (u) d x \\ ⩽ A ‖ u ‖_{γ}^{γ} ⩽ A (C^{})^{γ} ‖ \nabla u ‖_{2}^{γ} = A (C^{})^{γ} ‖ \nabla u ‖_{2}^{γ - 2} ‖ \nabla u ‖_{2}^{2}, \\ \frac{1}{γ} (‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2}) & ⩽ \frac{1}{2} (‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2}) ⩽ \int_{Ω} F (u) d x ⩽ A ‖ u ‖_{γ}^{γ} \\ ⩽ A (C_{})^{γ} ‖ u ‖_{U_{0}}^{γ} = A (C_{})^{γ} ‖ u ‖_{U_{0}}^{γ - 2} ‖ u ‖_{U_{0}}^{2} . \end{aligned}$
We claim that

(i) if $μ = 0$ ,
$\begin{aligned} ‖ \nabla u ‖_{2} \equiv 0 or ‖ \nabla u ‖_{2} ⩾ r_{1} (1) = {(\frac{1}{γ A (C^{})^{γ}})}^{\frac{1}{γ - 2}}, \end{aligned}$
(ii) if $0 < μ < 1$ ,
$\begin{aligned} ‖ \nabla u ‖_{2} \equiv 0 or ‖ \nabla u ‖_{2} ⩾ r_{1} (1) = {(\frac{1}{γ A {(C^{})}^{γ}})}^{\frac{1}{γ - 2}}, \\ ‖ u ‖_{U_{0}} \equiv 0 or ‖ u ‖_{U_{0}} ⩾ r_{2} (1) = {(\frac{μ}{γ A {(C_{})}^{γ}})}^{\frac{1}{γ - 2}}, \end{aligned}$
(iii) if $μ ⩾ 1$ ,
$\begin{aligned} ‖ \nabla u ‖_{2} \equiv 0 or ‖ \nabla u ‖_{2} ⩾ r_{1} (1) = {(\frac{1}{γ A {(C^{})}^{γ}})}^{\frac{1}{γ - 2}}, \\ ‖ u ‖_{U_{0}} \equiv 0 or ‖ u ‖_{U_{0}} ⩾ r_{2} (1) = {(\frac{1}{γ A {(C_{})}^{γ}})}^{\frac{1}{γ - 2}}, \end{aligned}$
for all $t \in [0, T)$ . If $‖ \nabla u_{0} ‖_{2} = 0$ and $‖ u_{0} ‖_{U_{0}} = 0$ , then $‖ \nabla u ‖_{2} = 0$ and $‖ u ‖_{U_{0}} = 0$ for all $t \in (0, T)$ . Otherwise, there exists $τ \in (0, T)$ such that $0 < ‖ \nabla u (τ) ‖_{2} < r_{1} (1)$ and $0 < ‖ u (τ) ‖_{U_{0}} < r_{2} (1)$ , which leads to a contradiction. From Theorem 3.6(2), if $‖ \nabla u_{0} ‖_{2} ⩾ r_{1} (1)$ and $‖ u_{0} ‖_{U_{0}} ⩾ r_{2} (1)$ , we obtain $‖ \nabla u ‖_{2} ⩾ r_{1} (1)$ and $‖ u ‖_{U_{0}} ⩾ r_{2} (1)$ for $t \in (0, T)$ .

We can now demonstrate that there are blow-up solutions for problem (1.1). Theorem 3.8.
Let* $u_{0} \in X_{μ}^{s} (Ω)$ , $u_{1} \in L^{2} (Ω)$ . If $E (0) < d$ and $H (u_{0}) < 0$ , then any nontrivial solution of (1.1) blows up in finite time.
Proof.
Let $u = u (x, t)$ be a fixed nontrivial solution of problem (1.1), satisfying $E (0) < d$ and $H (u_{0}) < 0$ , and let T be the maximal existence time. Putting $J (t) = ‖ u ‖_{2}^{2}$ for $t \in [0, T)$ . Apparently, $J^{'} (t) = 2 (u_{t}, u)$ . Since
$\begin{aligned} \frac{1}{2} ‖ u_{t} ‖^{2} + \frac{1}{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} ‖ u ‖_{U_{0}}^{2} - \int_{Ω} F (u) d x = E (0), for all t \in [0, T) . \end{aligned}$
Assumption $(Z_{4})$ , Lemma 2.2 together with the Poincaré inequality yield that
$\begin{aligned} J^{″} (t) & = 2 ‖ u_{t} ‖_{2}^{2} + 2 (u_{t t}, u) \\ = 2 ‖ u_{t} ‖_{2}^{2} + 2 \int_{Ω}^{u f} (u) d x - 2 ‖ \nabla u ‖_{2}^{2} - 2 μ ‖ u ‖_{U_{0}}^{2} \\ ⩾ 2 ‖ u_{t} ‖_{2}^{2} + 2 p \int_{Ω}^{F} (u) d x - 2 ‖ \nabla u ‖_{2}^{2} - 2 μ ‖ u ‖_{U_{0}}^{2} \\ = 2 ‖ u_{t} ‖_{2}^{2} - 2 ‖ \nabla u ‖_{2}^{2} - 2 μ ‖ u ‖_{U_{0}}^{2} + 2 p (\frac{1}{2} ‖ u_{t} ‖_{2}^{2} + \frac{1}{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} ‖ u ‖_{U_{0}}^{2} - E (0)) \\ = (p + 2) ‖ u_{t} ‖_{2}^{2} + (p - 2) ‖ \nabla u ‖_{2}^{2} + (p - 2) μ ‖ u ‖_{U_{0}}^{2} - 2 p E (0) \end{aligned}$

$\begin{aligned} ⩾ (p + 2) ‖ u_{t} ‖_{2}^{2} + C_{2} (p - 2) ‖ u ‖_{2}^{2} + C_{1} (p - 2) μ ‖ u ‖_{2}^{2} - 2 p E (0) \end{aligned}$
(3.14)

$\begin{aligned} = (p + 2) ‖ u_{t} ‖_{2}^{2} + C_{2} (p - 2) J (t) + C_{1} (p - 2) μ J (t) - 2 p E (0), \end{aligned}$
(3.15)
where $C_{2} > 0$ is a constant in the Poincaré inequality and $C_{1}$ as defined in Lemma 2.2.

