Asymptotic Analysis and Blow-Up of Solution for a q-Kirchhoff-Type Equation with Nonlinear Boundary Damping and Source Terms with Variable Exponents

Abstract

In this study, we examine a q-Kirchhoff-type equation defined on a bounded domain, incorporating nonlinear boundary damping and source terms with variable exponents. Assuming appropriate conditions on the initial data and the variable exponent functions, we establish the global existence of solutions. Subsequently, we derive a general decay estimate for these solutions. Lastly, we demonstrate that solutions with negative initial energy exhibit finite-time blow-up.

1. Introduction

In this paper, we study the following initial boundary value problem

$$\left\{\begin{array}{cc}{v}_{tt}-{\mathrm{M}}_1\left({\parallel \nabla v\parallel }_q^q\right){\Delta }_{q}v-{\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right){\Delta }_q{v}_t=0,\hfill & x\in \mathsf{\Omega },t>0,\hfill \\ v(x,t)=0,\hfill & x\in {\mathsf{\Gamma }}_0,t>0,\hfill \\ {\displaystyle {\mathrm{M}}_1\left({\parallel \nabla v\parallel }_q^q\right){|\nabla v|}^{q-2}\frac{\partial v}{\partial \nu }+{\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right)|\nabla v_t{|}^{q-2}\frac{\partial v_t}{\partial \nu }+{|v_t|}^{m\left(x\right)-2}{v}_t}\hfill & \\ ={|v|}^{k\left(x\right)-2}v,\hfill & x\in {\mathsf{\Gamma }}_1,t>0,\hfill \\ v(x,0)=v_0\left(x\right),v_t(x,0)=v_1\left(x\right),\hfill & x\in \mathsf{\Omega },\hfill \end{array}\right.$$

(1)

where $\mathsf{\Omega }\subset {\mathbb{R}}^n(n\ge 1)$ is a bounded domain with a smooth boundary $\partial \mathsf{\Omega }={\mathsf{\Gamma }}_0\cup {\mathsf{\Gamma }}_1$, $\mathrm{mes}\left({\mathsf{\Gamma }}_0\right)>0,{\mathsf{\Gamma }}_0\cap {\mathsf{\Gamma }}_1=\varnothing$, ${\displaystyle \frac{\partial v}{\partial \nu }}$ denotes the unit outer normal derivative, $q>2$, $v_0,v_1$ are given functions belonging to suitable spaces. The operator ${\Delta }_{q}v$ is the classical q-Laplacian given by ${\Delta }_{q}v=\mathrm{div}\left({|\nabla v|}^{q-2}\nabla q\right)$. The function ${\mathrm{M}}_1$ is given by ${\mathrm{M}}_1\left(s\right)={\zeta }_1+s{\zeta }_2$, where ${\zeta }_1$ and ${\zeta }_2$ are positive constants. The function ${\mathrm{M}}_2$ will be specified later. The exponents $k(·)$ and $m(·)$ are given measurable functions on $\mathsf{\Omega }$ and satisfy

$$2\le k_1\le k\left(x\right)\le k_2<q^{\partial },$$

(2)

$$2\le m_1\le m\left(x\right)\le m_2<q^{\partial },$$

(3)

where

$$\begin{array}{ccc}\hfill k_1=ess\underset{x\in \mathsf{\Omega }}{inf}k\left(x\right),& & k_2=ess\underset{x\in \mathsf{\Omega }}{sup}k\left(x\right),\hfill \\ \hfill m_1=ess\underset{x\in \mathsf{\Omega }}{inf}m\left(x\right),& & m_2=ess\underset{x\in \mathsf{\Omega }}{sup}m\left(x\right),\hfill \end{array}$$

$$q^{\partial }=\left\{\begin{array}{cc}{\displaystyle \frac{(n-1)q}{n-q},}\hfill & \mathrm{if}q<n<2q-1,\hfill \\ \infty ,\hfill & \mathrm{if}q\ge n.\hfill \end{array}\right.$$

(4)

and the log-Hölder continuity condition

$$|k\left(x\right)-k\left(y\right)|\le -{\displaystyle \frac{A}{log|x-y|}}\mathrm{and}|m\left(x\right)-m\left(y\right)|\le -{\displaystyle \frac{B}{log|x-y|}},$$

(5)

for $x,y\in \mathsf{\Omega }$, with $|x-y|<\delta ,A,B>0$ and $0<\delta <1$.

Remark 1. 

In (4), we have added the condition $n<2q-1$ which was not present in the original result (see Lemma 5 below). This additional assumption ensures that, when we later use the condition $k_1>2q$ in the following sections, the inequality $2q<{\displaystyle \frac{(n-1)q}{n-q}}$ holds true.

This initial boundary value problem (1) describes a nonlinear hyperbolic partial differential equation with nonlocal, nonlinear damping and source terms, and mixed boundary conditions. It models the dynamics of a vibrating medium (e.g., membrane, plate, or elastic body) whose mechanical properties depend nonlinearly on the global energy (or gradient norm) of the system a feature common in materials with nonlinear strain-dependent stiffness or viscoelastic damping.

The function $v(x,t)$ represents the displacement of the medium at position x and time t. Think of it as the vertical deflection of a membrane or the deformation of an elastic body.

The terms $v_{tt}(x,t),$ ${\Delta }_{q}v=$ =div $\left({|\nabla v|}^{q-2}\nabla v\right)$ denote respectively the acceleration term and the $q$-Laplacian which arises in modeling materials with nonlinear stress–strain relationships. For $q>2$, it describes media where the resistance to deformation increases more than linearly with strain (e.g., certain polymers, soils, or biological tissues).

The term $M_1\left({\parallel \nabla v\parallel }_q^q\right){\Delta }_{q}v$ is a nonlocal elastic restoring force [1]. The coefficient $M_1\left(s\right)={\zeta }_1+s{\zeta }_2$ depends on the total $q$-energy of the gradient ${\parallel \nabla v\parallel }_q^q.$ This means the stiffness of the material increases with the overall deformation energy modeling strain-hardening materials or systems with nonlocal feedback (e.g., Kirchhoff-type strings or plates where tension depends on the entire deformation). The term ${\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right){\Delta }_q{v}_t$ is a nonlocal, nonlinear damping term. It acts on the velocity gradient $\nabla v_t,$ and its strength depends again on the total deformation energy [2].

The boundary condition in Equation (1), is a dynamic boundary condition coupling the interior dynamics with boundary interactions, where $M_1{|\nabla v|}^{q-2}{\displaystyle \frac{\partial v}{\partial \nu }}$ is the nonlinear elastic flux (stress) normal to the boundary. It’s the boundary counterpart of the internal $q$-Laplacian force, and $M_2{|\nabla v_t|}^{q-2}{\displaystyle \frac{\partial v_t}{\partial \nu }}$ is the nonlinear damping flux at the boundary, depending on the gradient of velocity. The right-hand side ${|v|}^{k\left(x\right)-2}v$ is a nonlinear, $x$-dependent source term on the boundary. This could represent external forcing, boundary feedback, or even boundary reactions (e.g., in combustion or growth models) (see [3,4,5] for more details).

We begin by reviewing existing literature relevant to problem (1). When boundary conditions are omitted, the equation reduces to the classical Kirchhoff-type model, which has been widely investigated in the literature with well-established results on existence, nonexistence, and decay rates of solutions. Let us recall some works related to problem (1) in the case where $q=2$. When ${\mathsf{\Gamma }}_1$ is empty, Chueshov [6] investigated problem (1) in the presence of a source term. The author proved the existence and uniqueness of weak solutions and analyzed the long-term dynamics of the system. In a similar study, Ding et al. [7] established the existence of a global attractor. More generally, Lazo [8] considered an extension of the problem by replacing the Laplace operator with a positive self-adjoint operator A defined on a Hilbert space H, and proved the existence of a global solution using the Galerkin method. Now, when $q>2$, Ouaoua et al. [9] studied a problem similar to (1) by taking ${\mathsf{\Gamma }}_1=$, $M_2=1$, a damping term of the form $|u_t{|}^{m\left(x\right)-2}{u}_t$, and a source term with variable exponent. They proved the global existence of the solution with positive initial energy and established a stability result. Messaoudi [10] analyzed the equation

$$v_{tt}-{\Delta }_{q}v-\Delta v_t+|v_t{|}^{m-1}{v}_t={|v|}^{k-1}v,$$

(6)

establishing decay properties of its solutions through a blend of perturbed energy and potential well techniques. Wu and Xue [11] revisited the same equation and derived uniform energy decay estimates using multiplier methods. Pişkin [12] further explored both energy decay and finite-time blow-up phenomena for (6). Mokeddem and Mansour [13] extended these contributions by proving global existence and providing precise estimates for the energy decay rate. Additionally, Ouaoua et al. [14] examined the following related equation:

$$v_{tt}-M\left({\parallel \nabla \parallel }_q^q\right){\Delta }_{q}v+|v_t{|}^{m-2}{v}_t={|v|}^{r-2}v,$$

(7)

where they established the global existence of solutions using potential energy and Nehari functionals. The stability of the equation was proved using Komornik’s integral inequality. Recently, Gheraibia et al. [15] considered the following p-Kirchhoff type hyperbolic equation with damping terms, dynamic boundary conditions, and source term acting on the boundary

$$|v_t{|}^{\rho }{v}_{tt}-M\left({\parallel \nabla v\parallel }_q^q\right){\Delta }_{q}v-\sigma \left(t\right)\left(\Delta v_t-v_t\right)=0,$$

(8)

where $\rho >0,M\left(s\right)={\xi }_1+{\xi }_{2}s,{\xi }_1,{\xi }_2>0$, and $\sigma$ is a nonincreasing positive function. Under suitable assumptions on the function $\sigma$ and the initial data, the authors proved the global existence of solutions and established a general decay of energy by using Martinez’s integral inequalities [16].

