Global existence,nonexistence,and decay of solutions for a wave equation of p -Laplacian type with weak and p -Laplacian damping,nonlinear boundary delay and source terms

Abstract

In this paper, we consider the initial boundary value problem for the p-Laplacian equation with weak and p-Laplacian damping terms, nonlinear boundary, delay and source terms acting on the boundary. By introducing suitable energy and perturbed Lyapunov functionals, we prove global existence, finite time blow up and asymptotic behavior of solutions in cases $p > 2$ and $p = 2$ . To our best knowledge, there is no results of the p-Laplacian equation with a nonlinear boundary delay term.

Keywords

strong damping delay term nonlinear boundary conditions global existence general decay finite time blow up

1. Introduction

In this paper, we study initial boundary value problem of the p-Laplacian equation with weak and p-Laplacian damping terms, nonlinear boundary, delay and source terms acting on the boundary $\begin{array}{l} (1) & \{\begin{matrix} u_{t t} - Δ_{p} u - Δ_{p} u_{t} + u_{t} = 0, & x \in Ω, t > 0, \\ u (x, t) = 0, & x \in Γ_{0}, t > 0, \\ | \nabla u |^{p - 2} \frac{\partial u}{\partial ν} + | \nabla u_{t} |^{p - 2} \frac{\partial u_{t}}{\partial ν} + μ_{1} | u_{t} (x, t) |^{m - 2} u_{t} (x, t) \\ + μ_{2} | u_{t} (x, t - τ) |^{m - 2} u_{t} (x, t - τ) = | u |^{k - 2} u, & x \in Γ_{1}, t > 0, \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & x \in Ω, \\ u (x, t - τ) = f_{0} (x, t - τ), & x \in Γ_{1}, t > 0, \end{matrix} \end{array}$ where $Ω \subset R^{n} (n ⩾ 1)$ , $\partial Ω = Γ_{0} \cup Γ_{1}$ , $mes (Γ_{0}) > 0, Γ_{0} \cap Γ_{1} = \emptyset$ , $\frac{\partial u}{\partial ν}$ denotes the unit outer normal derivative, $p, m, k, > 2, μ_{1}$ are positive constants, $μ_{2}$ is a real number, $τ > 0$ represents the time delay, and $u_{0}, u_{1}, f_{0}$ are given functions belonging to suitable spaces. The operator $Δ_{p} u$ is the classical p-Laplacian given by $Δ_{p} u = div (| \nabla u |^{p - 2} \nabla u)$ .

The wave equation of p-Laplacian type with $p = 2$ , which has been extensively studied and results concerning existence, asymptotic behavior and blow-up (nonexistence) have been established, see for instance [4, 5, 11, 12, 17, 19, 27, 30] and the references therein.

When $p \neq 2$ . Messaoudi [20] considered the following quasilinear hyperbolic equations with nonlinear damping and source terms $\begin{array}{l} (2) & u_{t t} - div (| \nabla u |^{p - 2} \nabla u) - Δ u_{t} + | u_{t} |^{q - 1} u_{t} = | u |^{p - 1} u . \end{array}$ and studied decay of solutions by using the techniques combination of the perturbed energy and potential well methods. Wu and Xue [31] studied (2) and proved the uniform energy decay rates of the solutions by utilizing the multiplier method. Pişkin [26] investigated the energy decay and blow-up of solutions to (2).

Mokeddem and Mansour [22] considered the initial boundary value problem for the following nonlinear wave equation of p-Laplacian type with a weak nonlinear dissipation $\begin{array}{l} (3) & u_{t t} - div (| \nabla u |^{p - 2} \nabla u) - a div (| \nabla u_{t} |^{m - 2} \nabla u_{t}) = b | u |^{r - 2} u . \end{array}$ They established the global existence and decay rate estimate of the energy. Messaoudi et al. [21] studied (3) with nonlinear damping term and established blow-up result of solutions with negative initial energy. Recently, Pereira et al. [25] studied (3) with $m = p$ and proved existence of the global solutions by using Faedo–Galerkin’s method and the decay of energy based on the method of Nakao.

In recent years, wave equation with boundary damping and source terms have been studied by many authors. Vitillaro [28] studied the following initial boundary value problem $\begin{array}{l} (4) & \{\begin{matrix} u_{t t} - Δ u = 0, & in Ω \times (0, \infty), \\ u (x, t) = 0, & on Γ_{0} \times (0, \infty), \\ u_{ν} = - | u_{t} |^{m - 2} u_{t} + | u |^{p - 2} u, & on Γ_{1} \times (0, \infty), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & x \in Ω . \end{matrix} \end{array}$ He showed that the presence of the superlinear damping term $- | u_{t} |^{m - 2} u_{t}$ , when $2 ⩽ p ⩽ m$ , implies the global existence of solutions for arbitrary initial data, in opposition with the nonexistence phenomenon occurring when $m = 2 < p$ . Zhang and Hu [32] proved the asymptotic behavior of the solutions of problem (4), where the initial data is inside a stable set. For the related works of wave equations with a nonlinear boundary source and damping terms, we also refer other works [2, 3, 8, 29] and the references therein.

For the p-Laplacian equation with boundary conditions. Kass and Rammaha [16] considered the following quasilinear wave equation of p-Laplacian type $\begin{array}{l} (5) & \{\begin{matrix} u_{t t} - Δ_{p} u - Δ u_{t} = 0, & in Ω \times (0, T), \\ | \nabla u |^{p - 2} \partial_{ν} u + | u |^{p - 2} u + \partial_{ν} u_{t} + u_{t} = f (u), & on Γ \times (0, T), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & on Ω, \end{matrix} \end{array}$ where $2 < p < 3$ . They established the local and global existence of the solutions by using Faedo–Galerkin method. Jeong et al. [13] studied the following quasilinear wave equation with acoustic boundary conditions $\begin{array}{l} (6) & u_{t t} - Δ u_{t} - div (| \nabla u |^{α - 2} \nabla u) - div (| \nabla u_{t} |^{β - 2} \nabla u) + a | u |^{m - 2} u = b | u |^{p - 2} u, \end{array}$ and proved the global nonexistence of solution with negative initial energy.