(1) If $E (0) ⩽ 0$ , then (3.14) derives the following inequality
$\begin{aligned} J^{″} (t) ⩾ (p + 2) ‖ u_{t} ‖_{2}^{2} for t \in [0, T) . \end{aligned}$
(3.16)
So there exists a sufficiently large $t_{}$ such that $J^{'} (t_{}) > 0$ and $J (t_{}) > 0$ ,
$\begin{aligned} J^{- α} (t_{}) > 0, {(J^{- α} (t_{}))}^{^{'}} = \frac{- α}{J^{α + 1} (t_{})} J^{'} (t_{}) < 0, for α = \frac{p - 2}{4} . \end{aligned}$
Since $J (t) = ‖ u (t) ‖_{2}^{2}$ , we combine with Cauchy–Schwarsz inequality to obtain
$\begin{aligned} {(J^{'} (t))}^{2} = 4 (u_{t}, u)^{2} ⩽ 4 ‖ u_{t} ‖_{2}^{2} ‖ u ‖_{2}^{2}, \end{aligned}$
then
$\begin{aligned} J (t) J^{″} (t) - \frac{p + 2}{4} {(J^{'} (t))}^{2} ⩾ (p + 2) ‖ u_{t} ‖_{2}^{2} ‖ u ‖_{2}^{2} - (p + 2) ‖ u_{t} ‖_{2}^{2} ‖ u ‖_{2}^{2} = 0, \end{aligned}$
i.e.
$\begin{aligned} J (t) J^{″} (t) - (α + 1) {(J^{'} (t))}^{2} ⩾ 0. \end{aligned}$
From this, we have
$\begin{aligned} {(J^{- α} (t))}^{^{″}} = \frac{- α}{J^{α + 2} (t)} (J^{″} (t) J (t) - (α + 1) (J^{'} (t))^{2}) ⩽ 0. \end{aligned}$
Therefore, for any $t ⩾ t_{}$ , $J^{- α} (t)$ is a monotonically decreasing concave function, and there exists T such that
$\begin{aligned} lim_{t \to T} J^{- α} (t) = 0, \\ lim_{t \to T} J (t) = + \infty . \end{aligned}$

(2) If $0 < E (0) < d$ , we set $σ \in (σ_{1}, σ_{2})$ and $t \in [0, T)$ . Then, we can know $u (x, t) \in V_{σ}$ from Theorem 3.1 and Corollary 3.5, where $σ_{1}$ , $σ_{2}$ are the two roots of equation $d (σ) = E (0)$ . By $u (x, t) \in V_{σ}$ , we have $H_{σ} (u) < 0$ . According to Lemma 2.7, we get $‖ \nabla u ‖_{2} > r_{1} (σ)$ and $‖ u ‖_{U_{0}} > r_{2} (σ)$ for $σ \in (σ_{1}, σ_{2})$ and $t \in [0, T)$ . And by the invariant sign of $H_{σ} (u)$ , for any $t \in [0, T)$ , we can know $H_{σ_{2}} (u) ⩽ 0$ . Hence, $‖ \nabla u ‖_{2} ⩾ r_{1} (σ)$ and $‖ u ‖_{U_{0}} ⩾ r_{2} (σ)$ . Furthermore,
$\begin{aligned} J^{″} (t) & = 2 ‖ u_{t} ‖_{2}^{2} + 2 \int_{Ω} u f (u) d x - 2 ‖ \nabla u ‖_{2}^{2} - 2 μ ‖ u ‖_{U_{0}}^{2} \\ ⩾ 2 \int_{Ω} u f (u) d x - 2 ‖ \nabla u ‖_{2}^{2} - 2 μ ‖ u ‖_{U_{0}}^{2} + 2 σ_{2} ‖ \nabla u ‖_{2}^{2} + 2 μ σ_{2} ‖ u ‖_{U_{0}}^{2} \\ - 2 σ_{2} ‖ \nabla u ‖_{2}^{2} - 2 μ σ_{2} ‖ u ‖_{U_{0}}^{2} \\ = 2 (σ_{2} - 1) ‖ \nabla u ‖_{2}^{2} + 2 (σ_{2} - 1) μ ‖ u ‖_{U_{0}}^{2} - 2 H_{σ_{2}} (u) \\ ⩾ 2 (σ_{2} - 1) r_{1}^{2} (σ_{2}) + 2 (σ_{2} - 1) μ r_{2}^{2} (σ_{2}) \\ > 0, for t \in [0, T), \end{aligned}$
and
$\begin{aligned} J^{'} (t) ⩾ 2 (σ_{2} - 1) r_{1}^{2} (σ_{2}) t + 2 (σ_{2} - 1) μ r_{2}^{2} (σ_{2}) t + J^{'} (0), for t \in [0, T) . \end{aligned}$
This indicates that there exists a $t_{}$ in $[0, T)$ and $t ⩾ t_{}$ such that $J^{'} (t) > 0$ ,
$\begin{aligned} J (t) ⩾ J^{'} (t_{}) (t - t_{}) + J (t_{}) . \end{aligned}$
That is, for sufficiently large $t \in [t_{}, T)$ , we can obtain
$\begin{aligned} C_{2} (p - 2) ‖ u ‖_{2}^{2} + C_{1} (p - 2) μ ‖ u ‖_{2}^{2} = C_{2} (p - 2) J (t) + C_{1} (p - 2) μ J (t) > 2 p E (0) . \end{aligned}$
Using (3.16) and the aforementioned inequality, we obtain
$\begin{aligned} J^{″} (t) ⩾ (p + 2) ‖ u_{t} ‖_{2}^{2} for t \in [t_{*}, T) . \end{aligned}$
(3.17)

Repeating the last part of the proof of (1) proves the assertion.

4. .The critical energy level

The existence of global weak solutions and blow-up in finite time for problem (1.1) with critical energy level $H (u_{0}) ⩾ 0$ and $E (0) = d$ are the topics of discussion in this section. It is assumed that the function f satisfies assumption $(Z)$ throughout. To study the blow-up of solutions in finite time, we introduce the unstable set

\begin{aligned} N = {u \in X_{μ}^{s} (Ω) | H (u) < 0} . \end{aligned}

Lemma 4.1.
(Invariance of the set $N$ when $E (0) = d$ ) Let $u_{0} \in X_{μ}^{s} (Ω)$ , $u_{1} \in L^{2} (Ω)$ , and suppose that $\int_{Ω} u_{0} u_{1} d x > 0$ . If $E (0) = d$ and $u_{0} \in N$ , then all (weak) solutions of problem (1.1) belong to $N$ .
Proof.
Let $u = u (x, t)$ be any weak solution of problem (1.1) satisfying $E (0) = d$ and $u_{0} \in N$ . We need to prove that $u \in N$ for $t \in [0, T)$ , only given that $H (u) < 0$ for $t \in [0, T)$ . We argue by contradiction. Suppose that there exists a $t^{0} \in (0, T)$ such that $H (u (t^{0})) = 0$ and $H (u (t)) < 0$ for any $t \in [0, t^{0})$ . Combining the energy equation and the definition of d, we can obtain
$\begin{aligned} d ⩽ G (u (t^{0})) ⩽ \frac{1}{2} {‖ u_{t} (t^{0}) ‖}_{2}^{2} + G (u (t^{0})) = E (t^{0}) = E (0) = d, \end{aligned}$
which yields that $d = G (u (t^{0}))$ and
$\begin{aligned} {‖ u_{t} (t^{0}) ‖}_{2}^{2} = 0. \end{aligned}$
(4.1)

