Global existence and asymptotic behavior of solutions to fractional ( p,q ) -Laplacian equations

Abstract

The aim of this paper is to consider the following fractional parabolic problem $\begin{array}{l} \{\begin{matrix} u_{t} + {(- Δ)}_{p}^{α} u + {(- Δ)}_{q}^{β} u = f (x, u) & (x, t) \in Ω \times (0, \infty), \\ u = 0 & (x, t) \in (R^{N} ∖ Ω) \times (0, \infty), \\ u (x, 0) = u_{0} (x) & x \in Ω, \end{matrix} \end{array}$ where $Ω \subset R^{N}$ is a bounded domain with Lipschitz boundary, ${(- Δ)}_{p}^{α}$ is the fractional p-Laplacian with $0 < α < 1 < p < \infty$ , ${(- Δ)}_{q}^{β}$ is the fractional q-Laplacian with $0 < β < α < 1 < q < p < \infty$ , $r > 1$ and $λ > 0$ . The global existence of nonnegative solutions is obtained by combining the Galerkin approximations with the potential well theory. Then, by virtue of a differential inequality technique, we give a decay estimate of solutions.

Keywords

global existence decay estimates

1. Introduction and the main results

In this paper, we discuss the global existence and asymptotical behavior of solutions for the following fractional p-Kirchhoff problem $\begin{array}{l} (1.1) & \{\begin{matrix} u_{t} + {(- Δ)}_{p}^{α} u + {(- Δ)}_{q}^{β} u = f (x, u) & (x, t) \in Ω \times (0, \infty), \\ u = 0 & (x, t) \in (R^{N} ∖ Ω) \times (0, \infty), \\ u (x, 0) = u_{0} (x) & x \in Ω, \end{matrix} \end{array}$ where $Ω \subset R^{N}$ is a bounded domain with Lipschitz boundary, ${(- Δ)}_{p}^{α}$ is the fractional p-Laplacian with $0 < α < 1 < p < \infty$ , ${(- Δ)}_{q}^{β}$ is the fractional q-Laplacian with $0 < β < 1 < q < \infty$ , which is nonlinear nonlocal operator defined on smooth functions by $\begin{array}{rcl} {(- Δ)}_{p}^{α} φ (x) = 2 lim_{ϵ \to 0^{+}} \int_{R^{N} ∖ B_{ϵ} (x)} \frac{| φ (x) - φ (y) |^{p - 2} (φ (x) - φ (y))}{| x - y |^{N + p α}} d y, x \in R^{N} . \end{array}$ This definition is consistent, up to a normalization constant depending on N and s. Throughout this paper $B_{ϵ} (x)$ denotes the ball in $R^{N}$ centered at $x \in R^{N}$ with radius $ϵ > 0$ . For the bascic properties and definitions of fractional operators and fractional Sobolev spaces, we refer the readers to [9,15,19,25].

Related to the function $f : Ω \times R \to R$ we assume that it is a continuous function, with $f (x, ξ) = 0$ for $ξ ⩽ 0$ and $x \in Ω$ and satisfies the following conditions: $(f_{1})$

there exists $p_{s}^{*} > r > p$ such that for any $ε > 0$ there exists $C_{ε} > 0$ such that $\begin{matrix} | f (x, z) | ⩽ ε | z |^{q - 1} + C_{ε} | z |^{r - 1} . \end{matrix}$

(f_{2})

There exists $μ > p$ such that, $\begin{matrix} f (x, z) z ⩾ μ F (x, z), \forall (x, z) \in Ω \times R, \end{matrix}$ where $F (x, z) = \int_{0}^{z} f (x, σ) d σ$ .

The fractional operators and related differential equations have important applications in many areas such as physics [17], mechanics chemistry, population dynamic [6,7], anomalous diffusion [5,33] and so on. Fractional diffusion equations are used to describe macroscopic transport and usually result in superdiffusion phenomenon. So far, the works on elliptic type problems involving the fractional Laplacian and its variants are quite large, here we just list a few, see [2,10,23–25,27,31,34,36,37] and the references cited there.

To the best of our knowledge, there are some papers to study parabolic problems involving fractional p-Laplacian. Pucci, Xiang and Zhang in [28] discussed the initial boundary value problem of following fractional p-Kirchhoff equation of parabolic type $\begin{array}{l} (1.2) & \{\begin{matrix} u_{t} + M (\iint_{R^{2 N}} \frac{| u (x, t) - u (y, t) |^{p}}{| x - y |^{N + s p}} d x d y) {(- Δ)}_{p}^{s} u = f (x, t), (x, t) \in Ω \times (0, \infty), \\ u = 0, (x, t) \in (R^{N} ∖ Ω) \times (0, \infty), \\ u (x, 0) = u_{0} (x), x \in Ω, \end{matrix} \end{array}$ where $M : [0, \infty) \to (0, \infty)$ is a continuous satisfying some assumptions. The authors obtained the well-posedness of problem (1.2) by employing the sub-differential approach. Further, the authors studied the asymptotical behaviors and finite time extinction property of solutions. In [26], Pan, Zhang and Cao considered the following degenerate Kirchhoff problem involving the fractional p-Laplacian $\begin{array}{l} \{\begin{matrix} u_{t} + {(\iint_{R^{2 N}} \frac{| u (x, t) - u (y, t) |^{p}}{| x - y |^{N + s p}} d x d y)}^{λ - 1} {(- Δ)}_{p}^{s} u = | u |^{q - 2} u, (x, t) \in Ω \times (0, \infty), \\ u = 0, (x, t) \in (R^{N} ∖ Ω) \times (0, \infty), \\ u (x, 0) = u_{0} (x), x \in Ω, \end{matrix} \end{array}$ where $1 < p < N / s$ , $1 ⩽ λ < N / (N - s p)$ and $p < q < N p / (N - s p)$ . The existence of a global solution was given by the potential well theory combined with the Galerkin method. See also [11,18,35] for an application of the potential well theory in fractional problems. Xiang, Rădulescu and Zhang [22] studied the following Kirchhoff problem $\begin{array}{l} \{\begin{matrix} u_{t} + M (\iint_{R^{2 N}} \frac{| u (x, t) - u (y, t) |^{2}}{| x - y |^{N + 2 s}} d x d y) {(- Δ)}^{s} u = | u |^{p - 2} u, (x, t) \in Ω \times (0, \infty), \\ u = 0, (x, t) \in (R^{N} ∖ Ω) \times (0, \infty), \\ u (x, 0) = u_{0} (x), x \in Ω . \end{matrix} \end{array}$ The local existence of nonnegative solutions was obtained by the Galerkin method and the global nonexistence of solutions was also investigated. In [38], Xiang and Yang studied the extinction and non-extinction of the following problem $\begin{array}{l} \{\begin{matrix} u_{t} + M ({[u]}_{s, p}^{p}) {(- Δ)}_{p}^{s} u = λ | u |^{r - 2} u - μ | u |^{q - 2} u & (x, t) \in Ω \times (0, \infty), \\ u = 0 & (x, t) \in (R^{N} ∖ Ω) \times (0, \infty), \\ u (x, 0) = u_{0} (x) & x \in Ω . \end{matrix} \end{array}$ Recently, Boudjeriou [4] considered the following fractional p-Laplacian problem with logarithmic growth $\begin{array}{l} \{\begin{matrix} u_{t} + {(- Δ)}_{p}^{s} u + | u |^{p - 2} u = | u |^{p - 2} u log (| u |) & (x, t) \in Ω \times (0, \infty), \\ u = 0 & (x, t) \in (R^{N} ∖ Ω) \times (0, \infty), \\ u (x, 0) = u_{0} (x) & x \in Ω . \end{matrix} \end{array}$ By using the Galerkin method and potential well theory, the author established the local existence of solutions and proved that the local solutions blow up in finite time with arbitrary negative initial energy and suitable initial values. For the asymptotical behaviors of solutions of fractional diffusion equations, we also refer to [13,16].

