Global Existence, General Decay, and Blow up of Solution for a p-Biharmonic Equation of Hyperbolic Type with Delay and Acoustic Boundary Conditions

Abstract

The objective of this work is to investigate the global existence, general decay and blow-up results for a class of p-Biharmonic-type hyperbolic equations with delay and acoustic boundary conditions. The global existence of solutions has been obtained by potential well theory and the general decay result of energy has been established, in which the exponential decay and polynomial decay are only special cases, by using the multiplier techniques combined with a nonlinear integral inequality given by Komornik. Finally, the blow-up of solutions is established with positive initial energy. To our knowledge, the global existence, general decay, and blow-up result of solutions to p-Biharmonic-type hyperbolic equations with delay and acoustic boundary conditions has not been studied.

1. Introduction

In this paper, we study the initial boundary value problem for a fourth-order wave equation with a delay and acoustic boundary conditions

$$\left\{\begin{array}{cc}{\left|u_t\right|}^{\sigma }{u}_{tt}+{\Delta }_p^{2}u-\Delta u_t=0,\hfill & x\in \mathsf{\Omega },t>0,\hfill \\ u(x,t)={\displaystyle \frac{𝜕u}{𝜕\nu }}(x,t)=0,\hfill & x\in {\mathsf{\Gamma }}_0,t>0,\hfill \\ \Delta u(x,t)=0,\hfill & x\in {\mathsf{\Gamma }}_1,t>0,\hfill \\ {\displaystyle \frac{𝜕v}{𝜕\nu }}(x,t)-{\displaystyle \frac{𝜕\Delta v}{𝜕\nu }}(x,t)+{\mu }_1{\left|u_t(x,t)\right|}^{m-2}{u}_t(x,t)\hfill \\ +{\mu }_2|u_t{(x,t-\tau )|}^{m-2}{u}_t(x,t-\tau )=h\left(x\right)y_t\left(t\right)+{\left|u\right|}^{r-2}u,\hfill & x\in {\mathsf{\Gamma }}_1,t>0,\hfill \\ u_t(x,t)+p\left(x\right)y_t\left(t\right)+q\left(x\right)y\left(t\right)=0,\hfill & x\in {\mathsf{\Gamma }}_1,t>0,\hfill \\ v(x,0)=u_0\left(x\right),v_t(x,0)=u_1\left(x\right),\hfill & x\in \mathsf{\Omega },\hfill \\ y(x,0)=y_0\left(x\right),\hfill & x\in \mathsf{\Omega },\hfill \\ u(x,t-\tau )=f_0(x,t-\tau ),\hfill & x\in {\mathsf{\Gamma }}_1,t>0,\hfill \end{array}\right.$$

(1)

where $\mathsf{\Omega }$ is a regular and bounded domain ${\mathbb{R}}^n(n\ge 1)$ with a smooth boundary $𝜕\mathsf{\Omega }={\mathsf{\Gamma }}_0\cup {\mathsf{\Gamma }}_1$, $\mathrm{mes}\left({\mathsf{\Gamma }}_0\right)>0,{\mathsf{\Gamma }}_0\cap {\mathsf{\Gamma }}_1=\varnothing$. Here, we mean by ${\displaystyle \frac{𝜕u}{𝜕\nu }}$ the unit outer normal derivative, $p\sigma ,p,m,r,{\mu }_1>0$, ${\mu }_2$ is a real number, and the parameter $\tau >0$ is the time delay. The functions $h,p,q$ are essentially bounded, and $u_0,u_1,f_0$ represent the initial data. The fourth-order operator ${\Delta }_p^2$ is the well-known p-Biharmonic operator, which is given by

$${\Delta }_p^{2}v=\Delta \left({|\Delta v|}^{p-2}\Delta v\right).$$

The fourth-order equation is an equation involving the partial derivatives of a function where the highest-order derivative is of order four. In [1], the authors discussed this type of problem to study traveling waves in suspension bridges. For more physical background on the fourth-order equations, please see [2,3,4,5,6] and the references therein.

First, we recall some results related to problem (1) without acoustic boundary conditions, when $p=2$ has been extensively studied and also the existence, nonexistence, and decay rate have been established. For example, in [7], the following was considered

$$u_{tt}-{\Delta }^{2}u+{\left|u_t\right|}^{m-2}{u}_t={\left|u\right|}^{r-2}u,$$

(2)

where $m,r\ge 2$ and an existence result was established, showing that the solution continues to exist globally if $m\ge r$, and blows up in finite time if $m<r$ and the initial energy is negative. Wu and Tsai [8] proved the global existence and blow-up of the solution to the problem (2). This result was later improved by Chen and Zhou [9]. Recently, Zhang et al. [10] considered (2) when the nonlinear damping term $|u_t{|}^{m-2}{u}_t$ is replaced by the nonlinear averaged damping ${\parallel u_t\parallel }_m^m{u}_t$. Recently, Zhang et al. [10] considered (2) when the nonlinear damping term $|u_t{|}^{m-2}{u}_t$ is replaced by the nonlinear averaged damping ${\parallel u_t\parallel }_m^m{u}_t$.

When $p>2$, Ferreira et al. [11] studied the next nonlinear beam equation with the existence of strong damping with the $p\left(x\right)$-Biharmonic operator

$$u_{tt}+{\Delta }_{p\left(x\right)}^{2}u-\Delta u_t=f(x,t,u).$$

(3)

The existence of local solutions is proved by applying the well-known Faedo–Galerkin’s method and then the behavior of solutions is discussed by using the Nakao lemma under certain conditions on $p(·)$. Butakın and Pişkin [12] investigated the global existence in the times and nonexistence of solutions with negative initial energy to similar $p\left(x\right)$-Biharmonic equation nonlinear sources.

As for delayed systems with many areas of use in scientific and practical life, many researchers have addressed the subject in an in-depth study, and several results have been found, including [13], where the following system was considered:

$$u_{tt}-\Delta u+{\mu }_1{u}_t+{\mu }_2{u}_t(t-\tau )=0.$$

(4)

The authors obtained certain new stability results for $0<{\mu }_2<{\mu }_1$ and then it was extended to the variable delay in [14]. A nonlinear delayed wave equation was considered in [15] as

$$u_{tt}-\mathrm{div}\left({|\nabla u|}^{m-2}\nabla u\right)+{\mu }_1{u}_t+{\mu }_1{u}_t(t-\tau )=b{\left|u\right|}^{p-2}u.$$

(5)

The authors established the blow-up phenomenon for solutions in the case $p\ge m$. In [16], the following delayed equation

$$u_{tt}-{\Delta }_{p}u-{\Delta }_p{u}_t+u_t=0,$$

(6)

was considered with nonlinear boundary conditions and a source term acting on the boundary. The authors proved the global existence in time of solutions and established the decay rate result. Furthermore, the blow-up of the solutions in finite time with negative initial energy was shown. Recently, Kamache et al. [17] investigated the global existence, decay rate, and the blow-up of solutions to a Kirchhoff-type equation with nonlinear boundary delay and source terms.

In [18], the authors introduced acoustic boundary conditions as

$$\left\{\begin{array}{cc}{𝜕}_{\nu }u=y_t,\hfill & \mathrm{on}{\mathsf{\Gamma }}_0,\times (0,\infty ),\hfill \\ \lambda u_t+m\left(x\right)y_{tt}+p\left(x\right)y_t+q\left(x\right)y=0,\hfill & \mathrm{on}{\mathsf{\Gamma }}_0,\times (0,\infty ).\hfill \end{array}\right.$$

(7)

here $\lambda >0$, $p,m,$ and $q$ are non-negative functions. Beale [19,20] investigated the global existence and regularity of solutions in a Hilbert space for the linear wave equation with (7) using the semigroup approach. In [21], a nonlinear wave equation subject to (7) was considered and studied on ${\mathsf{\Gamma }}_1$ and Dirichlet boundary conditions on ${\mathsf{\Gamma }}_0$. The authors in [22] showed global solvability and decay rate estimates for a linear wave equation with acoustic boundary conditions. Park and Park [23] investigated the qualitative properties of solutions for wave equations in viscoelastic media with acoustic boundary conditions, which were improved in [24]. The authors in [25] considered a nonlinear $q\left(x\right)$-Laplacian equation with a time delay and variable exponents, and then the blow up of solutions was shown. Then, the decay result was achieved by using a Komornik integral inequality; see [26].

In the present paper, we are concerned with the quantitative studies (the global existence and uniqueness) and qualitative properties (stability and blow-up results) to (1) with weak and strong damping terms. We begin by stating the needed results and tools to prove our next results in Section 2. Our first main results regarding the global existence and general decay rate are stated and proved in Section 3 by using the energy method together with Komornik’s lemma. In Section 4, under suitable conditions, we established a finite time blow-up of the solution by using the concavity method.