Time delays arise in many applications because, in most instances, physical, chemical, biological, thermal, and economic phenomena naturally not only depend on the present state but also on some past occurrences. The PDEs with time delay effects have become active area of research, see for instance [6, 7, 18, 23, 24] and the references therein. Nicaise and Pignotti [23] considered the following wave equation with a linear boundary term $\begin{array}{l} (7) & \{\begin{matrix} u_{t t} - Δ u + μ_{1} u_{t} + μ_{2} u_{t} (t - τ) = 0, & in Ω \times (0, \infty), \\ u (x, t) = 0, & on Γ_{0} \times (0, \infty), \\ \frac{\partial u}{\partial ν} = - μ_{1} u_{t} (x, t) - μ_{2} u_{t} (x, t - τ), & on Γ_{1} \times (0, \infty), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & in Ω, \\ u (x, t - τ) = f_{0} (x, t - τ), & in Γ_{1} \times (0, \infty), \end{matrix} \end{array}$ They obtained some stability results in the case $0 < μ_{2} < μ_{1}$ . Then, they extended the result to the time-dependent delay case in the work of Nicaise and Pignotti [24]. Kafini and Messaoudi [15] studied the following nonlinear wave equation with delay $\begin{array}{l} (8) & u_{t t} - div (| \nabla u |^{m - 2} \nabla u) + μ_{1} u_{t} + μ_{1} u_{t} (t - τ) = b | u |^{p - 2} u . \end{array}$ They established the blow-up result of solutions with negative initial energy and $p > m$ . Recently, Jeong et al. [14] studied (6) with time delay and acoustic boundary conditions, they proved under suitable conditions and for negative initial energy, a global nonexistence of solutions.

Inspired by the previous works [9, 10, 14], in this paper, we consider the initial boundary value problem for the p-Laplacian equation with weak and p-Laplacian damping terms, nonlinear boundary conditions, delay and source terms acting on the boundary, and we will concern the global existence, general decay, and blow-up result of solutions in cases $p > 2$ and $p = 2$ .

Our paper is organized as follows. In Section 2, we give some notations and preliminary lemmas. In Section 3, we prove the global existence of solution. Section 4 and Section 5 are studied to the general decay and blow-up of solutions, respectively. The last section, we prove the global existence, decay, and blow up of solutions with $p = 2$ .

2. Preliminaries

In this section we give some notation for function spaces and some preliminary lemmas. We denote by $‖ u ‖_{q}$ , $‖ u ‖_{q, Γ_{1}}$ , and $‖ u ‖_{1, q}$ to the usual $L^{q} (Ω)$ norm, $L^{q} (Γ_{1})$ norm, and $‖ u ‖_{W^{1, q} (Ω)}$ , respectively. Specially we introduce the set $\begin{array}{l} H_{Γ_{0}}^{1} (Ω) = {u \in H^{1} (Ω) | u_{| Γ_{0}} = 0} and W_{Γ_{0}}^{1, q} (Ω) = {u \in W^{1, q} (Ω) | u_{| Γ_{0}} = 0} . \end{array}$

To state and prove our results, we need the following assumptions:

$2 < p < m < k$ .

$k > 2, if n = 1, 2, 2 < k ⩽ \frac{2 (n - 1)}{n - 2}, if n ⩾ 3$ .

$| μ_{2} | < μ_{1}$ .

Let

c_{q}

be the optimal constant of Sobolev imbedding which satisfies the inequality

\begin{array}{l} (9) & ‖ u ‖_{q} ⩽ c_{q} ‖ \nabla u ‖_{2}, \forall u \in H^{1} (Ω), \end{array}

According to

(A_{3})

, we recall the trace Sobolev embedding inequality

\begin{array}{l} H_{Γ_{0}}^{1} (Ω) ↪ L^{q} (Ω), \end{array}

and the embedding inequality

\begin{array}{l} (10) & ‖ u ‖_{q, Γ_{1}} ⩽ c_{*} ‖ \nabla u ‖_{2}, \end{array}

where

c_{*}

is a positive constant.

To deal with the time delay term, motivated by Nicaise and Pignotti [23], we introduce a new variable $\begin{array}{l} (11) & z (x, ρ, t) = u_{t} (x, t - τ ρ), x \in Γ_{1}, ρ \in (0, 1), t > 0 \end{array}$ which gives us $\begin{array}{l} (12) & τ z_{t} (x, ρ, t) + z_{ρ} (x, ρ, t) = 0, in Γ_{1} \times (0, 1) \times (0, \infty) . \end{array}$ Then, problem (1) is equivalent to $\begin{array}{l} (13) & \{\begin{matrix} u_{t t} - Δ_{p} u - Δ_{p} u_{t} + u_{t} = 0, & x \in Ω, t > 0, \\ u (x, t) = 0, & x \in Γ_{0}, t > 0, \\ | \nabla u |^{p - 2} \frac{\partial u}{\partial ν} + | \nabla u_{t} |^{p - 2} \frac{\partial u_{t}}{\partial ν} + μ_{1} | u_{t} (x, t) |^{m - 2} u_{t} (x, t) \\ + μ_{2} | z (1, t) |^{m - 2} z (1, t) = | u |^{k - 2} u, & x \in Γ_{1}, t > 0, \\ τ z_{t} (ρ, t) + z_{ρ} (ρ, t) = 0, & x \in Γ_{1}, ρ \in (0, 1), t > 0, \\ z (ρ, 0) = f_{0} (- τ ρ), & x \in Γ_{1}, ρ \in (0, 1), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & x \in Ω, \end{matrix} \end{array}$ Let ξ be a positive constant satisfying $\begin{array}{l} (14) & τ (m - 1) | μ_{2} | ⩽ ξ ⩽ τ (m μ_{1} - | μ_{2} |) . \end{array}$ We first state a local existence theorem that can be established by combining arguments [1, 9].