Inspired by above cited papers, in this paper, we considered the global existence and asymptotical behavior of solutions of problem (1.1). As introduced in [8], one of the motivation of studying problems like (1.1) is related to the reaction–diffusion equation $\begin{array}{l} (1.3) & u_{t} - div (D (u) \nabla u) = c (x, u), \end{array}$ where $D (u) = | \nabla u |^{p - 2} + | \nabla u |^{q - 2}$ . Equation like (1.3) appears in many fields such as biophysics, chemical reaction design, and plasma physics. For the applications of $(p, q)$ -Laplacian in nonlinear elasticity, we refer to [21]. In recent years, the existence and multiplicity of stationary solutions of problem (1.1) have been received much more attention, we refer to [1,12,14]. To the best of our knowledge, this paper is the first time to deal with parabolic type problems involving the fractional $(p, q)$ -Laplacian.

In order to obtain the existence of weak solutions for (1.1), we consider the space $\begin{matrix} W_{0}^{α, p} (Ω) = {u \in L^{p} (Ω) : {[u]}_{α, p} < \infty, u = 0 a.e. in R^{N} ∖ Ω}, \end{matrix}$ where the $(α, p)$ -Gagliardo seminorm is defined as $\begin{matrix} {[u]}_{α, p} = {(\iint_{R^{2 N}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + α p}} d x d y)}^{1 / p} . \end{matrix}$ Equip $W_{0}^{α, p} (Ω)$ with the norm $\begin{matrix} ‖ u ‖ : = {[u]}_{α, p} . \end{matrix}$ Then $(W_{0}^{α, p} (Ω), ‖ \cdot ‖)$ is a reflexive Banach space. The fractional critical exponent is defined by $\begin{matrix} p_{α}^{*} = \{\begin{matrix} \frac{N p}{N - α p} & if α p < N, \\ \infty & if α p ⩾ N . \end{matrix} \end{matrix}$

Hereafter, we denote by $⟨ \cdot, \cdot ⟩$ the duality pairing between ${(W^{α, p} (Ω))}^{'}$ and $W^{α, p} (Ω)$ , where ${(W^{α, p} (Ω))}^{'}$ denotes the dual space of $W^{α, p} (Ω)$ . Associated with equation (1.1) we have the energy functional $E : W_{0}^{α, p} (Ω) \to R$ given by $\begin{matrix} (1.4) & E (u) = \frac{1}{p} {[u]}_{α, p}^{p} + \frac{1}{q} {[u]}_{β, q}^{q} - \int_{Ω} F (x, u) d x, \end{matrix}$ where $F (x, r) = \int_{0}^{r} f (x, σ) d σ$ . By $(f_{1})$ , we have $\begin{matrix} (1.5) & | F (x, z) | ⩽ ε | z |^{q} + C_{ε} | z |^{r}, \forall (x, z) \in Ω \times R, \forall ε > 0 . \end{matrix}$ In view of the fractional Sobolev embedding theorem, one can verify that E is well defined, of class $C^{1} (W_{0}^{α, p} (Ω), R)$ and $\begin{matrix} ⟨ E^{'} (u), v ⟩ = {⟨ u, v ⟩}_{α, β, p, q} - \int_{Ω} f (x, u) v d x, \forall u, v \in W_{0}^{α, p} (Ω), \end{matrix}$ where $\begin{array}{l} {⟨ u, v ⟩}_{α, β, p, q} & = \iint_{R^{2 N}} \frac{| u (x) - u (y) |^{p - 2} (u (x) - u (y))}{| x - y |^{N + α p}} (v (x) - v (y)) d x d y \\ + \iint_{R^{2 N}} \frac{| u (x) - u (y) |^{q - 2} (u (x) - u (y))}{| x - y |^{N + β q}} (v (x) - v (y)) d x d y . \end{array}$ The potential well associated with problem (1.1) is defined as $\begin{matrix} (1.6) & W = {u \in W_{0}^{α, p} (Ω) : E (u) < d, I (u) > 0} \cup {0}, \end{matrix}$ where $I : W_{0}^{α, p} (Ω) \to R$ is the Nehari functional given by $\begin{matrix} (1.7) & I (u) = {[u]}_{α, p}^{p} + {[u]}_{β, q}^{q} - \int_{Ω} f (x, u) u d x . \end{matrix}$ Related to the functional E, we give the well-known Nehari manifold. $\begin{matrix} N = {u \in W_{0}^{α, p} (Ω) : I (u) = 0} . \end{matrix}$ Define $\begin{matrix} (1.8) & d = inf_{u \in N} E (u) . \end{matrix}$ Before introducing our main results, we first give two definitions.

Definition 1.1 (Weak solution).