2. Preliminaries

We state certain results on the function spaces and some preliminary lemmas.

Let

$$W_{{\mathsf{\Gamma }}_0}^{k,q}\left(\mathsf{\Omega }\right)=\left\{u\in W^{k,q}\left(\mathsf{\Omega }\right)|u_{|{\mathsf{\Gamma }}_0}=0\right\},k=1,2.$$

be the subspaces of the classical Sobolev spaces $W^{k,q}\left(\mathsf{\Omega }\right),i=1,2$ of real-valued functions of order one. Let $c_p$ and $c_{*}$ be the smallest positive constants such that

$${\parallel u\parallel }_q\le c_p{\parallel \nabla u\parallel }_p,\forall u\in W_{{\mathsf{\Gamma }}_0}^{1,p}\left(\mathsf{\Omega }\right),$$

(8)

and

$${\parallel \nabla u\parallel }_q\le c_{\varrho }{\parallel \Delta u\parallel }_p,\forall u\in W_{{\mathsf{\Gamma }}_0}^{2,p}\left(\mathsf{\Omega }\right),$$

(9)

where

$${\parallel \nabla u\parallel }_q={\left({\int }_{\mathsf{\Omega }}{|\nabla v|}^{q}dx\right)}^{1/q},$$

and

$${\parallel \Delta u\parallel }_q={\left({\int }_{\mathsf{\Omega }}{|\Delta u|}^{q}dx\right)}^{1/q}.$$

By the trace theory, $\exists c_{\varsigma }>0$ depends only on ${\mathsf{\Gamma }}_1$, where

$${\parallel u\parallel }_{q,{\mathsf{\Gamma }}_1}\le c_{\varsigma }{\parallel \nabla u\parallel }_p,\forall u\in W_{{\mathsf{\Gamma }}_0}^{1,p}\left(\mathsf{\Omega }\right).$$

(10)

The following assumptions are needed:

• $\left(H_1\right)$$2<p<r<m.$
• $\left(H_2\right)$$r>2,\mathrm{if}n=1,2,2<m\le {\displaystyle \frac{2(n-1)}{n-2}},\mathrm{if}n\ge 3$.
• $\left(H_3\right)$$\left|{\mu }_2\right|<{\mu }_1.$
• $\left(H_4\right)$$2<p<m<r.$
• The assumptions $H_1,H_2$ and $H_4$ are necessary for the embedding of the spaces $L^p$, $L^m$, and $L^r$. The first two assumptions are used for the well-posedness. The assumption $H_3$ is necessary to compare the weights of the nonlinear damping with/without delay. The assumption $H_4$ is needed for the embedding for the blow-up phenomenon.
• To deal with the time delay term, as in [13], we introduce a new variable

  $$z(x,\kappa ,t)=u_t(x,t-\tau \kappa ),x\in {\mathsf{\Gamma }}_1,\kappa \in (0,1),t>0$$

  (11)

  which means,

  $$\tau z_t(x,\kappa ,t)+z_{\kappa }(x,\kappa ,t)=0,\mathrm{in}{\mathsf{\Gamma }}_1\times (0,1)\times (0,\infty ).$$

  (12)

Then, system (1) becomes

$$\left\{\begin{array}{cc}{\left|u_t\right|}^{\sigma }{u}_{tt}+{\Delta }_p^{2}u-\Delta u_t=0,\hfill & x\in \mathsf{\Omega },t>0,\hfill \\ u(x,t)={\displaystyle \frac{𝜕u}{𝜕\nu }}(x,t)=0,\hfill & x\in {\mathsf{\Gamma }}_0,t>0,\hfill \\ \Delta u(x,t)=0,\hfill & x\in {\mathsf{\Gamma }}_1,t>0,\hfill \\ {\displaystyle \frac{𝜕v}{𝜕\nu }}(x,t)-{\displaystyle \frac{𝜕\Delta v}{𝜕\nu }}(x,t)+{\mu }_1{\left|u_t(x,t)\right|}^{m-2}{u}_t(x,t)\hfill \\ +{\mu }_2{\left|z(1,t)\right|}^{m-2}z(1,t)=h\left(x\right)y_t\left(t\right)+{\left|u\right|}^{r-2}u,\hfill & x\in {\mathsf{\Gamma }}_1,t>0,\hfill \\ u_t(x,t)+p\left(x\right)y_t\left(t\right)+q\left(x\right)y\left(t\right)=0,\hfill & x\in {\mathsf{\Gamma }}_1,t>0,\hfill \\ \tau z_t(\kappa ,t)+z_{\kappa }(\kappa ,t)=0,\hfill & x\in {\mathsf{\Gamma }}_1,\kappa \in (0,1),t>0,\hfill \\ v(x,0)=u_0\left(x\right),v_t(x,0)=u_1\left(x\right),\hfill & x\in \mathsf{\Omega },\hfill \\ y(x,0)=y_0\left(x\right),\hfill & x\in {\mathsf{\Gamma }}_1,\hfill \\ z(\kappa ,0)=f_0(-\tau \kappa ),\hfill & x\in {\mathsf{\Gamma }}_1,\kappa \in (0,1),\hfill \end{array}\right.$$

(13)

Next, the total energy functional associated with system (13) is defined as

$$\begin{array}{cc}E\left(t\right)\hfill & ={\displaystyle \frac{1}{\sigma +2}}{\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}+{\displaystyle \frac{1}{p}}{\parallel \Delta u\parallel }_p^p+{\displaystyle \frac{\zeta }{m}}{\int }_0^1{\parallel z(\kappa ,t)\parallel }_{m,{\mathsf{\Gamma }}_1}^{m}d\rho +{\displaystyle \frac{1}{2}}{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma },\hfill \\ \hfill & -{\displaystyle \frac{1}{r}}{\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^r,\hfill \end{array}$$

(14)

where $\zeta >0$, verifying

$$\tau (m-1)|{\mu }_2|<\zeta <\tau (m{\mu }_1-|{\mu }_2\left|\right).$$

(15)

Lemma 1. 

Let $(v,y,z)$ be a solution of (13). Then, the energy functional $E\left(t\right)$ is decreasing and satisfies

$$E^{\prime }\left(t\right)\le -{\parallel \nabla u_t\parallel }_2^2-m_0\left({\parallel u_t\parallel }_{m,{\mathsf{\Gamma }}_1}^m+{\parallel z(1,t)\parallel }_{m,{\mathsf{\Gamma }}_1}^m\right)-{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)p\left(x\right)y_t^2\left(t\right)d\mathsf{\Gamma }\le 0$$

(16)

Proof. 

Multiplying (13)₁ by $u_t$, integrating over $\mathsf{\Omega }$ and exploiting (1)₄, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}\left\{{\displaystyle \frac{1}{\sigma +2}}{\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}+{\displaystyle \frac{1}{p}}{\parallel \Delta u\parallel }_p^p-{\displaystyle \frac{1}{r}}{\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^r\right\}-{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)u_t\left(t\right)y_t\left(t\right)d\mathsf{\Gamma }\hfill \\ \hfill & =-{\mu }_1{\parallel u_t\parallel }_{m,{\mathsf{\Gamma }}_1}^m-{\mu }_2{\int }_{{\mathsf{\Gamma }}_1}{\left|z(1,t)\right|}^{m-2}z(1,t)u_{t}d\mathsf{\Gamma }.\hfill \end{array}$$

(17)

On the other hand, by (13)₅, we have

$$-{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)u_t\left(t\right)y_t\left(t\right)d\mathsf{\Gamma }={\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)p\left(x\right)y_t^2\left(t\right)d\mathsf{\Gamma }+{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y\left(t\right)y_t\left(t\right)d\mathsf{\Gamma }.$$

(18)

Multiplying (13)₆ by ${\zeta \left|z\right|}^{m-2}z$ and integrating over ${\mathsf{\Gamma }}_1\times (0,1)$, we obtain

$$\begin{array}{cc}{\displaystyle \frac{\zeta }{m}}{\displaystyle \frac{d}{dt}}{\int }_{{\mathsf{\Gamma }}_1}{\int }_0^1{\left|z(\kappa ,t)\right|}^{m}d\kappa d\mathsf{\Gamma }\hfill & =-{\displaystyle \frac{\zeta }{m\tau }}{\int }_{{\mathsf{\Gamma }}_1}{\int }_0^1{\displaystyle \frac{𝜕}{𝜕\kappa }}{\left|z(\kappa ,t)\right|}^{m}d\kappa d\mathsf{\Gamma }\hfill \\ \hfill & ={\displaystyle \frac{\zeta }{m\tau }}\left({\parallel u_t\parallel }_{m,{\mathsf{\Gamma }}_1}^m-{\parallel z(1,t)\parallel }_{m,{\mathsf{\Gamma }}_1}^m\right).\hfill \end{array}$$