Theorem 2.1.
Let $(A_{1})$ – $(A_{3})$ hold. Then, for every $u_{0} \in W_{Γ_{0}}^{1, p} (Ω)$ , $u_{1} \in W_{Γ_{0}}^{1, p} (Ω)$ and $f_{0} \in L^{2} ((Γ_{1}) \times (0, 1))$ . Then, the problem admits a unique local solution in the class $\begin{array}{l} u \in C ([0, T]; W_{Γ_{0}}^{1, p} (Ω)), u_{t} \in C ([0, T]; W_{Γ_{0}}^{1, p} (Ω)) \cap L^{m} ([0, T] \times Γ_{1}) . \end{array}$

Now, we define the energy associated with problem (1) by $\begin{array}{l} (15) & E (t) = \frac{1}{2} ‖ u_{t} ‖_{2}^{2} + \frac{1}{p} ‖ \nabla u ‖_{p}^{p} + \frac{ξ}{m} \int_{0}^{1} {‖ z (ρ, t) ‖}_{m, Γ_{1}}^{m} d ρ - \frac{1}{k} ‖ u ‖_{k, Γ_{1}}^{p} . \end{array}$
Lemma 2.2.
Let u be a solution of problem ( 1 ). Then, $\begin{array}{l} (16) & E^{'} (t) ⩽ - ‖ \nabla u_{t} ‖_{p}^{p} - ‖ u_{t} ‖_{2}^{2} - c (‖ u_{t} ‖_{m, Γ_{1}}^{m} + {‖ z (1, t) ‖}_{m, Γ_{1}}^{m}) ⩽ 0 . \end{array}$
Proof.
Multiplying the first equation in (13) by $u_{t}$ and integrating over Ω, we have $\begin{array}{l} \frac{d}{d t} [\frac{1}{2} ‖ u_{t} ‖_{2}^{2} + \frac{1}{p} ‖ \nabla u ‖_{p}^{p} - \frac{1}{k} ‖ u ‖_{k, Γ_{1}}^{k}] \\ (17) & = - ‖ \nabla u_{t} ‖_{p}^{p} - ‖ u_{t} ‖_{2}^{2} - μ_{1} ‖ u_{t} ‖_{m, Γ_{1}}^{m} - μ_{2} \int_{Γ_{1}} {| z (1, t) |}^{m - 2} z (1, t) u_{t} d x . \end{array}$ Multiplying the second equation in (13) by $ξ | z |^{m - 2} z$ and integrating over $Γ_{1} \times (0, 1)$ , we obtain $\begin{array}{l} - ξ \frac{d}{d t} \int_{Γ_{1}} \int_{0}^{1} {| z (ρ, t) |}^{m - 2} z (ρ, t) d ρ d Γ & = - \frac{ξ}{m τ} \int_{Γ_{1}} \int_{0}^{1} \frac{\partial}{\partial ρ} {| z (ρ, t) |}^{m} d ρ d Γ \\ (18) & = \frac{ξ}{m τ} (‖ u_{t} ‖_{m}^{m} - {‖ z (1, t) ‖}_{m}^{m}) . \end{array}$ By using Young’s inequality, we have $\begin{array}{l} (19) & - μ_{2} \int_{Ω} {| z (1, t) |}^{m - 2} z (1, t) u_{t} d x ⩽ \frac{(m - 1) | μ_{2} |}{m} {‖ z (1, t) ‖}_{m}^{m} + \frac{| μ_{2} |}{m} ‖ u_{t} ‖_{m}^{m} . \end{array}$ Combining (17), (18), and (19), we obtain $\begin{array}{l} (20) & E^{'} (t) ⩽ - ‖ \nabla u_{t} ‖_{p}^{p} - ‖ u_{t} ‖_{2}^{2} - c (‖ u_{t} ‖_{m, Γ_{1}}^{m} + {‖ z (1, t) ‖}_{m, Γ_{1}}^{m}) ⩽ 0, \end{array}$ where $c = min {μ_{1} - \frac{ξ}{m τ} - \frac{| μ_{2} |}{m}, \frac{ξ}{m τ} - \frac{(m - 1) | μ_{2} |}{m}}$ , which is positive by (14) □

Similar in [15], we can get the following lemma that is needed later. Lemma 2.3.
Then there exists a positive constant $\begin{array}{l} ‖ u ‖_{k, Γ_{1}}^{s} ⩽ C_{1} (‖ u ‖_{k, Γ_{1}}^{k} + ‖ \nabla u ‖_{p}^{p}), \end{array}$ for any $u \in W^{1, p} (Ω)$ and $p ⩽ s ⩽ k$ .

3. Global existence

In this section, we will prove that the solutions established in Theorem 2.1 are global in time. For this purpose, we define the functionals $\begin{array}{l} (21) & I (t) = I (u (t)) = ‖ \nabla u ‖_{p}^{p} + ξ \int_{0}^{1} {‖ z (ρ, t) ‖}_{m, Γ_{1}}^{m} d ρ - ‖ u ‖_{k, Γ_{1}}^{k}, \end{array}$ and $\begin{array}{l} (22) & J (t) = J (u (t)) = \frac{1}{p} ‖ \nabla u ‖_{p}^{p} + \frac{ξ}{m} \int_{0}^{1} {‖ z (ρ, t) ‖}_{m, Γ_{1}}^{m} d ρ - \frac{1}{k} ‖ u ‖_{k, Γ_{1}}^{k} . \end{array}$ Then, it is obvious that $\begin{array}{l} (23) & E (t) = \frac{1}{2} ‖ u_{t} ‖_{2}^{2} + J (t) . \end{array}$ In order to show our result, we first establish the following lemma.

