Diversity of internal structures in inhibited epoxy primers

Anthony E. Hughes; Y. Sam Yang; Simon G. Hardin; Andrew Tulloh; Yudan Wang; You He; Anthony E. Hughes; Y. Sam Yang; Simon G. Hardin; Andrew Tulloh; Yudan Wang; You He

doi:10.3934/matersci.2015.4.379

AIMS Materials Science

2015, Volume 2, Issue 4: 379-391. doi: 10.3934/matersci.2015.4.379

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Research article Special Issues

Diversity of internal structures in inhibited epoxy primers

1.
Institute for Frontier Materials, Deakin University, Waurn Ponds 3216 Victoria Australia;
2.
CSIRO, Mineral Resources, Clayton 3169, Victoria, Australia;
3.
CSIRO, Manufacturing, Private Bag 10, Clayton 3169 Victoria Australia;
4.
School of Chemistry, University of Melbourne, Parkville 3010 Victoria Australia;
5.
Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai, P. R. China

Received: 03 June 2015 Accepted: 27 September 2015 Published: 16 October 2015

Computed tomography is making a significant impact in the field of materials science in recent years. In this paper the authors report on advances made in three areas of characterization and also identified where further research needs to be focused. First we report on a new approach to data analysis called “Data Constrained Modelling (DCM)” in which compositional tomography can be undertaken rather than adsorption or phase contrast tomography. This is achieved by collecting X-ray CT data at different energies and then combining the datasets to reconstruct 3D compositional tomography. Second, on the application of this approach to inhibited primers typical of those used in the aerospace industry. Aerospace primers are effectively composite materials containing inorganic phases which are bound together with a polymer. Understanding the materials science of these systems requires information over several orders of magnitude in length-scale. In this paper we report on how DCM can be used to extend our understanding at the smaller length scales at the limits of resolution of the technique. The third and final advance is in extending the approach to include 4-dimensional studies. In this case we examine the primer before and after leaching. This process causes changes in the primer which can be both detected and quantified using the above approach.

Keywords:

Citation: Anthony E. Hughes, Y. Sam Yang, Simon G. Hardin, Andrew Tulloh, Yudan Wang, You He. Diversity of internal structures in inhibited epoxy primers[J]. AIMS Materials Science, 2015, 2(4): 379-391. doi: 10.3934/matersci.2015.4.379

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Abstract

1. Introduction

In this paper, we study the initial boundary value problem of the nonlinear viscoelastic hyperbolic problem with variable exponents:

$\begin{equation} \begin{cases} u_{tt}+\triangle^{2}u+\triangle^{2}u_{tt}- \int_{0}^{t}g(t-\tau)\triangle^{2}u(\tau)d\tau+|u_{t}|^{m(x)-2}u_{t} = |u|^{p(x)-2}u, \quad & (x, t) \in \Omega\times(0, T), \\ u(x, t) = \frac{\partial u}{\partial \nu}(x, t) = 0, \quad & (x, t) \in \partial \Omega\times(0, T), \\ u(x, 0) = u_{0}(x), \ u_{t}(x, 0) = u_{1}(x), \quad & x \in \Omega, \\ \end{cases} \end{equation}$ $

(1.1)

where $ \Omega\subset R^{n}(n\geq1) $ is a bounded domain in $ R^n $ with a smooth boundary $ \partial \Omega $, $ \nu $ is the unit outer normal to $ \partial\Omega $, the exponents $ m(x) $ and $ p(x) $ are continuous functions on $ \overline{\Omega} $ with the logarithmic module of continuity:

$\begin{equation} \forall x, y\in\Omega, \, |x-y| < 1, \, |m(x)-m(y)|+|p(x)-p(y)|\leq\omega(|x-y|), \end{equation}$ $

(1.2)

where

$\begin{equation} \lim\limits_{\tau\rightarrow0^{+}}sup\, \omega(\tau)\ln\frac{1}{\tau} = C < \infty. \end{equation}$ $

(1.3)

In addition to this condition, the exponents satisfy the following:

$\begin{equation} 2\leq m^{-}: = ess \inf\limits_{x\in\Omega}m(x)\leq m(x) \leq m^{+}: = ess \sup\limits_{x\in\Omega}m(x) < \frac{2(n-2)}{n-4}, \end{equation}$ $

(1.4)

$\begin{equation} 2\leq p^{-}: = ess \inf\limits_{x\in\Omega}p(x)\leq p(x) \leq p^{+}: = ess \sup\limits_{x\in\Omega}p(x) < \frac{2(n-2)}{n-4}, \end{equation}$ $

(1.5)

$ g: R^{+} \, \rightarrow R^{+} $ is a $ C^{1} $ function satisfying

$\begin{equation} g(0) > 0, \ g'(\tau)\leq0, \, 1-\int_{0}^{\infty}g(\tau)d\tau = l > 0. \end{equation}$ $

(1.6)

The equation of Problem $ (1.1) $ arises from the modeling of various physical phenomena such as the viscoelasticity and the system governing the longitudinal motion of a viscoelastic configuration obeying a nonlinear Boltzmann's model, or electro-rheological fluids, viscoelastic fluids, processes of filtration through a porous medium, and fluids with temperature-dependent viscosity and image processing which give rise to equations with nonstandard growth conditions, that is, equations with variable exponents of nonlinearities. More details on these problems can be found in previous studies ^{[1,2,3,4,5,6]}.

When $ m(x) $ and $ p(x) $ are constants, Messaoudi ^[7] discussed the nonlinear viscoelastic wave equation

$\begin{equation} \nonumber u_{tt}-\triangle u+\int_{0}^{t}g(t-\tau)\triangle u(\tau)d\tau\\+|u_{t}|^{m-2}u_{t} = |u|^{p-2}u, \end{equation}$ $

he proved that any weak solution with negative initial energy blows up in finite time if $ p > m $, and a global existence result for $ p\leq m $. The results were improved later by Messaoudi ^[8], where the blow-up result in finite time with positive initial energy was obtained. Moreover, Song ^[9] showed the finite-time blow-up of some solutions whose initial data had arbitrarily high initial energy. In the same year, Song ^[10] studied the initial-boundary value problem

$\begin{equation} \nonumber |u_{t}|^{\rho}u_{tt}-\triangle u+\int_{0}^{t}g(t-\tau)\triangle u(\tau)d\tau\\+|u_{t}|^{m-2}u_{t} = |u|^{p-2}u, \end{equation}$ $

and proved the nonexistence of global solutions with positive initial energy. Cavalcanti, Domingos, and Ferreira ^[11] were concerned with the non-linear viscoelastic equation

$\begin{equation} \nonumber |u_{t}|^{\rho}u_{tt}-\triangle u-\triangle u_{tt}+\int_{0}^{t}g(t-\tau)\triangle u(\tau)d\tau\\-\gamma\triangle u_{t} = 0, \end{equation}$ $

and proved the global existence of weak solutions. Moreover, they obtained the uniform decay rates of the energy by assuming a strong damping $ \triangle u_{t} $ acting in the domain and providing the relaxation function which decays exponentially.

In 2017, Messaoudi ^[12] considered the following nonlinear wave equation with variable exponents:

$\begin{equation} \nonumber u_{tt}-\triangle u+a|u_{t}|^{m(x)-2}u_{t} = b|u|^{p(x)-2}u, \end{equation}$ $

where $ a, b $ are positive constants. By using the Faedo$ - $Galerkin method, the existence of a unique weak solution is established under suitable assumptions on the variable exponents $ m(x) $ and $ p(x) $. Then this paper also proved the finite-time blow-up of solutions and gave a two-dimensional numerical example to illustrate the blow up result. Park ^[13] showed the blow up of solutions for a viscoelastic wave equation with variable exponents

$\begin{equation} \nonumber u_{tt}-\triangle u+\int_{0}^{t}g(t-s)\triangle u (s)ds+a|u_{t}|^{m(x)-2}u_{t} = b|u|^{p(x)-2}u, \end{equation}$ $

where the exponents of nonlinearity $ p(x) $ and $ m(x) $ are given functions and $ a, b > 0 $ are constants. For nonincreasing positive function $ g $, they prove the blow-up result for the solutions with positive initial energy as well as nonpositive initial energy. Alahyane ^[14] discussed the nonlinear viscoelastic wave equation with variable exponents

$\begin{equation} \nonumber u_{tt}-\triangle u+\int_{0}^{t}g(t-\tau)\triangle u (\tau)d\tau+\mu u_{t} = |u|^{p(x)-2}u, \end{equation}$ $

where $ \mu $ is a nonnegative constant and the exponent of nonlinearity $ p(x) $ and $ g $ are given functions. Under arbitrary positive initial energy and specific conditions on the relaxation function $ g $, they prove a finite-time blow-up result and give some numerical applications to illustrate their theoretical results. Ouaoua and Boughamsa ^[15] considered the following boundary value problem:

$\begin{equation} \nonumber u_{tt}+\triangle^{2}u-\triangle u+|u_{t}|^{m(x)-2}u_{t} = |u|^{p(x)-2}u, \end{equation}$ $

the authors established the local existence by using the Faedo$ - $Galerkin method with positive initial energy and suitable conditions on the variable exponents $ m(x) $ and $ r(x) $. In addition, they also proved that the local solution is global and obtained the stability estimate of the solution. Ding and Zhou ^[16] considered a Timoshenko-type equation

$\begin{equation} \nonumber u_{tt}+\triangle^{2} u-M(||\nabla u||_{2}^{2})\triangle u+|u_{t}|^{p(x)-2}u_{t} = |u|^{q(x)-2}u, \end{equation}$ $

they prove that the solutions blow up in finite time with positive initial energy. Therefore, the existence of finite-time blow-up solutions with arbitrarily high initial energy is established, and the upper and lower bounds of the blow-up time are derived. More related references can be found in ^{[17,18,19,20,21,22]}.

Motivated by ^[7,13,14], we considered the existence of the solutions and their blow-up for the nonlinear damping and viscoelastic hyperbolic problem with variable exponents. Our aim in this work is to prove the existence of the weak solutions and to find sufficient conditions on $ m(x) $ and $ p(x) $ for which the blow-up takes place.

This article consists of three sections in addition to the introduction. In Section 2, we recall the definitions and properties of $ L^{p(x)}(\Omega) $ and the Sobolev spaces $ W^{1, p(x)}(\Omega) $. In Section 3, we prove the existence of weak solutions for Problem (1.1). In Section 4, we state and prove the blow-up result for solutions with positive initial energy as well as nonpositive initial energy.

2. Preliminaries

In this section, we review some results regarding Lebesgue and Sobolev spaces with variable exponents first. All of these results and a comprehensive study of these spaces can be found in ^[23]. Here $ (\cdot, \cdot) $ and $ \langle \cdot, \cdot \rangle $ denote the inner product in space $ L^{2}(\Omega) $ and the duality pairing between $ H^{-2}(\Omega) $ and $ H_{0}^{2}(\Omega) $.