We say that $u \in L^{\infty} (0, T; W_{0}^{α, p} (Ω))$ is a weak solution of problem (1.1) if, $u_{t} \in L^{2} (0, T; L^{2} (Ω))$ and the following equalities hold

$\int_{Ω} u_{t} (t) v d x + {⟨ u (t), v ⟩}_{α, β, p, q} = \int_{Ω} f (x, u (t)) v d x$ for each $v \in W_{0}^{α, p} (Ω)$ and a.e. time $0 < t < T$ , and

$u (0) = u_{0}$ .

Definition 1.2 (Maximal existence time).

Let $u (t)$ be a solution of problem (1.1). We define the maximal existence time $T_{max}$ of u as follows: $\begin{matrix} T_{max} = sup {t > 0 : u = u (t) exists on [0, T]} . \end{matrix}$

If $T_{max} < \infty$ we say that the solution of (1.1) blows up in finite time and $T_{max}$ is the blow-up time.

If $T_{max} = \infty$ , we say that the solution is global.

The proof of the following Theorem 1.2 relies on the potential well method which was introduced by Sattinger in [30].

Theorem 1.1.
Let $0 < β < α < 1$ and $1 < q ⩽ p < \frac{N}{α}$ . Assume that f satisfies $(f_{1})$ – $(f_{2})$ , $0 ⩽ u_{0} \in W_{0}^{α, p} (Ω)$ and the following condition holds true: $\begin{matrix} (1.9) & E (u_{0}) < d and I (u_{0}) > 0 . \end{matrix}$ Then problem ( 1.1 ) admits a global positive weak solution $u = u (t)$ . Moreover, $\begin{matrix} (1.10) & \int_{0}^{t} {‖ u_{τ} (τ) ‖}_{2}^{2} d τ + E (u (t)) ⩽ E (u_{0}), a.e. t \in [0, + \infty) . \end{matrix}$

The following theorem shows the asymptotic behavior of global solutions to problem (1.1).
Theorem 1.2.
Assume that the assumptions given in Theorem 1.1 are satisfied. If $\begin{matrix} E (u_{0}) < (\frac{μ - p}{p μ}) {(\frac{1}{C_{λ_{1}} C_{r}^{r}})}^{\frac{p}{r - p}}, \end{matrix}$ then we have the following estimates: for $p > 2$ $\begin{matrix} {‖ u (t) ‖}_{2} ⩽ ‖ u_{0} ‖_{2} {(\frac{p}{2 + (p - 2) χ t})}^{\frac{1}{p - 2}}, \forall t > 0; \end{matrix}$ and for $p = 2$ $\begin{array}{l} {‖ u (t) ‖}_{2} ⩽ ‖ u_{0} ‖_{2} e^{\frac{1}{2} (1 - χ t)}, \forall t > 0, \end{array}$ where $χ = 2 C_{2}^{- p} (1 - C_{λ_{1}} C_{r}^{r} ({(\frac{p μ}{μ - p})}^{\frac{r - p}{p}} {(E (u_{0}))}^{\frac{r - p}{p}}))$ and $C_{2}$ , $C_{r}$ denote the embedding constants from $W_{0}^{α, p} (Ω)$ to $L^{2} (Ω)$ and $L^{r} (Ω)$ , respectively.

The rest of the paper is organized as follows. In Section 2, we will give some preliminary lemmas. In Section 3, the global existence of solutions for problem (1.1) is obtained by the Galerkin method and the potential well theory. A decay estimates of solutions for problem (1.1) is given in Section 4.
2. Preliminaries

In this section, we provide some basic results which will be used in the next sections.

Lemma 2.1 ([3, Lemma 2.2]).

Let $0 < β < α < 1$ , $1 < q ⩽ p$ and let Ω be a smooth bounded domain in $R^{N}$ with $N > α p$ . Then $W_{0}^{α, p} (Ω)$ can be continuously embedded in $W_{0}^{β, q} (Ω)$ and there exists $C = C (| Ω |, N, p, q, α, β) > 0$ such that $\begin{array}{l} {[u]}_{β, q} ⩽ C {[u]}_{α, p} for all u \in W_{0}^{α, p} (Ω) . \end{array}$

In order to show the existence of global solutions to problem (1.1), we give some necessary properties of the operator $L : W_{0}^{α, p} (Ω) \to {(W_{0}^{α, p} (Ω))}^{'}$ defined by $\begin{matrix} ⟨ L (u), v ⟩ = {⟨ u, v ⟩}_{α, β, p, q} . \end{matrix}$