(19)

Thanks to Young’s inequality, we obtain

$$-{\mu }_2{\int }_{\mathsf{\Omega }}{\left|z(1,t)\right|}^{m-2}z(1,t)u_{t}dx\le {\displaystyle \frac{(m-1)|{\mu }_2|}{m}}{\parallel z(1,t)\parallel }_{m,{\mathsf{\Gamma }}_1}^m+{\displaystyle \frac{|{\mu }_2|}{m}}{\parallel u_t\parallel }_{m,{\mathsf{\Gamma }}_1}^m.$$

(20)

Combining (18)–(20), and (17), we obtain

$$E^{\prime }\left(t\right)\le -{\parallel \nabla u_t\parallel }_2^2-m_0\left({\parallel u_t\parallel }_{m,{\mathsf{\Gamma }}_1}^m+{\parallel z(1,t)\parallel }_{m,{\mathsf{\Gamma }}_1}^m\right)-{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)p\left(x\right)y_t^2\left(t\right)d\mathsf{\Gamma }\le 0,$$

(21)

where

$$m_0=min\left\{{\mu }_1-{\displaystyle \frac{\zeta }{m\tau }}-{\displaystyle \frac{|{\mu }_2|}{m}},{\displaystyle \frac{\zeta }{m\tau }}-{\displaystyle \frac{(m-1)|{\mu }_2|}{m}}\right\},$$

which is positive by (15). □

Now, the local existence result for (1) is stated in the next theorem; its proof follows that in [27,28,29,30].

Theorem 1. 

Assume that $\left(H_1\right)$–$\left(H_3\right)$ hold. Then, $\forall (u_0,u_1)\in W_{{\mathsf{\Gamma }}_0}^{2,p}\left(\mathsf{\Omega }\right)\times L^2\left(\mathsf{\Omega }\right)$ and $f_0\in L^2(\left({\mathsf{\Gamma }}_1\right)\times (0,1))$, the system (1) has a unique local solution in the class

$$\begin{array}{c}\hfill u\in L^{\infty }\left([0,T];W_{{\mathsf{\Gamma }}_0}^{2,p}\left(\mathsf{\Omega }\right)\right),\\ \hfill u_t\in L^{\infty }\left([0,T];L^2\left(\mathsf{\Omega }\right)\right)\cap L^m\left([0,T];{\mathsf{\Gamma }}_1\right)\\ \hfill u_t\in L^{\infty }\left([0,T];L^2\left(\mathsf{\Omega }\right)\right),\\ \hfill y,y_t\in L^2\left([0,T];L^2\left({\mathsf{\Gamma }}_1\right)\right).\end{array}$$

The next lemma is used to prove the decay rate of the solution.

Lemma 2 

([31]). Let $E\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ be a non-increasing function. Assume that $\omega ,\zeta >0$ such that

$${\int }_S^{+\infty }{E}^{\omega +1}\left(t\right)dt\le =cE\left(S\right),\forall S>0.$$

Thus,

$$E\left(t\right)\le \left\{\begin{array}{cc}{\displaystyle \frac{cE\left(0\right)}{{(1+t)}^{1/\omega }}},\hfill & t>0,\mathrm{if}\omega >0,\hfill \\ cE\left(0\right)e^{-\zeta t},\hfill & t>0,\mathrm{if}\omega =0.\hfill \end{array}\right.$$

Lemma 3 

([15]). Suppose that $\left(H_1\right)$ holds, then there exists a positive constant $C>1$, depending on Ω only such that

$${\parallel v\parallel }_r^s\le C_1\left({\parallel v\parallel }_r^r+{\parallel \Delta v\parallel }_p^p\right),$$

for any $u\in W_{{\mathsf{\Gamma }}_0}^{k,q}$ and $p\le s\le r$.

3. The Existence of Global Solution

Let us define the functionals

$$I\left(t\right)={\parallel \Delta u\parallel }_p^p+\zeta {\int }_0^1{\parallel z(\kappa ,t)\parallel }_{m,{\mathsf{\Gamma }}_1}^{m}d\kappa +{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }-{\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^r,$$

(22)

and

$$J\left(t\right)={\displaystyle \frac{1}{p}}{\parallel \Delta u\parallel }_p^p+{\displaystyle \frac{\zeta }{m}}{\int }_0^1{\parallel z(\kappa ,t)\parallel }_{m,{\mathsf{\Gamma }}_1}^{m}d\kappa -{\displaystyle \frac{1}{r}}{\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^r+{\displaystyle \frac{1}{2}}{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }.$$

(23)

Then, we have

$$E\left(t\right)={\displaystyle \frac{1}{\sigma +2}}{\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}+J\left(t\right).$$

(24)

Lemma 4. 

Let $\left(H_1\right)$–$\left(H_3\right)$ hold, and for any $(v_0,v_1)\in H_0^1\left(\mathsf{\Omega }\right)\times L^2\left(\mathsf{\Omega }\right)$, where

$$I\left(0\right)>0\mathrm{and}\gamma =c_{\varsigma }^r{c}_{\varrho }^r{\left[{\displaystyle \frac{rp}{r-p}}E\left(0\right)\right]}^{\frac{r-p}{p}}<1,$$

(25)

then,

$$I\left(t\right)>0,\forall t>0.$$

(26)

Proof. 

Since $I\left(0\right)>0$, then by the continuity of u, $\exists T_1>0$ where

$$I\left(t\right)\ge 0,\forall t\in \left[0,T_1\right].$$

(27)

This, together with (22), (23), (25), and $\left(H_1\right)$, gives

$$\begin{array}{cc}J\left(t\right)\hfill & ={\displaystyle \frac{1}{r}}I\left(t\right)+{\displaystyle \frac{r-p}{rp}}[{\parallel \Delta u\parallel }_p^p+\zeta {\displaystyle \frac{p(r-m)}{m(r-p)}}{\int }_0^1{\parallel z(\kappa ,t)\parallel }_{m,{\mathsf{\Gamma }}_1}^{m}d\kappa \hfill \\ \hfill & +{\displaystyle \frac{p(r-2)}{2(r-p)}}{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }]\hfill \\ \hfill & \ge {\displaystyle \frac{r-p}{rp}}\left[{\parallel \Delta u\parallel }_p^p+{\displaystyle \frac{p(r-2)}{2(r-p)}}{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }\right].\hfill \end{array}$$

(28)

Applying (24), (28), and (16), we obtain

$${\parallel \Delta u\parallel }_p^p\le {\displaystyle \frac{rp}{r-p}}J\left(t\right)\le {\displaystyle \frac{rp}{r-p}}E\left(t\right)\le {\displaystyle \frac{rp}{r-p}}E\left(0\right),\forall t\in \left[0,T_1\right].$$

(29)

Using (9), (10), $\left(H_1\right)$, (29), and (25), we obtain

$$\begin{array}{cc}{\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^k\hfill & \le c_{\varsigma }^r{\parallel \nabla u\parallel }_p^r\le c_{\varsigma }^r{c}_{\varrho }^r{\parallel \Delta u\parallel }_p^r=c_{\varsigma }^r{c}_{\varrho }^r{\parallel \Delta u\parallel }_p^p{\parallel \Delta u\parallel }_p^{r-p}\hfill \\ \hfill & \le c_{\varsigma }^r{c}_{\varrho }^r{\left[{\displaystyle \frac{rp}{r-p}}E\left(0\right)\right]}^{\frac{r-p}{p}}{\parallel \Delta u\parallel }_p^p={\gamma \parallel \Delta u\parallel }_p^p<{\parallel \Delta u\parallel }_p^p,\forall t\in [0,T_1].\hfill \end{array}$$

(30)

Then we have

$$I\left(t\right)>0,\forall t\in [0,T_1].$$

Repeating this procedure, then $T_1$ can be extended to T. □

Theorem 2. 

Under the conditions of Lemma 4 hold, the local solution of (1) is global in time

Proof. 

It suffices to prove that

$${\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}+{\parallel \Delta u\parallel }_p^p+{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma },$$

is bounded independently of t. Exploiting Lemma 1, (24) and (28), we obtain

$$\begin{array}{cc}E\left(0\right)\hfill & \ge E\left(t\right)={\displaystyle \frac{1}{\sigma +2}}{\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}\hfill \\ \hfill & \ge {\displaystyle \frac{1}{\sigma +2}}{\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}+{\displaystyle \frac{r-p}{rp}}\left[{\parallel \Delta u\parallel }_p^p+{\displaystyle \frac{p(r-2)}{2(r-p)}}{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }\right],\hfill \end{array}$$

(31)

which means,

$${\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}+{\parallel \Delta u\parallel }_p^p+{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }\le KE\left(0\right)<\infty ,$$

(32)

where K is a positive constant depending only on $r,p$ and $\sigma$. The proof is complete. □

Here, the decay rate of the global solution is stated and proved by using Komornik’s method.