The variable exponent Lebesgue space $ L^{p(x)}(\Omega) $ is defined by

$ L^{p(x)}(\Omega) = \left\{u(x): u\, \, {\rm is\, \, \rm measurable\, \, \rm in}\, \, \Omega, \ \rho_{p(x)}(u) = \int_{\Omega}|u|^{p(x)}dx < \infty\right\}, $

this space is endowed with the norm

$ \|u\|_{p(x)} = \mbox{inf}\ \left\{\lambda > 0:\int_{\Omega}\Big|\frac{u(x)}{\lambda}\Big|^{p(x)}\mathrm{d}x\leq1\right\}. $

The variable exponent Sobolev space $ W^{1, p(x)}(\Omega) $ is defined by

$ W^{1, p(x)}(\Omega) = \left\{u\in L^{p(x)}(\Omega)\ \, {\rm such \, \, \rm that}\, \, \nabla u \, \, {\rm exists\, \, \rm and}\, \, |\nabla u|\in L^{p(x)}(\Omega)\right\}, $

the corresponding norm for this space is

$ \|u\|_{1, p(x)} = \|u\|_{{p(x)}}+\|\nabla u\|_{{p(x)}}, $

define$ \ W_0^{1, p(x)}(\Omega) $ as the closure of $ \ C_0^\infty(\Omega) $ with respect to the$ \ W^{1, p(x)}(\Omega) $ norm. The spaces $ \ L^{p(x)}(\Omega), W^{1, p(x)}(\Omega) $ and $ W_0^{1, p(x)}(\Omega) $ are separable and reflexive Banach spaces when $ 1 < p^-\leq p^+ < \infty $, where $ p^-: = ess\inf\limits_{\Omega} p(x) $ and $ p^+: = ess\sup\limits_{\Omega} p(x). $ As usual, we denote the conjugate exponent of $ p(x) $ by $ p'(x) = p(x)/(p(x)-1) $ and the Sobolev exponent by

$ p^*(x) = \left\{

$\begin{array}{ll} \frac{np(x)}{n-kp(x)}, \; \; \; \; \mbox{if} \ p(x) < n, \\ \infty, \; \; \; \; \; \; \; \; \; \; \; \mbox{if} \ p(x)\geq n. \end{array}$ \right. $

Lemma 2.1. If $ p_{1}(x), \ p_{2}(x)\in C_{+}(\overline{\Omega}) = \{h\in C(\overline{\Omega}):\min\limits_{x\in \overline{\Omega}} h(x) > 1\} $, $ p_{1}(x)\leq p_{2}(x) $ for any $ x\in\Omega $, then there exists the continuous embedding $ L^{p_{2}(x)}(\Omega)\hookrightarrow L^{p_{1}(x)}(\Omega) $, whose norm does not exceed $ |\Omega|+1 $.

Lemma 2.2. Let $ p(x), \ q(x)\in C_{+}(\overline{\Omega}). $ Assuming that $ q(x) < p^{\ast}(x) $, there is a compact and continuous embedding $ W^{k, p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega). $

Lemma 2.3. (Hölder's inequality) ^[24] For any $ u\in L^{p(x)}(\Omega) $ and $ v\in L^{q(x)}(\Omega) $, then the following inequality holds:

$ \left|\int_{\Omega}uvdx\right|\leq(\frac{1}{p^{-}}+\frac{1}{q^{-}})||u||_{p(x)}||v||_{q(x)}\leq2||u||_{p(x)}||v||_{q(x)}. $

Lemma 2.4. For $ u\in L^{p(x)}(\Omega) $, the following relations hold:

$ u\neq0 \Rightarrow \Big(\|u\|_{p(x)} = \lambda\Leftrightarrow\rho_{p(x)}(\frac{u}{\lambda}) = 1\Big ), $

$ \|u\|_{p(x)} < 1( = 1; > 1)\Leftrightarrow\rho_{p(x)}(u) < 1( = 1; > 1), $

$ \|u\|_{p(x)} > 1\Rightarrow \|u\|_{p(x)}^{p^{-}}\leq\rho_{p(x)}(u)\leq\|u\|_{p(x)}^{p^+}, $

$ \|u\|_{p(x)} < 1\Rightarrow \|u\|_{p(x)}^{p^+}\leq\rho_{p(x)}(u)\leq\|u\|_{p(x)}^{p^-}. $

Next, we give the definition of the weak solution to Problem $ (1.1) $.

Definition 2.1. A function u(x, t) is called a weak solution for Problem $ (1.1) $, if $ u\in C(0, T;H_{0}^{2}(\Omega)) $ $ \cap C^{1}(0, T;H_{0}^{2}(\Omega))\cap C^{2}(0, T;H^{-2}(\Omega)) $ with $ u_{tt}\in L^{2}(0, T;H_{0}^{2}(\Omega)) $ and u satisfies the following conditions:

(1) For every $ \omega \in H_{0}^{2}(\Omega) $ and for $ a.e. \, t\in (0, T) $

$ \langle u_{tt}, \omega\rangle+(\triangle u, \triangle \omega)+(\triangle u_{tt}, \triangle \omega)-\int_{0}^{t}g(t-\tau)(\triangle u(\tau), \triangle \omega)d\tau\\+(|u_{t}|^{m(x)-2}u_{t}, \omega) = (|u|^{p(x)-2}u, \omega), $

(2) $ u(x, 0) = u_{0}(x)\in H_{0}^{2}(\Omega), \, u_{t}(x, 0) = u_{1}(x)\in H_{0}^{2}(\Omega). $

3. The local existence of weak solution

In this section, we prove the existence of a weak solution for Problem $ (1.1) $ by making use of the Faedo–Galerkin method and the contraction mapping principle. For a fixed $ T > 0 $, we consider the space $ \mathscr H = C(0, T;H_{0}^{2}(\Omega))\cap C^{1}(0, T;H_{0}^{2}(\Omega)) $ with the norm $ ||v||_{\mathscr H}^{2} = \max\limits_{0\leq t\leq T} (||\triangle v_{t}||_{2}^{2}+l||\triangle v||_{2}^{2}) $.

Lemma 3.1. Assume that $ (1.4) $, $ (1.5) $, and $ (1.6) $ hold, let $ (u_{0}, u_{1})\in H_{0}^{2}(\Omega)\times H_{0}^{2}(\Omega) $, for any $ T > 0 $, $ v\in \mathscr H $, then there exists $ u\in C(0, T;H_{0}^{2}(\Omega))\cap C^{1}(0, T;H_{0}^{2}(\Omega))\cap C^{2}(0, T;H^{-2}(\Omega)) $ with $ \ u_{tt}\in L^{2}(0, T;H_{0}^{2}(\Omega)) $ satisfying

$\begin{equation} \begin{cases} u_{tt}+\triangle^{2}u+\triangle^{2}u_{tt}- \int_{0}^{t}g(t-\tau)\triangle^{2}u(\tau)d\tau+|u_{t}|^{m(x)-2}u_{t} = |v|^{p(x)-2}v, \quad & (x, t) \in \Omega\times(0, T), \\ u(x, t) = \frac{\partial u}{\partial\nu}(x, t) = 0, \quad & (x, t) \in \partial \Omega\times(0, T), \\ u(x, 0) = u_{0}(x), \ u_{t}(x, 0) = u_{1}(x), \quad & x \in \Omega.\\ \end{cases} \end{equation}$ $

(3.1)

Proof. Let $ \{\omega_{j}\}_{j = 1}^{\infty} $ be the orthogonal basis of $ H\mathcal{}_{0}^{2}(\Omega) $, which is the standard orthogonal basis in $ L^{2}(\Omega) $ such that

$ -\triangle\omega_{j} = \lambda_{j}\omega_{j} \ \ \rm in\ \ \Omega, \, \, \omega_{j} = 0 \ \ \rm on\ \ \partial \Omega, $

we denote by $ V_{k} = {\rm span}\{\omega_{1}, \omega_{2}, \cdot\cdot\cdot, \omega_{k}\} $ the subspace generated by the first $ k $ vectors of the basis $ \{\omega_{j}\}_{j = 1}^{\infty} $. By normalization, we have $ ||\omega_{j}||_{2} = 1 $. For all $ k\geq1 $, we seek $ k $ functions $ c_{1}^{k}(t), c_{2}^{k}(t), \ldots, c_{k}^{k}(t)\in C^{2}[0, T] $ such that

$ u^{k}(x, t) = \sum\limits_{j = 1}^{k}c_{j}^{k}(t)\omega_{j}(x), $

satisfying the following approximate problem

$\begin{equation} \begin{cases} (u_{tt}^{k}, \omega_{i})+(\triangle u^{k}, \triangle \omega_{i})+(\triangle u_{tt}^{k}, \triangle \omega_{i})- \int_{0}^{t}g(t-\tau)(\triangle u^{k}, \triangle \omega_{i})d\tau\\ +(|u_{t}^{k}|^{m(x)-2}u_{t}^{k}, \omega_{i}) = \int_{\Omega}|v|^{p(x)-2}v\omega_{i}dx, \\ u^{k}(0) = u_{0}^{k}, \ \ \ u_{t}^{k}(0) = u_{1}^{k}, \ \ \ \ \ i = 1, 2, \ldots k, \end{cases} \end{equation}$ $

(3.2)

where

$ u_{0}^{k} = \sum\limits_{i = 1}^{k}(u_{0}, \omega_{i})\omega_{i}\rightarrow u_{0}\ \ \ \rm in\ \ H_{0}^{2}(\Omega), $

$ u_{1}^{k} = \sum\limits_{i = 1}^{k}(u_{1}, \omega_{i})\omega_{i}\rightarrow u_{1}\ \ \ \rm in\ \ H_{0}^{2}(\Omega), $

thus, $ (3.2) $ generates the initial value problem for the system of second-order differential equations with respect to $ c_{i}^{k}(t) $:

$\begin{equation} \begin{aligned} \begin{cases} (1+\lambda_{i}^{2})c_{itt}^{k}(t)+\lambda_{i}^{2}c_{i}^{k}(t) = G_{i}(c_{1t}^{k}(t), \ldots, c_{kt}^{k}(t))+g_{i}(c_{i}^{k}(t)), \ \ \ &i = 1, 2, \ldots, k, \\ c_{i}^{k}(0) = \int_{\Omega}u_{0}\omega_{i}dx, \ \ \ \ \ \ c_{it}^{k}(0) = \int_{\Omega}u_{1}\omega_{i}dx, \ \ \ \ \ \ \ \ \ \ \ \ \ \ &i = 1, 2, \ldots, k. \end{cases} \end{aligned} \end{equation}$ $

(3.3)

where

$\begin{equation} \begin{aligned} \nonumber G_{i}(c_{1t}^{k}(t), \ldots, c_{kt}^{k}(t)) = - \int_{\Omega}|\sum\limits_{j = 1}^{k}c_{jt}^{k}(t)\omega_{j}(x)|^{m(x)-2}\sum\limits_{j = 1}^{k}c_{jt}^{k}(t)\omega_{j}(x)\omega_{i}(x)dx, \end{aligned} \end{equation}$ $

and

$\begin{equation} \begin{aligned} \nonumber g_{i}(c_{i}^{k}(t)) = \lambda_{i}^{2}\int_{0}^{t}g(t-\tau)c_{i}^{k}(\tau)d\tau+\int_{\Omega}|v|^{p(x)-2}v\omega_{i}dx, \end{aligned} \end{equation}$ $

by Peano's Theorem, we infer that the Problem $ (3.3) $ admits a local solution $ c_{i}^{k}(t)\in C^{2}[0, T] $.