Lemma 2.2.
The operator $L : W_{0}^{α, p} (Ω) \to {(W_{0}^{α, p} (Ω))}^{'}$ is monotone and $\begin{matrix} {‖ L (u) ‖}_{{(W_{0}^{α, p} (Ω))}^{'}} ⩽ {[u]}_{α, p}^{p - 1} + C {[u]}_{β, q}^{q - 1} . \end{matrix}$
Proof.
Let $u, v \in W_{0}^{α, p} (Ω)$ . Then we have $\begin{array}{l} ⟨ L (u) - L (v), u - v ⟩ & = ⟨ L (u), u ⟩ + ⟨ L (v), v ⟩ - ⟨ L (u), v ⟩ - ⟨ L (v), u ⟩ \\ (2.1) & = {[u]}_{α, p}^{p} + {[u]}_{β, q}^{q} + {[v]}_{α, p}^{p} + {[v]}_{β, q}^{q} - ⟨ L (u), v ⟩ - ⟨ L (v), u ⟩ . \end{array}$ Using the following inequality $\begin{matrix} | a^{p - 2} a b | ⩽ \frac{p - 1}{p} | a |^{p} + \frac{1}{p} | b |^{p}, \forall p ⩾ 1, \end{matrix}$ we deduce $\begin{matrix} - ⟨ L (u), v ⟩ ⩾ \frac{1 - p}{p} {[u]}_{α, p}^{p} - \frac{1}{p} {[v]}_{α, p}^{p} + \frac{1 - q}{q} {[u]}_{β, q}^{q} - \frac{1}{q} {[v]}_{β, q}^{q} \end{matrix}$ and $\begin{matrix} - ⟨ L (v), u ⟩ ⩾ \frac{1 - p}{p} {[v]}_{α, p}^{p} - \frac{1}{p} {[u]}_{α, p}^{p} + \frac{1 - q}{q} {[v]}_{β, q}^{q} - \frac{1}{q} {[u]}_{β, q}^{q} . \end{matrix}$ Inserting the last inequalities in (2.1), it yields $\begin{matrix} ⟨ L (u) - L (v), u - v ⟩ ⩾ 0, \forall u, v \in W_{0}^{α, p} (Ω) . \end{matrix}$ It remains to show that $‖ L u ‖_{{(W_{0}^{α, p} (Ω))}^{'}} ⩽ {[u]}_{α, p}^{p - 1} + C {[u]}_{β, q}^{q - 1}$ . By the Hölder inequality, we have $\begin{array}{rcl} ⟨ L (u), v ⟩ & = & \iint_{R^{2 N}} \frac{| u (x) - u (y) |^{p - 2} (u (x) - u (y))}{| x - y |^{N (1 - \frac{1}{p}) + (p - 1) α}} \frac{(v (x) - v (y))}{| x - y |^{\frac{N}{p} + α}} d x d y \\ + \iint_{R^{2 N}} \frac{| u (x) - u (y) |^{q - 2} (u (x) - u (y))}{| x - y |^{N (1 - \frac{1}{q}) + (q - 1) β}} \frac{(v (x) - v (y))}{| x - y |^{\frac{N}{q} + β}} d x d y \\ ⩽ & {[u]}_{α, p}^{p - 1} {[v]}_{α, p} + {[u]}_{β, q}^{q - 1} {[v]}_{β, q}, \end{array}$ which together with Lemma 2.1 leads to $\begin{array}{l} ⟨ L (u), v ⟩ ⩽ ({[u]}_{α, p}^{p - 1} + C {[u]}_{β, q}^{q - 1}) {[v]}_{α, p} . \end{array}$ The proof is now complete. □
Lemma 2.3.
The operator $L : W_{0}^{α, p} (Ω) \to {(W_{0}^{α, p} (Ω))}^{'}$ is hemicontinuous.
Proof.
We are going to prove that the map $t \mapsto ⟨ L (u + t v), w ⟩$ is continuous on $[0, 1]$ for all $u, v, w \in W_{0}^{α, p} (Ω)$ , i.e, $\begin{matrix} lim_{t \to 0} ⟨ L (u + t v), w ⟩ = ⟨ L (u), w ⟩ \end{matrix}$ for all $w \in W_{0}^{α, p} (Ω)$ . Observe that $\begin{array}{rcl} ⟨ L (u + t v), w ⟩ = {⟨ u + t v, w ⟩}_{α, β, p, q} . \end{array}$ We define $G_{t} : [0, 1] \to R$ by $\begin{array}{l} G_{t} (x, y) & = \frac{| u (x) + t v (x) - u (y) - t v (y) |^{p - 2} (u (x) + t v (x) - u (y) - t v (y)) (w (x) - w (y))}{| x - y |^{N + α p}} \\ + \frac{| u (x) + t v (x) - u (y) - t v (y) |^{q - 2} (u (x) + t v (x) - u (y) - t v (y)) (w (x) - w (y))}{| x - y |^{N + β q}} \end{array}$ and set $\begin{array}{l} G (x, y) & = \frac{| u (x) - u (x) |^{p - 2} (u (x) - u (y)) (w (x) - w (y))}{| x - y |^{N + α p}} \\ + \frac{| u (x) - u (y) |^{q - 2} (u (x) - u (y)) (w (x) - w (y))}{| x - y |^{N + β q}} . \end{array}$ By Young’s inequality, one can show that $\begin{array}{l} | G_{t} (x, y) | & ⩽ \frac{p - 1}{p} \frac{| u (x) + t v (x) - u (y) - t v (y) |^{p}}{| x - y |^{N + α p}} + \frac{1}{p} \frac{| w (x) - w (y) |^{p}}{| x - y |^{N + α p}} \\ + \frac{q - 1}{q} \frac{| u (x) + t v (x) - u (y) - t v (y) |^{q}}{| x - y |^{N + β q}} + \frac{1}{q} \frac{| w (x) - w (y) |^{q}}{| x - y |^{N + β q}} \\ ⩽ \frac{(p - 1) 2^{p - 1}}{p} \frac{(| u (x) - u (y) |^{p} + | v (x) - v (y) |^{p})}{| x - y |^{N + α p}} + \frac{1}{p} \frac{| w (x) - w (y) |^{p}}{| x - y |^{N + α p}} \\ + \frac{(q - 1) 2^{q - 1}}{q} \frac{(| u (x) - u (y) |^{q} + | v (x) - v (y) |^{q})}{| x - y |^{N + β q}} + \frac{1}{q} \frac{| w (x) - w (y) |^{q}}{| x - y |^{N + β q}} \\ : = h (x, y) \in L^{1} (R^{2 N}) . \end{array}$ Obviously, ${lim}_{t \to 0} G_{t} (x, y) = G (x, y)$ . Thus, by the Lebesgue dominated convergence theorem, we obtain the desired result. □

The following lemma is taken from [39, Remark 3.2.2, p. 118].
Lemma 2.4.
Assume that the operator L satisfies the properties in Lemmas 2.2 and 2.3 . Moreover, it possesses the following property: $\begin{matrix} (2.2) & \{\begin{matrix} u_{m} \to u weakly in L^{p} (0, T, W_{0}^{α, p} (Ω)), (1 < p < \infty) \\ L (u_{m}) \to χ weakly in L^{p^{'}} (0, T, {(W_{0}^{α, p} (Ω))}^{'}) with \frac{1}{p^{'}} + \frac{1}{p} = 1, \\ {lim sup}_{m \to \infty} \int_{0}^{T} ⟨ L (u_{m}), u_{m} ⟩ d t ⩽ \int_{0}^{T} ⟨ χ, u ⟩ d t, \end{matrix} \end{matrix}$ then, $\begin{matrix} χ = L u . \end{matrix}$
Proposition 2.5 ([32, Proposition 1.2]).

Let V be a Banach space which is dense and continuously embedded in the Hilbert space H. We identify $H = H^{'}$ so that $V ↪ H = H^{'} ↪ V^{'}$ . Then the Banach space $W_{p} = {u \in L^{p} (0, T, V), u^{'} \in L^{p^{'}} (0, T, V^{'})}$ is contained in $C ([0, T], H)$ . Moreover, if $u \in W_{p}$ then $‖ u (t) ‖_{L^{2} (Ω)}$ is absolutely continuous on $[0, T]$ , we have $\begin{matrix} \frac{d}{d t} {‖ u (t) ‖}_{L^{2} (Ω)}^{2} = 2 {(u (t), u (t))}_{L^{2} (Ω)}, a.e. on [0, T], \end{matrix}$ and there is a constant $C > 0$ such that $\begin{matrix} ‖ u ‖_{C (0, T, H)} ⩽ C ‖ u ‖_{W_{p}}, for all u \in W_{p} . \end{matrix}$

To get the compactness of approximate solutions, we need the following Lions–Aubin lemma.