Theorem 3. 

Under the conditions of Lemma 4, there exist two positive constants ${\gamma }_0$ and ${\gamma }_1$ where

$$\left\{\begin{array}{cc}E\left(t\right)\le {\displaystyle \frac{{\gamma }_0}{{(1+t)}^{1/\sigma }}},\hfill & t>0,\mathrm{if}\sigma >0,\hfill \\ E\left(t\right)\le {\gamma }_0{e}^{-{\gamma }_{1}t},\hfill & t>0,\mathrm{if}\sigma =0.\hfill \end{array}\right.$$

Proof. 

Multiplying (1)₁ by $u{E}^{\varpi }\left(t\right)(\varpi >0)$ and integrating over $\mathsf{\Omega }\times (S,T)$, to have

$${\int }_S^T{E}^{\varpi }\left(t\right){\int }_{\mathsf{\Omega }}\left[{\left|u_t\right|}^{\sigma }{u}_{tt}+{\Delta }_p^{2}u-\Delta u_t\right]udxdt=0.$$

(33)

By integration by parts, we see that

$$\begin{array}{cc}\hfill & {\int }_S^T{E}^{\varpi }{\int }_{\mathsf{\Omega }}\left[{\left|u_t\right|}^{\sigma }{u}_{tt}+{|\Delta u|}^p+\nabla u_t\nabla u\right]dxdt\hfill \\ \hfill & +{\int }_S^T{E}^{\varpi }{\int }_{{\mathsf{\Gamma }}_1}\left[{\mu }_1|u_t{|}^{m-2}{u}_{t}u+{\mu }_2{\left|z(1,t)\right|}^{m-2}z(1,t)u-{\left|v\right|}^r-h\left(x\right)y_t\left(t\right)\right]d\mathsf{\Gamma }dt=0.\hfill \end{array}$$

(34)

On the other hand, we have

$$\begin{array}{cc}& {\int }_S^T{E}^{\varpi }\left(t\right){\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\sigma }{u}_{tt}udxdt\hfill \\ & ={\left[{\displaystyle \frac{1}{\sigma +1}}{E}^{\varpi }\left(t\right){\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\sigma }{u}_{t}udx\right]}_S^T-{\displaystyle \frac{1}{\sigma +1}}{\int }_S^T{E}^{\varpi }\left(t\right){\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\sigma +2}dxdt\hfill \\ & -{\displaystyle \frac{\varpi }{\sigma +1}}{\int }_S^T{E}^{\prime }{E}^{\varpi -1}\left(t\right){\int }_{\mathsf{\Omega }}{\left|v_t\right|}^{\sigma }{v}_{t}vdxdt,\hfill \end{array}$$

Then, inequality (34) becomes

$$\begin{array}{cc}\hfill & {\int }_S^T{E}^{\varpi }\left(t\right){\parallel \Delta u\parallel }_p^{p}dt\hfill \\ \hfill & =-{\left[{\displaystyle \frac{1}{\sigma +1}}{E}^{\varpi }\left(t\right){\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\sigma }{u}_{t}udx\right]}_S^T+{\displaystyle \frac{1}{\sigma +1}}{\int }_S^T{E}^{\varpi }\left(t\right){\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\sigma +2}dxdt\hfill \\ \hfill & +{\displaystyle \frac{\varpi }{\sigma +1}}{\int }_S^T{E}^{\prime }{E}^{\varpi -1}\left(t\right){\int }_{\mathsf{\Omega }}{\left|v_t\right|}^{\sigma }{v}_{t}vdx-{\int }_S^T{E}^{\varpi }\left(t\right){\int }_{\mathsf{\Omega }}\nabla u_t\nabla udxdt\hfill \\ \hfill & -{\mu }_1{\int }_S^T{E}^{\varpi }\left(t\right){\int }_{{\mathsf{\Gamma }}_1}|u_t{|}^{m-2}{u}_{t}ud\mathsf{\Gamma }dt-{\mu }_2{\int }_S^T{E}^{\varpi }\left(t\right){\int }_{{\mathsf{\Gamma }}_1}{\left|z(1,t)\right|}^{m-2}z(1,t)ud\mathsf{\Gamma }dt\hfill \\ \hfill & +{\int }_S^T{E}^{\varpi }\left(t\right){\parallel v\parallel }_{r,{\mathsf{\Gamma }}_1}^{r}dt+{\int }_S^T{E}^{\varpi }\left(t\right){\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)v\left(t\right)y_t\left(t\right)d\mathsf{\Gamma }dt.\hfill \end{array}$$

(35)

Applying (13) to the term ${\parallel \Delta u\parallel }_p^p$ on the left-hand side of (35), we get

$$\begin{array}{cc}\hfill & p{\int }_S^T{E}^{\varpi +1}\left(t\right)dt\hfill \\ \hfill & =-{\left[{\displaystyle \frac{1}{\sigma +1}}{E}^{\varpi }\left(t\right){\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\sigma }{u}_{t}udx\right]}_S^T+\left({\displaystyle \frac{p}{\sigma +2}}+{\displaystyle \frac{1}{\sigma +1}}\right){\int }_S^T{E}^{\varpi }\left(t\right){\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}dt\hfill \\ \hfill & +{\displaystyle \frac{\varpi }{\sigma +1}}{\int }_S^T{E}^{\prime }\left(t\right)E^{\varpi -1}\left(t\right){\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\sigma }{u}_{t}udx+{\int }_S^T{E}^{\varpi }{\int }_{\mathsf{\Omega }}\nabla u_t\nabla udxdt\hfill \\ \hfill & -{\mu }_1{\int }_S^T{E}^{\varpi }\left(t\right){\int }_{{\mathsf{\Gamma }}_1}|u_t{|}^{m-2}{u}_{t}ud\mathsf{\Gamma }dt-{\mu }_2{\int }_S^T{E}^{\varpi }\left(t\right){\int }_{{\mathsf{\Gamma }}_1}{\left|z(1,t)\right|}^{m-2}z(1,t)ud\mathsf{\Gamma }dt\hfill \\ \hfill & +\left(1-{\displaystyle \frac{p}{r}}\right){\int }_S^T{E}^{\varpi }\left(t\right){\parallel v\parallel }_{r,{\mathsf{\Gamma }}_1}^{r}dt+{\int }_S^T{E}^{\varpi }\left(t\right){\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)v\left(t\right)y_t\left(t\right)d\mathsf{\Gamma }dt\hfill \\ \hfill & +p{\displaystyle \frac{\xi }{m}}{\int }_S^T{E}^{\varpi }\left(t\right){\int }_0^1{\parallel z(\rho ,t)\parallel }_{m,{\mathsf{\Gamma }}_1}^{m}d\rho dt+{\displaystyle \frac{p}{2}}{\int }_S^T{E}^{\varpi }\left(t\right){\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }.\hfill \end{array}$$

(36)

The right-hand side terms in (36) will be estimated. Exploiting Hölder’s inequality, (8), (9), and (14), we obtain

$$\begin{array}{ccc}\hfill I_1& =& {\left[E^{\varpi }\left(t\right){\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\sigma }{u}_{t}udx\right]}_S^T\hfill \\ & \le & {\left[E^{\varpi }\left(t\right)\left(\eta {\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}+c_{\eta }{\parallel u\parallel }_{\sigma +2}^{\sigma +2}\right)\right]}_S^T\hfill \\ & \le & {\left[E^{\varpi }\left(t\right)\left(\eta {\parallel u_t\parallel }_{\rho +2}^{\rho +2}+c_p^{\rho +2}{c}_{\varrho }^{\rho +2}{c}_{\eta }{\parallel \Delta u\parallel }_p^{\rho +2}\right)\right]}_S^T\hfill \\ & \le & {\left[c_1{E}^{\varpi +1}\left(t\right)+c_2{E}^{\varpi +\frac{\sigma +2}{p}}\left(t\right)\right]}_S^T\hfill \\ & \le & c_1{E}^{\varpi +1}\left(S\right)+c_2{E}^{\kappa +\frac{\rho +2}{p}}\left(S\right)\hfill \\ & \le & c_{3}E\left(S\right),\hfill \end{array}$$