Variable exponent problems arise naturally in numerous mathematical models across applied sciences—including viscoelastic and electro-rheological fluids, filtration processes in porous media, and fluids whose viscosity depends on temperature, among other things. For further reading, we refer interested readers to [17,18] and the references therein. Messaoudi et al. [19] considered the following nonlinear wave equation with variable exponents

$$v_{tt}-a\Delta v+|v_t{|}^{m\left(x\right)-2}{v}_t=b{|v|}^{k\left(x\right)-2}v,$$

(9)

and used the Faedo Galerkin method to establish the existence of a unique weak local solution. They also proved that the solutions with negative initial energy blow up in finite time. Ghegal et al. [20] studied (9) and, under suitable conditions on the initial data and the variable exponents, the authors used a stable-set method to prove a global existence result. Then, by applying an integral inequality due to Komornik, they obtained the stability result. Park and Kang [21] considered the following equation

$$v_{tt}-\Delta v+\alpha \left(t\right)\left[v_t+{|v_t|}^{m\left(x\right)-2}{v}_t\right]={|v|}^{k\left(x\right)-2}v,$$

(10)

with variable sources and acoustic boundary conditions and established a decay result by using the multiplier method. Messaoudi et al. [22] considered the following nonlinear equation:

$$v_{tt}-\mathrm{div}\left({|\nabla v|}^{q\left(x\right)-2}\nabla v\right)+{|v_t|}^{m\left(x\right)-2}{v}_t=0,$$

(11)

and proved the decay results for the solution under suitable assumptions. Also, the authors gave two numerical applications to illustrate the theoretical results. This result was later improved by Messaoudi [23]. Li et al. [24] studied (11) with variable sources and established various decay rates of the energy solution. Moreover, the authors presented some numerical examples to illustrate their results. In [25,26], the authors established a blow-up result for solutions to equation considered in [24].

In recent years, wave equations incorporating boundary damping and source terms have attracted considerable attention in the mathematical literature. Vitillaro [27] studied the following initial boundary value problem:

$$\left\{\begin{array}{cc}{v}_{tt}-\Delta v=0,\hfill & x\in \mathsf{\Omega },t>0,\hfill \\ v(x,t)=0,\hfill & x\in {\mathsf{\Gamma }}_0,t>0,\hfill \\ v_{\nu }=-|v_t{|}^{m-2}{v}_t+{|v|}^{k-2}v,\hfill & x\in {\mathsf{\Gamma }}_1,t>0,\hfill \\ v(x,0)=v_0\left(x\right),v_t(x,0)=v_1\left(x\right),\hfill & x\in \mathsf{\Omega }.\hfill \end{array}\right.$$

(12)

and demonstrated that when $2\le k\le m,$ the superlinear damping term $-|v_t{|}^{m-2}{v}_t$, ensures global existence for arbitrary initial data in contrast to the blow-up behavior observed when $m=2<k$. Zhang and Hu [28] further examined the asymptotic behavior of solutions to (12), assuming initial data lies within a stable set. Kass and Rammaha [29] considered the following q-Laplacian-type quasilinear wave equation:

$$\left\{\begin{array}{cc}{v}_{tt}-{\Delta }_{q}v-\Delta v_t=0,\hfill & \mathrm{in}\mathsf{\Omega }\times (0,T),\hfill \\ {|\nabla v|}^{q-2}{\partial }_{\nu }v+{|v|}^{q-2}v+{\partial }_{\nu }{v}_t+v_t=f\left(u\right),\hfill & \mathrm{on}\mathsf{\Gamma }\times (0,T),\hfill \\ v(x,0)=v_0\left(x\right),v_t(x,0)=v_1\left(x\right),\hfill & x\in \mathsf{\Omega },\hfill \end{array}\right.$$

(13)

where $2<q<3$. They established the local and global existence of the solutions by using Faedo–Galerkin method. Boumaza and Gheraibia [30] considered the following equation:

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

(14)

with nonlinear boundary, delay, and source terms acting on the boundary, they proved the global existence of solutions and establish the decay rate result by. Furthermore, they proved the blow-up of the solutions in finite time with negative initial energy. Recently, Kamache et al. [31] investigated global existence, decay rate, and the blow-up of solutions to a Kirchhoff-type equation with nonlinear boundary delay and source terms. We also cite recent works that address problems involving variable exponents [32,33,34].

Motivated by the aforementioned research, this work focuses on analyzing the global existence, general decay rates, and conditions for finite-time blow-up of solutions to a q-Kirchhoff-type equation in a bounded domain, featuring nonlinear damping and source terms on the boundary with variable exponents.

The rest of this article is organized as follows. In Section 2, we give some notations and material needed for this work. In Section 3, we establish a global existence result. In Section 4, we state and prove the decay rate. In Section 5, we show the blow-up of solutions.

2. Preliminaries

2.1. Function Spaces

In this part, we list and recall some well-known results and facts from the theory of the Sobolev spaces with variable exponent (See [17,35,36,37]).

Let $p:\mathsf{\Omega }\to [1,+\infty ]$ be a measurable function. We define the Lebesgue space with variable exponent $p(·)$ by

$$L^{p(·)}\left(\mathsf{\Omega }\right)=\left\{v:\mathsf{\Omega }\to \mathbb{R}\mathrm{measurable}\mathrm{and}{\varrho }_{p(·)}\left(v\right)<\infty \right\},$$

where ${\displaystyle {\varrho }_{p(·)}\left(v\right)={\int }_{\mathsf{\Omega }}{|v|}^{p\left(x\right)}dx}$.

The space $L^{p(·)}\left(\mathsf{\Omega }\right)$ is equipped with the Luxemburg-type norm

$${\parallel v\parallel }_{p(·)}=inf\left\{{\displaystyle \lambda >0:{\int }_{\mathsf{\Omega }}{\left|\frac{v}{\lambda }\right|}^{p\left(x\right)}dx\le 1}\right\}.$$

Next, we define the variable-exponent Sobolev space $W^{1,p(.)}\left(\mathsf{\Omega }\right)$ by

$$W^{1,p(.)}\left(\mathsf{\Omega }\right)=\left\{v\in L^{p(.)}\left(\mathsf{\Omega }\right)\mathrm{such}\mathrm{that}\nabla v\mathrm{exists}\mathrm{and}|\nabla v|{\in }^{p(.)}\left(\mathsf{\Omega }\right)\right\}.$$

Endowed with the norm

$${\parallel v\parallel }_{1,p(.)}:={\parallel v\parallel }_{W^{1,p(.)}\left(\mathsf{\Omega }\right)}={\parallel v\parallel }_{p(.)}+{\parallel \nabla v\parallel }_{p(.)},$$

$W^{1,p(.)}\left(\mathsf{\Omega }\right)$ is a Banach space. The space $W_0^{1,p(.)}\left(\mathsf{\Omega }\right)$ is defined as the closure of $C_0^{\infty }\left(\mathsf{\Omega }\right)$ in $W^{1,p(.)}\left(\mathsf{\Omega }\right)$ with respect to the norm ${\parallel v\parallel }_{1,p(.)}$, and will be equipped by the norm

$${\parallel v\parallel }_{W_0^{1,p(.)}\left(\mathsf{\Omega }\right)}={\parallel \nabla v\parallel }_{p(.)}.$$

Lemma 1 

(Poincaré inequality [17]). Let Ω be a bounded domain of ${\mathbb{R}}^n$ and $p(·)$ satisfies (5). Let $p_1=ess{inf}_{x\in \mathsf{\Omega }}p\left(x\right)$ and $p_2=ess{sup}_{x\in \mathsf{\Omega }}p\left(x\right)$. Then,

$${\parallel v\parallel }_{L^{p(·)}\left(\mathsf{\Omega }\right)}\le c{\parallel \nabla v\parallel }_{L^{p(·)}\left(\mathsf{\Omega }\right)}\mathrm{for}\mathrm{all}v\in W_0^{1,p(.)}\left(\mathsf{\Omega }\right),$$

where c is a positive constant that depends on $p_1$, $p_2$, and Ω.

Lemma 2 

([17,37]). If $p(.)\in C\left(\overline{\mathsf{\Omega }}\right)$ and $q:\mathsf{\Omega }\to [1,\infty )$ is a measurable function such that

$$ess\underset{x\in \mathsf{\Omega }}{inf}\left(p^{*}\left(x\right)-q\left(x\right)\right)>0\mathrm{with}{p}^{*}\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{np\left(x\right)}{ess{sup}_{x\in \mathsf{\Omega }}\left(n-p\left(x\right)\right)},}\hfill & \mathrm{if}{p}_2<n,\hfill \\ \infty ,\hfill & \mathrm{if}{p}_2>n.\hfill \end{array}\right.$$

Then, the embedding $W_0^{1,p(.)}\left(\mathsf{\Omega }\right)\hookrightarrow L^{q(.)}\left(\mathsf{\Omega }\right)$ is continuous and compact.

