1 Introduction

In this work, we discuss a viscoelastic von Karman equation with strong damping and variable exponent source terms,

$$\begin{aligned}& w_{tt} + \Delta ^{2} w - \int ^{t}_{0} h(t-s) \Delta ^{2} w(s) \,ds - \Delta w_{t} \\& \quad = \bigl[w, \chi ( w ) \bigr] + \vert w \vert ^{q(x)-2} w \quad \text{in } \Omega \times (0,T), \end{aligned}$$
(1.1)
$$\begin{aligned}& \Delta ^{2} \chi ( w ) = - [w,w] \quad \text{in } \Omega \times (0,T), \end{aligned}$$
(1.2)
$$\begin{aligned}& w= \frac{\partial w}{\partial \nu } = 0, \quad\quad \chi ( w ) = \frac{\partial \chi ( w ) }{\partial \nu } = 0 \quad \text{on } \partial \Omega \times (0, T), \end{aligned}$$
(1.3)
$$\begin{aligned}& w(0)=w_{0}, \quad\quad w_{t}(0)=w_{1} \quad \text{in } \Omega , \end{aligned}$$
(1.4)

where \(\Omega \subset {\mathbb{R}}^{2}\) is a bounded domain with sufficiently smooth boundary ∂Ω, \(\nu = (\nu _{1}, \nu _{2})\) is the unit normal vector outward to ∂Ω, the differentiable kernel function h defined on \([0, \infty ) \) satisfies \(h(0) >0\), \(h(t) \geq 0\), \(h'(t) \leq 0\), and

$$ 1- \int ^{\infty }_{0} h(s)\,ds : = l >0 . $$

The Von Karman bracket \([\cdot ,\cdot ]\) is given as

$$ [ y ,v ] = y_{x_{1} x_{1}} v_{x_{2} x_{2}} + y_{x_{2} x_{2}} v_{x_{1} x_{1}} - 2 y_{x_{1} x_{2}} v_{x_{1} x_{2}}, $$

here \(x=(x_{1}, x_{2}) \in \Omega \). The exponent function \(q(\cdot )\) is measurable and verifies

$$ \bigl\vert q(\overline{x})-q(x) \bigr\vert \leq - \frac{a}{\log \vert \overline{x} - x \vert } \quad \text{for all } \overline{x}, x \in \Omega \text{ with } \vert \overline{x} -x \vert < \kappa , $$
(1.5)

where \(a >0\) and \(0 < \kappa < 1\), and

$$ 2 \leq q_{1} : = \operatorname{ess} \inf _{x\in \Omega } q(x) \leq q(x) \leq q_{2} :=\operatorname{ess} \sup_{x\in \Omega } q(x) < \infty . $$

The von Karman equations (1.1)–(1.4) model a nonlinear elastic plate by describing the transversal displacement w and the Airy-stress function \(\chi (w)\). Von Karman equation also arises in many applications such as bifurcation theory and shells. The type of von Karman equations

$$\begin{aligned} w_{tt} + \Delta ^{2} w - \int ^{t}_{0} h(t-s) \Delta ^{2} w(s) \,ds + g(x, w_{t}) = \bigl[w, \chi ( w ) \bigr] \end{aligned}$$
(1.6)

associated with different boundary conditions has been intensively treated about existence and stability (see [5, 8, 9, 16–18] and the references therein). When \(h=g=0 \), the authors of [8] studied the unique existence of global solution. When \(g(x, u_{t}) = a(x) u_{t} \) and \(h=0\), Horn and Lasiecka [9] discussed energy decay estimates. When \(g=0 \), Park et al. [18] obtained the general decay behavior of solutions.