Next, we define an auxiliary function $J (t) = ‖ u ‖_{2}^{2}$ , and directly
$\begin{aligned} J^{'} (t) = 2 (u, u_{t}), \\ J^{″} (t) = 2 ‖ u_{t} ‖_{2}^{2} + 2 (u, u_{t t}) = 2 ‖ u_{t} ‖_{2}^{2} - 2 H (u), for any t \in [0, t^{0}) . \end{aligned}$

Hence, we have $J^{'} (0) = 2 (u_{0}, u_{1}) > 0$ and $J^{″} (t) > 0$ for any $t \in [0, t^{0})$ . Futhermore, $J^{'} (t)$ is strictly monotonically increasing on the interval $[0, t^{0})$ , and $J^{'} (t^{0}) = 2 (u (t^{0}), u_{t} (t^{0})) > J^{'} (0) > 0$ , i.e. $u_{t} (t^{0}) \neq 0$ , which contradicts (4.1). So this proves the conclusion.
Theorem 4.1.
Let $u_{0} \in X_{μ}^{s} (Ω)$ , $u_{1} \in L^{2} (Ω)$ , and suppose that $E (0) = d$ and $H (u_{0}) ⩾ 0$ . If
$\begin{aligned} p B ‖ u_{0} ‖_{p}^{p} ⩾ γ A ‖ u_{0} ‖_{γ}^{γ}, \end{aligned}$
(4.2)
where A, B are defined in Lemma 2.4, then u is a global solution of problem (1.1) for any $t \in [0, \infty)$ , with $u \in L^{\infty} (0, \infty; X_{μ}^{s} (Ω), u_{t} \in L^{\infty} (0, \infty; L^{2} (Ω))$ and $u (t) \in \bar{W}$ .
Proof.
The proof is split into two different cases.

(1) If $‖ \nabla u_{0} ‖_{2} \neq 0$ and $‖ u_{0} ‖_{U_{0}} \neq 0$ . Let $θ_{m} = 1 - \frac{1}{m}$ , $u_{0 m} (x) = θ_{m} u_{0} (x)$ , $m = 2, 3, \dots$ . Setting
$\begin{aligned} u (x, 0) = u_{0 m} and u_{t} (x, 0) = u_{1} . \end{aligned}$
We consider the following initial value problem
$\begin{aligned} {\begin{cases} u_{t t} - Δ u + μ {(- Δ)}^{s} u = f (u), & in Ω \times R^{+}, \\ u (x, 0) = u_{0 m}, u_{t} (x, 0) = u_{1}, & in Ω, \\ u (x, t) = 0, & in (R^{N} ∖ Ω) \times R_{0}^{+} . \end{cases} \end{aligned}$
(4.3)
Lemma 2.6(2) and (2.1) imply that there exists an unique positive constant $θ^{}$ such that
$\begin{aligned} 0 & = {\frac{\partial K (θ u_{0})}{θ} |}_{θ = θ^{}} = θ^{} ‖ \nabla u_{0} ‖_{2}^{2} + θ^{} μ ‖ u_{0} ‖_{U_{0}}^{2} - \int_{Ω} f (θ^{} u_{0}) u_{0} d x \\ = θ^{} H (u_{0}) + θ^{} \int_{Ω} f (u_{0}) u_{0} d x - \int_{Ω} f (θ^{} u_{0}) u_{0} d x . \end{aligned}$
Since $H (u_{0}) ⩾ 0$ , Lemmas 2.3, 2.4 and $(Z_{4})$ yield
$\begin{aligned} 0 & ⩾ θ^{} \int_{Ω} f (u_{0}) u_{0} d x - \frac{1}{θ^{}} \int_{Ω} f (θ^{} u_{0}) θ^{} u_{0} d x \\ ⩾ θ^{} \int_{Ω} p F (u_{0}) d x - \frac{1}{θ^{}} \int_{Ω} p F (θ^{} u_{0}) d x \\ ⩾ θ^{} \int_{Ω} B p {| u_{0} |}^{p} d x - \frac{1}{θ^{}} \int_{Ω} p A {| θ^{} u_{0} |}^{γ} d x \\ ⩾ θ^{} p B ‖ u_{0} ‖_{p}^{p} - (θ^{})^{γ - 1} γ A ‖ u_{0} ‖_{γ}^{γ} . \end{aligned}$
From the assumption (4.2), we get
$\begin{aligned} θ^{} ⩾ {(\frac{p B ‖ u_{0} ‖_{p}^{p}}{γ A ‖ u_{0} ‖_{γ}^{γ}})}^{\frac{1}{γ - 2}} ⩾ 1. \end{aligned}$
Since $0 < θ_{m} < 1 ⩽ θ^{}$ , by Lemma 2.6 we have
$\begin{aligned} H (u_{0 m}) & = H (θ_{m} u_{0}) > 0, \\ G (u_{0 m}) & = \frac{1}{2} ‖ \nabla u_{0 m} ‖_{2}^{2} + \frac{μ}{2} ‖ u_{0 m} ‖_{U_{0}}^{2} - \int_{Ω} F (u_{0 m}) d x \\ ⩾ \frac{1}{2} ‖ \nabla u_{0 m} ‖_{2}^{2} + \frac{μ}{2} ‖ u_{0 m} ‖_{U_{0}}^{2} - \frac{1}{p} \int_{Ω} u_{0 m} f (u_{0 m}) d x \\ ⩾ (\frac{1}{2} - \frac{1}{p}) (‖ \nabla u_{0 m} ‖_{2}^{2} + μ ‖ u_{0 m} ‖_{U_{0}}^{2}) + \frac{1}{p} H (u_{0 m}) \\ > 0, \end{aligned}$
and
$\begin{aligned} G (u_{0 m}) = G (θ_{m} u_{0}) < G (u_{0}) . \end{aligned}$
Furthermore,
$\begin{aligned} 0 < E_{m} (0) = \frac{1}{2} ‖ u_{1} ‖_{2}^{2} + G (u_{0 m}) < \frac{1}{2} ‖ u_{1} ‖_{2}^{2} + G (u_{0}) = E (0) = d . \end{aligned}$
On the other hand, Theorem 3.2 states that for any m, there is a global solution of problem (4.3) $u_{m}$ , with $u_{m} \in L^{\infty} (0, \infty; X_{μ}^{s} (Ω))$ , $u_{m t} \in L^{\infty} (0, \infty; L^{2} (Ω))$ and $u_{m} \in W$ for $t \in [0, \infty)$ . For $t \in R_{0}^{+}$ , satisfying
$\begin{aligned} (u_{m t}, v) + \int_{0}^{t} (\nabla u_{m}, \nabla v) d τ + μ \int_{0}^{t} {⟨ u_{m}, v ⟩}_{U_{0}} d τ = \int_{0}^{t} (f (u_{m}), v) d τ + (u_{1}, v), \\ for v \in X_{μ}^{s} (Ω), \end{aligned}$
(4.4)
and
$\begin{aligned} \frac{1}{2} ‖ u_{m t} ‖_{2}^{2} + G (u_{m}) = E_{m} (0) < d . \end{aligned}$
(4.5)
According to $u_{m} \in W$ for $t \in [0, \infty)$ , it holds
$\begin{aligned} G (u_{m}) & ⩾ (\frac{1}{2} - \frac{1}{p}) (‖ \nabla u_{m} ‖_{2}^{2} + μ ‖ u_{m} ‖_{U_{0}}^{2}) + \frac{1}{p} H (u_{m}) > (\frac{1}{2} - \frac{1}{p}) (‖ \nabla u_{m} ‖_{2}^{2} + μ ‖ u_{m} ‖_{U_{0}}^{2}), \end{aligned}$
which together with (4.5) yields
$\begin{aligned} \frac{1}{2} ‖ u_{m t} ‖_{2}^{2} + \frac{p - 2}{2 p} (‖ \nabla u_{m} ‖_{2}^{2} + μ ‖ u_{m} ‖_{U_{0}}^{2}) < d . \end{aligned}$
Hence, from $(Z_{4})$ we can obtain
$\begin{aligned} ‖ \nabla u_{m} ‖_{2}^{2} < \frac{2 p}{p - 2} d, ‖ u_{m} ‖_{U_{0}}^{2} < \frac{2 p}{μ (p - 2)} d, and ‖ u_{m t} ‖_{2}^{2} ⩽ 2 d . \end{aligned}$
Similar to Theorem 3.2, we get
$\begin{aligned} {‖ f (u_{m}) ‖}_{q}^{q} ⩽ {\begin{cases} {(γ A)}^{q} {(C^{})}^{γ} {(\frac{2 p d}{p - 2})}^{\frac{γ}{2}}, & μ = 0, \\ 2 {(γ A)}^{q} Λ^{γ} {(\frac{2 p d}{μ (p - 2)})}^{\frac{γ}{2}}, & 0 < μ < 1, \\ 2 {(γ A)}^{q} Λ^{γ} {(\frac{2 p d}{p - 2})}^{\frac{γ}{2}}, & μ ⩾ 1, \end{cases} \end{aligned}$
where $Λ = max {C^{}, C_{}}$ , $q = \frac{γ}{γ - 1}$ and $t \in [0, \infty)$ .