Lemma 3 

(Hölder inequality [17]). Suppose that $p,q,s\ge 1$ be measurable functions defined on Ω and satisfy

$${\displaystyle \frac{1}{s\left(x\right)}}={\displaystyle \frac{1}{p\left(x\right)}}+{\displaystyle \frac{1}{q\left(x\right)}}.$$

If $f\in L^{p(·)}\left(\mathsf{\Omega }\right)$ and $g\in L^{q(·)}\left(\mathsf{\Omega }\right)$, then $fg\in L^{s(·)}\left(\mathsf{\Omega }\right)$ and

$${\parallel fg\parallel }_{L^{s(·)}\left(\mathsf{\Omega }\right)}\le {2\parallel f\parallel }_{L^{p(·)}\left(\mathsf{\Omega }\right)}{\parallel g\parallel }_{L^{q(·)}\left(\mathsf{\Omega }\right)}.$$

Lemma 4 

([17]). If p is a measurable function and $1\le p_1\le p\left(x\right)\le p_2<\infty$, then, we have

$$min\left\{{\parallel v\parallel }_{p\left(x\right)}^{p_1},{\parallel v\parallel }_{p\left(x\right)}^{p_2}\right\}\le {\varrho }_{p(·)}\left(v\right)\le max\left\{{\parallel v\parallel }_{p\left(x\right)}^{p_1},{\parallel v\parallel }_{p\left(x\right)}^{p_2}\right\},\forall v\in L^{p\left(x\right)}.$$

(15)

Lemma 5 

([38]). Let Ω be a bounded domain in ${\mathbb{R}}^n$ with boundary Γ. Suppose that $p\in C^0\left(\overline{\mathsf{\Omega }}\right)$ with $p_1>1$. If $k\in C^0\left(\mathsf{\Gamma }\right)$ satisfies the condition

$$1\le k\left(x\right)<p^{\partial }\left(x\right),\forall x\in \mathsf{\Gamma },$$

then, there is a compact boundary trace embeddding $W^{1,p(.)}\left(\mathsf{\Omega }\right)\hookrightarrow L^{k(.)}\left(\mathsf{\Gamma }\right)$, where

$$p^{\partial }\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{(n-1)p\left(x\right)}{n-p\left(x\right)},}\hfill & \mathrm{if}p\left(x\right)<n,\hfill \\ \infty ,\hfill & \mathrm{if}p\left(x\right)\ge n.\hfill \end{array}\right.$$

2.2. Mathematical Hypotheses

In this section we give some notation for function spaces and some preliminary lemmas. We denote by ${\parallel v\parallel }_r$ and ${\parallel v\parallel }_{r,{\mathsf{\Gamma }}_1}$ the usual $L^r\left(\mathsf{\Omega }\right)$ and $L^r\left({\mathsf{\Gamma }}_1\right)$ norms, respectively. Moreover, we denote by $(·,·)$ and ${(·,·)}_{\mathsf{\Gamma }1}$ the standard inner products in $L^2\left(\mathsf{\Omega }\right)$ and $L^2\left({\mathsf{\Gamma }}_1\right)$, respectively.

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

H₁. 

${\mathrm{M}}_2$ is a $C^1$-function and there exists a constant $m_0$ such that ${\mathrm{M}}_2\left(s\right)\ge m_0$ for all $s\ge 0$.

H₂. 

Assumptions on $v_0,v_1$. Assume that $(v_0,v_1)\in W_{{\mathsf{\Gamma }}_0}^{1,q}\left(\mathsf{\Omega }\right)\times L^2\left(\mathsf{\Omega }\right)$ satisfying the compatibility conditions

$${\displaystyle {\mathrm{M}}_1\left(\parallel \nabla v_0{\parallel }_q^q\right)|\nabla v_0{|}^{q-2}\frac{\partial v_0}{\partial \nu }+{\mathrm{M}}_2\left(\parallel \nabla v_0{\parallel }_q^q\right)|\nabla v_1{|}^{q-2}\frac{\partial v_1}{\partial \nu }+|v_1{|}^{m\left(x\right)-2}{v}_1={|v_0|}^{k\left(x\right)-2}{v}_0,\mathrm{on}{\mathsf{\Gamma }}_1.}$$

Let $c_{\varrho }$ be the optimal constant of Sobolev imbedding which satisfies the inequality

$${\parallel v\parallel }_r\le c_{\varrho }{\parallel \nabla v\parallel }_r,\forall v\in W_{{\mathsf{\Gamma }}_0}^{1,r}\left(\mathsf{\Omega }\right),$$

(16)

where

$${\parallel \nabla v\parallel }_r={\left[{\int }_{\mathsf{\Omega }}{|\nabla v|}^r\right]}^{1/r}\mathrm{and}{W}_{{\mathsf{\Gamma }}_0}^{1,r}\left(\mathsf{\Omega }\right)=\left\{u\in W^{1,r}\left(\mathsf{\Omega }\right)|u_{|{\mathsf{\Gamma }}_0}=0\right\}.$$

According to (2), (3), and Lemma 5, we recall the trace Sobolev embeddings

$$W_{{\mathsf{\Gamma }}_0}^{1,q}\left(\mathsf{\Omega }\right)\hookrightarrow L^{k(.)}\left({\mathsf{\Gamma }}_1\right),L^{m(.)}\left({\mathsf{\Gamma }}_1\right),$$

and the embedding inequalities

$${\parallel v\parallel }_{k(.),{\mathsf{\Gamma }}_1}\le c_{*}{\parallel \nabla v\parallel }_q{,\mathrm{and}\parallel v\parallel }_{m(.),{\mathsf{\Gamma }}_1}\le c_{*}{\parallel \nabla v\parallel }_q,\forall v\in W_{{\mathsf{\Gamma }}_0}^{1,q}\left(\mathsf{\Omega }\right),$$

(17)

where $c_{*}$ is a positive constant that depends on $q,n$, and the log-Hölder modulus.

Definition 1. 

Let $T>0$. A function v defined on $\mathsf{\Omega }\times (0,T)$ is called a weak solution of problem (1) if

$$\begin{array}{cc}\hfill & v\in L^{\infty }\left((0,T);W_{{\mathsf{\Gamma }}_0}^{1,q}\left(\mathsf{\Omega }\right)\right),\hfill \\ \hfill & v_t\in L^{\infty }\left((0,T);L^2\left(\mathsf{\Omega }\right)\right)\cap L^q\left((0,T);W_{{\mathsf{\Gamma }}_0}^{1,q}\left(\mathsf{\Omega }\right)\right),\hfill \\ \hfill & v_{tt}\in L^{\infty }\left((0,T);W^{-1,q^{\prime }}\left(\mathsf{\Omega }\right)\right),\hfill \end{array}$$

and it satisfies the variational equation

$$\begin{array}{cc}\hfill & \langle v_{tt},w\rangle +{\mathrm{M}}_1\left({\parallel \nabla v\parallel }_q^q\right){(|\nabla v|}^{q-2}\nabla v,\nabla w)+{\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right)(|\nabla v_t{|}^{q-2}\nabla v_t,\nabla w)\hfill \\ \hfill & +(|v_t{|}^{m\left(x\right)-2}{v}_t{,w)}_{{\mathsf{\Gamma }}_1}={(|v|}^{k\left(x\right)-2}{v,w)}_{{\mathsf{\Gamma }}_1},\forall w\in W_{{\mathsf{\Gamma }}_0}^{1,q}\left(\mathsf{\Omega }\right),\hfill \end{array}$$

where $q^{\prime }$ is the conjugate of q (i.e., ${\displaystyle \frac{1}{q}}+{\displaystyle \frac{1}{q^{\prime }}}=1$) and $\langle ·,·\rangle$ is the duality pairing between $W^{-1,q^{\prime }}\left(\mathsf{\Omega }\right)$ and $W_{{\mathsf{\Gamma }}_0}^{1,q}\left(\mathsf{\Omega }\right)$.

Next, we give the local existence theorem that can be established by combining the following arguments [39,40,41,42]:

Theorem 1. 

Let $v_0\in W_{{\mathsf{\Gamma }}_0}^{1,q}\left(\mathsf{\Omega }\right)$, $v_1\in L^2\left(\mathsf{\Omega }\right)$. Assume that $\left(H_1\right)-\left(H_2\right)$ hold and the exponents k and m satisfy the conditions (2)–(5). Then, the problem (1) admits a local weak solution.

Remark 2. 

Our main tool for proving existence is the Faedo–Galerkin method, which consists of constructing approximate solutions and then deriving a priori estimates to guarantee the convergence of these approximations. The proof is organized as follows: In the first step, we define an approximate problem in a finite-dimensional space, which admits a solution. In the second step, we establish the necessary a priori estimates. Finally, in the third step, we pass to the limit in the approximations by using the compactness of certain embeddings in Sobolev spaces.

Now, we define the energy functional associated with problem (1) by

$${\displaystyle E\left(t\right)=\frac{1}{2}\parallel v_t{\parallel }_2^2+\frac{{\zeta }_1}{q}{\parallel \nabla v\parallel }_q^q+\frac{{\zeta }_2}{2q}{\parallel \nabla v\parallel }_q^{2q}-{\int }_{{\mathsf{\Gamma }}_1}\frac{{|v|}^{k\left(x\right)}}{k\left(x\right)}d\mathsf{\Gamma }.}$$

(18)

Lemma 6. 