Proposition 2.6 ([32, Proposition 1.3]).

Let $B_{0}$ , B, $B_{1}$ be three reflexive Banach spaces with $B_{0} \subset B \subset B_{1}$ . Assume that $B_{0} ↪ B$ is compact and $B ↪ B_{1}$ is continuous. Let $1 < p < \infty$ , $1 < q < \infty$ , and define $\begin{matrix} W = {v : v \in L^{p} (0, T; B_{0}), v_{t} \in L^{q} (0, T; B_{1})} . \end{matrix}$ Then the embedding $W ↪ L^{p} (0, T, B)$ is compact.

At the end of this section, we consider the problem (1.1) in the stationary case. We point out that if we replace u in the following by $u (t)$ for any $t \in [0, T)$ , all the facts are still valid.

Lemma 2.7.
The functional E restricts to $N$ is bounded from below.
Proof.
For each $u \in N$ , we have $\begin{array}{l} {[u]}_{α, p}^{p} + {[u]}_{β, q}^{q} = \int_{Ω} f (x, u) u d x . \end{array}$ Then $\begin{matrix} E (u) = \frac{1}{p} {[u]}_{α, p}^{p} + \frac{1}{q} {[u]}_{β, q}^{q} - \int_{Ω} F (x, u) d x - \frac{1}{μ} ({[u]}_{α, p}^{p} + {[u]}_{β, q}^{q}) + \frac{1}{μ} \int_{Ω} f (x, u) u d x . \end{matrix}$ Using $(f_{2})$ , we deduce $\begin{matrix} E (u) ⩾ (\frac{1}{p} - \frac{1}{μ}) {[u]}_{α, p}^{p} + (\frac{1}{q} - \frac{1}{μ}) {[u]}_{β, q}^{q} ⩾ 0 . \end{matrix}$ Thus, we obtain the desired result. □
Lemma 2.8.
The real number d given by ( 1.8 ) is positive, that is, $d > 0$ .
Proof.
Arguing by contradiction, we assume that $d = 0$ . Then, there is $(u_{n}) \subset N$ such that $E (u_{n}) \to 0$ . In view of the proof of Lemma 2.7, we have ${[u_{n}]}_{s, p}^{p}, {[u_{n}]}_{β, q}^{q} \to 0$ as $n \to \infty$ . Thus, it follows from the condition $(f_{1})$ that $\begin{array}{rcl} | \int_{Ω} f (x, u_{n}) u_{n} d x | & ⩽ & ε C {[u_{n}]}_{β, q}^{q} + C_{ε} {[u_{n}]}_{α, p}^{r}, \forall ε > 0 . \end{array}$ On the other hand, we have $\begin{matrix} {[u_{n}]}_{α, p}^{p} + {[u_{n}]}_{β, q}^{q} = \int_{Ω} f (x, u_{n}) u_{n} d x ⩽ ε C {[u_{n}]}_{β, q}^{q} + C_{ε} {[u_{n}]}_{α, p}^{r}, \forall ε > 0 . \end{matrix}$ Taking $ε = \frac{1}{C}$ , we get $\begin{matrix} {[u_{n}]}_{α, p}^{p} ⩽ C_{ε} {[u_{n}]}_{α, p}^{r} . \end{matrix}$ Since $u_{n} \neq 0$ and $r > q$ , it follows that $\begin{matrix} {[u_{n}]}_{α, p}^{r - p} ⩾ \frac{1}{C_{ε}} > 0, \end{matrix}$ which yields a contradiction by letting $n \to \infty$ . □

3. Global existence of solutions

In this section by means of the Galerkin method combined with the potential well theory, we establish the existence of global solutions to the problem (1.1). By (1.9) and $(f_{2})$ , we observe that $\begin{matrix} E (u_{0}) ⩾ (\frac{1}{p} - \frac{1}{μ}) {[u_{0}]}_{α, p}^{p} + (\frac{1}{q} - \frac{1}{μ}) {[u_{0}]}_{β, q}^{q} + \frac{1}{μ} I (u_{0}) > 0 . \end{matrix}$

Proof of Theorem 1.1.
We divide the proof into its naturally arising steps.

Step 1 Galerkin approximation.

Since $p ⩾ \frac{2 N}{N + 2 α}$ , we have the Gelfand triple $\begin{matrix} W_{0}^{α, p} (Ω) ↪^{c, d} L^{2} (Ω) ↪^{c, d} {(W_{0}^{α, p} (Ω))}^{'} . \end{matrix}$ Here, $↪^{c, d}$ denotes a dense and compact embedding, and ${(W_{0}^{α, p} (Ω))}^{'}$ is the dual space of $W_{0}^{α, p} (Ω)$ . Let ${V_{n}}_{n \in N}$ be a Galerkin scheme of the separable Banach space $W_{0}^{α, p} (Ω)$ , i.e, $\begin{matrix} (3.1) & V_{n} = span {e_{1}, e_{2}, \dots, e_{n}}, {\overline{⋃_{n \in N} V_{n}}}^{‖ \cdot ‖} = W_{0}^{α, p} (Ω), \end{matrix}$ with ${e_{j}}_{j = 1}^{\infty}$ is an orthonormal basis in $L^{2} (Ω)$ . Let $u_{0} \in W_{0}^{α, p} (Ω)$ . Then we can find $u_{0 n} \in V_{n}$ such that $\begin{matrix} (3.2) & u_{n} (0) = u_{0 n} \to u_{0} strongly in W_{0}^{α, p} (Ω) as n \to \infty . \end{matrix}$ For each n, we look for the approximate solutions $u_{n} (x, t) = \sum_{j = 1}^{n} g_{j n} (t) e_{j} (x)$ , where the unknown functions $g_{j n}$ satisfy the following identities: $\begin{matrix} (3.3) & \int_{Ω} u_{n t} (t) e_{j} d x + ⟨ L (u_{n} (t)), e_{j} ⟩ = \int_{Ω} f (x, u_{n} (t)) e_{j} d x, j \in {1, \dots, n}, \end{matrix}$ with the initial conditions $\begin{matrix} (3.4) & u_{n} (0) = u_{0 n} . \end{matrix}$ Then (3.3)–(3.4) is equivalent to the following initial value problem for a system of nonlinear ordinary differential equations on $g_{j n}$ : $\begin{matrix} (3.5) & \{\begin{matrix} g_{j n}^{'} (t) = H_{j} (g (t)), j = 1, 2, \dots, n, t \in [0, t_{0}], \\ g_{j n} (0) = a_{j n}, j = 1, 2, \dots, n, \end{matrix} \end{matrix}$ where $H_{j} (g (t)) = - ⟨ L (u_{n}), e_{j} ⟩ + \int_{Ω} f (x, u_{n}) e_{j} d x$ and $a_{j n} = \int_{Ω} u_{0 n} e_{j} d x$ . By the Peano Theorem, there is $t_{0, n} > 0$ depending on $| a_{j n} |$ such that problem (3.5) admits a unique local solution $g_{j n} \in C^{1} ([0, t_{0, n}])$ . Hereafter, we will assume that $[0, T_{0, n})$ is the maximal interval of existence of the solution $u_{n} (t)$ .