Hence, as $m \to \infty$ , there exist u, ι and a subsequence of $(u_{m})_{m}$ (still denoted by $(u_{m})_{m}$ ) such that
$\begin{aligned} u_{m} \overset{}{⇀} u in L^{\infty} (0, \infty; X_{μ}^{s} (Ω)), \\ u_{m t} \overset{}{⇀} u_{t} in L^{\infty} (0, \infty; L^{2} (Ω)), \\ f (u_{m}) \overset{}{⇀} ι in L^{\infty} (0, \infty; L^{q} (Ω)) \end{aligned}$
and
$\begin{aligned} u_{n} \to u a . e . in Ω \times R_{0}^{+}, \\ f (u_{n}) \to f (u) a . e . in Ω \times R_{0}^{+} . \end{aligned}$
Furthermore, Lemma 1.3 in [17] yields that $ι = f (u)$ . Letting $m \to \infty$ in (4.4), we get
$\begin{aligned} (u_{t}, v) + \int_{0}^{t} (\nabla u, \nabla v) d τ + μ \int_{0}^{t} ⟨ u, v ⟩_{U_{0}} d τ = \int_{0}^{t} (f (u), v) d τ + (u_{1}, v), v \in X_{μ}^{s} (Ω), t ⩾ 0. \end{aligned}$

This suggests that, since $u_{t} (x, 0) = u_{1}$ and $u (x, 0) = u_{0}$ are evident, u satisfies problem (4.3). Therefore, the global solution to problem (1.1) is $u = u (x, t)$ .