(37)

and

$$\begin{array}{cc}{I}_2\hfill & ={\int }_S^T{E}^{\varpi }\left(t\right){\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}dt\le {\int }_S^T{E}^{\varpi }\left(t\right)\left(KE\left(0\right)\right)dt\le {\int }_S^T\left[E^{\varpi +1}\left(t\right)+c_{\eta }{\left(KE\left(0\right)\right)}^{\varpi }\right]dt\hfill \\ \hfill & \le \eta {\int }_S^T{E}^{\varpi +1}\left(t\right)dt+c_4{\left(KE\left(0\right)\right)}^{\varpi }\le \eta {\int }_S^T{E}^{\varpi +1}\left(t\right)dt+c_{5}E\left(S\right).\hfill \end{array}$$

(38)

Similar to (37), we have

$$\begin{array}{cc}{I}_3\hfill & ={\int }_S^T{E}^{\prime }{E}^{\varpi -1}\left(t\right){\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\sigma }{u}_{t}udx\le {\int }_S^T\left(-E^{\prime }\left(t\right)\right)\left[c_6{E}^{\varpi }\left(t\right)+c_7{E}^{\varpi +\frac{\sigma +2}{p}-1}\left(t\right)\right]dt\hfill \\ \hfill & \le {\left[c_6{E}^{\varpi +1}\left(t\right)+c_7{E}^{\varpi +\frac{\sigma +2}{p}}\left(t\right)\right]}_T^S\le c_{8}E\left(S\right).\hfill \end{array}$$

(39)

Again, exploiting Hölder’s inequality, (8), (9), and the definition of energy $E\left(t\right)$, we get

$$\begin{array}{cc}{I}_4\hfill & ={\int }_S^T{E}^{\varpi }\left(t\right){\int }_{{\mathsf{\Gamma }}_1}{\left|u_t\right|}^{m-2}{u}_{t}ud\mathsf{\Gamma }dt\le {\int }_S^T{E}^{\varpi }\left(t\right)\left(\eta {\parallel u\parallel }_{m,{\mathsf{\Gamma }}_1}^m+c_{\eta }{\parallel u_t\parallel }_{m,{\mathsf{\Gamma }}_1}^m\right)dt\hfill \\ \hfill & \le \eta c_{\varsigma }^m{\int }_S^T{E}^{\varpi }\left(t\right){\parallel \Delta \parallel }_p^{m}dt+c_{\eta }{\int }_S^T{E}^{\varpi }\left(t\right)\left(-E^{\prime }\left(t\right)\right)dt\hfill \\ \hfill & \le \eta c_9{\int }_S^T{E}^{\varpi }\left(t\right){\parallel \Delta \parallel }_p^{p}dt+c_{10}{E}^{\varpi +1}\left(S\right)\le \eta c_9{\int }_S^T{E}^{\varpi +1}\left(t\right)dt+c_{11}E\left(S\right).\hfill \end{array}$$

(40)

Similar to (40), we have

$$I_5={\int }_S^T{E}^{\varpi }\left(t\right){\int }_{{\mathsf{\Gamma }}_1}{\left|z(1,t)\right|}^{m-2}z(1,t)ud\mathsf{\Gamma }dt\le \eta c_{12}{\int }_S^T{E}^{\varpi +1}\left(t\right)dt+c_{13}E\left(S\right).$$

(41)

Using Young’s inequality, $\left(H_4\right)$, (8), (9), and the definition of energy $E\left(t\right)$, we get

$$\begin{array}{cc}{I}_6\hfill & ={\int }_S^T{E}^{\varpi }\left(t\right)\left|{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)u\left(t\right)y_t\left(t\right)d\mathsf{\Gamma }\right|dt\hfill \\ \hfill & \le {\displaystyle \frac{{\parallel h\parallel }_{\infty }^{1/2}{\parallel p\parallel }_{\infty }^{1/2}}{p_0}}{\int }_S^T{E}^{\varpi }\left(t\right){\left({\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)p\left(x\right)y_t^2\left(t\right)d\mathsf{\Gamma }\right)}^{1/2}{\left({\int }_{{\mathsf{\Gamma }}_1}{u}^{2}d\mathsf{\Gamma }\right)}^{1/2}dt\hfill \\ \hfill & \le \eta {\int }_S^T{E}^{\varpi }\left(t\right){\parallel u\parallel }_{2,{\mathsf{\Gamma }}_1}^{2}dt+{\displaystyle \frac{{\parallel h\parallel }_{\infty }{\parallel p\parallel }_{\infty }}{4\eta p_0^2}}{\int }_S^T{E}^{\varpi }\left(t\right){\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)p\left(x\right)y_t^2\left(t\right)d\mathsf{\Gamma }dt\hfill \\ \hfill & \le c_{14}\eta {\int }_S^T{E}^{\varpi }\left(t\right){\parallel \Delta u\parallel }_p^{2}dt+c_{15}{\int }_S^T{E}^{\varpi }\left(t\right)\left(-E^{\prime }\left(t\right)\right)dt\hfill \\ \hfill & \le c_{14}\eta {\int }_S^T{E}^{\varpi }\left(t\right){\left[kE\left(0\right)\right]}^{\frac{2}{p}}dt+c_{15}{E}^{\varpi +1}\left(S\right){\int }_S^T{E}^{\varpi }\left(t\right)\left(-E^{\prime }\left(t\right)\right)dt\hfill \\ \hfill & \le c_{16}\eta {\int }_S^T\left[E^{\varpi +1}\left(t\right)+{\left(kE\left(0\right)\right)}^{\frac{2\varpi }{p}}\right]dt+c_{15}{E}^{\varpi +1}\left(S\right)\hfill \\ \hfill & \le c_{17}\eta {\int }_S^T{E}^{\varpi +1}\left(t\right)dt+c_{18}E\left(S\right).\hfill \end{array}$$

Similarly, we have

$$\begin{array}{cc}{I}_7\hfill & ={\int }_S^T{E}^{\varpi }\left(t\right){\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }dt\le c_{19}{\int }_S^T{E}^{\varpi }\left(t\right)\left[KE\left(0\right)\right]dt\hfill \\ \hfill & \le c_{20}\eta {\int }_S^T{E}^{\varpi +1}\left(t\right)dt+c_{21}E\left(S\right),\hfill \end{array}$$

(42)

and

$$\begin{array}{cc}{I}_8\hfill & =p{\displaystyle \frac{\xi }{m}}{\int }_S^T{E}^{\varpi }\left(t\right){\int }_0^1{\parallel z(\rho ,t)\parallel }_{m,{\mathsf{\Gamma }}_1}^{m}d\rho dt\le c_{22}{\int }_S^T{E}^{\varpi }\left(t\right)\left[KE\left(0\right)\right]dt\hfill \\ \hfill & \le c_{23}\eta {\int }_S^T{E}^{\varpi +1}\left(t\right)dt+c_{24}E\left(S\right)\hfill \end{array}$$

For the last term of (36),

$$I_9=\left(1-{\displaystyle \frac{p}{r}}\right){\int }_S^T{E}^{\varpi }\left(t\right){\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^{r}dt\le \gamma \left(1-{\displaystyle \frac{p}{r}}\right){\int }_S^T{E}^{\varpi }\left(t\right){\parallel \Delta u\parallel }_p^{p}dt\le \gamma p{\int }_S^T{E}^{\sigma +1}\left(t\right)dt.$$

(43)

Inserting (37)–(43) into (36), we get

$$\left\{p(1-\gamma )-\eta c_{25}\right\}{\int }_S^T{E}^{\varpi +1}\left(t\right)\le c_{26}E\left(S\right).$$

Since $0<\gamma <1$, then $p(1-\gamma )>0$, we choose $\eta$ small enough such that

$$p(1-\gamma )-\eta c_{25}>0.$$

Then, inequality (36) becomes

$${\int }_S^T{E}^{\varpi +1}\left(t\right)\le \lambda E\left(S\right).$$

(44)

By taking T to infinity, we get

$${\int }_S^{\infty }{E}^{\varpi +1}\left(t\right)\le \lambda E\left(S\right).$$

Therefore, Komornik’s lemma provides the desired result. □

4. Finite Time Blow up of Solution

With positive initial energy $E\left(0\right)<E_1$, we prove the finite time blow up of (1). Let us introduce

$$\kappa ={\left[c_{\varsigma }^r{c}_{\varrho }^r\right]}^{\frac{1}{r}},{\xi }_1={\kappa }^{\frac{-r}{r-p}},E_1=\left[{\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{r}}\right]{\xi }_1^p.$$

Lemma 5. 