Let v be a solution of problem (1). Then,

$${\displaystyle E^{\prime }\left(t\right)\le -{\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right)\parallel \nabla v_t{\parallel }_q^q-{\int }_{{\mathsf{\Gamma }}_1}{|v_t|}^{m\left(x\right)}d\mathsf{\Gamma }\le 0.}$$

(19)

Proof. 

Multiplying the first equation in (1) by $v_t$ and integrating over $\mathsf{\Omega }$, we get (19). □

Theorem 2 

(Komornik, [43]). Let $E:[0,\infty )\to [0,\infty )$ be a non-increasing function and assume that there exist two constants $c>0$ and $\varpi >0$ such that

$${\displaystyle {\int }_t^{\infty }{E}^{1+\varpi }\left(s\right)ds\le cE\left(t\right),\forall t\ge 0.}$$

Then, we have, for all $t\ge 0$ and some positive constants c and ζ

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

3. Global Existence

In this section, we will prove that the solution established in Theorem 1 is global in time. For this purpose, we set

$${\displaystyle I\left(t\right)={\zeta }_1{\parallel \nabla v\parallel }_q^q+{\zeta }_2{\parallel \nabla v\parallel }_q^{2q}-{\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }}$$

(20)

and

$${\displaystyle J\left(t\right)=\frac{{\zeta }_1}{q}{\parallel \nabla v\parallel }_q^q+\frac{{\zeta }_2}{2q}{\parallel \nabla v\parallel }_q^{2q}-{\int }_{{\mathsf{\Gamma }}_1}\frac{{|v|}^{k\left(x\right)}}{k\left(x\right)}d\mathsf{\Gamma }.}$$

(21)

Then, it is obvious that

$$E\left(t\right)={\displaystyle \frac{1}{2}}{\parallel v_t\parallel }_2^2+J\left(t\right).$$

(22)

In order to show our result, we first establish the following lemma.

Lemma 7. 

Assume that the conditions of Theorem 1 hold and $k_1>2q$. Let $v_0\in W_{{\mathsf{\Gamma }}_0}^{1,q}\left(\mathsf{\Omega }\right)$ and $v_1\in L^2\left(\mathsf{\Omega }\right)$ such that

$$I\left(0\right)>0\mathrm{and}{\alpha }_1+{\alpha }_2<1,$$

(23)

where

$$\begin{array}{cc}\hfill & {\alpha }_1=\gamma max\left\{{\displaystyle \frac{c_{*}^{k_1}}{{\zeta }_1}}{\left[{\displaystyle \frac{k_{1}q}{{\zeta }_1(k_1-q)}}\right]}^{{\displaystyle \frac{k_1-q}{q}}},{\displaystyle \frac{c_{*}^{k_2}}{{\zeta }_1}}{\left[{\displaystyle \frac{k_{1}q}{{\zeta }_1(k_1-q)}}\right]}^{{\displaystyle \frac{k_2-q}{q}}}\right\},\hfill \\ \hfill & {\alpha }_2=(1-\gamma )max\left\{{\displaystyle \frac{c_{*}^{k_1}}{{\zeta }_2}}{\left[{\displaystyle \frac{2k_{1}q}{{\zeta }_2(k_1-2q)}}\right]}^{{\displaystyle \frac{k_1-2q}{2q}}},{\displaystyle \frac{c_{*}^{k_2}}{{\zeta }_2}}{\left[{\displaystyle \frac{2k_{1}q}{{\zeta }_2(k_1-2q)}}\right]}^{{\displaystyle \frac{k_2-2q}{2q}}}\right\}.\hfill \end{array}$$

with $0<\gamma <1$. Then,

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

(24)

Proof. 

By continuity of v, there exist a time $T_{*}<T$ such that

$$I\left(t\right)\ge 0,\forall t\in [0,T_{*}].$$

(25)

From (21) and (25), we have

$$\begin{array}{cc}J\left(t\right)\hfill & {\displaystyle =\frac{{\zeta }_1}{q}{\parallel \nabla v\parallel }_q^q+\frac{{\zeta }_2}{2q}{\parallel \nabla v\parallel }_q^{2q}-\frac{1}{k_1}{\int }_{\mathsf{\Omega }}{|v|}^{k\left(x\right)}dx}\hfill \\ \hfill & {\displaystyle ={\zeta }_1\frac{(k_1-q)}{k_{1}q}{\parallel \nabla v\parallel }_q^q+{\zeta }_2\frac{(k_1-2q)}{k_{1}2q}{\parallel \nabla v\parallel }_q^{2q}+\frac{1}{k_1}I\left(t\right)}\hfill \\ \hfill & {\displaystyle \ge {\zeta }_1\frac{(k_1-q)}{k_{1}q}{\parallel \nabla v\parallel }_q^q+{\zeta }_2\frac{(k_1-2q)}{k_{1}2q}{\parallel \nabla v\parallel }_q^{2q}.}\hfill \end{array}$$

(26)

By using (19) and (22), we obtain for all $t\in [0,T_{*}]$

$${\displaystyle {\zeta }_1{\parallel \nabla v\parallel }_q^q\le \frac{k_{1}q}{(k_1-q)}E\left(t\right)\le \frac{k_{1}q}{(k_1-q)}E\left(0\right),}$$

(27)

and

$${\displaystyle {\zeta }_2{\parallel \nabla v\parallel }_q^{2q}\le \frac{k_{1}2q}{(k_1-2q)}E\left(t\right)\le \frac{k_{1}2q}{(k_1-2q)}E\left(0\right).}$$

(28)

Exploiting Lemma 4, (17) and the fact that $k_2\ge k_1>2q$, we have

$$\begin{array}{cc}{\displaystyle {\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }}\hfill & \le max\left\{{\parallel v\parallel }_{k(·),{\mathsf{\Gamma }}_1}^{k_1},{\parallel v\parallel }_{k(·),{\mathsf{\Gamma }}_1}^{k_2}\right\}\hfill \\ \hfill & =\left[\gamma +(1-\gamma )\right]max\left\{{\parallel v\parallel }_{k(·),{\mathsf{\Gamma }}_1}^{k_1},{\parallel v\parallel }_{k(·),{\mathsf{\Gamma }}_1}^{k_2}\right\}\hfill \\ \hfill & \le \left[\gamma +(1-\gamma )\right]max\left\{c_{*}^{k_1}{\parallel \nabla v\parallel }_q^{k_1},c_{*}^{k_2}{\parallel \nabla v\parallel }_q^{k_2}\right\}\hfill \\ \hfill & \le \gamma max\left\{c_{*}^{k_1}{\parallel \nabla v\parallel }_q^{k_1-q},c_{*}^{k_2}{\parallel \nabla v\parallel }_q^{k_2-q}\right\}{\parallel \nabla v\parallel }_q^q\hfill \\ \hfill & +(1-\gamma )max\left\{c_{*}^{k_1}{\parallel \nabla v\parallel }_q^{k_1-2q},c_{*}^{k_2}{\parallel \nabla v\parallel }_q^{k_2-2q}\right\}{\parallel \nabla v\parallel }_q^{2q}\hfill \\ \hfill & \le {\alpha }_1{\zeta }_1{\parallel \nabla v\parallel }_q^q+{\alpha }_2{\zeta }_2{\parallel \nabla v\parallel }_q^{2q}.\hfill \end{array}$$

(29)

Since ${\alpha }_1+{\alpha }_2<1$, we get

$${\displaystyle {\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }\le {\zeta }_1{\parallel \nabla v\parallel }_q^q+{\zeta }_2{\parallel \nabla v\parallel }_q^{2q},\forall t\in [0,T_{*}].}$$

(30)

Therefore, we conclude that

$$I\left(t\right)>0,\forall t\in [0,T_{*}].$$

By repeating the procedure, $T_{*}$ is extended to T. The proof is complete. □

Theorem 3. 

Under the assumptions of Lemma 7, the local solution of (1) is global.

Proof. 

From (19), (22), and (26), we have

$${\displaystyle E\left(0\right)\ge E\left(t\right)=\frac{1}{2}\parallel v_t{\parallel }_2^2+J\left(t\right)\ge \frac{1}{2}\parallel v_t{\parallel }_2^2+{\zeta }_1\frac{(k_1-q)}{k_{1}q}{\parallel \nabla v\parallel }_q^q,}$$

(31)

which means,

$$\parallel v_t{\parallel }_2^2+{\parallel \nabla v\parallel }_q^q\le KE\left(0\right)<\infty ,$$

(32)

where K is a positive constant that depends only on ${\zeta }_1,q$ and $k_1$. This implies that the local solution is global in time. □

4. Decay Result

In this section, we state and prove the decay result of solution to problem (1) by using the integral inequalities method ([43], Theorem 9.1).

Lemma 8. 

Suppose that the assumptions of Lemma 7 and $m_1>q$ hold, then there exists a positive constant μ such that

$${\int }_{{\mathsf{\Gamma }}_1}{|v|}^{m\left(x\right)}d\mathsf{\Gamma }\le \mu {\parallel \nabla v\parallel }_q^q.$$

(33)

Proof. 