Lemma 3.1.
Assume that $(A_{1})$ – $(A_{3})$ hold, and for any $u_{0} \in W_{Γ_{0}}^{1, p} (Ω)$ , $u_{1} \in W_{Γ_{0}}^{1, p} (Ω)$ and $f_{0} \in L^{2} ((Γ_{1}) \times (0, 1))$ , such that $\begin{array}{l} (24) & I (0) > 0 and β = c_{}^{k} C^{p} {[\frac{k p}{(k - p)} E (0)]}^{\frac{k - p}{p}} < 1, \end{array}$ then,* $\begin{array}{l} (25) & I (t) > 0, \forall t > 0 . \end{array}$
Proof.
Since $I (0) > 0$ , then by continuity of u, there exist a time $T_{} < T$ such that $\begin{array}{l} (26) & I (t) ⩾ 0, \forall t \in [0, T_{}] . \end{array}$ It follows from (21), (22) and (26), gives that $\begin{array}{l} (27) & J (t) = \frac{1}{k} I (t) + \frac{(k - p)}{k p} [‖ \nabla u ‖_{p}^{p} + ξ \frac{p (k - m)}{m (k - p)} \int_{0}^{1} {‖ z (ρ, t) ‖}_{m, Γ_{1}}^{m} d ρ] ⩾ \frac{(k - p)}{k p} ‖ \nabla u ‖_{p}^{p} . \end{array}$ By using (16) and (23), we obtain $\begin{array}{l} (28) & ‖ \nabla u ‖_{p}^{p} ⩽ \frac{k p}{(k - p)} J (t) ⩽ \frac{k p}{(k - p)} E (t) ⩽ \frac{k p}{(k - p)} E (0), \forall t \in [0, T_{}] . \end{array}$ Exploiting (10), (23), $(A_{1})$ and (24), we obtain $\begin{array}{l} ‖ u ‖_{k, Γ_{1}}^{k} & ⩽ c_{}^{k} ‖ \nabla u ‖_{2}^{k} ⩽ c_{}^{k} C^{p} ‖ \nabla u ‖_{p}^{k} = c_{}^{k} C^{p} ‖ \nabla u ‖_{p}^{p} ‖ \nabla u ‖_{p}^{k - p} \\ (29) & ⩽ c_{}^{k} C^{p} {[\frac{k p}{(k - p)} E (0)]}^{\frac{k - p}{p}} ‖ \nabla u ‖_{p}^{p} = β ‖ \nabla u ‖_{p}^{p} < ‖ \nabla u ‖_{p}^{p}, \forall t \in [0, T_{}] . \end{array}$ Therefore, we conclude that $\begin{array}{l} I (t) > 0, \forall t \in [0, T_{}] . \end{array}$ By repeating the procedure, $T_{}$ is extended to T. The proof is complete. □
Theorem 3.2.
Assume that the conditions of Lemma 3.1 hold, then the solution ( 1 ) is global and bounded.
Proof.
It suffices to show that $\begin{array}{l} ‖ u_{t} ‖_{2}^{2} + ‖ \nabla u ‖_{p}^{p}, \end{array}$ is bounded independently of t. By using (16) and (28), we obtain $\begin{array}{l} (30) & E (0) ⩾ E (t) = \frac{1}{2} ‖ u_{t} ‖_{2}^{2} + J (t) ⩾ \frac{1}{2} ‖ u_{t} ‖_{2}^{2} + \frac{(k - p)}{k p} ‖ \nabla u ‖_{p}^{p}, \end{array}$ which means, $\begin{array}{l} (31) & ‖ u_{t} ‖_{2}^{2} + ‖ \nabla u ‖_{p}^{p} ⩽ C E (0), \end{array}$ where C is a positive constant. The proof is complete. □

4. General decay

In this section, we state and prove the decay result of solution to problem (1) by constructing a suitable Lyapunov functional. For this goal, we set $\begin{array}{l} (32) & F (t) : = E (t) + ε \int_{Ω} u u_{t} d x + \frac{ε}{2} ‖ u ‖_{2}^{2}, \end{array}$ where ε is a positive constant to be specified later.