On the other hand, problems with variable exponent source have been attracting great interest [13–15, 21]. Such problems appear in physical phenomena such as nonlinear elastics [22], electrorheological fluids [20], stationary thermorheological viscous flows of non-Newtonian fluids [2], and image precessing [1]. Recently, Messaoudi et al. [15] considered wave equations with source and damping terms of variable exponent,

$$\begin{aligned} w_{tt} - \Delta w + a \vert w_{t} \vert ^{\gamma (x)-2} w_{t} = b \vert w \vert ^{q(x)-2} w. \end{aligned}$$

They obtained the local existence of solutions under appropriate conditions on \(\gamma (\cdot )\) and \(q(\cdot )\) by utilizing the Faedo–Galerkin’s technique and the contraction mapping theorem. Furthermore, they showed that the solution with negative initial energy blows up in a finite time. Later, Park and Kang [19] improved and complemented the result of [15] by obtaining a blow-up result of solution with certain positive initial energy for a wave equation of memory type. We also refer to a recent work [4] for a nonlinear diffusion system involving variable exponents dependent on spatial and time variables and cross-diffusion terms. At this point, it is worthwhile to mention that there is little work concerning global nonexistence of solutions for viscoelastic von Karman equations with variable source effect. Particularly, there is no literature concerning blow-up results of solutions with high initial energy for the equations. Thus, in this article, we establish blow-up results of solutions with three levels of initial energy such as non-positive initial energy, certain positive initial energy, and high initial energy. Furthermore, we estimate not only the upper bound but also the lower bound of the blow-up time. These are inspired by the ideas of [23], where the authors proved blow-up results of solutions with high initial energy and estimated bounds of existence time of solutions for wave equation with logarithmic nonlinear source term.

The outline of this article is here. In Sect. 2, we state some definitions, notations, and auxiliary lemmas. In Sect. 3, we construct blow-up results and obtain bounds of the blow-up time.

2 Preliminaries

We denote \(( y , v ) = \int _{\Omega } y(x) v(x) \,dx\), \(\Vert y \Vert _{2}^{2} = ( y , y ) \). Generally, we denote by \(\Vert \cdot \Vert _{Y} \) the norm of a space Y. For simplicity, we write \(\Vert \cdot \Vert _{L^{p}(\Omega )}\) by \(\Vert \cdot \Vert _{p} \). If there is no ambiguity, we will omit the variables t and x. Let \(B_{1} \) and \(B_{2}\) be the constants with

$$ \Vert y \Vert _{2} \leq B_{1} \Vert \nabla y \Vert _{2} \quad \text{for } y \in H^{1}_{0} (\Omega ), \quad\quad \Vert \nabla y \Vert _{2} \leq B_{2} \Vert \Delta y \Vert _{2} \quad \text{for } y \in H^{2}_{0} (\Omega ) . $$
(2.1)

For a measurable function \(p : \Omega \subset {\mathbb{R}}^{n} \to [1, \infty ] \), the Lebesgue space

$$ L^{p(\cdot )}(\Omega ) = \biggl\{ y : \Omega \to {\mathbb{R}} \Bigm| y \text{ is measurable in } \Omega , \int _{\Omega } \bigl\vert \delta y(x) \bigr\vert ^{p(x)} \,dx < \infty \text{ for some } \delta >0 \biggr\} $$

is a Banach space equipped with Luxembourg-type norm

$$ \Vert y \Vert _{p(\cdot )} = \inf \biggl\{ \delta > 0 \Bigm| \int _{\Omega } \biggl\vert \frac{ y (x)}{\delta } \biggr\vert ^{p(x)} \,dx \leq 1 \biggr\} . $$

Lemma 2.1

([3])

If \({ 1 < p_{1} := \operatorname{ess} \inf_{x\in \Omega } p(x) \leq p(x) \leq p_{2} :=\operatorname{ess} \sup_{x\in \Omega } p(x) < \infty , } \) then

$$ \min \bigl\{ \Vert y \Vert _{p(\cdot )}^{p_{1}} , \Vert y \Vert _{p(\cdot )}^{p_{2}} \bigr\} \leq \int _{\Omega } \bigl\vert y (x) \bigr\vert ^{p(x)} \,dx \leq \max \bigl\{ \Vert y \Vert _{p(\cdot )}^{p_{1}} , \Vert y \Vert _{p(\cdot )}^{p_{2} } \bigr\} $$

for any \(y \in L^{p(\cdot )}(\Omega )\).

Since \(\dim (\Omega ) =2 \), the embedding \({ H^{2}_{0}(\Omega ) \hookrightarrow L^{q(\cdot )}( \Omega )} \) (\(2 \leq q(x) < \infty \)) is continuous and compact [11]. We let B be the constant of the embedding inequality

$$ \Vert y \Vert _{ q (\cdot )} \leq B \Vert \Delta y \Vert _{2} \quad \text{for } y \in H^{2}_{0} (\Omega ). $$
(2.2)

See [6, 7, 11] for more properties of a Lebesgue space with variable exponent.