Step 2 A priori estimates.

Multiplying the jth equation in (3.3) by $g_{j n}^{'} (t)$ and summing over j from 1 to n, we obtain $\begin{matrix} (3.6) & {‖ u_{n t} (t) ‖}_{2}^{2} + \frac{d}{d t} E (u_{n} (t)) = 0, t \in [0, T_{0 n}) . \end{matrix}$ Integrating (3.6) over $(0, t)$ yields $\begin{matrix} (3.7) & \int_{0}^{t} {‖ u_{n τ} (τ) ‖}_{2}^{2} d τ + E (u_{n} (t)) = E (u_{0 n}), t \in [0, T_{0, n}) . \end{matrix}$ Since $u_{0 n}$ converges to $u_{0}$ strongly in $W_{0}^{α, p} (Ω)$ , by the continuity of E it follows that $\begin{matrix} E (u_{0 n}) \to E (u_{0}), as n \to + \infty . \end{matrix}$ From the assumption that $E (u_{0}) < d$ , we have $E (u_{0 n}) < d$ , for sufficiently large n. This combined with (3.7) yields $\begin{matrix} (3.8) & \int_{0}^{t} {‖ u_{n τ} (τ) ‖}_{2}^{2} d τ + E (u_{n} (t)) < d, t \in [0, T_{0, n}), \end{matrix}$ for sufficiently large n. We will show that $T_{0, n} = + \infty$ and $\begin{matrix} (3.9) & u_{n} (t) \in W, \forall t ⩾ 0, \end{matrix}$ for sufficiently large n. Suppose by contradiction that $u_{n} (t_{1}) \notin W$ for some $t_{1} \in [0, T_{0, n})$ . Let $t_{} \in [0, T_{0, n})$ be the smallest time for which $u_{n} (t_{}) \notin W$ . Then, by continuity of $u_{n} (t)$ , we get $u_{n} (t_{}) \in \partial W$ . Hence, it turns out that $\begin{matrix} (3.10) & E (u_{n} (t_{})) = d \end{matrix}$ or $\begin{matrix} (3.11) & I (u_{n} (t_{})) = 0 . \end{matrix}$ It is clear that (3.10) could not occur by (3.8) while if (3.11) holds. Then, by the definition of d, we have $\begin{matrix} E (u_{n} (t_{})) ⩾ inf_{u \in N} E (u) = d, \end{matrix}$ which contradicts (3.8). Consequently, (3.9) holds.

Combining (3.8) and (3.9), we deduce $\begin{matrix} \frac{1}{μ} I (u_{n} (t)) + (\frac{1}{p} - \frac{1}{μ}) {[u_{n} (t)]}_{α, p}^{p} + (\frac{1}{q} - \frac{1}{μ}) {[u_{n} (t)]}_{β, q}^{q} < d, t \in [0, T_{0, n}) . \end{matrix}$ It follows from (3.8) and (3.9) that $\begin{matrix} (3.12) & (\frac{1}{p} - \frac{1}{μ}) {[u_{n} (t)]}_{α, p}^{p} + (\frac{1}{p} - \frac{1}{μ}) {[u_{n} (t)]}_{β, q}^{q} < d, \int_{0}^{t} {‖ u_{n τ} (τ) ‖}_{2} d τ < d, \end{matrix}$ for all $t \in [0, T_{0, n})$ . Therefore, we conclude that $T_{0, n} = + \infty$ . Actually, if $T_{0, n} < + \infty$ , then we must have ${lim}_{t \to T_{0, n}^{-}} ‖ u_{n} (t) ‖ = + \infty$ , see for example [29, Lemma 2.4, p. 48].

Step 3 The limit process.

From (3.12), we get the existence of a function u and a subsequence of ${u_{n}}_{n = 1}^{\infty}$ still denoted by ${u_{n}}_{n = 1}^{\infty}$ such that $\begin{matrix} (3.13) & \{\begin{matrix} u_{n} ⇀^{} u & weakly star in L^{\infty} (0, + \infty; W_{0}^{α, p} (Ω) \cap W_{0}^{β, q} (Ω)), \\ u_{n t} ⇀ u_{t} & weakly in L^{2} (0, + \infty; L^{2} (Ω)), \\ L (u_{n}) ⇀^{} Ξ & weakly star in L^{\infty} (0, + \infty; {(W_{0}^{α, p} (Ω))}^{*}) . \end{matrix} \end{matrix}$ Moreover, from Proposition 2.6, for all $T > 0$ we have $\begin{matrix} u_{n} \to u in L^{2} ([0, T], L^{2} (Ω)) . \end{matrix}$ Furthermore, we get $\begin{matrix} (3.14) & u_{n} (x, t) \to u (x, t) a.e. x \in Ω, \forall t ⩾ 0 . \end{matrix}$ Using (3.14) and $(f_{1})$ , we deduce from the Vitali convergence theorem that $\begin{array}{l} f (x, u_{n}) \to f (x, u) strongly in L^{\frac{r}{r - 1}} (Ω) . \end{array}$ This implies that $\begin{matrix} (3.15) & lim_{n \to \infty} \int_{0}^{t} \int_{Ω} f (x, u_{n} (τ)) w_{j} d x d τ = \int_{0}^{t} \int_{Ω} f (x, u (τ)) w_{j} d x d τ, \forall t > 0 . \end{matrix}$ Moreover, from (3.13) we have $\begin{matrix} lim_{n \to \infty} \int_{0}^{t} ⟨ L (u_{n} (τ)), w_{j} ⟩ d τ = \int_{0}^{t} ⟨ Ξ, w_{j} ⟩ d τ, \forall t > 0, \end{matrix}$ and $\begin{matrix} lim_{n \to \infty} \int_{0}^{t} \int_{Ω} u_{n τ} (τ) w_{j} d x d τ = \int_{0}^{t} \int_{Ω} u_{τ} (τ) w_{j} d x d τ, \forall t > 0 . \end{matrix}$ For any fixed j, letting $n \to \infty$ in (3.3), we obtain $\begin{array}{rcl} (3.16) & \int_{0}^{t} \int_{Ω} u_{τ} (τ) w_{j} d x d τ + \int_{0}^{t} ⟨ Ξ, w_{j} ⟩ d τ = \int_{0}^{t} \int_{Ω} f (x, u (τ)) w_{j} d x d τ . \end{array}$ Since $V_{m}$ is dense in $W_{0}^{α, p} (Ω)$ , it follows that $\begin{matrix} (3.17) & \int_{Ω} u_{t} (t) v d x + ⟨ Ξ, v ⟩ = \int_{Ω} f (x, u (t)) v d x, for a.e. t \in (0, + \infty) \end{matrix}$ and for all $v \in W_{0}^{α, p} (Ω)$ . From (3.13) and [39, see, Lemma 3.1.7], we obtain $\begin{matrix} u_{n} (0) \to u (0) weakly in L^{2} (Ω) . \end{matrix}$ However, by (3.2) we know that $u_{n} (0) \to u_{0}$ in $W_{0}^{α, p} (Ω)$ , in particular $u_{n} (0) \to u_{0}$ in $L^{2} (Ω)$ , and so, $u (0) = u_{0}$ . This shows that u satisfies the first initial condition.