Lemma 4.1.
Let u be a solution of problem ( 1 ). Then, there exist two positive constants $α_{1}$ and $α_{2}$ depending on ε such that $\begin{array}{l} (33) & α_{1} E (t) ⩽ F (t) ⩽ α_{2} E (t) . \end{array}$
Proof.
It is easy to see that $F (t)$ and $E (t)$ are equivalent. So we omit it here. □
Theorem 4.2.
Let $u_{0} \in W_{Γ_{0}}^{1, p} (Ω)$ , $u_{1} \in W_{Γ_{0}}^{1, p} (Ω)$ and $f_{0} \in L^{2} ((Γ_{1}) \times (0, 1))$ . Assume that $(A_{1})$ – $(A_{3})$ hold, then there exist two positive constant K and ω such that $\begin{array}{l} E (t) ⩽ K e^{- ω t}, t ⩾ 0 . \end{array}$
Proof.
Taking a derivative of (32) with respect to t, using (1) and (16), we obtain $\begin{array}{l} F^{'} (t) = & E^{'} (t) + ε ‖ u_{t} ‖_{2}^{2} + ε \int_{Ω} u u_{t t} d x + ε \int_{Ω} u u_{t} d x \\ ⩽ & - ‖ \nabla u_{t} ‖_{p}^{p} - (1 - ε) ‖ u_{t} ‖_{2}^{2} - c ‖ u_{t} ‖_{m, Γ_{1}}^{m} - c {‖ z (1, t) ‖}_{m, Γ_{1}}^{m} \\ - ε ‖ \nabla u ‖_{p}^{p} + ε ‖ u ‖_{k, Γ_{1}}^{k} - ε \int_{Ω} | \nabla u_{t} |^{p - 2} \nabla u_{t} \nabla u d x \\ (34) & - μ_{1} ε \int_{Γ_{1}} | u_{t} |^{m - 2} u_{t} u d Γ - μ_{2} ε \int_{Γ_{1}} {| z (1, t) |}^{m - 2} z (1, t) u d Γ . \end{array}$ Applying Hölder’s and Young’s inequalities, we have $\begin{array}{l} (35) & \int_{Ω} | \nabla u_{t} |^{p - 2} \nabla u_{t} \nabla u d x ⩽ \frac{δ^{p}}{p} ‖ \nabla u ‖_{p}^{p} + \frac{p - 1}{p} δ^{- \frac{p}{p - 1}} ‖ \nabla u_{t} ‖_{p}^{p} . \end{array}$ Similar to (35), we have $\begin{array}{l} (36) & μ_{1} \int_{Γ_{1}} | u_{t} |^{m - 2} u_{t} u d Γ ⩽ \frac{μ_{1}^{m} δ^{m}}{m} ‖ u ‖_{m, Γ_{1}}^{m} + \frac{m - 1}{m} δ^{- \frac{m}{m - 1}} ‖ u_{t} ‖_{m, Γ_{1}}^{m}, \end{array}$ and $\begin{array}{l} (37) & μ_{2} \int_{Γ_{1}} {| z (1, t) |}^{m - 2} z (1, t) u d Γ ⩽ \frac{μ_{2}^{m} δ^{m}}{m} ‖ u ‖_{m, Γ_{1}}^{m} + \frac{m - 1}{m} δ^{- \frac{m}{m - 1}} {‖ z (1, t) ‖}_{m, Γ_{1}}^{m} . \end{array}$ A substitution of (35)–(37) into (34), we obtain $\begin{array}{l} F^{'} (t) ⩽ & - {1 - ε \frac{p - 1}{p} δ^{- \frac{p}{p - 1}}} ‖ \nabla u_{t} ‖_{p}^{p} - {c - ε \frac{m - 1}{m} δ^{- \frac{m}{m - 1}}} ‖ u_{t} ‖_{m, Γ_{1}}^{m} - {1 - ε} ‖ u_{t} ‖_{2}^{2} \\ - {c - ε \frac{m - 1}{m} δ^{- \frac{m}{m - 1}}} {‖ z (1, t) ‖}_{m, Γ_{1}}^{m} - ε {1 - \frac{δ^{p}}{p}} ‖ \nabla u ‖_{p}^{p} + ε ‖ u ‖_{k, Γ_{1}}^{k} \\ (38) & + ε \frac{(μ_{1}^{m} + μ_{2}^{m})}{m} δ^{m} ‖ u ‖_{m, Γ_{1}}^{m} . \end{array}$ On the other hand, similar to (29), we have, for $m > p$ , $\begin{array}{l} (39) & ‖ u ‖_{m, Γ_{1}}^{m} ⩽ c_{}^{m} ‖ \nabla u ‖_{2}^{m} ⩽ c_{}^{m} C^{p} ‖ \nabla u ‖_{p}^{m} = c_{*}^{m} C^{p} ‖ \nabla u ‖_{p}^{p} ‖ \nabla u ‖_{p}^{m - p} ⩽ c_{p} ‖ \nabla u ‖_{p}^{p} . \end{array}$ Combining (39) and (38), we obtain $\begin{array}{l} F^{'} (t) ⩽ & - {1 - ε \frac{p - 1}{p} δ^{- \frac{p}{p - 1}}} ‖ \nabla u_{t} ‖_{p}^{p} - {c - ε \frac{m - 1}{m} δ^{- \frac{m}{m - 1}}} ‖ u_{t} ‖_{m, Γ_{1}}^{m} - {1 - ε} ‖ u_{t} ‖_{2}^{2} \\ (40) & - {c - ε \frac{m - 1}{m} δ^{- \frac{m}{m - 1}}} {‖ z (1, t) ‖}_{m, Γ_{1}}^{m} - ε {1 - \frac{δ^{p}}{p} - {\tilde{c}}_{p} δ^{m}} ‖ \nabla u ‖_{p}^{p} + ε ‖ u ‖_{k, Γ_{1}}^{k} . \end{array}$ where ${\tilde{c}}_{p} = (c_{p} (μ_{1}^{m} + μ_{2}^{m}) / m)$ .

At this point, we choose δ small enough such that $\begin{array}{l} 1 - \frac{δ^{p}}{p} - {\tilde{c}}_{p} δ^{m} > 0 . \end{array}$ For any fixed δ, we choose ε so small satisfying $\begin{array}{l} 1 - ε \frac{p - 1}{p} δ^{- \frac{p}{p - 1}} > 0, c - ε \frac{m - 1}{m} δ^{- \frac{m}{m - 1}} > 0, 1 - ε > 0 . \end{array}$ Then inequality (40) becomes $\begin{array}{l} (41) & F^{'} (t) ⩽ - κ ε E (t), \forall t > 0 . \end{array}$ From (33), we have $\begin{array}{l} (42) & F^{'} (t) ⩽ - κ ε E (t) ⩽ - ω ε F (t), \forall t > 0 . \end{array}$ A simple integration of (42), leads to $\begin{array}{l} (43) & F (t) ⩽ c_{5} e^{- ω t}, \forall t > 0 . \end{array}$ Again (33), gives $\begin{array}{l} (44) & E (t) ⩽ K e^{- ω t}, \forall t > 0 . \end{array}$ This completes the proof. □

5. Blow-up

In this section, we state and prove the blow-up result of solution to problem (1) with negative initial energy.