$ \textbf{The} $ $ \textbf{first} $ $ \textbf{estimate}. $ Multiplying $ (3.2) $ by $ c_{it}^{k}(t) $ and summing with respect to $ i $, we arrive at the relation

$\begin{eqnarray} \begin{aligned} &\frac{d}{dt}(\frac{1}{2}||u_{t}^{k}||_{2}^{2}+\frac{1}{2}||\triangle u^{k}||_{2}^{2}+\frac{1}{2}||\triangle u_{t}^{k}||_{2}^{2})+\int_{\Omega}|u_{t}^{k}|^{m(x)}dx-\int_{0}^{t}g(t-\tau)\int_{\Omega}\triangle u^{k}(\tau)\triangle u_{t}^{k}dxd\tau\\ = &\int_{\Omega}|v|^{p(x)-2}vu_{t}^{k}dx. \end{aligned} \end{eqnarray}$ $

(3.4)

By simple calculation, we have

$\begin{eqnarray} \begin{aligned} &-\int_{0}^{t}g(t-\tau)\int_{\Omega}\Delta u^{k}(\tau)\Delta u_{t}^{k}dxd\tau\\ & = \frac{1}{2}\frac{d}{dt}(g\diamond \bigtriangleup u^{k}) -\frac{1}{2}(g'\diamond\bigtriangleup u^{k})-\frac{1}{2}\frac{d}{dt}\int_{0}^{t}g(\tau)d\tau||\Delta u^{k}||_{2}^{2}+\frac{1}{2}g(t)||\triangle u^{k}||_{2}^{2}, \end{aligned} \end{eqnarray}$ $

(3.5)

where

$ (\varphi\diamond\bigtriangleup\psi) = \int_{0}^{t}\varphi(t-\tau)||\triangle\psi(t)-\triangle\psi(\tau)||_{2}^{2}d\tau, $

inserting $ (3.5) $ into $ (3.4) $, using Hölder's inequality and Young's inequality, we obtain

$\begin{eqnarray} \begin{aligned} &\frac{d}{dt}\left[\frac{1}{2}||u_{t}^{k}||_{2}^{2}+\frac{1}{2}||\triangle u_{t}^{k}||_{2}^{2}+\frac{1}{2}(g\diamond\bigtriangleup u^{k})+\frac{1}{2}(1-\int_{0}^{t}g(\tau)d\tau)||\Delta u^{k}||_{2}^{2}\right]\\ = &\frac{1}{2}(g'\diamond\bigtriangleup u^{k})-\frac{1}{2}g(t)||\triangle u^{k}||_{2}^{2}+\int_{\Omega}|v|^{p(x)-2}vu_{t}^{k}dx-\int_{\Omega}|u_{t}^{k}|^{m(x)}dx\\ \leq&\int_{\Omega}|v|^{p(x)-2}vu_{t}^{k}dx\leq\left\||v|^{p(x)-2}v\right\|_{2}||u_{t}^{k}||_{2}\\ \leq&\frac{\eta}{2}\int_{\Omega}|v|^{2(p(x)-1)}dx+\frac{1}{2\eta}||u_{t}^{k}||_{2}^{2}, \end{aligned} \end{eqnarray}$ $

(3.6)

using the embedding $ H_{0}^{2}(\Omega)\hookrightarrow L^{2(p(x)-1)}(\Omega) $ and Lemma $ 2.4 $, we easily obtain

$\begin{eqnarray} \begin{aligned} \int_{\Omega}|v|^{2(p(x)-1)}dx&\leq\max\left\{||v||_{2(p(x)-1)}^{2(p^{-}-1)}, ||v||_{2(p(x)-1)}^{2(p^{+}-1)}\right\}\\ &\leq C \max\left\{||\triangle v||_{2}^{2(p^{-}-1)}, ||\triangle v||_{2}^{2(p^{+}-1)}\right\}\\ &\leq C, \end{aligned} \end{eqnarray}$ $

(3.7)

where $ C $ is a positive constant. We denote by $ C $ various positive constants that may be different at different occurrences.

Combining $ (3.6) $ and $ (3.7) $, we obtain

$\begin{eqnarray} \begin{aligned} \nonumber &\frac{d}{dt}\left[\frac{1}{2}||u_{t}^{k}||_{2}^{2}+\frac{1}{2}||\triangle u_{t}^{k}||_{2}^{2}+\frac{1}{2}(g\diamond\bigtriangleup u^{k})+\frac{1}{2}(1-\int_{0}^{t}g(\tau)d\tau)||\Delta u^{k}||_{2}^{2}\right]\\ \leq&\frac{\eta}{2}C+\frac{1}{2\eta}||u_{t}^{k}||_{2}^{2}, \end{aligned} \end{eqnarray}$ $

by Gronwall's inequality, there exists a positive constant $ C_{T} $ such that

$\begin{eqnarray} \begin{aligned} ||u_{t}^{k}||_{2}^{2}+||\triangle u_{t}^{k}||_{2}^{2}+(g\diamond\bigtriangleup u^{k})+l||\Delta u^{k}||_{2}^{2}\leq C_{T}, \end{aligned} \end{eqnarray}$ $

(3.8)

therefore, there exists a subsequence of $ \{u^{k}\}_{k = 1}^{\infty} $, which we still denote by $ \{u^{k}\}_{k = 1}^{\infty} $, such that

$\begin{eqnarray} \begin{aligned} &u^{k} \mathop\rightharpoonup \limits^{\ast} u\ weakly\ star\ in\ L^{\infty}(0, T;H_{0}^{2}(\Omega)), \\ &u_{t}^{k} \mathop\rightharpoonup \limits^{\ast} u_{t}\ weakly\ star\ in\ L^{\infty}(0, T;H_{0}^{2}(\Omega)), \\ &u^{k} \rightharpoonup u\ weakly\ in\ L^{2}(0, T;H_{0}^{2}(\Omega)), \\ &u_{t}^{k} \rightharpoonup u_{t}\ weakly\ in\ L^{2}(0, T;H_{0}^{2}(\Omega)). \end{aligned} \end{eqnarray}$ $

(3.9)

$ \textbf{The} $ $ \textbf{second} $ $ \textbf{estimate}. $ Multiplying $ (3.2) $ by $ c_{itt}^{k}(t) $ and summing with respect to $ i $, we obtain

$\begin{eqnarray} \begin{aligned} &||u_{tt}^{k}||^{2}_{2}+||\Delta u_{tt}^{k}||_{2}^{2}+\frac{d}{dt}(\int_{\Omega}\frac{1}{m(x)}|u_{t}^{k}|^{m(x)}dx)\\ = &-\int_{\Omega}\triangle u^{k}\triangle u_{tt}^{k}dx+\int_{0}^{t}g(t-\tau)\int_{\Omega}\Delta u^{k}(\tau)\Delta u_{tt}^{k}dxd\tau+\int_{\Omega}|v|^{p(x)-2}vu_{tt}^{k}dx. \end{aligned} \end{eqnarray}$ $

(3.10)

Note that we have the estimates for $ \varepsilon > 0 $

$\begin{eqnarray} \begin{aligned} \left|\int_{\Omega}\triangle u^{k}\triangle u_{tt}^{k}dx\right|\leq\varepsilon||\triangle u_{tt}^{k}||_{2}^{2}+\frac{1}{4\varepsilon}||\triangle u^{k}||_{2}^{2}, \\ \end{aligned} \end{eqnarray}$ $

(3.11)

$\begin{eqnarray} \begin{aligned} \int_{\Omega}|v|^{p(x)-2}vu_{tt}^{k}dx&\leq\left\||v|^{p(x)-2}v\right\|_{2}\left\|u_{tt}^{k}\right\|_{2}\leq\varepsilon||u_{tt}^{k}||_{2}^{2}+\dfrac{1}{4\varepsilon}\int_{\Omega}|v|^{2(p(x)-1)}dx, \end{aligned} \end{eqnarray}$ $

(3.12)

and

$\begin{eqnarray} \begin{aligned} &\left|\int_{0}^{t}g(t-\tau)\int_{\Omega}\triangle u^{k}(\tau)\triangle u_{tt}^{k}dxd\tau\right|\\ \leq&\frac{1}{4\varepsilon}\int_{\Omega}(\int_{0}^{t}g(t-\tau)\triangle u^{k}(\tau)d\tau)^{2}dx+\varepsilon||\triangle u_{tt}^{k}||_{2}^{2}\\ \leq&\varepsilon||\triangle u_{tt}^{k}||_{2}^{2}+\dfrac{1}{4\varepsilon}\int_{0}^{t}g(s)ds\int_{0}^{t}g(t-\tau)\int_{\Omega}|\triangle u^{k}(\tau)|^{2}dxd\tau\\ \leq&\varepsilon||\triangle u_{tt}^{k}||_{2}^{2}+\dfrac{(1-l)g(0)}{4\varepsilon}\int_{0}^{t}||\triangle u^{k}(\tau)||_{2}^{2}d\tau, \end{aligned} \end{eqnarray}$ $

(3.13)

similar to $ (3.6) $ and $ (3.7) $, from $ H_{0}^{2}(\Omega)\hookrightarrow L^{2}(\Omega) $, we have

$\begin{eqnarray} \int_{\Omega}|v|^{p(x)-2}vu_{tt}^{k}dx\leq \varepsilon C||\triangle u_{tt}^{k}||_{2}^{2}+\dfrac{C}{4\varepsilon}. \end{eqnarray}$ $

(3.14)

Taking into account $ (3.10)-(3.14) $, we obtain

$\begin{eqnarray} \begin{aligned} &||u_{tt}^{k}||_{2}^{2}+(1-2\varepsilon-C\varepsilon)||\triangle u_{tt}^{k}||_{2}^{2}+\dfrac{d}{dt}(\int_{\Omega}\frac{1}{m(x)}|u_{t}^{k}|^{m(x)}dx)\\ \leq&\frac{1}{4\varepsilon}||\triangle u^{k}||_{2}^{2}+\frac{(1-l)g(0)}{4\varepsilon}\int_{0}^{t}||\triangle u^{k}(\tau)||_{2}^{2}d\tau+\frac{C}{4\varepsilon}, \end{aligned} \end{eqnarray}$ $

(3.15)

integrating $ (3.15) $ over $ (0, t) $, we obtain

$\begin{eqnarray} \begin{aligned} &\int_{0}^{t}||u_{tt}^{k}||_{2}^{2}d\tau+(1-2\varepsilon-C\varepsilon)\int_{0}^{t}||\triangle u_{tt}^{k}||_{2}^{2}d\tau+\int_{\Omega}\frac{1}{m(x)}|u_{t}^{k}|^{m(x)}dx\\ \leq&\frac{C}{4\varepsilon}\int_{0}^{t}(||\triangle u^{k}||_{2}^{2}+\int_{0}^{\tau}||\triangle u^{k}(s)||_{2}^{2}ds)d\tau+C_{T}, \end{aligned} \end{eqnarray}$ $

(3.16)

taking $ \varepsilon $ small enough in (3.16), for some positive constant $ C_{T} $, we obtain

$\begin{eqnarray} \begin{aligned} \int_{0}^{t}||u_{tt}^{k}||_{2}^{2}d\tau+\int_{0}^{t}||\triangle u_{tt}^{k}||_{2}^{2}d\tau\leq C_{T}, \end{aligned} \end{eqnarray}$ $