Let $(u,z,y)$ be a solution of problem (13). Assume that $E\left(0\right)<E_1$ and

$${\left[{\parallel \Delta u_0\parallel }_p\right]}^{1/p}>{\xi }_1.$$

(45)

Then $\exists {\xi }_2>{\xi }_1$ where

$${\left[{\parallel \Delta u\parallel }_p^p\right]}^{1/p}>{\xi }_2,$$

(46)

and

$${\left[{\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^r\right]}^{1/r}\ge \kappa {\xi }_2,\forall t\ge 0.$$

(47)

Proof. 

From the definition of energy $E\left(t\right)$, we get

$$\begin{array}{cc}E\left(t\right)\hfill & \ge {\displaystyle \frac{1}{p}}{\parallel \Delta u\parallel }_p^p-{\displaystyle \frac{1}{r}}{\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^r\ge {\displaystyle \frac{1}{p}}{\parallel \Delta u\parallel }_p^p-{\displaystyle \frac{1}{r}}{c}_{\varsigma }^r{c}_{\varrho }^r{\parallel \Delta u\parallel }_p^r\hfill \\ \hfill & \ge {\displaystyle \frac{1}{p}}{\parallel \Delta u\parallel }_p^p-{\displaystyle \frac{{\kappa }_0^r}{r}}{\left({\parallel \Delta u\parallel }_p^p\right)}^{\frac{r}{p}}={\displaystyle \frac{1}{p}}{\xi }^p-{\displaystyle \frac{{\kappa }_0^r}{r}}{\xi }^r=\gamma \left(\xi \right),\hfill \end{array}$$

(48)

where $\xi ={\left[{\parallel \Delta u\parallel }_p^p\right]}^{1/p}$. It is easy to see that $\gamma$ is increasing for $0<\xi <{\xi }_1$, and decreasing for $\xi >{\xi }_1$, $\gamma \left(\xi \right)\to -\infty$ as $\xi \leftarrow +\infty$, and

$$\gamma \left({\xi }_1\right)={\displaystyle \frac{1}{p}}{\xi }_1^p-{\displaystyle \frac{{\kappa }^r}{r}}{\xi }_1^r=E_1,$$

Then, since $E\left(0\right)<E_1$ we have $\exists {\xi }_2>{\xi }_1$ such that $\gamma \left({\xi }_2\right)=E\left(0\right)$.

If we set ${\xi }_0={\left({\parallel \Delta u_0\parallel }_p\right)}^{1/p}$, then by (48), we have $\gamma \left({\xi }_0\right)\le E\left(0\right)=\gamma \left({\xi }_2\right)$, then ${\xi }_0>{\xi }_2$.

Now, to show (46), we use the proof by contradiction

$${\left[{\parallel \Delta u\left(t_0\right)\parallel }_p^p\right]}^{1/p}<{\xi }_2,t_0>0.$$

By the continuity of ${\parallel \Delta u\parallel }_p^p$, we may choose $t_0$ so that

$${\left[{\parallel \Delta u\left(t_0\right)\parallel }_p^p\right]}^{1/p}>{\xi }_1.$$

By (48), we have

$$E\left(t_0\right)\ge \gamma \left[{\left({\parallel \Delta u\left(t_0\right)\parallel }_p^p\right)}^{1/p}\right]>\gamma \left({\xi }_2\right)=E\left(0\right),$$

which is impossible since $E\left(t\right)\le E\left(0\right),\forall t\in [0,T)$. Then (46) is proved.

To show (47), exploiting (13), (14), and (46), we get

$${\displaystyle \frac{1}{r}}{\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^r\ge {\displaystyle \frac{1}{p}}{\parallel \Delta u\parallel }_p^p-E\left(0\right)\ge {\displaystyle \frac{1}{p}}{\xi }_2^p-E\left(0\right)\ge {\displaystyle \frac{1}{p}}{\xi }_2^p-\gamma \left({\xi }_2\right)={\displaystyle \frac{{\kappa }^r}{r}}{\xi }_2^r$$

(49)

□

Theorem 4. 

Assume that $\left(H_2\right)$–$\left(H_4\right)$ hold. If $E\left(0\right)<E_1$, then the solution of (1) blows up in finite time.

Proof. 

Set

$$H\left(t\right)=E_1-E\left(t\right).$$

(50)

Then (14), (16), and (50) give

$$H^{\prime }\left(t\right)=-E^{\prime }\left(t\right)\ge {\parallel \nabla u_t\parallel }_2^2+c_0\left({\parallel u_t\parallel }_{m,{\mathsf{\Gamma }}_1}^m+{\parallel z(1,t)\parallel }_{m,{\mathsf{\Gamma }}_1}^m\right)+{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)p\left(x\right)y_t^2\left(t\right)d\mathsf{\Gamma }\ge 0,$$

(51)

and

$$\begin{array}{cc}\hfill & 0<H\left(0\right)\le H\left(t\right)\hfill \\ \hfill & =E_1-{\displaystyle \frac{1}{\sigma +2}}{\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}-{\displaystyle \frac{1}{p}}{\parallel \Delta u\parallel }_p^p-{\displaystyle \frac{\xi }{m}}{\int }_0^1{\parallel z(\rho ,t)\parallel }_{m,{\mathsf{\Gamma }}_1}^{m}d\rho -{\displaystyle \frac{1}{2}}{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }\hfill \\ \hfill & +{\displaystyle \frac{1}{r}}{\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^r.\hfill \end{array}$$

(52)

Exploiting (46), we get

$$\begin{array}{cc}\hfill & E_1-{\displaystyle \frac{1}{\sigma +2}}{\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}-{\displaystyle \frac{1}{p}}{\parallel \Delta u\parallel }_p^p-{\displaystyle \frac{\xi }{m}}{\int }_0^1{\parallel z(\rho ,t)\parallel }_{m,{\mathsf{\Gamma }}_1}^{m}d\rho -{\displaystyle \frac{1}{2}}{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }\hfill \\ \hfill & <E_1-{\displaystyle \frac{1}{p}}{\parallel \Delta u\parallel }_p^p<E_1-{\displaystyle \frac{1}{p}}{\xi }_1^p=-{\displaystyle \frac{1}{r}}{\xi }_1^p<0.\hfill \end{array}$$

(53)

Hence, we deduce

$$0<H\left(0\right)\le H\left(t\right)\le {\displaystyle \frac{1}{r}}{\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^r,\forall t>0.$$

(54)

Next, we define

$$\begin{array}{cc}F\left(t\right)\hfill & =H^{1-\zeta }\left(t\right)+{\displaystyle \frac{\epsilon }{1+\sigma }}{\int }_{\mathsf{\Omega }}|u_t{|}^{\sigma }{u}_{t}udx+{\displaystyle \frac{\epsilon }{2}}{\parallel \nabla u\parallel }_2^2-{\displaystyle \frac{\epsilon }{2}}{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)p\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }\hfill \\ \hfill & -\epsilon {\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)u\left(t\right)y\left(t\right)d\mathsf{\Gamma },\hfill \end{array}$$

(55)

where $\epsilon >0$ is a small positive constant. In addition, let

$$0<\sigma \le min\left\{{\displaystyle \frac{p-m}{p(m-1)}},{\displaystyle \frac{p-2}{2p}}\right\}.$$

(56)

Taking the derivative of $F\left(t\right)$, we get

$$\begin{array}{cc}{F}^{\prime }\left(t\right)\hfill & =(1-\zeta )H^{-\zeta }\left(t\right)H^{\prime }\left(t\right)+{\displaystyle \frac{\epsilon }{1+\sigma }}{\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}+\epsilon {\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\sigma }{u}_{tt}udx+\epsilon {\int }_{\mathsf{\Omega }}\nabla u_t\nabla udx\hfill \\ \hfill & -\epsilon {\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)p\left(x\right)y\left(t\right)y_t\left(t\right)d\mathsf{\Gamma }-\epsilon {\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)u_t\left(t\right)y\left(t\right)d\mathsf{\Gamma }\hfill \\ \hfill & -\epsilon {\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)u\left(t\right)y_t\left(t\right)d\mathsf{\Gamma }.\hfill \end{array}$$

(57)

By using (1), estimate (57) becomes

$$\begin{array}{cc}{F}^{\prime }\left(t\right)\hfill & =(1-\zeta )H^{-\zeta }\left(t\right)H^{\prime }\left(t\right)+{\displaystyle \frac{\epsilon }{1+\sigma }}{\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}-\epsilon {\parallel \Delta v\parallel }_p^p+\epsilon {\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^r\hfill \\ \hfill & -{\mu }_1{\int }_{{\mathsf{\Gamma }}_1}|u_t{|}^{m-2}{u}_{t}ud\mathsf{\Gamma }-{\mu }_2{\int }_{{\mathsf{\Gamma }}_1}{\left|z(1,t)\right|}^{m-2}z(1,t)ud\mathsf{\Gamma }\hfill \\ \hfill & +\epsilon {\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }.\hfill \end{array}$$