Exploiting Lemma 4, (17), and (27), we have

$$\begin{array}{cc}{\displaystyle {\int }_{{\mathsf{\Gamma }}_1}{|v|}^{m\left(x\right)}d\mathsf{\Gamma }}\hfill & \le max\left\{{\parallel v\parallel }_{m(·),{\mathsf{\Gamma }}_1}^{m_1},{\parallel v\parallel }_{m(·),{\mathsf{\Gamma }}_1}^{m_2}\right\}\hfill \\ \hfill & \le max\left\{c_{*}^{m_1}{\parallel \nabla v\parallel }_q^{m_1},c_{*}^{m_2}{\parallel \nabla v\parallel }_q^{m_2}\right\}\hfill \\ \hfill & \le max\left\{c_{*}^{m_1}{\parallel \nabla v\parallel }_q^{m_1-q},c_{*}^{m_2}{\parallel \nabla v\parallel }_q^{m_2-q}\right\}{\parallel \nabla v\parallel }_q^q\le \mu {\parallel \nabla v\parallel }_q^q.\hfill \end{array}$$

(34)

□

Theorem 4. 

Assume that the conditions of Lemma 8 hold. Then, there exist two positive constant λ and ζ such that

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

Proof. 

Multiplying the first equation in (1) by $v{E}^{\rho }\left(t\right)(\rho >0)$ and integrating over $\mathsf{\Omega }\times (S,T)$, we have

$${\displaystyle {\int }_S^T{E}^{\rho }\left(t\right){\int }_{\mathsf{\Omega }}\left[v_{tt}-{\mathrm{M}}_1\left({\parallel \nabla v\parallel }_q^q\right){\Delta }_{q}v-{\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right){\Delta }_q{v}_t\right]vdxdt=0}$$

(35)

which gives

$$\begin{array}{cc}\hfill & {\displaystyle {\int }_S^T{E}^{\rho }\left(t\right){\int }_{\mathsf{\Omega }}\left[{\displaystyle \frac{d}{dt}\left(v_{t}v\right)-|v_t{|}^2+{\zeta }_1{|\nabla v|}^q+{\zeta }_2{|\nabla v|}^{2q}+{\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right){|\nabla v_t|}^{q-2}\nabla v_t\nabla v}\right]dxdt}\hfill \\ \hfill & {\displaystyle ={\int }_S^T{E}^{\rho }\left(t\right){\int }_{{\mathsf{\Gamma }}_1}\left[{|v|}^{k\left(x\right)}-{|v_t|}^{m\left(x\right)-2}{v}_{t}v\right]d\mathsf{\Gamma }dt.}\hfill \end{array}$$

(36)

On the other hand, we have

$${\displaystyle \frac{d}{dt}\left[E^{\rho }\left(t\right){\int }_{\mathsf{\Omega }}{v}_{t}vdx\right]=\rho E^{\prime }\left(t\right)E^{\rho -1}\left(t\right){\int }_{\mathsf{\Omega }}{v}_{t}vdx+E^{\rho }\left(t\right)\frac{d}{dt}\left[{\int }_{\mathsf{\Omega }}{v}_{t}vdx\right].}$$

Then, (36) becomes

$$\begin{array}{cc}\hfill & {\displaystyle {\int }_S^T{E}^{\rho }\left(t\right)\left[{\displaystyle {\zeta }_1{\parallel \nabla v\parallel }_q^q+{\zeta }_2{\parallel \nabla v\parallel }_q^{2q}-{\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }}\right]dt}\hfill \\ \hfill & {\displaystyle =\rho {\int }_S^T{E}^{\rho -1}\left(t\right)E^{\prime }\left(t\right){\int }_{\mathsf{\Omega }}{v}_{t}vdx-{\int }_S^T\frac{d}{dt}\left[E^{\rho }\left(t\right){\int }_{\mathsf{\Omega }}{v}_{t}vdx\right]dt}\hfill \\ \hfill & {\displaystyle +{\int }_S^T{E}^{\rho }\left(t\right)\parallel v_t{\parallel }_2^{2}dt-{\int }_S^T{E}^{\rho }\left(t\right){\int }_{{\mathsf{\Gamma }}_1}{|v_t|}^{m\left(x\right)-2}{v}_{t}vd\mathsf{\Gamma }dt}\hfill \\ \hfill & {\displaystyle -{\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right){\int }_S^T{E}^{\rho }\left(t\right){\int }_{\mathsf{\Omega }}{|\nabla v_t|}^{q-2}\nabla v_t\nabla vdxdt.}\hfill \end{array}$$

(37)

From (29), and the definition of $E\left(t\right)$, we have

$${\displaystyle {\zeta }_1{\parallel \nabla v\parallel }_q^q+{\zeta }_2{\parallel \nabla v\parallel }_q^{2q}-{\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }\ge \left(1-{\alpha }_1-{\alpha }_2\right)\left({\zeta }_1{\parallel \nabla v\parallel }_q^q+{\zeta }_2{\parallel \nabla v\parallel }_q^{2q}\right),}$$

and

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{q}\left({\zeta }_1{\parallel \nabla v\parallel }_q^q+{\zeta }_2{\parallel \nabla v\parallel }_q^{2q}\right)}\hfill \\ \hfill & {\displaystyle \ge \frac{{\zeta }_1}{q}{\parallel \nabla v\parallel }_q^q+\frac{{\zeta }_2}{2q}{\parallel \nabla v\parallel }_q^{2q}=E\left(t\right)-\frac{1}{2}\parallel v_t{\parallel }_2^2+{\int }_{{\mathsf{\Gamma }}_1}\frac{{|v|}^{k\left(x\right)}}{k\left(x\right)}d\mathsf{\Gamma }\ge E\left(t\right)-\frac{1}{2}{\parallel v_t\parallel }_2^2.}\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill & {\displaystyle q\left(1-{\alpha }_1-{\alpha }_2\right){\int }_S^T{E}^{\rho +1}\left(t\right)dt}\hfill \\ \hfill & \le \underset{I_1}{\underbrace{{\displaystyle \rho {\int }_S^T{E}^{\rho -1}\left(t\right)E^{\prime }\left(t\right){\int }_{\mathsf{\Omega }}{v}_{t}vdx}}}-\underset{I_2}{\underbrace{{\displaystyle {\int }_S^T\frac{d}{dt}\left[E^{\rho }\left(t\right){\int }_{\mathsf{\Omega }}{v}_{t}vdx\right]dt}}}\hfill \\ \hfill & +\underset{I_3}{\underbrace{{\displaystyle \left({\displaystyle \frac{q\left(1-{\alpha }_1-{\alpha }_2\right)}{2}+1}\right){\int }_S^T{E}^{\rho }\left(t\right){\parallel v_t\parallel }_2^{2}dt}}}-\underset{I_4}{\underbrace{{\displaystyle {\int }_S^T{E}^{\rho }\left(t\right){\int }_{{\mathsf{\Gamma }}_1}{|v_t|}^{m\left(x\right)-2}{v}_{t}vd\mathsf{\Gamma }dt}}}\hfill \\ \hfill & -\underset{I_5}{\underbrace{{\displaystyle {\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right){\int }_S^T{E}^{\rho }\left(t\right){\int }_{\mathsf{\Omega }}{|\nabla v_t|}^{q-2}\nabla v_t\nabla vdxdt}}}.\hfill \end{array}$$

(38)

In what follows, we will estimate the right-hand-side terms in (38). By Young’s inequality, we have

$$\begin{array}{cc}|I_1|\hfill & {\displaystyle \le \frac{\rho }{2}{\int }_S^T\left(-E^{\prime }\left(t\right)\right)E^{\rho -1}\left(t\right)\left[\parallel v_t{\parallel }_2^2+{\parallel v\parallel }_2^2\right]dt}\hfill \\ \hfill & {\displaystyle \le \frac{\rho }{2}{\int }_S^T\left(-E^{\prime }\left(t\right)\right)E^{\rho -1}\left(t\right)\left[\parallel v_t{\parallel }_2^2+c_{\varrho }{\parallel \nabla v\parallel }_q^2\right]dt}\hfill \\ \hfill & {\displaystyle \le c_1\rho {\int }_S^T\left(-E^{\prime }\left(t\right)\right)E^{\rho -1}\left(t\right)\left[\parallel v_t{\parallel }_2^2+{\left({\parallel \nabla v\parallel }_q^q\right)}^{\frac{2}{q}}\right]dt}\hfill \\ \hfill & {\displaystyle \le c_1\rho {\int }_S^T\left(-E^{\prime }\left(t\right)\right)\left[E^{\rho }\left(t\right)+E^{\rho +\frac{2}{q}-1}\left(t\right)\right]dt}\hfill \\ \hfill & \le c_1(E^{\rho +1}\left(S\right)+E^{\rho +{\displaystyle \frac{2}{q}}}\left(S\right)).\hfill \end{array}$$

(39)

The second term can be estimated as follows:

$$\begin{array}{cc}|I_2|\hfill & {\displaystyle \le \left|{\displaystyle E^{\rho }\left(T\right){\int }_{\mathsf{\Omega }}v\left(T\right)v_t\left(T\right)dx}\right|+\left|{\displaystyle E^{\rho }\left(S\right){\int }_{\mathsf{\Omega }}v\left(S\right)v_t\left(S\right)dx}\right|}\hfill \\ \hfill & \le c_2(E^{\rho +1}\left(T\right)+E^{\rho +{\displaystyle \frac{2}{q}}}\left(T\right))+c_3(E^{\rho +1}\left(S\right)+E^{\rho +{\displaystyle \frac{2}{q}}}\left(S\right))\hfill \\ \hfill & \le c_4(E^{\rho +1}\left(S\right)+E^{\rho +{\displaystyle \frac{2}{q}}}\left(S\right)).\hfill \end{array}$$