(3.17)

we observe that estimate $ (3.17) $ implies that there exists a subsequence of $ \{u^{k}\}_{k = 1}^{\infty} $, which we still denote by $ \{u^{k}\}_{k = 1}^{\infty} $, such that

$\begin{eqnarray} &u_{tt}^{k} \rightharpoonup u_{tt}\ weakly\ in\ L^{2}(0, T;H_{0}^{2}(\Omega)). \end{eqnarray}$ $

(3.18)

In addition, from $ (3.9) $, we have

$\begin{eqnarray} (u_{tt}^{k}, \omega_{i}) = \frac{d}{dt}(u_{t}^{k}, \omega_{i}) \mathop\rightharpoonup \limits^{\ast} \frac{d}{dt}(u_{t}, \omega_{i}) = (u_{tt}, \omega_{i})\ \ weakly \ \ star\ \ in \ \ L^{\infty}(0, T;H^{-2}(\Omega)). \end{eqnarray}$ $

(3.19)

Next, we will deal with the nonlinear term. Combining $ (3.9) $, $ (3.18) $, and Aubin–Lions theorem ^[25], we deduce that there exists a subsequence of $ \{u^{k}\}_{k = 1}^{\infty} $ such that

$\begin{eqnarray} &u_{t}^{k}\rightarrow u_{t}\ strongly\ in\ C(0, T;L^{2}(\Omega)), \end{eqnarray}$ $

(3.20)

then

$\begin{eqnarray} |u_{t}^{k}|^{m(x)-2}u_{t}^{k}\rightarrow |u_{t}|^{m(x)-2}u_{t} \ \ a.e.\ (x, t)\in\Omega\times(0, T), \end{eqnarray}$ $

(3.21)

using the embedding $ H_{0}^{2}(\Omega)\hookrightarrow L^{2(m(x)-1)}(\Omega) $ and Lemma $ 2.4 $, we have

$\begin{eqnarray} \begin{aligned} \left\||u_{t}^{k}|^{m(x)-2}u_{t}^{k}\right\|_{2}^{2} = \int_{\Omega}|u_{t}^{k}|^{2(m(x)-1)}dx\leq\max\left\{||\triangle u_{t}^{k}||_{2}^{2(m^{-}-1)}, ||\triangle u_{t}^{k}||_{2}^{2(m^{+}-1)}\right\}\leq C, \end{aligned} \end{eqnarray}$ $

(3.22)

hence, using $ (3.21) $ and $ (3.22) $, we obtain

$\begin{eqnarray} &|u_{t}^{k}|^{m(x)-2}u_{t}^{k}\mathop\rightharpoonup \limits^{\ast} |u_{t}|^{m(x)-2}u_{t}\ \ weakly \ \ star\ in \ L^{\infty}(0, T;L^{2}(\Omega)). \end{eqnarray}$ $

(3.23)

Setting up $ k\rightarrow \infty $ in $ (3.2) $, combining with $ (3.9) $, $ (3.18) $, $ (3.19) $, and $ (3.23) $, we obtain

$ \langle u_{tt}, \omega \rangle+(\triangle u, \triangle \omega)+(\triangle u_{tt}, \triangle \omega)-\int_{0}^{t}g(t-\tau)(\triangle u(\tau), \triangle \omega)d\tau+(|u_{t}|^{m(x)-2}u_{t}, \omega) = (|v|^{p(x)-2}v, \omega). $

To handle the initial conditions. From $ (3.9) $ and Aubin–Lions theorem, we can easily get $ u^{k}\rightarrow u $ in $ C(0, T;L^{2}(\Omega)) $, thus $ u^{k}(0)\rightarrow u(0) $ in $ L^{2}(\Omega) $, and we also have that $ u^{k}(0) = u_{0}^{k}\rightarrow u_{0} $ in $ H_{0}^{2}(\Omega) $, hence $ u(0) = u_{0} $ in $ H_{0}^{2}(\Omega) $. Similarly, we get that $ u_{t}(0) = u_{1} $.

Uniqueness. Suppose that $ (3.1) $ has solutions $ u $ and $ z $, then $ \omega = u-z $ satisfies

$\begin{equation} \begin{cases} \nonumber \omega_{tt}+\triangle^{2}\omega+\triangle^{2}\omega_{tt}- \int_{0}^{t}g(t-\tau)\triangle^{2}\omega(\tau)d\tau+|u_{t}|^{m(x)-2}u_{t}-|z_{t}|^{m(x)-2}z_{t} = 0, & (x, t) \in \Omega\times(0, T), \\ \omega(x, t) = \frac{\partial\omega}{\partial\nu}(x, t) = 0, &(x, t) \in \partial \Omega\times(0, T), \\ \omega(x, 0) = 0, \ \omega_{t}(x, 0) = 0, &x \in \Omega.\\ \end{cases} \end{equation}$ $

Multiplying the first equation of Problem $ (3.1) $ by $ \omega_{t} $ and integrating over $ \Omega $, we have

$\begin{equation} \begin{split} \nonumber &\frac{1}{2}\frac{d}{dt}\left[||\omega_{t}||_{2}^{2}+(1-\int_{0}^{t}g(\tau)d\tau)||\triangle \omega||_{2}^{2}+||\triangle \omega_{t}||_{2}^{2}+(g\diamond \triangle \omega)\right]+\frac{1}{2}g(t)||\triangle\omega||_{2}^{2}\\ = &-\int_{\Omega}\left(|u_{t}|^{m(x)-2}u_{t}-|z_{t}|^{m(x)-2}z_{t}\right)(u_{t}-z_{t})dx+\frac{1}{2}(g'\diamond\bigtriangleup\omega), \end{split} \end{equation}$ $

from the inequality

$\begin{eqnarray} (|a|^{m(x)-2}a-|b|^{m(x)-2}b)(a-b)\geq0, \end{eqnarray}$ $

(3.24)

for all $ a, b\in R^{n} $ and a.e. $ x\in\Omega $, we obtain

$ ||\omega_{t}||_{2}^{2}+l||\triangle \omega||_{2}^{2}+||\triangle \omega_{t}||_{2}^{2} = 0, $

which implies that $ \omega = 0 $. This completes the proof. □

Theorem 3.1. Assume that $ (1.4) $ and $ (1.6) $ hold, let the initial date $ (u_{0}, u_{1})\in H_{0}^{2}(\Omega)\times H_{0}^{2}(\Omega) $, and

$ 2\leq p^{-}\leq p(x)\leq p^{+}\leq\frac{2(n-3)}{n-4}, $

then there exists a unique local solution of Problem $ (1.1) $.

Proof. For any $ T > 0 $, consider $ M_{T} = \{u\in\mathscr H:u(0) = u_{0}, u_{t}(0) = u_{1}, ||u||_{\mathscr H}\leq M\} $. Lemma $ 3.1 $ implies that for $ \forall v\in M_{T} $, there exists $ u = S(v) $ such that $ u $ is the unique solution to Problem $ 3.1 $. Next, we prove that for a suitable $ T > 0 $, $ S $ is a contractive map satisfying $ S(M_{T})\subset M_{T} $.

Multiplying the first equation of the Problem $ (3.1) $ by $ u_{t} $ and integrating it over $ (0, t) $, we obtain

$\begin{eqnarray} \begin{aligned} &||u_{t}||_{2}^{2}+||\triangle u_{t}||_{2}^{2}+(g\diamond\bigtriangleup u)+l||\Delta u||_{2}^{2}\\ \leq&||u_{1}||_{2}^{2}+||\triangle u_{1}||_{2}^{2}+||\Delta u_{0}||_{2}^{2}+2\int_{0}^{t}\int_{\Omega}|v|^{p(x)-2}vu_{t}dxd\tau, \end{aligned} \end{eqnarray}$ $

(3.25)

using Hölder's inequality and Young's inequality, we have

$\begin{eqnarray} \begin{aligned} \nonumber \left|\int_{\Omega}|v|^{p(x)-2}vu_{t}dx\right|&\leq\gamma||u_{t}||_{2}^{2}+\frac{1}{4\gamma}\int_{\Omega}|v|^{2p(x)-2}dx\\ &\leq\gamma||u_{t}||_{2}^{2}+\frac{1}{4\gamma}\left[\int_{\Omega}|v|^{2p^{-}-2}dx+\int_{\Omega}|v|^{2p^{+}-2}dx\right]\\ &\leq \gamma||u_{t}||_{2}^{2}+\frac{C}{4\gamma}\left[||\triangle v||_{2}^{2p^{-}-2}+||\triangle v||_{2}^{2p^{+}-2}\right], \end{aligned} \end{eqnarray}$ $

thus, $ (3.25) $ becomes

$\begin{eqnarray} \begin{aligned} \nonumber &||u_{t}||_{2}^{2}+||\triangle u_{t}||_{2}^{2}+l||\Delta u||_{2}^{2}\\ \leq& \lambda_{0}+2\int_{0}^{t}\int_{\Omega}|v|^{p(x)-2}vu_{t}dxd\tau\\ \leq& \lambda_{0}+2\gamma T \sup\limits_{(0, T)}||u_{t}||_{2}^{2}+\frac{TC}{2\gamma}\sup\limits_{(0, T)}\left[||\triangle v||_{2}^{2p^{-}-2}+||\triangle v||_{2}^{2p^{+}-2}\right], \end{aligned} \end{eqnarray}$ $

hence, we have

$\begin{eqnarray} \begin{aligned} \nonumber &\sup\limits_{(0, T)}||u_{t}||_{2}^{2}+ \sup\limits_{(0, T)}||\triangle u_{t}||_{2}^{2}+l \sup\limits_{(0, T)}||\Delta u||_{2}^{2}\\ \leq& \lambda_{0}+2\gamma T \sup\limits_{(0, T)}||u_{t}||_{2}^{2}+\frac{TC}{2\gamma}\sup\limits_{(0, T)}\left[||v||_{\mathscr H}^{2p^{-}-2}+||v||_{\mathscr H}^{2p^{+}-2}\right], \end{aligned} \end{eqnarray}$ $

where $ \lambda_{0} = ||u_{1}||_{2}^{2}+||\triangle u_{1}||_{2}^{2}+||\Delta u_{0}||_{2}^{2} $, choosing $ \gamma = \frac{1}{2T} $ such that

$\begin{eqnarray} \begin{aligned} \nonumber ||u||_{\mathscr H}^{2}\leq \lambda_{0}+T^{2}C\sup\limits_{(0, T)}\left[||v||_{\mathscr H}^{2p^{-}-2}+||v||_{\mathscr H}^{2p^{+}-2}\right]. \end{aligned} \end{eqnarray}$ $

For any $ v\in M_{T} $, by choosing $ M $ large enough so that

$\begin{eqnarray} ||u||_{\mathscr H}^{2}\leq\lambda_{0}+2T^{2}CM^{2(p^{+}-1)}\leq M^{2}, \end{eqnarray}$ $

and $ T > 0 $, sufficiently small so that

$ T\leq\sqrt{\frac{M^{2}-\lambda_{0}}{2CM^{2(p^{+}-1)}}}, $

we obtain $ ||u||_{\mathscr H}\leq M $, which shows that $ S(M_{T})\subset M_{T} $.