(58)

From Hölder and Young’s inequalities, $\left(H_2\right)$ and $\left(H_3\right)$, we have

$${\int }_{{\mathsf{\Gamma }}_1}|u_t{|}^{m-2}{u}_{t}ud\mathsf{\Gamma }\le {\displaystyle \frac{{\delta }_1^m}{m}}{\parallel u\parallel }_{m,{\mathsf{\Gamma }}_1}^m+{\displaystyle \frac{m-1}{mc}}{\delta }_1^{-\frac{m}{m-1}}{H}^{\prime }\left(t\right),$$

(59)

and

$${\int }_{{\mathsf{\Gamma }}_1}{\left|z(1,t)\right|}^{m-2}z(1,t)ud\mathsf{\Gamma }\le {\displaystyle \frac{{\delta }_1^m}{m}}{\parallel u\parallel }_{m,{\mathsf{\Gamma }}_1}^m+{\displaystyle \frac{m-1}{mc}}{\delta }_1^{-\frac{m}{m-1}}{H}^{\prime }\left(t\right).$$

(60)

Inserting (59) and (60) into (58), we get

$$\begin{array}{cc}{F}^{\prime }\left(t\right)\hfill & \ge \left\{(1-\zeta )H^{-\zeta }\left(t\right)-\epsilon \left({\mu }_1+{\mu }_2\right)\frac{2(m-1)}{mc}{\delta }_1^{-\frac{m}{m-1}}\right\}{H}^{\prime }\left(t\right)+{\displaystyle \frac{\epsilon }{1+\sigma }}{\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}\hfill \\ \hfill & -\epsilon {\parallel \Delta v\parallel }_p^p+\epsilon {\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^r+\epsilon {\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }-\epsilon \left({\mu }_1+{\mu }_2\right){\displaystyle \frac{{\delta }_1^m}{m}}{\parallel u\parallel }_{m,{\mathsf{\Gamma }}_1}^m.\hfill \end{array}$$

(61)

By the definition of $H\left(t\right)$ and $E\left(t\right)$, we get, for constant $N>0$,

$$\begin{array}{cc}{F}^{\prime }\left(t\right)\hfill & =\left\{(1-\zeta )H^{-\zeta }\left(t\right)-\epsilon \left({\mu }_1+{\mu }_2\right)\frac{2(m-1)}{mc}{\delta }_1^{-\frac{m}{m-1}}\right\}{H}^{\prime }\left(t\right)+N\epsilon H\left(t\right)\hfill \\ \hfill & +\epsilon \left({\displaystyle \frac{N}{2+\sigma }}+{\displaystyle \frac{}{1+\sigma }}\right){\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}+\epsilon \left\{{\displaystyle \frac{N}{p}}-1\right\}{\parallel \Delta v\parallel }_p^p+\epsilon \left\{1-{\displaystyle \frac{N}{r}}\right\}{\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^r\hfill \\ \hfill & +\left\{{\displaystyle \frac{N}{2}}+1\right\}{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }+{\displaystyle \frac{\xi }{m}}{\int }_0^1{\parallel z(\rho ,t)\parallel }_{m,{\mathsf{\Gamma }}_1}^{m}d\rho \hfill \\ \hfill & -\epsilon N{E}_1-\epsilon \left({\mu }_1+{\mu }_2\right){\displaystyle \frac{{\delta }_1^m}{m}}{\parallel u\parallel }_{m,{\mathsf{\Gamma }}_1}^m.\hfill \end{array}$$

(62)

Taking

$$\delta ={\left[{\theta }_1{H}^{-\zeta }\left(t\right)\right]}^{-\frac{m-1}{m}},$$

where ${\theta }_1>0$. We see that

$$\begin{array}{cc}{F}^{\prime }\left(t\right)\hfill & =\left\{(1-\zeta )-\epsilon \left({\mu }_1+{\mu }_2\right){\displaystyle \frac{2(m-1)}{mc}}{\theta }_1\right\}{H}^{-\zeta }\left(t\right)H^{\prime }\left(t\right)+N\epsilon H\left(t\right)\hfill \\ \hfill & +\epsilon \left({\displaystyle \frac{N}{2+\sigma }}+{\displaystyle \frac{1}{1+\sigma }}\right){\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}+\epsilon \left\{{\displaystyle \frac{N}{p}}-1\right\}{\parallel \Delta v\parallel }_p^p+\epsilon \left\{1-{\displaystyle \frac{N}{r}}\right\}{\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^r\hfill \\ \hfill & +\left\{{\displaystyle \frac{N}{2}}+1\right\}{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }+{\displaystyle \frac{\xi }{m}}{\int }_0^1{\parallel z(\rho ,t)\parallel }_{m,{\mathsf{\Gamma }}_1}^{m}d\rho \hfill \\ \hfill & -\epsilon N{E}_1-\epsilon \left({\mu }_1+{\mu }_2\right){\displaystyle \frac{{\theta }_1^{1-m}}{m}}{H}^{\zeta (m-1)}\left(t\right){\parallel u\parallel }_{m,{\mathsf{\Gamma }}_1}^m.\hfill \end{array}$$

(63)

The last term in (63) can be estimated as follows: exploiting (9), (12), (32), $\left(H_3\right)$, we have

$$\begin{array}{ccc}\hfill H^{\zeta (m-1)}\left(t\right){\parallel u\parallel }_{m,{\mathsf{\Gamma }}_1}^m& \le & c_r^m{H}^{\zeta (m-1)}\left(t\right){\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^m\hfill \\ & \le & {\displaystyle \frac{c_r^m}{r^{\zeta (m-1)}}}{\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^{\zeta r(m-1)+m}.\hfill \end{array}$$

(64)

Applying (56) and Lemma 3, for $s=\zeta r(m-1)+m\le r$, we obtain

$${\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^{\zeta r(m-1)+m}\le c_1\left({\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^r+{\parallel \Delta v\parallel }_p^p\right).$$

(65)

Substituting (65) into (63), we get

$$\begin{array}{cc}{F}^{\prime }\left(t\right)\hfill & =\left\{(1-\zeta )-\epsilon \left({\mu }_1+{\mu }_2\right){\displaystyle \frac{2(m-1)}{mc}}{\theta }_1\right\}{H}^{-\zeta }\left(t\right)H^{\prime }\left(t\right)+\epsilon \left({\displaystyle \frac{N}{2+\sigma }}+{\displaystyle \frac{1}{1+\sigma }}\right){\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}\hfill \\ \hfill & +\epsilon \left\{{\displaystyle \frac{N}{p}}-1-c_m{\theta }_1^{1-m}\right\}{\parallel \Delta v\parallel }_p^p+\epsilon \left\{1-{\displaystyle \frac{N}{r}}-{\displaystyle \frac{N{E}_1}{{\kappa }^r{\xi }_2^r}}-c_m{\theta }_1^{1-m}\right\}{\parallel u\parallel }_{r,{\mathsf{\Gamma }}_1}^r\hfill \\ \hfill & +\left\{{\displaystyle \frac{N}{2}}+1\right\}{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }+{\displaystyle \frac{\xi }{m}}{\int }_0^1{\parallel z(\rho ,t)\parallel }_{m,{\mathsf{\Gamma }}_1}^{m}d\rho +N\epsilon H\left(t\right),\hfill \end{array}$$

(66)

where $c_1=\left(c_1{m}_2{c}_{*}^k{c}_{\varrho }^k{\left(KE\left(0\right)\right)}^{k/2}\right)/(H^{\frac{k}{p}}\left(0\right)p^{\sigma (k-1)+\frac{k}{p}})$.