Next step is to prove that $\begin{matrix} (3.18) & Ξ = {(- Δ)}_{p}^{α} u + {(- Δ)}_{q}^{β} u . \end{matrix}$ Indeed, multiplying (3.3) by $g_{j n} (t)$ and summing up from 1 to n, afterward integrating over $(0, T)$ , it leads to $\begin{matrix} (3.19) & \int_{0}^{T} ⟨ L (u_{n} (t)), u_{n} (t) ⟩ d t = - \frac{1}{2} {‖ u_{n} (T) ‖}_{2}^{2} + \frac{1}{2} {‖ u_{n} (0) ‖}_{2}^{2} + \int_{0}^{T} \int_{Ω} f (x, u_{n} (t)) u_{n} (t) d x d t . \end{matrix}$ Using the Vitali convergence theorem, one can show that $\begin{matrix} \int_{0}^{T} \int_{Ω} f (x, u_{n}) u_{n} d x d t \to \int_{0}^{T} \int_{Ω} f (x, u) u d x d t . \end{matrix}$ Letting $n \to \infty$ in (3.19) we obtain $\begin{array}{l} \underset{m \to \infty}{lim sup} \int_{0}^{T} ⟨ L (u_{n} (t)), u_{n} (t) ⟩ d t \\ (3.20) & = - \frac{1}{2} {‖ u (T) ‖}_{2}^{2} + \frac{1}{2} ‖ u_{0} ‖_{2}^{2} + \int_{0}^{T} \int_{Ω} f (x, u (t)) u (t) d x d t . \end{array}$ Choosing u with v in (3.17), afterward integrating over $(0, T)$ and using Proposition 2.5, we obtain $\begin{matrix} (3.21) & \int_{0}^{T} ⟨ Ξ, u (t) ⟩ d t = - \frac{1}{2} {‖ u (T) ‖}_{2}^{2} + \frac{1}{2} ‖ u_{0} ‖_{2}^{2} + \int_{0}^{T} \int_{Ω} f (x, u (t)) u (t) d x d t . \end{matrix}$ Combining (3.19) and (3.20), we get $\begin{matrix} \underset{m \to \infty}{lim sup} \int_{0}^{T} ⟨ L (u_{m} (t)), u_{m} (t) ⟩ d t = \int_{0}^{T} ⟨ Ξ, u (t) ⟩ d t . \end{matrix}$ Invoking Lemma 2.4, we conclude that (3.18) holds. Replacing (3.18) in (3.17), it yields $\begin{matrix} (3.22) & \int_{Ω} u_{t} (t) v d x + {⟨ u (t), v ⟩}_{α, β, p, q} = \int_{Ω} f (x, u (t)) v d x, a.e. in (0, + \infty) . \end{matrix}$ We now show that the solution u satisfies the following energy inequality $\begin{matrix} \int_{0}^{t} {‖ u_{τ} (τ) ‖}_{2}^{2} d τ + E (u (t)) ⩽ E (u_{0}), a.e. t ⩾ 0 . \end{matrix}$ To this end, let θ be the nonnegative function which belongs to $C ([0, T])$ . From (3.7) we have $\begin{matrix} (3.23) & \int_{0}^{T} θ (t) d t \int_{0}^{T} {‖ u_{n τ} (τ) ‖}_{2}^{2} d τ + \int_{0}^{T} E (u_{n} (t)) θ (t) d t = \int_{0}^{T} E (u_{n} (0)) θ (t) d t . \end{matrix}$ The right-hand side of (3.23) converges to $\begin{matrix} \int_{0}^{T} E (u_{0}) θ (t) d t \end{matrix}$ as $n \to \infty$ . The second term in the left-hand side $\int_{0}^{T} E (u_{n} (t)) θ (t) d t$ is lower semicontinuous with respect to the weak topology of $W_{0}^{α, p} (Ω)$ . Hence $\begin{matrix} (3.24) & \int_{0}^{T} E (u (t)) θ (t) d t ⩽ \underset{n \to \infty}{lim inf} \int_{0}^{T} E (u_{n} (t)) θ (t) d t . \end{matrix}$ Therefore, we obtain $\begin{matrix} \int_{0}^{T} θ (t) d t \int_{0}^{t} {‖ u_{τ} (τ) ‖}_{2}^{2} d τ + \int_{0}^{T} E (u (t)) θ (t) d t ⩽ \int_{0}^{T} E (u_{0}) θ (t) d t . \end{matrix}$ Since θ and T were arbitrarily chosen we conclude $\begin{matrix} \int_{0}^{t} {‖ u_{τ} (τ) ‖}_{2}^{2} d τ + E (u (t)) ⩽ E (u_{0}), a.e. t ⩾ 0 . \end{matrix}$

Finally, we show that the solutions of problem (1.1) are nonnegative. Using $f (x, ξ) ⩽ 0$ for all $ξ ⩽ 0$ and choosing $v = - u^{-} : = - max {- u, 0}$ in Definition 1.1, it leads to $\begin{array}{l} \frac{1}{2} \frac{d}{d t} \int_{Ω} {| u_{t} {(t)}^{-} |}^{2} d x + {⟨ u (t), - u^{-} ⟩}_{α, β, p, q} = 0 . \end{array}$ This implies that $u^{-} = 0$ a.e. in Ω and for all $t > 0$ . Thus, $u ⩾ 0$ a.e. in $Ω \times (0, \infty)$ . In conclusion, the proof is complete. □

4. Decay estimates of solutions

Now we turn our attention to the asymptotic behavior of the global solutions as $t \to + \infty$ . To this aim, we first recall the following lemma which due to Martinez [20].