(40)

Furthermore, by Young’s inequality and $\left(H_1\right)$, we have, for any $\eta >0$,

$$\begin{array}{cc}|I_3|\hfill & {\displaystyle \le c_5{\int }_S^T{E}^{\rho }\left(t\right){\parallel \nabla v_t\parallel }_q^{2}dt\le c_6{\int }_S^T{E}^{\rho }\left(t\right){\left(\parallel \nabla v_t{\parallel }_q^q\right)}^{\frac{2}{q}}dt}\hfill \\ \hfill & {\displaystyle \le \frac{c_6}{m_0^{\frac{2}{q}}}{\int }_S^T{E}^{\rho }\left(t\right){\left({\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right){\parallel \nabla v_t\parallel }_q^q\right)}^{\frac{2}{q}}dt\le \frac{c_6}{m_0^{\frac{2}{q}}}{\int }_S^T{E}^{\rho }\left(t\right){\left(-E^{\prime }\left(t\right)\right)}^{\frac{2}{q}}dt}\hfill \\ \hfill & {\displaystyle \le c_7\eta {\int }_S^T{E}^{\rho +1}\left(t\right)dt+c_{8}E\left(S\right).}\hfill \end{array}$$

(41)

For the fourth term in (38). By using Young’s inequality with $q\left(x\right)={\displaystyle \frac{m\left(x\right)}{m\left(x\right)-1}}$ and $q^{\prime }\left(x\right)=m\left(x\right)$. So, for a.e. $x\in {\mathsf{\Gamma }}_1,\eta >0$, and

$$c_{\eta }\left(x\right)={\eta }^{1-m\left(x\right)}{\left(m\left(x\right)\right)}^{-m\left(x\right)}{\left(m\left(x\right)-1\right)}^{m\left(x\right)-1}$$

we have

$$\begin{array}{cc}{\displaystyle {\int }_{\mathsf{\Omega }}{|v_t|}^{m\left(x\right)-2}{v}_{t}vdx}\hfill & {\displaystyle \le \eta {\int }_{{\mathsf{\Gamma }}_1}{|v|}^{m\left(x\right)}d\mathsf{\Gamma }+{\int }_{{\mathsf{\Gamma }}_1}{c}_{\eta }\left(x\right){|v_t|}^{m\left(x\right)}d\mathsf{\Gamma }}\hfill \\ \hfill & {\displaystyle \le {\eta \mu \parallel \nabla v\parallel }_q^q+{\int }_{{\mathsf{\Gamma }}_1}{c}_{\eta }\left(x\right){|v_t|}^{m\left(x\right)}d\mathsf{\Gamma }}\hfill \\ \hfill & {\displaystyle \le {\eta \mu \parallel \nabla v\parallel }_q^q+c_{\eta }\left(m\right){\int }_{{\mathsf{\Gamma }}_1}{|v_t|}^{m\left(x\right)}d\mathsf{\Gamma },}\hfill \end{array}$$

where $c_{\eta }\left(m\right)={max}_{x\in {\mathsf{\Gamma }}_1}{c}_{\eta }\left(x\right)$. Then, we have

$$\begin{array}{cc}|I_4|\hfill & {\displaystyle \le \eta \mu {\int }_S^T{E}^{\rho }{\left(t\right)\parallel \nabla v\parallel }_q^{q}dt+c_{\eta }\left(m\right){\int }_S^T{E}^{\rho }\left(t\right){\int }_{{\mathsf{\Gamma }}_1}{|v_t|}^{m\left(x\right)}d\mathsf{\Gamma }dt}\hfill \\ \hfill & {\displaystyle \le \eta c_9{\int }_S^T{E}^{\rho +1}\left(t\right)dt+c_{\eta }\left(m\right)\epsilon {\int }_S^T{E}^{\rho +1}\left(t\right)\left(-E^{\prime }\left(t\right)\right)dt}\hfill \\ \hfill & {\displaystyle \le \eta c_9{\int }_S^T{E}^{\rho +1}\left(t\right)dt+c_{10}{E}^{\rho +1}\left(S\right).}\hfill \end{array}$$

(42)

Now, we are going to estimate the last term in (38). Since ${\parallel \nabla v\parallel }_q^q\le qE\left(0\right)/{\zeta }_1$, for all $t\ge 0$, taking

$$M_0=sup\{{\mathrm{M}}_2\left(s\right),0<s\le qE\left(0\right)/{\zeta }_1\},$$

we see that ${\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right)\le M_0<\infty$. Then, we have

$$\begin{array}{cc}|I_5|\hfill & {\displaystyle \le {\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right){\int }_S^T{E}^{\rho }\left(t\right){\int }_{\mathsf{\Omega }}{|\nabla v_t|}^{q-2}\nabla v_t\nabla vdxdt}\hfill \\ \hfill & {\displaystyle \le \eta M_0{\int }_S^T{E}^{\rho }\left(t\right){\parallel \nabla v\parallel }_q^{q}dt+c_{\eta }{\int }_S^T{E}^{\rho }\left(t\right)\left({\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right){\parallel \nabla v_t\parallel }_q^q\right)dt}\hfill \\ \hfill & {\displaystyle \le \eta c_{11}{\int }_S^T{E}^{\rho +1}\left(t\right)dt+c_{\eta }\left(m\right)\epsilon {\int }_S^T{E}^{\rho +1}\left(t\right)\left(-E^{\prime }\left(t\right)\right)dt}\hfill \\ \hfill & {\displaystyle \le \eta c_{11}{\int }_S^T{E}^{\rho +1}\left(t\right)dt+c_{12}{E}^{\rho +1}\left(S\right).}\hfill \end{array}$$

(43)

Inserting (39)–(42) in (38), we find

$$\left(q\left(1-{\alpha }_1-{\alpha }_2\right)-\eta c_{13}\right){\int }_S^T{E}^{\rho +1}\left(t\right)\le c_{14}E\left(S\right).$$

(44)

where $c_{14}=[(c_1+c_4)(E^{\rho }\left(0\right)+E^{\rho +{\displaystyle \frac{2}{q}}-1}\left(0\right))+(c_{10}+c_{12})E^{\rho }\left(0\right)+c_8+c_{11}]$.

Since ${\alpha }_1+{\alpha }_2<1$, we choose $\eta$ small enough such that

$$q\left(1-{\alpha }_1-{\alpha }_2\right)-\eta c_{13}>0.$$

Then,

$${\int }_S^T{E}^{\varpi +1}\left(t\right)\le c_{14}E\left(S\right).$$

Thus, by taking $T\to \infty$, we obtain

$${\displaystyle {\int }_s^{\infty }{E}^{\rho +1}\left(t\right)dt\le c_{14}E\left(s\right).}$$

(45)

Therefore, Komornik’s Lemma [43] implies the desired result. □

5. Global Non-Existence

In this section, we state and prove the global nonexistence of solution to problem (1) with negative initial energy.

Theorem 5. 

Let (2) and (3), $\left(H_1\right)-\left(H_2\right)$, $k_1>max\{m_1,2q\}$ and $E\left(0\right)<0$ hold. Then, the solution of problem (1) blows up in finite time.

Proof. 

Set

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

(46)

Using (18) and (19), we get

$$H^{\prime }\left(t\right)=-E^{\prime }\left(t\right)\ge 0,$$

(47)

and

$${\displaystyle 0<H\left(0\right)\le H\left(t\right)\le {\int }_{{\mathsf{\Gamma }}_1}\frac{{|v|}^{k\left(x\right)}}{k\left(x\right)}d\mathsf{\Gamma }\le \frac{1}{k_1}{\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }.}$$

(48)

Next, we define

$${\displaystyle F\left(t\right)=H^{1-\sigma }\left(t\right)+ϵ{\int }_{\mathsf{\Omega }}{v}_{t}vdx,}$$

(49)

where $ϵ$ is small positive constant and will be chosen later, and

$$0<\sigma \le min\left\{{\displaystyle \frac{q}{k_2(q-1)}},{\displaystyle \frac{k_1-m_2}{k_1(m_2-1)}},{\displaystyle \frac{q-2}{2q}}\right\}$$

(50)

By taking a derivative of $F\left(t\right)$ with respect to t and using (1), we get

$$\begin{array}{ccc}\hfill F^{\prime }\left(t\right)& =& {\displaystyle (1-\sigma )H^{-\sigma }\left(t\right)H^{\prime }\left(t\right)+ϵ\parallel v_t{\parallel }_2^2-ϵ{\zeta }_1{\parallel \nabla v\parallel }_q^q-ϵ{\xi }_2{\parallel \nabla v\parallel }_q^{2q}+ϵ{\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }}\hfill \\ & & {\displaystyle -ϵ{\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right){\int }_{\mathsf{\Omega }}|\nabla v_t{|}^{q-2}\nabla v_t\nabla vdx-ϵ{\int }_{{\mathsf{\Gamma }}_1}{|v_t|}^{m\left(x\right)-2}{v}_{t}vd\mathsf{\Gamma }.}\hfill \end{array}$$

(51)