Theorem 5.1.
Let $(A_{1})$ – $(A_{3})$ and $E (0) < 0$ holds. Then, the solution of problem ( 1 ) blows up in finite time.
Proof.
Let $\begin{array}{l} (45) & H (t) = - E (t), \end{array}$ then $E (0) < 0$ , and (16), gives $\begin{array}{l} (46) & H^{'} (t) = - E^{'} (t) ⩾ ‖ \nabla u_{t} ‖_{p}^{p} + c (‖ u_{t} ‖_{m, Γ_{1}}^{m} + {‖ z (1, t) ‖}_{m, Γ_{1}}^{m}) ⩾ 0 . \end{array}$ From (15) and (45), we obtain $\begin{array}{l} (47) & 0 < H (0) ⩽ H (t) ⩽ \frac{1}{k} ‖ u ‖_{k, Γ_{1}}^{k}, t \in [0, T) . \end{array}$ Next, we define $\begin{array}{l} (48) & F (t) = H^{1 - σ} (t) + ε \int_{Ω} u u_{t} d x + \frac{ε}{2} ‖ u ‖_{2}^{2}, \end{array}$ where $ε > 0$ is small constant and will be chosen later, and $\begin{array}{l} (49) & 0 < σ ⩽ min {\frac{k - m}{k (m - 1)}, \frac{k - p}{k (p - 1)}} . \end{array}$ Taking a derivative of (48) and using (1), we have $\begin{array}{l} F^{'} (t) = & (1 - σ) H^{- σ} (t) H^{'} (t) + ε ‖ u_{t} ‖_{2}^{2} + ε \int_{Ω} u u_{t t} d x + ε \int_{Ω} u u_{t} d x \\ = & (1 - σ) H^{- σ} (t) H^{'} (t) + ε ‖ u_{t} ‖_{2}^{2} - ε ‖ \nabla u ‖_{p}^{p} + ε ‖ u ‖_{k, Γ_{1}}^{k} - \int_{Ω} | \nabla u_{t} |^{p - 2} \nabla u_{t} \nabla u d x \\ (50) & - μ_{1} \int_{Γ_{1}} | u_{t} |^{m - 2} u_{t} u d Γ - μ_{2} \int_{Γ_{1}} {| z (1, t) |}^{m - 2} z (1, t) u d Γ . \end{array}$ Applying Young’s inequality, for $δ > 0$ , we have $\begin{array}{l} \int_{Ω} | \nabla u_{t} |^{p - 2} \nabla u_{t} \nabla u d x & ⩽ \frac{δ^{p}}{p} ‖ \nabla u ‖_{p}^{p} + \frac{p - 1}{p} δ^{- \frac{p}{p - 1}} ‖ \nabla u_{t} ‖_{p}^{p} \\ (51) & ⩽ \frac{δ^{p}}{p} ‖ \nabla u ‖_{p}^{p} + \frac{p - 1}{p} δ^{- \frac{p}{p - 1}} H^{'} (t) . \end{array}$ Similar to (51), we obtain, for any $η > 0$ , $\begin{array}{l} (52) & μ_{1} \int_{Γ_{1}} | u_{t} |^{m - 2} u_{t} u d Γ ⩽ \frac{μ_{1}^{m} η^{m}}{m} ‖ u ‖_{m, Γ_{1}}^{m} + \frac{m - 1}{m c} η^{- \frac{m}{m - 1}} H^{'} (t), \end{array}$ and $\begin{array}{l} (53) & μ_{2} \int_{Γ_{1}} {| z (1, t) |}^{m - 2} z (1, t) u d Γ ⩽ \frac{μ_{2}^{m} η^{m}}{m} ‖ u ‖_{m, Γ_{1}}^{m} + \frac{m - 1}{m c} η^{- \frac{m}{m - 1}} H^{'} (t) . \end{array}$ Combining these estimates (51)–(53) and (50), we get $\begin{array}{l} F^{'} (t) ⩾ & {(1 - σ) H^{- σ} (t) - ε \frac{2 (m - 1)}{m c} η^{- \frac{m}{m - 1}} - ε \frac{p - 1}{p} δ^{- \frac{p}{p - 1}}} H^{'} (t) + ε ‖ u_{t} ‖_{2}^{2} \\ (54) & + ε ‖ u ‖_{k, Γ_{1}}^{k} - ε ‖ \nabla u ‖_{p}^{p} - ε \frac{δ^{p}}{p} ‖ \nabla u ‖_{p}^{p} - ε \frac{(μ_{1}^{m} + μ_{2}^{m})}{m} η^{m} ‖ u ‖_{m, Γ_{1}}^{m} . \end{array}$ It follows from (15) and (45), for constant $M > 0$ , we see that $\begin{array}{l} F^{'} (t) ⩾ & {(1 - σ) H^{- σ} (t) - ε \frac{2 (m - 1)}{m c} η^{- \frac{m}{m - 1}} - ε \frac{p - 1}{p} δ^{- \frac{p}{p - 1}}} H^{'} (t) + ε {\frac{M}{2} + 1} ‖ u_{t} ‖_{2}^{2} \\ + ε {1 - \frac{M}{k}} ‖ u ‖_{k, Γ_{1}}^{k} + ε {\frac{M}{p} - 1} ‖ \nabla u ‖_{p}^{p} + \frac{M ξ}{m} \int_{0}^{1} {‖ z (ρ, t) ‖}_{m, Γ_{1}}^{m} d ρ \\ (55) & - ε \frac{δ^{p}}{p} ‖ \nabla u ‖_{p}^{p} - ε \frac{(μ_{1}^{m} + μ_{2}^{m})}{m} η^{m} ‖ u ‖_{m, Γ_{1}}^{m} + M ε H (t) . \end{array}$ Therefore, by choosing δ and η so that $\begin{array}{l} η = {(θ_{1} H^{- σ} (t))}^{- \frac{m - 1}{m}} and δ = {(θ_{2} H^{- σ} (t))}^{- \frac{p - 1}{p}}, \end{array}$ where $θ_{1}$ and $θ_{2}$ are positive constants to be specified later, we see that $\begin{array}{l} F^{'} (t) ⩾ & {(1 - σ) - ε μ} H^{- σ} (t) H^{'} (t) + ε {\frac{M}{2} + 1} ‖ u_{t} ‖_{2}^{2} + ε {\frac{M}{p} - 1} ‖ \nabla u ‖_{p}^{p} \\ + ε {1 - \frac{M}{k}} ‖ u ‖_{k, Γ_{1}}^{k} + \frac{M ξ}{m} \int_{0}^{1} {‖ z (ρ, t) ‖}_{m, Γ_{1}}^{m} d ρ + M ε H (t) \\ (56) & - ε \frac{(μ_{1}^{m} + μ_{2}^{m}) θ_{1}^{1 - m}}{m} H^{σ (m - 1)} (t) ‖ u ‖_{m, Γ_{1}}^{m} - ε \frac{θ_{2}^{1 - p}}{p} H^{σ (p - 1)} (t) ‖ u ‖_{p, Γ_{1}}^{p}, \end{array}$ were $μ = \frac{2 (m - 1)}{m c} θ_{1} + \frac{p - 1}{p} θ_{2}$ . By using $(A_{1})$ and (47), we have $\begin{array}{l} (57) & H^{σ (m - 1)} (t) ‖ u ‖_{m, Γ_{1}}^{m} ⩽ C_{m} H^{σ (m - 1)} (t) ‖ u ‖_{k, Γ_{1}}^{m} ⩽ \frac{C_{m}}{k^{σ (m - 1)}} ‖ u ‖_{k, Γ_{1}}^{σ k (m - 1) + m}, \end{array}$ and $\begin{array}{l} (58) & H^{σ (p - 1)} (t) ‖ u ‖_{p, Γ_{1}}^{p} ⩽ C_{p} H^{σ (p - 1)} (t) ‖ u ‖_{k, Γ_{1}}^{p} ⩽ \frac{C_{p}}{k^{σ (p - 1)}} ‖ u ‖_{k, Γ_{1}}^{σ k (p - 1) + p} . \end{array}$ Substituting (57)–(58) into (56), we have $\begin{array}{l} F^{'} (t) ⩾ & {(1 - σ) - ε μ} H^{- σ} (t) H^{'} (t) + ε {\frac{M}{2} + 1} ‖ u_{t} ‖_{2}^{2} + ε {\frac{M}{p} - 1} ‖ \nabla u ‖_{p}^{p} \\ + ε {1 - \frac{M}{k}} ‖ u ‖_{k, Γ_{1}}^{k} + \frac{M ξ}{m} \int_{0}^{1} {‖ z (ρ, t) ‖}_{m, Γ_{1}}^{m} d ρ + M ε H (t) \\ (59) & - ε \frac{(μ_{1}^{m} + μ_{2}^{m}) C_{m}}{k^{σ (m - 1)} m} θ_{1}^{1 - m} ‖ u ‖_{k, Γ_{1}}^{σ k (m - 1) + m} - ε \frac{C_{p}}{k^{σ (p - 1)} p} θ_{2}^{1 - p} ‖ u ‖_{k, Γ_{1}}^{σ k (p - 1) + p} . \end{array}$ From (49) and Lemma 2.3, for $s_{1} = σ k (m - 1) + m ⩽ k$ , we deduce $\begin{array}{l} (60) & ‖ u ‖_{k, Γ_{1}}^{σ k (m - 1) + m} ⩽ C_{1} (‖ u ‖_{k, Γ_{1}}^{k} + ‖ \nabla u ‖_{p}^{p}) . \end{array}$ Similarly, for $s_{2} = σ k (p - 1) + p ⩽ k$ , we have $\begin{array}{l} (61) & ‖ u ‖_{k, Γ_{1}}^{σ k (p - 1) + p} ⩽ C_{2} (‖ u ‖_{k, Γ_{1}}^{k} + ‖ \nabla u ‖_{p}^{p}) . \end{array}$ Combining (60)–(61) with (59), we get $\begin{array}{l} F^{'} (t) ⩾ & {(1 - σ) - ε μ} H^{- σ} (t) H^{'} (t) + ε {\frac{M}{2} + 1} ‖ u_{t} ‖_{2}^{2} \\ + ε {(\frac{M}{p} - 1) - a_{1} θ_{1}^{1 - m} - a_{2} θ_{2}^{1 - p}} ‖ \nabla u ‖_{p}^{p} \\ + ε {(1 - \frac{M}{k}) - a_{1} θ_{1}^{1 - m} - a_{2} θ_{2}^{1 - p}} ‖ u ‖_{k, Γ_{1}}^{k} \\ (62) & + \frac{M ξ}{m} \int_{0}^{1} {‖ z (ρ, t) ‖}_{m, Γ_{1}}^{m} d ρ + M ε H (t), \end{array}$ where $\begin{array}{l} a_{1} = C_{1} \frac{(μ_{1}^{m} + μ_{2}^{m}) C_{m}}{k^{σ (m - 1)} m}, a_{2} = \frac{C_{2} C_{p}}{k^{σ (p - 1)} p} . \end{array}$ At this point, first, we choose $p < M < k$ such that $\begin{array}{l} \frac{M}{p} - 1 > 0 and 1 - \frac{M}{k} > 0 . \end{array}$ For any fixed M, we choose $θ_{1}$ and $θ_{2}$ so large such that $\begin{array}{l} (\frac{M}{p} - 1) - a_{1} θ_{1}^{1 - m} - a_{2} θ_{2}^{1 - p} > 0 and (1 - \frac{M}{k}) - a_{1} θ_{1}^{1 - m} - a_{2} θ_{2}^{1 - p} > 0 . \end{array}$ Once $θ_{1}$ and $θ_{2}$ is fixed, we select $ε > 0$ small enough so that $\begin{array}{l} (1 - σ) - ε μ > 0 and F (0) = H^{1 - σ} (0) + ε \int_{Ω} u_{0} u_{1} d x + \frac{ε}{2} ‖ u_{0} ‖_{2}^{2} . \end{array}$ Then inequality (62) becomes $\begin{array}{l} (63) & F^{'} (t) ⩾ K (‖ u_{t} ‖_{2}^{2} + ‖ \nabla u ‖_{p}^{p} + ‖ u ‖_{k, Γ_{1}}^{k} + \int_{0}^{1} {‖ z (ρ, t) ‖}_{m, Γ_{1}}^{m} d ρ + H (t)), \end{array}$ where K is a positive constant. Consequently, we have $\begin{array}{l} (64) & F (t) ⩾ F (0) > 0, \forall t ⩾ 0 . \end{array}$