Lemma 4.1.

Let $H : R^{+} \to R^{+}$ be a nonincreasing function and σ is a nonnegative constant such that $\begin{matrix} \int_{t}^{+ \infty} H^{1 + σ} (s) d s ⩽ \frac{1}{ω} H^{σ} (0) H (t), \forall t ⩾ 0, \end{matrix}$ then we have

$H (t) ⩽ H (0) e^{1 - ω t}$ , for all $t ⩾ 0$ , whenever $σ = 0$ .

$H (t) ⩽ H (0) {(\frac{1 + σ}{1 + ω σ t})}^{1 / σ}$ , for all $t ⩾ 0$ , whenever $σ > 0$ .

Proof of Theorem 1.2.

Taking $v = u$ in Definition 1.1, it yields $\begin{matrix} (4.1) & - \frac{1}{2} \frac{d}{d t} \int_{Ω} u^{2} (t) d x = I (u (t)) . \end{matrix}$ Set $\begin{matrix} g (t) = \int_{Ω} u^{2} (t) d x . \end{matrix}$ From (4.1) and (3.9), we know that the function $g (t)$ is nonincreasing with respect to $t > 0$ . In view of the proof of Lemma 2.8, for any $ε > 0$ we deduce from $(f_{1})$ that $\begin{array}{l} I (u (t)) & = {[u (t)]}_{α, p}^{p} + {[u (t)]}_{β, q}^{q} - \int_{Ω} f (x, u (t)) u (t) d x \\ = {[u (t)]}_{α, p}^{p} + {[u (t)]}_{β, q}^{q} - ε \int_{Ω} {| u (t) |}^{q} d x - C_{ε} \int_{Ω} {| u (t) |}^{r} d x \\ ⩾ {[u (t)]}_{α, p}^{p} + {[u (t)]}_{β, q}^{q} - \frac{ε}{λ_{1}} {[u (t)]}_{β, q}^{q} - C_{ε} C_{r}^{r} {[u (t)]}_{α, p}^{r}, \forall t ⩾ 0, \end{array}$ where $λ_{1}$ is the first eigenvalue of the fractional q-Laplacian and $C_{r}$ denotes the embedding constant from $W_{0}^{α, p} (Ω)$ to $L^{r} (Ω)$ . Choose $ε = λ_{1}$ . Then $\begin{array}{l} I (u (t)) & ⩾ {[u (t)]}_{α, p}^{p} + {[u (t)]}_{β, q}^{q} - \frac{ε}{λ_{1}} {[u (t)]}_{β, q}^{q} - C_{ε} C_{r}^{r} {[u (t)]}_{α, p}^{r} \\ ⩾ {[u (t)]}_{α, p}^{p} (1 - C_{λ_{1}} C_{r}^{r} {[u (t)]}_{α, p}^{r - p}), \forall t ⩾ 0 . \end{array}$ Using (1.10), we have $\begin{array}{l} (\frac{1}{p} - \frac{1}{μ}) {[u (t)]}_{α, p}^{p} + (\frac{1}{q} - \frac{1}{μ}) {[u (t)]}_{β, q}^{q} + \frac{1}{μ} I (u (t)) ⩽ E (u_{0}) . \end{array}$ This together with $I (u) > 0$ yields that $\begin{array}{l} (\frac{1}{p} - \frac{1}{μ}) {[u]}_{α, p}^{p} ⩽ E (u_{0}) . \end{array}$ Thus, $\begin{array}{rcl} (4.2) & I (u (t)) ⩾ {[u (t)]}_{α, p}^{p} (1 - C_{λ_{1}} C_{r}^{r} ({(\frac{p μ}{μ - p})}^{\frac{r - p}{p}} {(E (u_{0}))}^{\frac{r - p}{p}})), \forall t ⩾ 0 . \end{array}$

On the other hand, by assumption, we have $\begin{matrix} E (u_{0}) < (\frac{μ - p}{p μ}) {(\frac{1}{C_{λ_{1}} C_{r}^{r}})}^{\frac{p}{r - p}} . \end{matrix}$ Integrating (4.1) over $(t, T)$ , one has $\begin{array}{l} g (t) - g (T) = 2 I (u (t)) . \end{array}$ Gathering this with the continuous embedding $W_{0}^{α, p} (Ω) ↪ L^{2} (Ω)$ , we deduce from (4.2) that $\begin{matrix} (4.3) & \int_{t}^{T} g {(t)}^{\frac{p - 2}{2} + 1} d t ⩽ \frac{1}{χ} g (t), \end{matrix}$ where $χ = 2 C_{2}^{- p} (1 - C_{λ_{1}} C_{r}^{r} ({(\frac{p μ}{μ - p})}^{\frac{r - p}{p}} {(E (u_{0}))}^{\frac{r - p}{p}}))$ and $C_{2}$ is the embedding constant from $W_{0}^{α, p} (Ω)$ to $L^{2} (Ω)$ . Letting $T \to + \infty$ in (4.3) and applying Lemma 4.1, we obtain $\begin{matrix} {‖ u (t) ‖}^{2} ⩽ ‖ u_{0} ‖_{2}^{2} {(\frac{p}{2 + (p - 2) χ t})}^{\frac{2}{p - 2}}, \forall t > 0, if p > 2 \end{matrix}$ and $\begin{matrix} {‖ u (t) ‖}^{2} ⩽ ‖ u_{0} ‖_{2}^{2} e^{(1 - χ t)}, \forall t > 0, if p = 2 . \end{matrix}$ Thus the proof is complete. □

Footnotes

Acknowledgements

This work was supported by College Students Innovation and Entrepreneurship Training Program (No. 202010059073).

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