It follows from the definition of $H\left(t\right)$ and $E\left(t\right)$, for a constant $\mu >0$, we obtain

$$\begin{array}{cc}{F}^{\prime }\left(t\right)\hfill & =(1-\sigma )H^{-\sigma }\left(t\right)H^{\prime }\left(t\right)+\mu ϵH\left(t\right)+ϵ\left[{\displaystyle \frac{\mu }{2}+1}\right]\parallel v_t{\parallel }_2^2+ϵ{\zeta }_1\left[{\displaystyle \frac{\mu }{q}-1}\right]{\parallel \nabla v\parallel }_q^q\hfill \\ \hfill & {\displaystyle +ϵ{\zeta }_2\left[{\displaystyle \frac{\mu }{2q}-1}\right]{\parallel \nabla v\parallel }_q^{2q}+\left[{\displaystyle 1-\frac{\mu }{k_1}}\right]{\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }-{\int }_{{\mathsf{\Gamma }}_1}{|v_t|}^{m\left(x\right)-2}{v}_{t}vd\mathsf{\Gamma }}\hfill \\ \hfill & {\displaystyle -{\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right){\int }_{\mathsf{\Omega }}{|\nabla v_t|}^{q-2}\nabla v_t\nabla vdx,}\hfill \end{array}$$

(52)

Applying Young’s inequality, for $\delta ,>0$, we have

$$\begin{array}{cc}\hfill & {\displaystyle {\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right){\int }_{\mathsf{\Omega }}{|\nabla v_t|}^{q-2}\nabla v_t\nabla vdx}\hfill \\ \hfill & {\displaystyle \le \frac{M_0}{q}{\delta }^q{\parallel \nabla v\parallel }_q^q+{\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right)\frac{q-1}{q}{\delta }^{-\frac{q}{q-1}}\parallel \nabla v_t{\parallel }_q^q\le \frac{M_0}{q}{\delta }^p{\parallel \nabla v\parallel }_q^q+\frac{q-1}{q}{\delta }^{-\frac{q}{q-1}}{H}^{\prime }\left(t\right).}\hfill \end{array}$$

(53)

and

$$\begin{array}{cc}{\displaystyle {\int }_{{\mathsf{\Gamma }}_1}{|v_t|}^{m\left(x\right)-2}{v}_{t}vd\mathsf{\Gamma }}\hfill & {\displaystyle \le {\int }_{{\mathsf{\Gamma }}_1}\frac{{\eta }^{m\left(x\right)}}{m\left(x\right)}{|v|}^{m\left(x\right)}d{\mathsf{\Gamma }}_1+{\int }_{\mathsf{\Gamma }}\frac{m\left(x\right)-1}{m\left(x\right)}{\eta }^{-\frac{m\left(x\right)}{m\left(x\right)-1}}{|v_t|}^{m\left(x\right)}d\mathsf{\Gamma }}\hfill \\ \hfill & {\displaystyle \le \frac{1}{m_1}{\int }_{{\mathsf{\Gamma }}_1}{\eta }^{m\left(x\right)}{|v|}^{m\left(x\right)}d\mathsf{\Gamma }+\frac{m_2-1}{m_2}{\int }_{{\mathsf{\Gamma }}_1}{\eta }^{-\frac{m\left(x\right)}{m\left(x\right)-1}}{|v_t|}^{m\left(x\right)}d\mathsf{\Gamma }.}\hfill \end{array}$$

(54)

Taking ${\delta }^{-{\displaystyle \frac{q}{q-1}}}={\theta }_1{H}^{-\sigma }\left(t\right)$ and ${\eta }^{-{\displaystyle \frac{m\left(x\right)}{m\left(x\right)-1}}}={\theta }_2{H}^{-\gamma }\left(t\right)$, where ${\theta }_1$ and ${\theta }_2$ are positive constants to be specified later, we see that

$${\displaystyle {\mathrm{M}}_2\left({\parallel \nabla v\parallel }_q^q\right){\int }_{\mathsf{\Omega }}|\nabla v_t{|}^{q-2}\nabla v_t\nabla vdx\le \frac{M_0}{q}{\theta }_1^{1-q}{H}^{\gamma (q-1)}{\parallel \nabla v\parallel }_q^q+\frac{q-1}{q}{\theta }_1{H}^{-\sigma }\left(t\right)H^{\prime }\left(t\right),}$$

(55)

and

$${\displaystyle {\int }_{{\mathsf{\Gamma }}_1}|v_t{|}^{m\left(x\right)-2}{v}_{t}vd\mathsf{\Gamma }\le \frac{{\theta }_2^{1-m_1}}{m_1}{H}^{\gamma (m_2-1)}{\int }_{{\mathsf{\Gamma }}_1}{|v|}^{m\left(x\right)}d\mathsf{\Gamma }+\frac{m_2-1}{m_2}{\theta }_2{H}^{-\sigma }\left(t\right)H^{\prime }\left(t\right).}$$

(56)

Inserting estimates (55) and (56) in (52), we get

$$\begin{array}{cc}{F}^{\prime }\left(t\right)\hfill & \ge \left\{(1-\sigma )-ϵ\theta \right\}{H}^{-\sigma }\left(t\right)H^{\prime }\left(t\right)+\mu ϵH\left(t\right)+ϵ\left[{\displaystyle \frac{\mu }{2}+1}\right]\parallel v_t{\parallel }_2^2+ϵ{\zeta }_1\left[{\displaystyle \frac{\mu }{q}-1}\right]{\parallel \nabla v\parallel }_q^q\hfill \\ \hfill & {\displaystyle +ϵ{\zeta }_2\left[{\displaystyle \frac{\mu }{2q}-1}\right]{\parallel \nabla v\parallel }_q^{2q}+\left[{\displaystyle 1-\frac{\mu }{k_1}}\right]{\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }-\frac{M_0}{q}{\theta }_1^{1-q}{H}^{\sigma (q-1)}\left(t\right){\parallel \nabla v\parallel }_q^q}\hfill \\ \hfill & {\displaystyle -\frac{{\theta }_2^{1-m_1}}{m_1}{H}^{\sigma (m_2-1)}\left(t\right){\int }_{{\mathsf{\Gamma }}_1}{|v|}^{m\left(x\right)}d\mathsf{\Gamma }.}\hfill \end{array}$$

(57)

where $\theta ={\displaystyle \frac{q-1}{q}}{\theta }_1+{\displaystyle \frac{m_2-1}{m_2}}{\theta }_2$.

For the last two terms of the right-hand side of (57), we set,

$${\mathsf{\Gamma }}_{1,1}=\{x\in \mathsf{\Omega }:|v_t|<1\}\mathrm{and}{\mathsf{\Gamma }}_{1,2}=\{x\in \mathsf{\Omega }:|v_t|\ge 1\}.$$

(58)

Then, we have

$$\begin{array}{cc}\hfill & {\displaystyle {\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }={\int }_{{\mathsf{\Gamma }}_{1,1}}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }+{\int }_{{\mathsf{\Gamma }}_{1,2}}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }}\hfill \\ \hfill & {\displaystyle \le {\int }_{{\mathsf{\Gamma }}_{1,1}}{|v|}^{k_1}d\mathsf{\Gamma }+{\int }_{{\mathsf{\Gamma }}_{1,2}}{|v|}^{k_2}d\mathsf{\Gamma }\le c\left[{\displaystyle {\left({\int }_{{\mathsf{\Gamma }}_1}{|v|}^{q}dx\right)}^{\frac{k_1}{q}}+{\left({\displaystyle {\int }_{{\mathsf{\Gamma }}_1}{|v|}^{q}dx}\right)}^{\frac{k_2}{q}}}\right]}\hfill \\ \hfill & \le c\left[{\left({\parallel \nabla v\parallel }_q^q\right)}^{{\displaystyle \frac{k_1}{q}}}+{\left({\parallel \nabla v\parallel }_q^q\right)}^{{\displaystyle \frac{k_2}{q}}}\right],\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\displaystyle {\int }_{{\mathsf{\Gamma }}_1}{|v|}^{m\left(x\right)}d\mathsf{\Gamma }={\int }_{{\mathsf{\Gamma }}_{1,1}}{|v|}^{m\left(x\right)}d\mathsf{\Gamma }+{\int }_{{\mathsf{\Gamma }}_{1,2}}{|v|}^{m\left(x\right)}d\mathsf{\Gamma }}\hfill \\ \hfill & {\displaystyle \le {\int }_{{\mathsf{\Gamma }}_{1,1}}{|v|}^{m_1}d\mathsf{\Gamma }+{\int }_{{\mathsf{\Gamma }}_{1,2}}{|v|}^{m_2}d\mathsf{\Gamma }\le {\left({\int }_{{\mathsf{\Gamma }}_{1,1}}{|v|}^{k_1}d\mathsf{\Gamma }\right)}^{\frac{m_1}{k_1}}+{\left({\displaystyle {\int }_{{\mathsf{\Gamma }}_{1,2}}{|v|}^{k_1}d\mathsf{\Gamma }}\right)}^{\frac{m_2}{k_1}}}\hfill \\ \hfill & {\displaystyle \le {\left({\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }\right)}^{\frac{m_1}{k_1}}+{\left({\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }\right)}^{\frac{m_2}{k_1}}.}\hfill \end{array}$$

where $c=max\{c_{*}^{k_1},c_{*}^{k_2}\}$. Applying (48), (50), and the algebraic inequality

$$A^{\nu }\le A+1\le (1+{\displaystyle \frac{1}{\beta }})\left(A+\beta \right),\forall A\ge 0,0<\nu \le 1,\beta \ge 0,$$

we obtain, for all $t\ge 0$,

$$\begin{array}{cc}{H}^{\sigma (q-1)}\left(t\right){\parallel \nabla v\parallel }_q^q\hfill & \le c_q{\left[{\left({\parallel \nabla v\parallel }_q^q\right)}^{{\displaystyle \frac{k_1}{q}}}+{\left({\parallel \nabla v\parallel }_q^q\right)}^{{\displaystyle \frac{k_2}{q}}}\right]}^{\sigma (q-1)}{\parallel \nabla v\parallel }_q^q\hfill \\ \hfill & \le c_q\left[{\left({\parallel \nabla v\parallel }_q^q\right)}^{\sigma {\displaystyle \frac{k_1(q-1)}{q}}}+{\left({\parallel \nabla v\parallel }_q^q\right)}^{\sigma {\displaystyle \frac{k_2(q-1)}{q}}}\right]{\parallel \nabla v\parallel }_q^q\hfill \\ \hfill & \le c_q\left[c_0\left({\parallel \nabla v\parallel }_q^q+H\left(0\right)\right)\right]{\parallel \nabla v\parallel }_q^q\hfill \\ \hfill & \le c_1\left({\parallel \nabla v\parallel }_q^{2q}+{\parallel \nabla v\parallel }_q^q\right),\hfill \end{array}$$