At this point, first, we choose $p<N<{\displaystyle \frac{r{\kappa }^r{\xi }_2^r}{{\kappa }^r{\xi }_2^r+E_1}}$ such that

$${\displaystyle \frac{N}{p}}-1>0\mathrm{and}1-N\left\{{\displaystyle \frac{1}{r}}+{\displaystyle \frac{E_1}{{\kappa }^r{\xi }_2^r}}\right\}>0.$$

When N is fixed, we choose ${\theta }_1$ to be large such that

$${\displaystyle \frac{N}{p}}-1-c_m{\theta }_1^{1-m}>0\mathrm{and}1-{\displaystyle \frac{N}{r}}-{\displaystyle \frac{N{E}_1}{{\kappa }^r{\xi }_2^r}}-c_m{\theta }_1^{1-m}>0.$$

Once ${\theta }_1$ is fixed, we select $\epsilon >0$ small enough so that

$$(1-\zeta )-\epsilon \left({\mu }_1+{\mu }_2\right){\displaystyle \frac{2(m-1)}{mc}}{\theta }_1>0\mathrm{and}F\left(0\right)>0.$$

Then, (66) becomes

$$F^{\prime }\left(t\right)\ge \kappa \left(H\left(t\right)+{\parallel v_t\parallel }_2^2+{\parallel \Delta v\parallel }_2^2+{\parallel \nabla v\parallel }_2^2+{\parallel v\parallel }_p^p+{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)\beta \left(x\right)y^2\left(t\right)d\mathsf{\Gamma }\right),$$

(67)

here $\kappa >0$. Then

$$F\left(t\right)\ge F\left(0\right)>0\forall t\ge 0.$$

(68)

On the other hand, by (55) and $h,p>0$ we obtain

$$F\left(t\right)\le H^{1-\zeta }\left(t\right)+{\displaystyle \frac{\epsilon }{1+\sigma }}{\int }_{\mathsf{\Omega }}|u_t{|}^{\sigma }{u}_{t}udx+{\displaystyle \frac{\epsilon }{2}}{\parallel \nabla u\parallel }_2^2-\epsilon {\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)u\left(t\right)y\left(t\right)d\mathsf{\Gamma }.$$

Thus

$$F^{\frac{1}{1-\zeta }}\left(t\right)\le c_2\left(H\left(t\right)+{\left({\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\sigma }{u}_{t}udx\right)}^{\frac{1}{1-\zeta }}+{\parallel \nabla u\parallel }_2^{\frac{2}{1-\zeta }}+{\left({\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)u\left(t\right)y\left(t\right)d\mathsf{\Gamma }\right)}^{\frac{1}{1-\zeta }}\right).$$

(69)

In what follows, we will estimate the right-hand side terms in (69). By applying Hölder’s and Young’s inequalities, we have

$${\left|{\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\sigma }{u}_{t}udx\right|}^{\frac{1}{1-\zeta }}\le c_3{\parallel u_t\parallel }_{\sigma +2}^{\frac{\varrho +1}{1-\zeta }}{\parallel u\parallel }_{\rho +2}^{\frac{1}{1-\zeta }}\le c_3{\parallel u_t\parallel }_{\sigma +2}^{\frac{\sigma +1}{1-\zeta }}{\parallel u\parallel }_r^{\frac{1}{1-\zeta }}\le c_3\left({\parallel u_t\parallel }_{\sigma +2}^{\nu \frac{\sigma +1}{1-\zeta }}+{\parallel u\parallel }_r^{\vartheta \frac{1}{1-\zeta }}\right)$$

(70)

for $\frac{1}{\nu }+\frac{1}{\vartheta }=1$. Taking $\nu =\frac{(1-\zeta )(\sigma +2)}{(\sigma +1)}>1$ which gives $\frac{\vartheta }{1-\zeta }=\frac{\sigma +2}{(1-\zeta )(\sigma +2)-(\sigma +1)}$.

Applying Lemma 2, for $s_1=\frac{\sigma +2}{(1-\zeta )(\sigma +2)-(\sigma +1)}<r$, we get

$${\left|{\int }_{\mathsf{\Omega }}{\left|u_t\right|}^{\sigma }{u}_{t}udx\right|}^{\frac{1}{1-\zeta }}\le c_4\left({\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}+{\parallel \Delta u\parallel }_p^p+{\parallel u\parallel }_r^r\right).$$

(71)

Applying Hölder’s inequality, we have

$$\begin{array}{cc}\left|{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)u\left(t\right)y\left(t\right)d\mathsf{\Gamma }\right|\hfill & =\left|{\int }_{{\mathsf{\Gamma }}_1}\frac{h\left(x\right)q\left(x\right)}{q\left(x\right)}u\left(t\right)y\left(t\right)d\mathsf{\Gamma }\right|\hfill \\ \hfill & \le \frac{{\parallel h\parallel }_{\infty }^{\frac{1}{2}}{\parallel p\parallel }_{\infty }^{\frac{1}{2}}}{p_0}{\left({\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)p\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }\right)}^{\frac{1}{2}}{\left({\int }_{{\mathsf{\Gamma }}_1}{u}^{2}d\mathsf{\Gamma }\right)}^{\frac{1}{2}},\hfill \end{array}$$

(72)

which implies

$${\left|{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)u\left(t\right)y\left(t\right)d\mathsf{\Gamma }\right|}^{\frac{1}{1-\zeta }}\le c_5\left\{{\left({\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }\right)}^{\frac{\vartheta }{2(1-\zeta )}}+{\left({\int }_{{\mathsf{\Gamma }}_1}{u}^{2}d\mathsf{\Gamma }\right)}^{\frac{\nu }{2(1-\zeta )}}\right\},$$

(73)

for $\frac{1}{\nu }+\frac{1}{\vartheta }=1$. Taking $\vartheta =2(1-\zeta )$ which gives $\frac{\nu }{1-\zeta }=\frac{2}{1-2\zeta }$. Then, we have

$${\left|{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)u\left(t\right)y\left(t\right)d\mathsf{\Gamma }\right|}^{\frac{1}{1-\zeta }}\le c_7\left\{{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)q\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }+{\parallel u\parallel }_{2,{\mathsf{\Gamma }}_1}^{\frac{1}{1-2\zeta }}\right\}.$$

(74)

On the other hand, we have

$${\parallel u\parallel }_{2,{\mathsf{\Gamma }}_1}^{\frac{1}{1-2\zeta }}\le c_{\varsigma }^{\frac{1}{1-2\zeta }}{\parallel \nabla u\parallel }_2^{\frac{1}{1-2\zeta }}\le c_{\varsigma }^{\frac{1}{1-2\sigma }}{c}_{\varrho }^{\frac{1}{1-2\sigma }}{c}_8^{\frac{1}{1-2\sigma }}{\parallel \Delta u\parallel }_p^{\frac{1}{1-2\zeta }}\le c_{\varsigma }^{\frac{1}{1-2\zeta }}{c}_{\varrho }^{\frac{1}{1-2\zeta }}{c}_8^{\frac{1}{1-2\zeta }}\frac{H\left(t\right)}{H\left(0\right)}.$$

(75)

Similarly, we have

$${\parallel \nabla u\parallel }_2^{\frac{2}{1-\zeta }}\le c_{\varrho }^{\frac{2}{1-\zeta }}{c}_8^{\frac{2}{1-\zeta }}{\left(KE\left(0\right)\right)}^{\frac{2}{p(1-\zeta })}\frac{H\left(t\right)}{H\left(0\right)},$$

(76)

and

$${\parallel u\parallel }_r^r\le c_{\varrho }^r{c}_p^r{\parallel \Delta v\parallel }_p^r\le c_{\varrho }^r{c}_p^r{\left(KE\left(0\right)\right)}^{\frac{r-p}{r}}{\parallel \Delta u\parallel }_p^p.$$

(77)

Combining (71), (74)–(77), and (69), we obtain

$$F^{\frac{1}{1-\zeta }}\left(t\right)\le \left(H\left(t\right)+{\parallel u_t\parallel }_{\sigma +2}^{\sigma +2}+{\parallel \Delta u\parallel }_p^p+{\int }_{{\mathsf{\Gamma }}_1}h\left(x\right)p\left(x\right)y^2\left(t\right)d\mathsf{\Gamma }\right).$$

(78)

From (67) and (78), we have

$$F^{\prime }\left(t\right)\ge \alpha F^{\frac{1}{1-\zeta }}\left(t\right),$$

(79)

where $\alpha >0$. Integrating (79) over $(0,t)$, we get

$$F^{\frac{\zeta }{1-\zeta }}\left(t\right)\ge \frac{1}{F^{\frac{-\zeta }{1-\zeta }}\left(0\right)-\frac{\alpha \zeta }{1-\zeta }t}.$$

Hence $F\left(t\right)$ blows up in finite time $T^{*}$ and

$$T^{*}=\frac{1-\zeta }{\alpha \zeta F^{\frac{\zeta }{1-\zeta }}\left(0\right)}.$$

The proof is complete. □

5. Conclusions

In this paper, we investigated the global existence, general decay, and blow-up results of solutions for a class of p-bi-harmonic type hyperbolic equations with delay and acoustic boundary conditions using an appropriate approach.

The effectiveness of the proposed approach lies in the presence of nonlinear damping with/without delay with a nonlinear source. This type of system is mathematically rich and relevant, and the approach through potential well theory and Komornik’s inequality is appreciated. To the best of our knowledge, the results are new and very important from the applied point of view. Despite it being entirely of a theoretical nature since the partial differential equations with biharmonic terms have practical relevance, especially in energy and industry, it also arises in the modeling of solid mechanics and micropolar fluids; see [32,33]. It will be interesting to consider the case of variable exponents as a future work direction, extending these results for wide classes of models.