(2) If $‖ \nabla u_{0} ‖_{2} = 0$ and $‖ u_{0} ‖_{U_{0}} = 0$ , it yields that $G (u_{0}) = 0$ and $\frac{1}{2} ‖ u_{1} ‖_{2}^{2} = E (0) = d$ . Putting $θ_{m} = 1 - \frac{1}{m}$ , $u_{1 m} = θ_{m} u_{1}$ , $m = 2, 3, \dots$ . Letting
$\begin{aligned} u (x, 0) = u_{0}, u_{t} (x, 0) = u_{1 m} . \end{aligned}$
We will consider the following initial value problem
$\begin{aligned} {\begin{cases} u_{t t} - Δ u + μ {(- Δ)}^{s} u = f (u), & in Ω \times R^{+}, \\ u (x, 0) = u_{0}, u_{t} (x, 0) = u_{1 m}, & in Ω, \\ u (x, t) = 0, & in (R^{N} ∖ Ω) \times R_{0}^{+} . \end{cases} \end{aligned}$
(4.6)
From $‖ \nabla u_{0} ‖_{2} = 0$ and $‖ u_{0} ‖_{U_{0}} = 0$ , we can obtain
$\begin{aligned} 0 < E_{m} (0) = \frac{1}{2} ‖ u_{1 m} ‖_{2}^{2} + G (u_{0}) = \frac{1}{2} ‖ θ_{m} u_{1} ‖_{2}^{2} = \frac{1}{2} θ_{m}^{2} ‖ u_{1} ‖_{2}^{2} < \frac{1}{2} ‖ u_{1} ‖_{2}^{2} = E (0) = d . \end{aligned}$
By Theorem 3.2, it suggests that problem (4.6) has a global solution $u_{m} = u_{m} (x, t)$ for each m, with $u_{m} \in L^{\infty} (0, \infty; X_{0}^{s})$ , $u_{m t} \in L^{\infty} (0, \infty; L^{2} (Ω))$ and $u_{m} \in W$ for any $t \in [0, \infty)$ , and satisfying (4.4) and (4.5). The remaining proof resembles to that in case (1).
Theorem 4.2.
Let $u_{0} \in X_{μ}^{s} (Ω)$ , $u_{1} \in L^{2} (Ω)$ , satisfying $\int_{Ω} u_{0} u_{1} d x > 0$ . If $E (0) = d$ and $u_{0} \in N$ , then existence time of problem (1.1) is finite.
Proof.
Let $u = u (x, t)$ be a fixed nontrivial solution of problem (1.1), satisfying $E (0) < d$ , $u_{0} \in N$ and $\int_{Ω} u_{0} u_{1} d x > 0$ . Assume that T is maximal existence time and $T = + \infty$ . Putting $J (t) = ‖ u ‖_{2}^{2}$ for $t \in [0, \infty)$ . Apparently,
$\begin{aligned} J^{'} (t) = 2 (u_{t}, u), J^{'} (0) = 2 (u_{0}, u_{1}) > 0, \end{aligned}$
and
$\begin{aligned} J^{″} (t) = 2 ‖ u_{t} ‖_{2}^{2} + 2 (u_{t t}, u) = 2 ‖ u_{t} ‖_{2}^{2} - 2 H (u), for t \in [0, \infty) . \end{aligned}$
Lemma 4.1 yields $u \in N$ , i.e. $H (u) < 0$ for $t \in [0, \infty)$ . Since
$\begin{aligned} d & = E (0) = E (t) = \frac{1}{2} ‖ u_{t} ‖_{2}^{2} + \frac{1}{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} ‖ u ‖_{U_{0}}^{2} - \int_{Ω} F (u) d x \\ ⩾ \frac{1}{2} ‖ u_{t} ‖_{2}^{2} + \frac{p - 2}{2 p} (‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2}) + \frac{1}{p} H (u) . \end{aligned}$
(4.7)
Hence, we obtain
$\begin{aligned} J^{″} (t) & = 2 ‖ u_{t} ‖_{2}^{2} - 2 H (u) \\ ⩾ 2 ‖ u_{t} ‖_{2}^{2} - 2 p d + p ‖ u_{t} ‖_{2}^{2} + (p - 2) ‖ \nabla u ‖_{2}^{2} + (p - 2) μ ‖ u ‖_{U_{0}}^{2} \\ = (p + 2) ‖ u_{t} ‖_{2}^{2} + (p - 2) ‖ \nabla u ‖_{2}^{2} + (p - 2) μ ‖ u ‖_{U_{0}}^{2} - 2 p d \\ ⩾ (p + 2) ‖ u_{t} ‖_{2}^{2} + C_{2} (p - 2) ‖ u ‖_{2}^{2} + C_{1} (p - 2) μ ‖ u ‖_{2}^{2} - 2 p d \\ = (p + 2) ‖ u_{t} ‖_{2}^{2} + C_{2} (p - 2) J (t) + C_{1} (p - 2) μ J (t) - 2 p d, \end{aligned}$
(4.8)
where the definitions of $C_{1}$ and $C_{2}$ refer to Theorem 3.8.

On the other hand, by $J^{″} (t) > 0$ for $t \in [0, \infty)$ , we can know $J^{'} (t)$ is strictly monotonically increasing on the interval $t \in [0, \infty)$ . Then, for any $t^{0} > 0$ and $t ⩾ t^{0}$ , we have $J^{'} (t) ⩾ J^{'} (t^{0}) > 0$ , and
$\begin{aligned} J (t) ⩾ J^{'} (t^{0}) (t - t^{0}) + J (t^{0}) ⩾ J^{'} (t^{0}) (t - t^{0}) . \end{aligned}$
When t is large enough, we can get $C_{2} (p - 2) J (t) + C_{1} (p - 2) μ J (t) > 2 p d$ . Furthermore, $J^{″} (t) ⩾ (p + 2) ‖ u_{t} ‖_{2}^{2}$ . It yields that
$\begin{aligned} J (t) J^{″} (t) - \frac{p + 2}{4} {(J^{'} (t))}^{2} ⩾ (p + 2) [‖ u_{t} ‖_{2}^{2} ‖ u ‖_{2}^{2} - (u_{t}, u)^{2}] ⩾ 0. \end{aligned}$
From this, we have
$\begin{aligned} {(J^{- α} (t))}^{^{″}} = \frac{- α}{J^{α + 2} (t)} [J^{″} (t) J (t) - (α + 1) (J^{'} (t))^{2}] ⩽ 0, for α = \frac{p - 2}{4} . \end{aligned}$
Then there exists a $T_{1} > 0$ such that
$\begin{aligned} lim_{t \to T_{1}} J^{- α} (t) = 0, \end{aligned}$
and
$\begin{aligned} lim_{t \to T_{1}} J (t) = + \infty, \end{aligned}$
which contradicts $T = + \infty$ .

5. .The supercritical energy level

In this section, we will consider the blow-up of the solution of problem (1.1) in finite time at the supercritical initial energy level. We assume that $f = f (u)$ satisfies assumption $(Z)$ .