Let $ v_{1}, v_{2}\in M_{T}, u_{1} = S(v_{1}), u_{2} = S(v_{2}), u = u_{1}-u_{2} $, then $ u $ satisfies

$\begin{equation} \begin{cases} \nonumber u_{tt}+\triangle^{2}u+\triangle^{2}u_{tt}- \int_{0}^{t}g(t-\tau)\triangle^{2}u(\tau)d\tau\\ +|u_{1t}|^{m(x)-2}u_{1t}-|u_{2t}|^{m(x)-2}u_{2t} = |v_{1}|^{p(x)-2}v_{1}-|v_{2}|^{p(x)-2}v_{2}, \quad & (x, t) \in \Omega\times(0, T), \\ u(x, t) = \frac{\partial u}{\partial\nu}(x, t) = 0, \quad & (x, t) \in \partial \Omega\times(0, T), \\ u(x, 0) = 0, \ u_{t}(x, 0) = 0, \quad & x \in \Omega.\\ \end{cases} \end{equation}$ $

Multiplying by $ u_{t} $ and integrating over $ \Omega\times(0, t) $, we obtain

$\begin{eqnarray} \begin{aligned} &\frac{1}{2}||u_{t}||_{2}^{2}+\frac{1}{2}(1-\int_{0}^{t}g(\tau)d\tau)||\triangle u||_{2}^{2}+\frac{1}{2}||\triangle u_{t}||_{2}^{2}+\frac{1}{2}(g\diamond\bigtriangleup u)\\ +&\int_{0}^{t}\int_{\Omega}\left[|u_{1t}|^{m(x)-2}u_{1t}-|u_{2t}|^{m(x)-2}u_{2t}\right](u_{1t}-u_{2t})dxd\tau\leq\int_{0}^{t}\int_{\Omega}(f(v_{1})-f(v_{2}))u_{t}dxd\tau, \end{aligned} \end{eqnarray}$ $

(3.26)

where $ f(v) = |v|^{p(x)-2}v $. From $ (1.6) $ and $ (3.24) $, we obtain

$\begin{eqnarray} \begin{aligned} \frac{1}{2}||u_{t}||_{2}^{2}+\frac{l}{2}||\triangle u||_{2}^{2}+\frac{1}{2}||\triangle u_{t}||_{2}^{2}+\frac{1}{2}(g\diamond\bigtriangleup u) \leq\int_{0}^{t}\int_{\Omega}(f(v_{1})-f(v_{2}))u_{t}dxd\tau. \end{aligned} \end{eqnarray}$ $

(3.27)

Now, we evaluate

$ I = \int_{\Omega}|(f(v_{1})-f(v_{2}))||u_{t}|dx = \int_{\Omega}|f'(\xi)||v||u_{t}|dx, $

where $ v = v_{1}-v_{2} $ and $ \xi = \alpha v_{1}+(1-\alpha)v_{2} $, $ 0\leq\alpha\leq1 $. Thanks to Young's inequality and Hölder's inequality, we have

$\begin{eqnarray} \begin{aligned} I&\leq\frac{\delta}{2}||u_{t}||_{2}^{2}+\frac{1}{2\delta}\int_{\Omega}|f'(\xi)|^{2}|v|^{2}dx\\ &\leq\frac{\delta}{2}||u_{t}||_{2}^{2}+\frac{(p^{+}-1)^{2}}{2\delta}\int_{\Omega}|\xi|^{2(p(x)-2)}|v|^{2}dx\\ &\leq\frac{\delta}{2}||u_{t}||_{2}^{2}+\frac{(p^{+}-1)^{2}}{2\delta}\left(\int_{\Omega}|v|^{\frac{2n}{n-2}}dx\right)^{\frac{n-2}{n}}\left[\int_{\Omega}|\xi|^{n(p(x)-2)}dx\right]^{\frac{2}{n}}\\ &\leq\frac{\delta}{2}||u_{t}||_{2}^{2}+\frac{(p^{+}-1)^{2}}{2\delta}\left(\int_{\Omega}|v|^{\frac{2n}{n-2}}dx\right)^{\frac{n-2}{n}}\left[\left(\int_{\Omega}|\xi|^{n(p^{+}-2)}dx\right)^{\frac{2}{n}}+\left(\int_{\Omega}|\xi|^{n(p^{-}-2)}dx\right)^{\frac{2}{n}}\right]\\ &\leq\frac{\delta}{2}||u_{t}||_{2}^{2}+\frac{(p^{+}-1)^{2}C}{2\delta}||\Delta v||_{2}^{2}\left[||\triangle \xi||_{2}^{2(p^{+}-2)}+||\triangle\xi||_{2}^{2(p^{-}-2)}\right]\\ &\leq\frac{\delta}{2}||u_{t}||_{2}^{2}+\frac{(p^{+}-1)^{2}C}{2\delta}||\Delta v||_{2}^{2}\left(M^{2(p^{+}-2)}+M^{2(p^{-}-2)}\right). \end{aligned} \end{eqnarray}$ $

(3.28)

Inserting $ (3.28) $ into $ (3.27) $, choosing $ \delta $ small enough, we obtain

$ ||u||_{\mathscr H}^{2}\leq\frac{(p^{+}-1)^{2}CT}{\delta}(M^{2(p^{-}-2)}+M^{2(p^{+}-2)})||v||_{\mathscr H}^{2}, $

taking $ T $ small enough so that $ \frac{(p^{+}-1)^{2}CT}{\delta}(M^{2(p^{-}-2)}+M^{2(p^{+}-2)}) < 1 $, we conclude

$ ||u||_{\mathscr H}^{2} = ||S(v_{1})-S(v_{2})||_{\mathscr H}^{2}\leq||v_{1}-v_{2}||_{\mathscr H}^{2}, $

thus, the contraction mapping principle ensures the existence of a weak solution to Problem $ (1.1) $. This completes the proof. □

4. The blow-up the solution

In this section, we show that the solution to Problem $ (1.1) $ blows up in finite time when the initial energy lies in positive as well as nonpositive. For this task, we define

$\begin{eqnarray} E(t) = \frac{1}{2}||u_{t}||_{2}^{2}+\frac{1}{2}(1-\int_{0}^{t}g(\tau)d\tau)||\triangle u||_{2}^{2}+\frac{1}{2}||\triangle u_{t}||_{2}^{2}+\frac{1}{2}(g\diamond \bigtriangleup u)-\int_{\Omega}\frac{1}{p(x)}|u|^{p(x)}dx, \end{eqnarray}$ $

(4.1)

by the definition of $ E(t) $, we also have

$\begin{eqnarray} E'(t) = -\int_{\Omega}|u_{t}|^{m(x)}dx+\frac{1}{2}(g'\diamond\bigtriangleup u)-\frac{1}{2}g(t)||\triangle u||_{2}^{2}\leq0. \end{eqnarray}$ $

(4.2)

Now, we set

$ B_{1} = \max\left\{1, \frac{B}{l^{\frac{1}{2}}}\right\}, \ \ \lambda_{1} = \left(B_{1}^{2}\right)^{\frac{-2}{p^{-}-2}}, \ \ E_{1} = (\frac{1}{2}-\frac{1}{p^{-}})(B_{1}^{2})^{\frac{-p^{-}}{p^{-}-2}}, $

and

$\begin{eqnarray} H(t) = E_{2}-E(t), \end{eqnarray}$ $

(4.3)

where the constant $ E_{2}\in(E(0), E_{1}) $ will be discussed later, and $ B $ is the best constant of the Sobolev embedding $ H_{0}^{2}(\Omega)\hookrightarrow L^{p(x)}(\Omega) $. It follows from $ (4.2) $ that

$\begin{eqnarray} H'(t) = -E'(t)\geq0, \end{eqnarray}$ $

(4.4)

and $ H(t) $ is a non$ - $decreasing function.

To prove Theorem $ 4.1 $, we need the following two lemmas:

Lemma 4.1. Suppose that $ (1.6) $ holds and the exponents $ m(x) $ and $ p(x) $ satisfy condition $ (1.4) $ and $ (1.5) $. Assume further that

$ E(0) < E_{1} \ \ and\ \ \lambda_{1} < \lambda(0) = B_{1}^{2}l||\triangle u_{0}||_{2}^{2}, $

then there exists a constant $ \lambda_{2} > \lambda_{1} $ such that

$\begin{eqnarray} B_{1}^{2}l||\triangle u||_{2}^{2}\geq\lambda_{2}, \ \ t\geq0. \end{eqnarray}$ $

(4.5)

Proof. Using $ (1.6) $, $ (4.1) $, Lemma $ 2.4 $, and the embedding $ H_{0}^{2}(\Omega)\hookrightarrow L^{p(x)}(\Omega) $, we find that

$\begin{eqnarray} \begin{aligned} E(t)&\geq\frac{1}{2}(1-\int_{0}^{t}g(\tau)d\tau)||\triangle u||_{2}^{2}-\int_{\Omega}\frac{1}{p(x)}|u|^{p(x)}dx\geq\frac{l}{2}||\triangle u||_{2}^{2}-\frac{1}{p^{-}}\int_{\Omega}|u|^{p(x)}dx\\ &\geq \frac{l}{2}||\triangle u||_{2}^{2}-\frac{1}{p^{-}}\max\left\{||u||_{p(x)}^{p^{-}}, ||u||_{p(x)}^{p^{+}}\right\}\\ &\geq \frac{l}{2}||\triangle u||_{2}^{2}-\frac{1}{p^{-}}\max\left\{B^{p^{-}}||\triangle u||_{2}^{p^{-}}, B^{p^{+}}||\triangle u||_{2}^{p^{+}}\right\}\\ &\geq \frac{l}{2}||\triangle u||_{2}^{2}-\frac{1}{p^{-}}\max\left\{B_{1}^{p^{-}}l^{\frac{p^{-}}{2}}||\triangle u||_{2}^{p^{-}}, B_{1}^{p^{+}}l^{\frac{p^{+}}{2}}||\triangle u||_{2}^{p^{+}}\right\}\\ &\geq\frac{1}{2B_{1}^{2}}\lambda-\frac{1}{p^{-}}\max\left\{\lambda^{\frac{p^{-}}{2}}, \lambda^{\frac{p^{+}}{2}}\right\}: = G(\lambda), \end{aligned} \end{eqnarray}$ $

(4.6)

where $ \lambda: = \lambda(t) = B_{1}^{2}l||\triangle u||_{2}^{2}. $ Analyzing directly the properties of $ G(\lambda) $, we deduce that $ G(\lambda) $ satisfies the following properties:

$ G'(\lambda) =

$\begin{cases}\frac{1}{2B_{1}^{2}}-\frac{p^{+}}{2p^{-}}\lambda^{\frac{p^{+}-2}{2}} < 0, \, \ \ &\lambda > 1, \\\frac{1}{2B_{1}^{2}}-\frac{1}{2}\lambda^{\frac{p^{-}-2}{2}}, \ &0 < \lambda < 1, \ \end{cases}$ $

$ G'_{+}(1) = \frac{1}{2B_{1}^{2}}-\frac{p^{+}}{2p^{-}} < 0, \ \ G'_{-}(1) = \frac{1}{2B_{1}^{2}}-\frac{1}{2} < 0, $