We now estimate $F {(t)}^{\frac{1}{1 - σ}}$ . $\begin{array}{l} (65) & F^{\frac{1}{1 - σ}} (t) ⩽ C_{ε} {H (t) + {(ε \int_{Ω} u u_{t} d x)}^{\frac{1}{1 - σ}} + ‖ u ‖_{2}^{\frac{2}{1 - σ}}} . \end{array}$ By Hölder’s and Young’s inequalities, we have $\begin{array}{l} {| \int_{Ω} u u_{t} d x |}^{\frac{1}{1 - σ}} ⩽ c_{σ} ‖ u ‖_{2}^{\frac{1}{1 - σ}} ‖ u_{t} ‖_{2}^{\frac{1}{1 - σ}} ⩽ c_{σ} ‖ u ‖_{p}^{\frac{1}{1 - σ}} ‖ u_{t} ‖_{2}^{\frac{1}{1 - σ}} ⩽ c_{σ} (‖ u ‖_{p}^{\frac{μ}{1 - σ}} + ‖ u_{t} ‖_{2}^{\frac{θ}{1 - σ}}) \end{array}$ for $\frac{1}{μ} + \frac{1}{θ} = 1$ . By taking $θ = 2 (1 - σ)$ which gives $\frac{μ}{1 - σ} = \frac{2}{1 - 2 σ}$ . Therefore, we have $\begin{array}{l} (66) & {| \int_{Ω} u u_{t} d x |}^{\frac{1}{1 - σ}} ⩽ C (‖ u ‖_{p}^{\frac{2}{1 - 2 σ}} + ‖ u_{t} ‖_{2}^{2}) . \end{array}$ It follows from (47) and (31) that $\begin{array}{l} ‖ u ‖_{p}^{\frac{2}{1 - 2 σ}} & ⩽ C ‖ \nabla u ‖_{p}^{\frac{2}{1 - 2 σ}} ⩽ C {(K E (0))}^{\frac{2}{p (1 - 2 σ)}} ⩽ C {(K E (0))}^{\frac{2}{p (1 - 2 σ)}} \frac{H (t)}{H (0)} \\ (67) & ⩽ \frac{C {(K E (0))}^{\frac{2}{p (1 - 2 σ)}}}{k H (0)} ‖ u ‖_{k, Γ_{1}}^{k} . \end{array}$ Similarly, we have $\begin{array}{l} (68) & ‖ u ‖_{2}^{\frac{2}{1 - σ}} ⩽ C ‖ \nabla u ‖_{p}^{\frac{2}{1 - σ}} ⩽ \frac{C {(K E (0))}^{\frac{2}{p (1 - σ)}}}{k H (0)} ‖ u ‖_{k, Γ_{1}}^{k} . \end{array}$ A substitution of (66)–(68) into (65), we have $\begin{array}{l} (69) & F^{\frac{1}{1 - σ}} (t) ⩽ C (‖ u_{t} ‖_{2}^{2} + ‖ u ‖_{k, Γ_{1}}^{k} + H (t)) . \end{array}$ It follows from (63) and (69), we find that $\begin{array}{l} (70) & F^{'} (t) ⩾ κ F^{\frac{1}{1 - σ}} (t), for t > 0, \end{array}$ where κ is a positive constant. A simple integration of (70) over $(0, t)$ yields $\begin{array}{l} F^{\frac{σ}{1 - σ}} (t) ⩾ \frac{1}{F^{- \frac{σ}{1 - σ}} (0) - \frac{κ σ t}{1 - σ}} . \end{array}$ Consequently, the solution of problem (1) blows up in finite time $T^{}$ , and $T^{} ⩽ \frac{1 - σ}{κ σ F^{\frac{σ}{1 - σ}} (0)}$ . □