Lemma 5.1.
(Invariance of the set $N$ when $E (0) > 0$ ) Let $u_{0} \in X_{μ}^{s} (Ω)$ , $u_{1} \in L^{2} (Ω)$ , satisfying $\int_{Ω} u_{0} u_{1} d x > 0$ . Suppose that $E (0) > 0$ , $u_{0} \in N$ and
$\begin{aligned} ‖ u_{0} ‖_{2}^{2} ⩾ b E (0), \end{aligned}$
(5.1)
where
$\begin{aligned} b = \frac{2 p}{(C_{2} + C_{1} μ) (p - 2)}, \end{aligned}$
and the definitions of $C_{1}$ and $C_{2}$ are given in Theorem 3.8. Then all solutions of problem (1.1) belong to $N$ .
Proof.
Assume that $t_{1} \in (0, T)$ is the first time such that $H (u (t_{1})) = 0$ and $H (u (t)) < 0$ for $t \in [0, t_{1})$ . Similar to Lemma 4.1, we introduce an auxiliary function $J (t) = ‖ u ‖_{2}^{2}$ , for $t \in [0, \infty)$ . And we get
$\begin{aligned} J^{'} (t) & = 2 (u, u_{t}), \\ J^{″} (t) & = 2 ‖ u_{t} ‖_{2}^{2} + 2 (u, u_{t t}) = 2 ‖ u_{t} ‖_{2}^{2} - 2 H (u) . \end{aligned}$
Repeating the steps of Lemma 4.1, we similarly obtain $J^{'} (t)$ is strictly monotonically increasing on the interval $[0, t_{1})$ , hence, $J^{'} (t) > J^{'} (0) = 2 (u_{0}, u_{1}) > 0$ , for $t \in [0, t_{1})$ . It suggests that $J (t)$ is strictly monotonically increasing on the interval $[0, t_{1})$ . For $t \in (0, t_{1})$ , combining with (5.1), we get
$\begin{aligned} J (t) = ‖ u ‖_{2}^{2} > J (0) = ‖ u_{0} ‖_{2}^{2} ⩾ \frac{2 p}{(C_{2} + C_{1} μ) (p - 2)} E (0) . \end{aligned}$
(5.2)
From the continuity of u, at $t = t_{1}$ , it follows that
$\begin{aligned} J (t_{1}) = {‖ u (t_{1}) ‖}_{2}^{2} > J (0) ⩾ \frac{2 p}{(C_{2} + C_{1} μ) (p - 2)} E (0) . \end{aligned}$
(5.3)
Since (4.7) and Lemma 2.2 give
$\begin{aligned} E (0) & = E (t) ⩾ \frac{1}{2} {‖ u_{t} (t_{1}) ‖}_{2}^{2} + \frac{p - 2}{2 p} ({‖ \nabla u (t_{1}) ‖}_{2}^{2} + μ {‖ u (t_{1}) ‖}_{U_{0}}^{2}) + \frac{1}{p} H (u (t_{1})) \\ = \frac{1}{2} {‖ u_{t} (t_{1}) ‖}_{2}^{2} + \frac{p - 2}{2 p} ({‖ \nabla u (t_{1}) ‖}_{2}^{2} + μ {‖ u (t_{1}) ‖}_{U_{0}}^{2}) \\ ⩾ \frac{1}{2} {‖ u_{t} (t_{1}) ‖}_{2}^{2} + \frac{p - 2}{2 p} (C_{2} {‖ u (t_{1}) ‖}_{2}^{2} + C_{1} μ {‖ u (t_{1}) ‖}_{2}^{2}) \\ ⩾ \frac{(C_{2} + C_{1} μ) (p - 2)}{2 p} {‖ u (t_{1}) ‖}_{2}^{2}, \end{aligned}$
(5.4)
this contradicts (5.3). This leads to the conclusion.
Theorem 5.1.
Let $u_{0} \in X_{μ}^{s} (Ω)$ , $u_{1} \in L^{2} (Ω)$ , satisfying $\int_{Ω} u_{0} u_{1} d x > 0$ . Suppose that $E (0) > 0$ and (5.1) holds. Then the corresponding solutions to problem (1.1) blow up in finite time.
Proof.
Assume that $u = u (x, t)$ is a global solution of problem (1.1). Lemma 5.1 tells us $u \in N$ . We argue by contradiction. In Lemma 5.1, we have proved that $J (t) > 0$ . And by the Cauchy–Schwarz inequality we can get
$\begin{aligned} {(J^{'} (t))}^{2} = 4 (u, u_{t})^{2} ⩽ 4 ‖ u ‖_{2}^{2} ‖ u_{t} ‖_{2}^{2}, \end{aligned}$
then
$\begin{aligned} J (t) J^{″} (t) - (1 + α) (J^{'} (t))^{2} \\ ⩾ (2 ‖ u_{t} ‖_{2}^{2} - 2 ‖ \nabla u ‖_{2}^{2} - 2 μ ‖ u ‖_{U_{0}}^{2} + 2 \int_{Ω} u f (u) d x) J (t) - 4 (1 + α) ‖ u ‖_{2}^{2} ‖ u_{t} ‖_{2}^{2} \\ = J (t) [- 2 (1 + 2 α) ‖ u_{t} ‖_{2}^{2} - 2 ‖ \nabla u ‖_{2}^{2} - 2 μ ‖ u ‖_{U_{0}}^{2} + 2 \int_{Ω} u f (u) d x], \end{aligned}$
where $α > 0$ . We define
$\begin{aligned} ζ (t) := - 2 (1 + 2 α) ‖ u_{t} ‖_{2}^{2} - 2 ‖ \nabla u ‖_{2}^{2} - 2 μ ‖ u ‖_{U_{0}}^{2} + 2 \int_{Ω} u f (u) d x, \end{aligned}$
which together with the definition of $E (0)$ implies that
$\begin{aligned} ζ (t) & ⩾ - 2 (1 + 2 α) ‖ u_{t} ‖_{2}^{2} - 2 ‖ \nabla u ‖_{2}^{2} - 2 μ ‖ u ‖_{U_{0}}^{2} + 2 p \int_{Ω} F (u) d x \\ = - 2 (1 + 2 α) ‖ u_{t} ‖_{2}^{2} - 2 ‖ \nabla u ‖_{2}^{2} - 2 μ ‖ u ‖_{U_{0}}^{2} + p ‖ u_{t} ‖_{2}^{2} + p ‖ \nabla u ‖_{2}^{2} + p μ ‖ u ‖_{U_{0}}^{2} - 2 p E (0) \\ = (p - 2 - 4 α) ‖ u_{t} ‖_{2}^{2} + (p - 2) (‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2}) - 2 p E (0) \\ ⩾ (p - 2 - 4 α) ‖ u_{t} ‖_{2}^{2} + C_{2} (p - 2) ‖ u ‖_{2}^{2} + C_{1} (p - 2) μ ‖ u ‖_{2}^{2} - 2 p E (0) . \end{aligned}$
(5.5)
Let $α = (p - 2) / 4 > 0$ , we get
$\begin{aligned} ζ (t) ⩾ (C_{2} + C_{1} μ) (p - 2) ‖ u ‖_{2}^{2} - 2 p E (0), \end{aligned}$
which, together with (5.2), implies that
$\begin{aligned} J (t) J^{″} (t) - (1 + α) {(J^{'} (t))}^{2} ⩾ J (t) ζ (t) ⩾ J (t) [(C_{2} + C_{1} μ) (p - 2) ‖ u ‖_{2}^{2} - 2 p E (0)] ⩾ 0. \end{aligned}$
The remainder of the proof is similar to that of Theorem 3.8, we have $lim_{t \to T} ‖ u (t) ‖_{2} = + \infty$ .