$ G'(\lambda_{1}) = 0, \ \ \ \ 0 < \lambda_{1} < 1. $

It is easily verified that $ G(\lambda) $ is strictly increasing for $ 0 < \lambda < \lambda_{1} $, strictly decreasing for $ \lambda_{1} < \lambda $, $ G(\lambda)\rightarrow -\infty $ as $ \lambda\rightarrow +\infty $, and $ G(\lambda_{1}) = E_{1} $. Since $ E(0) < E_{1} $, there exists a $ \lambda_{2} > \lambda_{1} $ such that $ G(\lambda_{2}) = E(0) $. By $ (4.6) $, we see that $ G(\lambda(0))\leq E(0) = G(\lambda_{2}) $, which implies $ \lambda(0)\geq\lambda_{2} $ since the condition $ \lambda(0) > \lambda_{1}. $ To prove $ (4.5) $, we suppose by contradiction that for some $ t_{0} > 0 $, $ \lambda_{t_{0}} = B_{1}^{2}l||\triangle u(t_{0})||_{2}^{2} < \lambda_{2} $. The continuity of $ B_{1}^{2}l||\triangle u||_{2}^{2} $ illustrates that we could choose $ t_{0} $ such that $ \lambda_{1} < \lambda_{t_{0}} < \lambda_{2} $, then we have $ E(0) = G(\lambda_{2}) < G(\lambda_{t_{0}})\leq E(t_{0}) $. This is a contradiction. The proof is completed. □

Lemma 4.2. Let the assumption in Lemma $ 4.1 $ be satisfied. For $ t\in[0, T) $, we have

$\begin{eqnarray} 0 < H(0)\leq H(t)\leq\frac{1}{p^{-}}\rho_{p(x)}(u). \end{eqnarray}$ $

Proof. $ (4.4) $ indicates that $ H(t) $ is nondecreasing with respect to $ t $, thus

$ H(t)\geq H(0) = E_{2}-E(0) > 0, \ \ \forall t\in[0, T). $

It follows from $ (1.6) $, $ (4.1) $, and Lemma $ 4.1 $ that

$\begin{eqnarray} \begin{aligned} \nonumber H(t)& = E_{2}-E(t)\\ & = E_{2}-\frac{1}{2}||u_{t}||_{2}^{2}-\frac{1}{2}(1-\int_{0}^{t}g(\tau)d\tau)||\triangle u||_{2}^{2}-\frac{1}{2}(g\diamond\triangle u)-\frac{1}{2}||\triangle u_{t}||_{2}^{2}+\int_{\Omega}\frac{1}{p(x)}|u|^{p(x)}dx\\ &\leq E_{1}-\frac{l}{2}||\triangle u||_{2}^{2}+\int_{\Omega}\frac{1}{p(x)}|u|^{p(x)}dx\leq E_{1}-\frac{1}{2B_{1}^{2}}\lambda_{2}+\int_{\Omega}\frac{1}{p(x)}|u|^{p(x)}dx\\ &\leq E_{1}-\frac{1}{2B_{1}^{2}}\lambda_{1}+\int_{\Omega}\frac{1}{p(x)}|u|^{p(x)}dx\leq\int_{\Omega}\frac{1}{p(x)}|u|^{p(x)}dx\leq\frac{1}{p^{-}}\rho_{p(x)}(u). \end{aligned} \end{eqnarray}$ $

The proof is completed.

Our blow-up result reads as follows:

Theorem 4.3. Suppose that

$ 2\leq m^{-}\leq m(x)\leq m^{+} < p^{-}\leq p(x)\leq p^{+}\leq\frac{2(n-3)}{n-4}, $

and

$\begin{eqnarray} 1-l = \int_{0}^{\infty}g(\tau)d\tau < \frac{\frac{p^{-}}{2}-1}{\frac{p^{-}}{2}-1+\frac{1}{2p^{-}}}, \end{eqnarray}$ $

(4.7)

hold, if the following conditions

$ E(0) < \frac{1}{2}(\frac{1}{2}-\frac{1}{p^{-}})\left(1-\frac{1}{p^{-}(p^{-}-2)}\frac{1-l}{l}\right)(B_{1}^{2})^{\frac{-p^{-}}{p^{-}-2}} \ \ and\ \ \lambda_{1} < \lambda(0) = B_{1}^{2}l||\triangle u_{0}||_{2}^{2}, $

are satisfied, then there exists $ T^{\ast} < +\infty $ such that

$\begin{eqnarray} \lim\limits_{t\rightarrow T^{\ast-}}\left(||u_{t}||_{2}^{2}+||\triangle u_{t}||_{2}^{2}+||\triangle u||_{2}^{2}+||u||_{p^{+}}^{p^{+}}\right) = +\infty. \end{eqnarray}$ $

(4.8)

Proof. Assume by contradiction that $ (4.8) $ does not hold true, then for $ \forall T^{\ast} < +\infty $ and all $ t\in[0, T^{\ast}] $, we get

$\begin{eqnarray} ||u_{t}||_{2}^{2}+||\triangle u_{t}||_{2}^{2}+||\triangle u||_{2}^{2}+||u||_{p^{+}}^{p^{+}}\leq C_{\ast}, \end{eqnarray}$ $

(4.9)

where $ C_{\ast} $ is a positive constant.

Now, we define $ L(t) $ as follows:

$\begin{eqnarray} L(t) = H^{1-\alpha}(t)+\epsilon\int_{\Omega}u_{t}udx+\epsilon\int_{\Omega}\triangle u_{t}\triangle udx, \end{eqnarray}$ $

(4.10)

where $ \varepsilon > 0 $, small enough to be chosen later, and

$\begin{eqnarray} 0\leq\alpha\leq\min\left\{\frac{p^{-}-m^{+}}{p^{-}(m^{+}-1)}, \frac{p^{-}-2}{2p^{-}}\right\}. \end{eqnarray}$ $

The remaining proof will be divided into two steps.

$ \textbf{Step} $ $ \textbf{1:} $ $ \textbf{Estimate} $ $ \textbf{for} $ $ \textbf{L'(t)}. $ By taking the derivative of $ (4.10) $ and using $ (1.1) $, we obtain

$\begin{eqnarray} \begin{aligned} \nonumber L'(t) = &(1-\alpha)H^{-\alpha}(t)\left[\int_{\Omega}|u_{t}|^{m(x)}dx-\frac{1}{2}(g'\diamond\bigtriangleup u)+\frac{1}{2}g(t)||\triangle u||_{2}^{2}\right]\\ &+\epsilon||u_{t}||_{2}^{2}+\epsilon\int_{\Omega}\triangle u_{tt}\triangle udx+\epsilon||\triangle u_{t}||_{2}^{2}\\ &-\epsilon||\triangle u||_{2}^{2}+\epsilon\int_{\Omega}\int_{0}^{t}g(t-\tau)\triangle u(\tau)d\tau\triangle udx\\ &-\epsilon\int_{\Omega}|u_{t}|^{m(x)-2}u_{t}udx+\epsilon\int_{\Omega}|u|^{p(x)}dx-\epsilon\int_{\Omega}\triangle u_{tt}\triangle udx\\ &\geq(1-\alpha)H^{-\alpha}(t)\int_{\Omega}|u_{t}|^{m(x)}dx+\epsilon||u_{t}||_{2}^{2}-\epsilon||\triangle u||_{2}^{2}\\ &+\epsilon\int_{\Omega}\int_{0}^{t}g(t-\tau)\triangle u(\tau)d\tau\triangle udx-\epsilon\int_{\Omega}|u_{t}|^{m(x)-2}u_{t}udx+\epsilon\int_{\Omega}|u|^{p(x)}dx+\epsilon||\Delta u_{t}||_{2}^{2}, \end{aligned} \end{eqnarray}$ $

applying Hölder's inequality and Young's inequality, we have

$\begin{eqnarray} \begin{aligned} \nonumber &\epsilon\int_{\Omega}\int_{0}^{t}g(t-\tau)\Delta u(\tau) \Delta u(t)d\tau dx\\ = &\epsilon\int_{\Omega}\int_{0}^{t}g(t-\tau)\Delta u(t)(\Delta u(\tau)-\Delta u(t))d\tau dx+\epsilon\int_{0}^{t}g(t-\tau)d\tau||\Delta u ||_{2}^{2}\\ \geq& -\epsilon\int_{0}^{t}g(t-\tau)||\Delta u(\tau)-\Delta u(t)||_{2}||\Delta u (t)||_{2}d\tau+\epsilon\int_{0}^{t}g(t-\tau)d\tau||\Delta u ||_{2}^{2}\\ \geq&-\epsilon\frac{p^{-}(1-\varepsilon_{1})}{2}(g\diamond \bigtriangleup u)+\epsilon\left(1-\frac{1}{2p^{-}(1-\varepsilon_{1})}\right)\int_{0}^{t}g(\tau)d\tau||\Delta u||_{2}^{2}, \end{aligned} \end{eqnarray}$ $

where $ 0 < \varepsilon_{1} < \frac{p^{-}-2}{p^{-}} $, then

$\begin{eqnarray} \begin{aligned} \nonumber L'(t)&\geq(1-\alpha)H^{-\alpha}(t)\int_{\Omega}|u_{t}|^{m(x)}dx+\epsilon||u_{t}||_{2}^{2}-\epsilon||\Delta u||_{2}^{2}\\ &+\epsilon||\Delta u_{t}||_{2}^{2}-\epsilon\int_{\Omega}|u_{t}|^{m(x)-2}u_{t}udx+\epsilon\int_{\Omega}|u|^{p(x)}dx\\ &-\epsilon\frac{p^{-}(1-\varepsilon_{1})}{2}(g\diamond \bigtriangleup u)+\epsilon\left(1-\frac{1}{2p^{-}(1-\varepsilon_{1})}\right)\int_{0}^{t}g(\tau)d\tau||\Delta u||_{2}^{2}, \end{aligned} \end{eqnarray}$ $

rewriting $ (4.7) $ to $ (\frac{p^{-}}{2}-1)l-\frac{1}{2p^{-}}(1-l) > 0 $, using $ (4.1) $ and $ (4.3) $ to substitute for $ (g\diamond \bigtriangleup u) $, choosing $ \varepsilon_{1} > 0 $ sufficiently small, we obtain