6. Global existence, decay, and blow up: Case $p = 2$

In this section, we prove the global existence, decay, and blow up of solutions for problem (1) in case $p = 2$ . The proof almost the same to the one of Theorems 3.2, 4.2, and 5.1. So we omit it here

To prove our results, we need the following assumption

$2 < m < k$ .

Now let us state the main results of this section as follows.

Theorem 6.1.

Assume that $(A_{1}^{'})$ , $(A_{2})$ – $(A_{3})$ hold and $p = 2$ . Then, for any $u_{0} \in H_{Γ_{0}}^{1} (Ω)$ and $u_{1} \in H_{Γ_{0}}^{1} (Ω)$ , and $f_{0} \in L^{2} ((Γ_{1}) \times (0, 1))$ the solution of problem ( 1 ) is global and bounded.

Theorem 6.2.

Assume that $(A_{1}^{'})$ , $(A_{2})$ – $(A_{3})$ hold and $p = 2$ . Let $u_{0} \in H_{Γ_{0}}^{1} (Ω)$ , $u_{1} \in H_{Γ_{0}}^{1} (Ω)$ , and $f_{0} \in L^{2} ((Γ_{1}) \times (0, 1))$ . Then, there exist two positive constant $K_{1}$ and $ω_{1}$ such that $\begin{array}{l} E (t) ⩽ K_{1} e^{- ω_{1} t}, t ⩾ 0 . \end{array}$

Theorem 6.3.

Let $(A_{1}^{'})$ and $(A_{2})$ – $(A_{3})$ and $E (0) < 0$ hold and $p = 2$ . Then, the solution of problem ( 1 ) blows up in finite time.

Remark 6.4.

A similar result provided in cases $p > 2$ and $p = 2$ , we can get these results for $m = 2$ .

Footnotes

Acknowledgements

The authors would like to thank the referees and the handling editor for their helpful suggestions upon which this paper was revised.

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