6. .Upper and lower bounds on the blow-up time

From now on, in order to prove the upper and lower bound estimates for the blow-up time of the solutions of problem (1.1), we consider a special nonlinear source term given by $f (u) = | u |^{p - 2} u$ . Before proving the upper and lower bounds on blow-up time for problem (1.1), we provide a necessary lemma.

Lemma 6.1.

Suppose that $Ψ \in C^{2} ([0, T])$ satisfies

\begin{aligned} Ψ Ψ_{t t} - α Ψ_{t}^{2} + β Ψ Ψ_{t} + γ Ψ ⩾ 0, Ψ (t) ⩾ 0, Ψ (0) > 0, \end{aligned}

and

\begin{aligned} Ψ_{t} (0) > \frac{β}{α - 1} Ψ (0), {(Ψ_{t} (0) - \frac{β}{α - 1} Ψ (0))}^{2} > \frac{2 γ}{2 α - 1} Ψ (0) \end{aligned}

where

α > 1

β ⩾ 0

γ ⩾ 0

, then

\begin{aligned} \underset{t \to T}{\lim \sup} Ψ (t) = + \infty, \end{aligned}

where

\begin{aligned} T & ⩽ \frac{Ψ^{1 - α} (0)}{S}, \\ S^{2} & = (α - 1)^{2} Ψ^{- 2 α} (0) [{(Ψ_{t} (0) - \frac{β}{α + 1} Ψ (0))}^{2} - \frac{2 γ}{2 α - 1} Ψ (0)] . \end{aligned}

Moreover,

\begin{aligned} Ψ (t) ⩾ \frac{e^{\frac{β t}{α - 1}}}{(Ψ^{1 - α} (0) - S t)^{\frac{1}{α - 1}}} . \end{aligned}

For the proof of this lemma we refer to a similar version, see Theorem 2.1 in [15].

Theorem 6.1.

Let $u_{0} \in X_{0}^{s} (Ω)$ , $u_{1} \in L^{2} (Ω)$ , satisfying $\int_{Ω} u_{0} u_{1} d x > 0$ . Suppose that $E (0) > 0$ and $(\int_{Ω} u_{0} u_{1} d x)^{2} > 2 E (0) ‖ u_{0} ‖_{2}^{2}$ , then there exists a $T^{*} > 0$ , with $T^{*} ⩽ \frac{1}{S} ‖ u_{0} ‖_{2}^{\frac{2 - p}{2}}$ , such that $\underset{t \to T^{*}}{\lim \sup} ‖ u (t) ‖_{2} = + \infty$ , where

\begin{aligned} S^{2} = \frac{(p - 2)^{2}}{4} ‖ u_{0} ‖_{2}^{- 2 + p} [{(\int_{Ω} u_{0} u_{1} d x)}^{2} - 2 E (0) ‖ u_{0} ‖_{2}^{2}] . \end{aligned}

Proof.

Multiplying equation (1.1) by u and integrating over Ω, we have

\begin{aligned} \int_{Ω} u_{t t} u d x + \int_{Ω} {| \nabla u |}^{2} d x + μ \iint_{Q} \frac{| u (x, t) - u (y, t) |^{2}}{| x - y |^{N + 2 s}} d x d y = \int_{Ω} {| u |}^{p} d x . \end{aligned}

Furthermore,

\begin{aligned} \frac{1}{2} \int_{Ω} \frac{d^{2}}{d t^{2}} {| u |}^{2} d x - \int_{Ω} {| u_{t} |}^{2} d x + ‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2} = ‖ u ‖_{p}^{p} . \end{aligned}

Let

Ψ (t) = \int_{Ω} | u |^{2} d x

and

Φ (t) = \int_{Ω} | u_{t} |^{2} d x

, then

\begin{aligned} \frac{1}{2} \frac{d^{2}}{d t^{2}} Ψ (t) - Φ (t) + ‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2} = ‖ u ‖_{p}^{p} . \end{aligned}

(6.1)

Similarly, multiplying equation (1.1) by

u_{t}

and integrating over Ω, we have

\begin{aligned} \frac{d}{d t} (\frac{1}{2} Φ (t) + \frac{1}{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} ‖ u ‖_{U_{0}}^{2}) = \frac{d}{d t} (\frac{1}{p} ‖ u ‖_{p}^{p}) . \end{aligned}

(6.2)

From 0 to t integration of (6.2), it yields

\begin{aligned} \frac{1}{2} Φ (t) + \frac{1}{2} ‖ \nabla u ‖_{2}^{2} + \frac{μ}{2} ‖ u ‖_{U_{0}}^{2} - E (0) = \frac{1}{p} ‖ u ‖_{p}^{p}, \end{aligned}

(6.3)

which together with (6.1) implies that

\begin{aligned} \frac{1}{2} \frac{d^{2}}{d t^{2}} Ψ (t) + p E (0) = \frac{p + 2}{2} Φ (t) + \frac{p - 2}{2} (‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2}) . \end{aligned}

Since

p > 2

, this equation can be written as

\begin{aligned} \frac{1}{2} Ψ_{t t} (t) + p E (0) ⩾ \frac{p + 2}{2} Φ (t) . \end{aligned}

From Cauchy–Schwarz inequality, we obtain

\begin{aligned} Ψ_{t}^{2} (t) = 4 {(\int_{Ω} u u_{t} d x)}^{2} ⩽ 4 ‖ u ‖_{2}^{2} ‖ u_{t} ‖_{2}^{2} = 4 Ψ (t) Φ (t) . \end{aligned}

Hence, we can have

\begin{aligned} Ψ_{t t} (t) Ψ (t) + 2 p E (0) Ψ (t) ⩾ (p + 2) Φ (t) Ψ (t) ⩾ \frac{p + 2}{4} Ψ_{t}^{2} (t), \end{aligned}

i.e.

\begin{aligned} Ψ_{t t} (t) Ψ (t) - \frac{p + 2}{4} Ψ_{t}^{2} (t) + 2 p E (0) Ψ (t) ⩾ 0. \end{aligned}

(6.4)

From this, it is easy to see that

\begin{aligned} α = \frac{p + 2}{4} > 1, β = 0, γ = 2 p E (0) > 0, \end{aligned}

then

\begin{aligned} \frac{2 γ}{2 α - 1} = 8 E (0), \frac{β}{α - 1} = 0, α - 1 = \frac{p - 2}{4}, 1 - α = \frac{2 - p}{4} . \end{aligned}

Finally, applying Lemma 5.1 we can get the desired conclusion.