(59)

where $c_0=2(1+{\displaystyle \frac{1}{H\left(0\right)}})$. Similarly, we have

$$\begin{array}{cc}\hfill & {\displaystyle H^{\sigma (m_2-1)}\left(t\right){\int }_{{\mathsf{\Gamma }}_1}{|v|}^{m\left(x\right)}d\mathsf{\Gamma }}\hfill \\ \hfill & \le c_m{\left[{\displaystyle {\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }}\right]}^{\varrho (m_2-1)}\left[{\displaystyle {\left({\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }\right)}^{\frac{m_1}{k_1}}+{\left({\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }\right)}^{\frac{m_2}{k_1}}}\right]\hfill \\ \hfill & \le c_m{\left[{\displaystyle {\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }}\right]}^{\varrho (m_2-1)+{\displaystyle \frac{m_1}{k_1}}}+c_m{\left[{\displaystyle {\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }}\right]}^{\varrho (m_2-1)+{\displaystyle \frac{m_2}{k_1}}}\hfill \\ \hfill & \le c_2\left[{\displaystyle {\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }+H\left(0\right)}\right]\le c_2\left[{\displaystyle {\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }+H\left(t\right)}\right],\hfill \end{array}$$

(60)

where $c_2=c_m(1+{\displaystyle \frac{1}{H\left(0\right)}})$.

Combining (59), (60), and (57), we get

$$\begin{array}{cc}{F}^{\prime }\left(t\right)\hfill & \ge \left\{(1-\sigma )-ϵ\theta \right\}{H}^{-\sigma }\left(t\right)H^{\prime }\left(t\right)+ϵ\left[{\displaystyle \frac{\mu }{2}+1}\right]{\parallel v_t\parallel }_2^2+ϵ\left[{\displaystyle \mu -\frac{{\theta }_2^{1-m_1}}{m_1}{c}_2}\right]H\left(t\right)\hfill \\ \hfill & +ϵ\left[{\displaystyle {\zeta }_1\left({\displaystyle \frac{\mu }{q}-1}\right)-\frac{M_0}{q}{\theta }_1^{1-q}{c}_1}\right]{\parallel \nabla v\parallel }_q^q\hfill \\ \hfill & {\displaystyle +ϵ\left[{\displaystyle {\zeta }_2\left({\displaystyle \frac{\mu }{2q}-1}\right)-\frac{M_0}{q}{\theta }_1^{1-q}{c}_1}\right]{\parallel \nabla v\parallel }_q^{2q}+\left[{\displaystyle 1-\frac{\mu }{k_1}-\frac{{\theta }_2^{1-m_1}}{m_1}{c}_2}\right]{\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }.}\hfill \end{array}$$

(61)

At this point, first, we choose $2q<\mu <k_1$ such that

$${\displaystyle \frac{\mu }{2q}-1>0,\frac{\mu }{q}-1>0,1-\frac{\mu }{k_1}>0.}$$

For any fixed $\mu$, we choose ${\theta }_1$ and ${\theta }_2$ so large such that

$${\displaystyle \mu -\frac{{\theta }_2^{1-m_1}}{m_1}{c}_2>0,{\zeta }_2\left({\displaystyle \frac{\mu }{2q}-1}\right)-\frac{M_0}{q}{\theta }_1^{1-q}{c}_1>0,}$$

and

$${\displaystyle {\zeta }_1\left({\displaystyle \frac{\mu }{q}-1}\right)-\frac{M_0}{q}{\theta }_1^{1-q}{c}_1>0,1-\frac{\mu }{k_1}-\frac{{\theta }_2^{1-m_1}}{m_1}{c}_2>0.}$$

Once ${\theta }_1$ and ${\theta }_2$ are fixed, we select an $\epsilon >0$ small enough so that

$$(1-\sigma )-\epsilon \theta >0\mathrm{and}F\left(0\right)=H^{1-\sigma }\left(0\right)+ϵ{\int }_{\mathsf{\Omega }}{v}_0{v}_{1}dx>0.$$

Then, inequality (61) becomes

$$F^{\prime }\left(t\right)\ge c_2\left({\displaystyle \parallel v_t{\parallel }_2^2+{\parallel \nabla v\parallel }_q^q+{\parallel \nabla v\parallel }_q^{2q}+{\int }_{{\mathsf{\Gamma }}_1}{|v|}^{k\left(x\right)}d\mathsf{\Gamma }+H\left(t\right)}\right),$$

(62)

where $c_2$ is a positive constant. Consequently, we have

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

(63)

On the other hand, from (49) we have

$$F^{{\displaystyle \frac{1}{1-\sigma }}}\left(t\right)\le c_{\sigma }\left[H\left(t\right)+{\left({\displaystyle {\int }_{\mathsf{\Omega }}{v}_{t}vdx}\right)}^{{\displaystyle \frac{1}{1-\sigma }}}\right].$$

(64)

Applying Hölder’s and Young’s inequalities, we have

$${\left|{\int }_{\mathsf{\Omega }}v{v}_{t}dx\right|}^{{\displaystyle \frac{1}{1-\sigma }}}\le c_3{\parallel v\parallel }_q^{{\displaystyle \frac{1}{1-\sigma }}}{\parallel u_t\parallel }_2^{{\displaystyle \frac{1}{1-\sigma }}}\le c_3\left({\parallel v\parallel }_q^{{\displaystyle \frac{\nu }{1-\sigma }}}+{\parallel u_t\parallel }_2^{{\displaystyle \frac{\vartheta }{1-\sigma }}}\right).$$

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

$${\left|{\int }_{\mathsf{\Omega }}v{v}_{t}dx\right|}^{{\displaystyle \frac{1}{1-\sigma }}}\le c_3\left({\parallel v\parallel }_q^{{\displaystyle \frac{2}{1-2\sigma }}}+{\parallel v_t\parallel }_2^2\right)\le c_4\left({\parallel \nabla v\parallel }_q^{{\displaystyle \frac{2}{1-2\sigma }}}+{\parallel v_t\parallel }_2^2\right).$$

(65)

Applying (50) and applying algebraic inequality, we get

$${\parallel \nabla v\parallel }_q^{{\displaystyle \frac{2}{1-2\sigma }}}\le {\left({\parallel \nabla v\parallel }_q^q\right)}^{{\displaystyle \frac{2}{q(1-2\sigma )}}}\le c_5\left({\parallel \nabla v\parallel }_q^q+H\left(0\right)\right)\le c_5\left({\parallel \nabla v\parallel }_q^q+H\left(t\right)\right),$$

(66)

where $c_5=(1+{\displaystyle \frac{1}{H\left(0\right)}})$. By substituting (65) and (66) into (64), we obtain

$$F^{{\displaystyle \frac{1}{1-\sigma }}}\left(t\right)\le c_6\left[H\left(t\right)+\parallel v_t{\parallel }_2^2+{\parallel \nabla v\parallel }_q^q\right].$$

(67)

It follows from (63) and (67) that we find that

$$F^{\prime }\left(t\right)\ge \varpi F^{\prime {\displaystyle \frac{1}{1-\sigma }}}\left(t\right),\mathrm{for}t>0,$$

(68)

where $\varpi$ is a positive constant. A simple integration of (68) over $(0,t)$ yields

$$F^{{\displaystyle \frac{\sigma }{1-\sigma }}}\left(t\right)\ge {\displaystyle \frac{1}{F^{-\frac{\sigma }{1-\sigma }}\left(0\right)-\frac{\varpi \sigma t}{1-\sigma }}}.$$

Consequently, the solution of problem (1) blows up in finite time $T^{*}$, and $T^{*}\le {\displaystyle \frac{1-\sigma }{\varpi \sigma F^{\frac{\sigma }{1-\sigma }}\left(0\right)}}$. □

6. Conclusions

The objective of this study is to investigate the global existence, general decay properties, and blow-up phenomena of solutions to a $q$-Kirchhoff-type equation equipped with nonlinear boundary damping and source terms involving variable exponents. Our analysis relies on the potential well method, the multiplier technique, and nonlinear integral inequalities pioneered by Komornik, which together provide a robust framework for establishing the desired results.