$\begin{eqnarray} \begin{aligned} L'(t)&\geq(1-\alpha)H^{-\alpha}(t)\int_{\Omega}|u_{t}|^{m(x)}dx+\epsilon p^{-}(1-\varepsilon_{1})H(t)+\left(\epsilon+\frac{\epsilon p^{-}(1-\varepsilon_{1})}{2}\right)(||u_{t}||_{2}^{2}+||\triangle u_{t}||_{2}^{2})\\ &+\epsilon\left\{\left(\frac{p^{-}(1-\varepsilon_{1})}{2}-1\right)(1-\int_{0}^{t}g(\tau)d\tau)-\frac{1}{2p^{-}(1-\varepsilon_{1})}\int_{0}^{t}g(\tau)d\tau\right\}||\triangle u||_{2}^{2}\\ &-\epsilon p^{-}(1-\varepsilon_{1})E_{2}-\epsilon\int_{\Omega}|u_{t}|^{m(x)-2}u_{t}udx+\epsilon\varepsilon_{1}\int_{\Omega}|u|^{p(x)}dx\\ &\geq (1-\alpha)H^{-\alpha}(t)\int_{\Omega}|u_{t}|^{m(x)}dx+\epsilon p^{-}(1-\varepsilon_{1})H(t)+\left(\epsilon+\frac{\epsilon p^{-}(1-\varepsilon_{1})}{2}\right)(||u_{t}||_{2}^{2}+||\triangle u_{t}||_{2}^{2})\\ &+\epsilon\frac{\left\{\left(\frac{p^{-}(1-\varepsilon_{1})}{2}-1\right)\frac{l}{2}-\frac{1}{2p^{-}(1-\varepsilon_{1})}\frac{1-l}{2}\right\}}{l}\frac{\lambda_{2}}{B_{1}^{2}}-\epsilon p^{-}(1-\varepsilon_{1})E_{2}-\epsilon\int_{\Omega}|u_{t}|^{m(x)-2}u_{t}udx\\ &+\epsilon\left\{\left(\frac{p^{-}(1-\varepsilon_{1})}{2}-1\right)\frac{l}{2}-\frac{1}{2p^{-}(1-\varepsilon_{1})}\frac{1-l}{2}\right\}||\triangle u||_{2}^{2}+\epsilon\varepsilon_{1}\int_{\Omega}|u|^{p(x)}dx.\\ &\geq (1-\alpha)H^{-\alpha}(t)\int_{\Omega}|u_{t}|^{m(x)}dx+\epsilon p^{-}(1-\varepsilon_{1})H(t)+\left(\epsilon+\frac{\epsilon p^{-}(1-\varepsilon_{1})}{2}\right)(||u_{t}||_{2}^{2}+||\triangle u_{t}||_{2}^{2})\\ &+\epsilon\frac{\left\{\left(\frac{p^{-}(1-\varepsilon_{1})}{2}-1\right)\frac{l}{2}-\frac{1}{2p^{-}(1-\varepsilon_{1})}\frac{1-l}{2}\right\}}{l}(B_{1}^{2})^{\frac{-p^{-}}{p^{-}-2}}-\epsilon p^{-}(1-\varepsilon_{1})E_{2}-\epsilon\int_{\Omega}|u_{t}|^{m(x)-2}u_{t}udx\\ &+\epsilon\left\{\left(\frac{p^{-}(1-\varepsilon_{1})}{2}-1\right)\frac{l}{2}-\frac{1}{2p^{-}(1-\varepsilon_{1})}\frac{1-l}{2}\right\}||\triangle u||_{2}^{2}+\epsilon\varepsilon_{1}\int_{\Omega}|u|^{p(x)}dx.\\ \end{aligned} \end{eqnarray}$ $

(4.11)

$ \textbf{Step} $ $ \textbf{1.1:} $ $ \textbf{Estimate} $ $ \textbf{for} $ $ \epsilon\frac{\left\{\left(\frac{p^{-}(1-\varepsilon_{1})}{2}-1\right)\frac{l}{2}-\frac{1}{2p^{-}(1-\varepsilon_{1})}\frac{1-l}{2}\right\}}{l}(B_{1}^{2})^{\frac{-p^{-}}{p^{-}-2}}-\epsilon p^{-}(1-\varepsilon_{1})E_{2}. $ It follows from the condition in Theorem $ 3.1 $ that

$ E(0) < \frac{1}{2}(\frac{1}{2}-\frac{1}{p^{-}})\left(1-\frac{1-l}{p^{-}(p^{-}-2)l}\right)(B_{1}^{2})^{\frac{-p^{-}}{p^{-}-2}} = \frac{(\frac{p^{-}}{2}-1)\frac{l}{2}-\frac{1}{2p^{-}}\frac{(1-l)}{2}}{lp^{-}}(B_{1}^{2})^{\frac{-p^{-}}{p^{-}-2}} < E_{1}, $

here, we can take $ \varepsilon_{1} > 0 $ sufficiently small and choose $ E_{2}\in(E(0), E_{1}) $ sufficiently close to $ E(0) $ such that

$\begin{eqnarray} \begin{aligned} &\epsilon\frac{\left(\frac{p^{-}(1-\varepsilon_{1})}{2}-1\right)\frac{l}{2}-\frac{1}{2p^{-}(1-\varepsilon_{1})}\frac{(1-l)}{2}}{l}(B_{1}^{2})^{\frac{-p^{-}}{p^{-}-2}}-\epsilon(1-\varepsilon_{1})p^{-}E_{2}\\ \geq&\epsilon\frac{\left(\frac{p^{-}(1-\varepsilon_{1})}{2}-1\right)\frac{l}{2}-\frac{1}{2p^{-}(1-\varepsilon_{1})}\frac{(1-l)}{2}}{l}(B_{1}^{2})^{\frac{-p^{-}}{p^{-}-2}}-\epsilon(1-\varepsilon_{1})p^{-}\frac{\left(\frac{p^{-}}{2}-1\right)\frac{l}{2}-\frac{1}{2p^{-}}\frac{(1-l)}{2}}{lp^{-}}(B_{1}^{2})^{\frac{-p^{-}}{p^{-}-2}}\\ \geq&0. \end{aligned} \end{eqnarray}$ $

(4.12)

Therefore, we obtain by combining $ (4.11) $ and $ (4.12) $,

(4.13)

$ \textbf{Step} $ $ \textbf{1.2:} $ $ \textbf{Estimate} $ $ \textbf{for} $ $ -\epsilon \int_{\Omega}|u_{t}|^{m(x)-2}u_{t}udx. $ Applying Young's inequality with $ \varepsilon_{2} > 1 $, the embedding $ L^{p(x)}(\Omega)\hookrightarrow L^{m(x)}(\Omega) $, Lemma $ 2.4 $ and Lemma $ 4.2 $, we easily have

$\begin{eqnarray} \begin{aligned} &\left|\int_{\Omega}|u_{t}|^{m(x)-2}u_{t}udx\right|\leq\int_{\Omega}|u_{t}|^{m(x)-1}H^{-\alpha\frac{m(x)-1}{m(x)}}(t)H^{\alpha\frac{m(x)-1}{m(x)}}(t)|u|dx\\ \leq&\varepsilon_{2}H^{-\alpha}(t)\int_{\Omega}|u_{t}|^{m(x)}dx+\frac{1}{\varepsilon_{2}^{m^{-}-1}}\int_{\Omega}|u|^{m(x)}H^{\alpha(m(x)-1)}(t)dx\\ \leq&\varepsilon_{2}H^{-\alpha}(t)\int_{\Omega}|u_{t}|^{m(x)}dx+\frac{2C_{1}^{\alpha(m^{-}-m^{+})}}{\varepsilon_{2}^{m^{-}-1}}H^{\alpha(m^{+}-1)}(t)\int_{\Omega}|u|^{m(x)}dx\\ \leq&\varepsilon_{2}H^{-\alpha}(t)\int_{\Omega}|u_{t}|^{m(x)}dx+\frac{C_{2}}{\varepsilon_{2}^{m^{-}-1}}H^{\alpha(m^{+}-1)}(t)\, \max\, \left\{||u||_{p(x)}^{m^{+}}, ||u||_{p(x)}^{m^{-}}\right\}, \end{aligned} \end{eqnarray}$ $

(4.14)

where $ C_{1} = \min\left\{H(0), 1\right\} $, $ C_{2} = 2(1+|\Omega|)^{m^{+}}C_{1}^{\alpha(m^{-}-m^{+})} $. Next, we have

$\begin{eqnarray} \begin{aligned} \nonumber ||u||_{p(x)}^{m^{+}}&\leq \max \, \left\{\left(\int_{\Omega}|u|^{p(x)}dx\right)^{\frac{m^{+}}{p^{+}}}, \left(\int_{\Omega}|u|^{p(x)}dx\right)^{\frac{m^{+}}{p^{-}}}\right\}\\ &\leq\max\left\{[p^{-}H(t)]^{\frac{m^{+}}{p^{+}}-\frac{m^{+}}{p^{-}}}, 1\right\}\left(\int_{\Omega}|u|^{p(x)}dx\right)^{\frac{m^{+}}{p^{-}}}, \end{aligned} \end{eqnarray}$ $

and

$\begin{eqnarray} \begin{aligned} \nonumber ||u||_{p(x)}^{m^{-}}\leq\max\left\{[p^{-}H(t)]^{\frac{m^{-}}{p^{+}}-\frac{m^{+}}{p^{-}}}, [p^{-}H(t)]^{\frac{m^{-}-m^{+}}{p^{-}}}\right\}\left(\int_{\Omega}|u|^{p(x)}dx\right)^{\frac{m^{+}}{p^{-}}}, \end{aligned} \end{eqnarray}$ $

which illustrate

$\begin{eqnarray} \max\left\{||u||_{p(x)}^{m^{+}}, ||u||_{p(x)}^{m^{-}}\right\}\leq C_{3}\left(\int_{\Omega}|u|^{p(x)}dx\right)^{\frac{m^{+}}{p^{-}}}, \end{eqnarray}$ $

where $ C_{3} = 2\min\left\{p^{-}H(0), 1\right\}^{\frac{m^{-}}{p^{+}}-\frac{m^{+}}{p^{-}}} $. Recalling $ 0 < \alpha\leq\frac{p^{-}-m^{+}}{p^{-}(m^{+}-1)} $ and Lemma $ 4.2 $, apparently,

$\begin{eqnarray} \begin{aligned} &H^{\alpha(m^{+}-1)}(t)\, \max\, \left\{||u||_{p(x)}^{m^{+}}, ||u||_{p(x)}^{m^{-}}\right\}\leq C_{3}H^{\alpha(m^{+}-1)}(t)\left(\int_{\Omega}|u|^{p(x)}dx\right)^{\frac{m^{+}}{p^{-}}}\\ \leq& C_{3}\frac{H^{\alpha(m^{+}-1)+\frac{m^{+}}{p^{-}}-1}(t)}{H^{\alpha(m^{+}-1)+\frac{m^{+}}{p^{-}}-1}(0)}H^{1-\frac{m^{+}}{p^{-}}}(t)H^{\alpha(m^{+}-1)+\frac{m^{+}}{p^{-}}-1}(0)\left(\int_{\Omega}|u|^{p(x)}dx\right)^{\frac{m^{+}}{p^{-}}}\\ \leq& C_{3}(\frac{1}{p^{-}})^{1-\frac{m^{+}}{p^{-}}}\left(\int_{\Omega}|u|^{p(x)}dx\right)^{1-\frac{m^{+}}{p^{-}}}H^{\alpha(m^{+}-1)+\frac{m^{+}}{p^{-}}-1}(0)\left(\int_{\Omega}|u|^{p(x)}dx\right)^{\frac{m^{+}}{p^{-}}}\\ \leq& C_{3}(\frac{1}{p^{-}})^{1-\frac{m^{+}}{p^{-}}}C_{1}^{\alpha(m^{+}-1)+\frac{m^{+}}{p^{-}}-1}\int_{\Omega}|u|^{p(x)}dx, \end{aligned} \end{eqnarray}$ $