Theorem 6.2.

Let $2 < p ⩽ (2_{s}^{*} + 2) / 2$ . Suppose that $E (0) > 0$ and the solutions $u (t)$ of problem (1.1) blow up in finite time. Then it holds that

\begin{aligned} T ⩾ \int_{M (0)}^{\infty} \frac{1}{p E (0) + M (t) + p ε^{p - 1} I E (0)^{p - 1} + p^{2 - p} ε^{p - 1} I M {(t)}^{p - 1}} d M (t), \end{aligned}

where

M (0) = ‖ u_{0} ‖_{p}^{p}

I = 2^{2 p - 4}

ε = max {C_{0}^{2}, μ {C_{0}^{^{'}}}^{2}}

and

C_{0}

C_{0}^{^{'}}

are the best embedding constants for

H^{1} (Ω) ↪ L^{2 p - 2} (Ω)

and

U_{0} ↪ L^{2 p - 2} (Ω)

, respectively.

Proof.

The definition of $E (t)$ can be written as

\begin{aligned} ‖ u_{t} ‖_{2}^{2} + ‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2} = 2 E (t) + \frac{2}{p} ‖ u ‖_{p}^{p} = 2 E (0) + \frac{2}{p} ‖ u ‖_{p}^{p} . \end{aligned}

Set

M (t) = \int_{Ω} | u |^{p} d x = ‖ u ‖_{p}^{p}

. By the Cauchy–Schwarz inequality, the mean value inequality and Lemma 2.2, we can yield

\begin{aligned} M^{'} (t) & = p \int_{Ω} {| u |}^{p - 2} u u_{t} d x ⩽ p {({\int_{Ω} | u |}^{2 p - 2} d x)}^{\frac{1}{2}} {({\int_{Ω} | u_{t} |}^{2} d x)}^{\frac{1}{2}} \\ ⩽ \frac{p}{2} ({\int_{Ω} | u |}^{2 p - 2} d x + \int_{Ω} {| u_{t} |}^{2} d x) \\ = \frac{p}{2} (‖ u_{t} ‖_{2}^{2} + ‖ u ‖_{2 p - 2}^{2 p - 2}) . \end{aligned}

(6.5)

Lemma 2.2 implies

U_{0} ↪ L^{2 p - 2} (Ω)

H^{1} (Ω) ↪ L^{2 p - 2} (Ω)

, and we set

C_{0} > 0

C_{0}^{^{'}} > 0

are the best embedding constants. Therefore,

\begin{aligned} ‖ u ‖_{2 p - 2}^{2 p - 2} ⩽ {C_{0}}^{2 p - 2} ‖ \nabla u ‖_{2}^{2 p - 2} + (μ C_{0}^{^{'}})^{2 p - 2} ‖ u ‖_{U_{0}}^{2 p - 2}, \end{aligned}

(6.6)

where

μ ⩾ 0

. By (6.5) and (6.6), we get

\begin{aligned} M^{'} (t) & ⩽ \frac{p}{2} (‖ u_{t} ‖_{2}^{2} + C_{0}^{2 p - 2} ‖ \nabla u ‖_{2}^{2 p - 2} + {(μ {C^{'}}_{0})}^{2 p - 2} ‖ u ‖_{U_{0}}^{2 p - 2}) \\ ⩽ \frac{p}{2} [‖ u_{t} ‖_{2}^{2} + ε^{p - 1} {(‖ \nabla u ‖_{2}^{2} + μ ‖ u ‖_{U_{0}}^{2})}^{p - 1}] \\ ⩽ \frac{p}{2} [2 E (t) + \frac{2}{p} ‖ u ‖_{p}^{p} + ε^{p - 1} {(2 E (t) + \frac{2}{p} ‖ u ‖_{p}^{p})}^{p - 1}] \\ = p E (t) + M (t) + \frac{1}{2} p ε^{p - 1} {(2 E (t) + \frac{2}{p} ‖ u ‖_{p}^{p})}^{p - 1} \\ ⩽ p E (t) + M (t) + p ε^{p - 1} 2^{p - 3} [{(2 E (t))}^{p - 1} + {(\frac{2}{p} M (t))}^{p - 1}] \\ = p E (0) + M (t) + p ε^{p - 1} 2^{2 p - 4} E (0)^{p - 1} + p^{2 - p} ε^{p - 1} 2^{2 p - 4} M {(t)}^{p - 1} \\ = p E (0) + M (t) + p ε^{p - 1} I E (0)^{p - 1} + p^{2 - p} ε^{p - 1} I M {(t)}^{p - 1}, \end{aligned}

(6.7)

where

ε = max {C_{0}^{2}, μ {C_{0}^{^{'}}}^{2}}

I = 2^{2 p - 4}

. Then (6.7) implies

\begin{aligned} \frac{M^{'} (t)}{p E (0) + M (t) + p ε^{p - 1} I E (0)^{p - 1} + p^{2 - p} ε^{p - 1} I M {(t)}^{p - 1}} ⩽ 1. \end{aligned}

Integrating the above equation with respect to t from 0 to T, we can get

\begin{aligned} \int_{0}^{T} \frac{M^{'} (t)}{p E (0) + M (t) + p ε^{p - 1} I E (0)^{p - 1} + p^{2 - p} ε^{p - 1} I M {(t)}^{p - 1}} d t ⩽ \int_{0}^{T} 1 d t . \end{aligned}

(6.8)

That is to say, (6.8) gives

\begin{aligned} T ⩾ \int_{M (0)}^{M (T)} \frac{1}{p E (0) + M (t) + p ε^{p - 1} I E (0)^{p - 1} + p^{2 - p} ε^{p - 1} I M {(t)}^{p - 1}} d M (t) . \end{aligned}

According to the definition of blow-up of solutions and the Sobolev embedding from

L^{p} (Ω)

into

L^{2} (Ω)

, we obtain

lim_{t \to T^{-}} M (t) = + \infty

. Then from (6.8) we can draw the final conclusion.

Footnotes

Acknowledgements

B. Zhang was supported by the National Natural Science Foundation of China (No. 12171152) and the Shandong Provincial Natural Science Foundation (No. ZR2023MA090).

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