(4.15)

it follows from $ (4.13) $, $ (4.14) $, and $ (4.15) $ that

$\begin{eqnarray} \begin{aligned} \nonumber L'(t)&\geq (1-\alpha-\epsilon\varepsilon_{2})H^{-\alpha}(t)\int_{\Omega}|u_{t}|^{m(x)}dx+\left(\epsilon+\frac{\epsilon p^{-}(1-\varepsilon_{1})}{2}\right)(||u_{t}||_{2}^{2}+||\triangle u_{t}||_{2}^{2})\\ &+\epsilon(1-\varepsilon_{1})p^{-}H(t)+\epsilon\left(\varepsilon_{1}-\frac{C_{1}^{\alpha(m^{+}-1)+\frac{m^{+}}{p^{-}}-1}C_{2}C_{3}(\frac{1}{p^{-}})^{1-\frac{m^{+}}{p^{-}}}}{\varepsilon_{2}^{m^{-}-1}}\right)\int_{\Omega}|u|^{p(x)}dx\\ &+\epsilon\left\{\left(\frac{p^{-}(1-\varepsilon_{1})}{2}-1\right)\frac{l}{2}-\frac{1}{2p^{-}(1-\varepsilon_{1})}\frac{1-l}{2}\right\}||\triangle u||_{2}^{2}, \end{aligned} \end{eqnarray}$ $

let us fix the constant $ \varepsilon_{2} $ so that

$ \varepsilon_{1} > \frac{C_{1}^{\alpha(m^{+}-1)+\frac{m^{+}}{p^{-}}-1}C_{2}C_{3}(\frac{1}{p^{-}})^{1-\frac{m^{+}}{p^{-}}}}{\varepsilon_{2}^{m^{-}-1}}, $

and then choose $ \epsilon $ so small that $ 1-\alpha > \epsilon\varepsilon_{1} $. Therefore, we obtain

$\begin{eqnarray} \begin{aligned} L'(t)\geq M_{1}\left(H(t)+||\triangle u||_{2}^{2}+||u_{t}||_{2}^{2}+||\triangle u_{t}||_{2}^{2}+\int_{\Omega}|u|^{p(x)}dx\right), \end{aligned} \end{eqnarray}$ $

(4.16)

where

$\begin{eqnarray} \begin{aligned} \nonumber M_{1} = \epsilon\min\left\{\left(1+\frac{ p^{-}(1-\varepsilon_{1})}{2}\right), (1-\varepsilon_{1})p^{-}, \varepsilon_{1}-\frac{C_{1}^{\alpha(m^{+}-1)+\frac{m^{+}}{p^{-}}-1}C_{2}C_{3}(\frac{1}{p^{-}})^{1-\frac{m^{+}}{p^{-}}}}{\varepsilon_{2}^{m^{-}-1}}, \right., \\ \left.\left(\frac{p^{-}(1-\varepsilon_{1})}{2}-1\right)\frac{l}{2}-\frac{1}{2p^{-}(1-\varepsilon_{1})}\frac{1-l}{2}\right\}. \end{aligned} \end{eqnarray}$ $

Inequalities $ (4.16) $ and Lemma $ 4.2 $ imply $ L(t)\geq L(0). $ Therefore, for a sufficiently small $ \epsilon $, we have

$ L(0) = H^{1-\alpha}(0)+\epsilon\int_{\Omega}u_{1}u_{0}dx+\epsilon\int_{\Omega}\triangle u_{1}\triangle u_{0}dx > 0. $

$ \textbf{Step} $ $ \textbf{2:} $ $ \textbf{A} $ $ \textbf{differential} $ $ \textbf{inequality} $ $ \textbf{for} $ $ \textbf{L(t)}. $ Applying Hölder's inequality, Young's inequality and the embedding $ L^{p(x)}(\Omega)\hookrightarrow L^{2}(\Omega) $, we easily obtain

$\begin{eqnarray} \begin{aligned} \left|\int_{\Omega}u_{t}udx\right|^{\frac{1}{1-\alpha}}&\leq\left(\left\|u_{t}\right\|_{2}\left\|u\right\|_{2}\right)^{\frac{1}{1-\alpha}}\leq (1+|\Omega|)^{\frac{1}{1-\alpha}}||u_{t}||_{2}^{\frac{1}{1-\alpha}}||u||_{p(x)}^{\frac{1}{1-\alpha}}\\ &\leq\frac{(1+|\Omega|)^{\frac{1}{1-\alpha}}}{\mu}||u_{t}||_{2}^{\frac{1}{1-\alpha}\mu}+\frac{(1+|\Omega|)^{\frac{1}{1-\alpha}}}{\nu}||u||_{p(x)}^{\frac{1}{1-\alpha}\nu}, \end{aligned} \end{eqnarray}$ $

(4.17)

where $ \frac{1}{\mu}+\frac{1}{\nu} = 1.\ $Choosing $ \mu = 2(1-\alpha) > 1 $, then $ \nu = \frac{2(1-\alpha)}{2(1-\alpha)-1}, \ $further, $ (4.17) $ can be rewritten as

$\begin{eqnarray} \begin{aligned} \left|\int_{\Omega}u_{t}udx\right|^{\frac{1}{1-\alpha}}\leq\frac{(1+|\Omega|)^{\frac{1}{1-\alpha}}}{\mu}||u_{t}||_{2}^{2}+\frac{(1+|\Omega|)^{\frac{1}{1-\alpha}}}{\nu}||u||_{p(x)}^{\frac{2}{2(1-\alpha)-1}}, \end{aligned} \end{eqnarray}$ $

(4.18)

recalling $ 0 < \alpha < \frac{p^{-}-2}{2p^{-}} $, we obtain

$\begin{eqnarray} \begin{aligned} &||u||_{p(x)}^{\frac{2}{2(1-\alpha)-1}} \leq \max \left\{(\int_{\Omega}|u|^{p(x)}dx)^{\frac{2}{p^{-}[2(1-\alpha)-1]}}, (\int_{\Omega}|u|^{p(x)}dx)^{\frac{2}{p^{+}[2(1-\alpha)-1]}}\right\}\\ \leq&\left\{\left[p^{-}H(t)\right]^{\frac{2-p^{-}[2(1-\alpha)-1]}{p^{-}[2(1-\alpha)-1]}}, [p^{-}H(t)]^{\frac{2-p^{+}[2(1-\alpha)-1]}{p^{+}[2(1-\alpha)-1]}}\right\}\int_{\Omega}|u|^{p(x)}dx\\ \leq& C_{4}\int_{\Omega}|u|^{p(x)}dx, \end{aligned} \end{eqnarray}$ $

(4.19)

with $ C_{4} = \min\{p^{-}H(0), 1\}^{\frac{2-p^{+}[2(1-\alpha)-1]}{p^{+}[2(1-\alpha)-1]}} $. Inserting $ (4.19) $ into $ (4.18) $, we obtain

$\begin{eqnarray} \begin{aligned} \left|\int_{\Omega}u_{t}udx\right|^{\frac{1}{1-\alpha}}\leq\frac{(1+|\Omega|)^{\frac{1}{1-\alpha}}}{\mu}||u_{t}||_{2}^{2}+\frac{(1+|\Omega|)^{\frac{1}{1-\alpha}}}{\nu}C_{4}\int_{\Omega}|u|^{p(x)}dx. \end{aligned} \end{eqnarray}$ $

(4.20)

We now estimate

$\begin{eqnarray} \begin{aligned} \left|\int_{\Omega}\triangle u_{t}\triangle udx\right|^{\frac{1}{1-\alpha}}\leq ||\triangle u_{t}||_{2}^{\frac{1}{1-\alpha}}||\Delta u||_{2}^{\frac{1}{1-\alpha}} \leq C^{\frac{1}{1-\alpha}}_{\ast}\leq\frac{C_{\ast}^{\frac{1}{1-\alpha}}}{H(0)}H(t), \end{aligned} \end{eqnarray}$ $

(4.21)

therefore, combining $ (4.20) $ and $ (4.21) $, we obtain

$\begin{eqnarray} \begin{aligned} L^{\frac{1}{1-\alpha}}(t)& = \left(H^{1-\alpha}(t)+\epsilon\int_{\Omega}u_{t}udx+\epsilon\int_{\Omega}\triangle u_{t}\triangle udx\right)^{\frac{1}{1-\alpha}}\\ &\leq M_{2}\left(H(t)+||u_{t}||_{2}^{2}+||\triangle u_{t}||_{2}^{2}+||\triangle u||_{2}^{2}+\int_{\Omega}|u|^{p(x)}dx\right), \end{aligned} \end{eqnarray}$ $

(4.22)

where

$\begin{eqnarray} \begin{aligned} \nonumber M_{2} = \max\left\{2^{\frac{1}{1-\alpha}}(2^{\frac{1}{1-\alpha}}+\epsilon^{\frac{1}{1-\alpha}}\frac{C_{\ast}^{\frac{1}{1-\alpha}}}{H(0)}), \ 2^{\frac{2}{1-\alpha}}\epsilon^{\frac{1}{1-\alpha}}\frac{(1+|\Omega|)\frac{1}{1-\alpha}}{\mu}, \ 2^{\frac{2}{1-\alpha}}\epsilon^{\frac{1}{1-\alpha}}\frac{(1+|\Omega|)\frac{1}{1-\alpha}}{\nu}C_{4}\right\}. \end{aligned} \end{eqnarray}$ $

Combining $ (4.16) $ and $ (4.22) $, we arrive at

$\begin{eqnarray} L'(t)\geq\frac{M_{1}}{M_{2}}L^{\frac{1}{1-\alpha}}(t), \, \, \forall t\geq 0. \end{eqnarray}$ $

(4.23)

A simple integration of $ (4.23) $ over $ (0, t) $ yields

$\begin{eqnarray} L^{\frac{\alpha}{1-\alpha}}(t)\geq\frac{1}{L^{\frac{\alpha}{\alpha-1}}(0)-\frac{M_{1}}{M_{2}}\frac{\alpha}{1-\alpha}t}, \end{eqnarray}$ $

this shows that $ L(t) $ blows up in finite time

$\begin{eqnarray} T^{\ast}\leq \frac{M_{2}}{M_{1}}\frac{1-\alpha}{\alpha}L^{\frac{\alpha}{\alpha-1}}(0), \end{eqnarray}$ $

furthermore, one gets from $ (4.22) $ that

$ \lim\limits_{t\rightarrow T^{\ast-}}\left(H(t)+||u_{t}||_{2}^{2}+||\triangle u_{t}||_{2}^{2}+||\triangle u||_{2}^{2}+\int_{\Omega}|u|^{p(x)}dx\right) = +\infty, $

it easily follows that

$ \int_{\Omega}|u|^{p(x)}dx\leq \int_{\{|u|\geq1\}}|u|^{p^{+}}dx+\int_{\{|u| < 1\}}|u|^{p^{-}}dx\leq||u||_{p^{+}}^{p^{+}}+|\Omega|, $

and using Lemma $ 4.2 $, we obtain

$ \lim\limits_{t\rightarrow T^{\ast-}}\left(||u_{t}||_{2}^{2}+||\triangle u_{t}||_{2}^{2}+||\triangle u||_{2}^{2}+||u||_{p^{+}}^{p^{+}}\right) = +\infty, $

this leads to a contradiction with $ (4.9) $. Thus, the solution to Problem $ (1.1) $ blows up in finite time. □

Author contributions

Ying Chu: Methodology, Wring-original draft, Writing-review editing; Bo Wen and Libo Cheng: Methodology, Writing-original draft.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors express their heartfelt thanks to the editors and referees who have provided some important suggestions. This work was supported by Science and Technology Development Plan Project of Jilin Province, China (20240101307JC).

Conflict of interest

The authors declare there is no conflict of interest.

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