Lemma 2.2

([9], Lemma 2.1)

If \(y \in H^{2}(\Omega )\), then \(\chi ( y ) \in W^{2, \infty }(\Omega ) \) and \(\Vert \chi ( y ) \Vert _{ W^{2, \infty }(\Omega ) } \leq a_{1} \Vert y \Vert ^{2}_{H^{2}( \Omega )}\).

Lemma 2.3

([8], p. 270)

If \(y \in H^{2}(\Omega )\) and \(v \in W^{2, \infty }(\Omega )\), then \([ y , v ] \in L^{2}(\Omega ) \) and \(\Vert [ y , v ] \Vert _{2} \leq a_{2} \Vert y \Vert _{H^{2}(\Omega )} \Vert v \Vert _{W^{2, \infty }(\Omega )}\).

Lemma 2.4

([5])

Let \(y_{1}, y_{2}, y_{3} \in H^{2}(\Omega )\). If at least one of them is an element of \(H^{2}_{0}(\Omega )\), then

$$ \int _{\Omega } [ y_{1}, y_{2}] y_{3} \,dx = \int _{\Omega } [ y_{1}, y_{3}] y_{2} \,dx. $$

By combining the arguments of [10, 19], for every \((w_{0}, w_{1}) \in H^{2}_{0}(\Omega ) \times L^{2}(\Omega )\), we can get a unique local solution w of problem (1.1)–(1.4) with \(w \in C(0,T; H^{2}_{0}(\Omega ) ) \cap C^{1} ( 0,T; L^{2}(\Omega )) \) and \(w_{t} \in L^{2}(0,T; H^{1}_{0}(\Omega ))\).

3 Blow-up results

In this section, we establish blow-up results of solutions with three levels of initial energy and estimate bounds of blow-up time. For this, we need ab auxiliary lemma.

Lemma 3.1

([12])

Let \(B(t)\) be a positive, twice differentiable function verifying

$$ B(t) B''(t) - ( 1+ \theta ) \bigl( B'(t) \bigr)^{2} \geq 0 $$

for \(t > 0 \), where θ is a positive constant. If \(B(0) >0\) and \(B'(0) >0\), then there exists a \(T_{1} \leq \frac{B(0)}{\theta B'(0)} \) with \({ \lim_{t \to T_{1}^{-} } B(t) = \infty } \).

Taking the scalar product (1.1) by \(w_{t}\) in \(L^{2}(\Omega )\) and using (1.3), we get

$$\begin{aligned}& \frac{d}{dt} \biggl( \frac{1}{2} \Vert w_{t} \Vert _{2}^{2} + \frac{1}{2} \Vert \Delta w \Vert _{2}^{2} - \int _{\Omega } \frac{ \vert w \vert ^{ q (x)} }{q(x)} \,dx \biggr) - \int ^{t}_{0} h (t-s) \bigl( \Delta w(s) , \Delta w_{t} (t) \bigr) \,ds \\& \quad = - \Vert \nabla w_{t} \Vert _{2}^{2} + \bigl( \bigl[w, \chi (w) \bigr] , w_{t} \bigr) . \end{aligned}$$

From Lemma 2.4, (1.2) and (1.3), we have

$$\begin{aligned} \bigl( \bigl[w, \chi (w) \bigr] , w_{t} \bigr) = \frac{1}{2} \biggl( \frac{d}{dt} [w, w ] , \chi (w) \biggr) = - \frac{1}{4} \frac{d}{dt} \bigl\Vert \Delta \chi (w) \bigr\Vert _{2}^{2} . \end{aligned}$$

Using this and the relation

$$\begin{aligned}& - \int ^{t}_{0} h (t-s) \bigl( \Delta w(s) , \Delta w_{t} (t) \bigr) \,ds \\& \quad = \frac{1}{2} \frac{d}{dt} \biggl\{ ( h \circ \Delta w ) - \biggl( \int ^{t}_{0} h (s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} \biggr\} + \frac{h(t)}{2} \Vert \Delta w \Vert _{2}^{2} - \frac{1}{2} \bigl( h' \circ \Delta w \bigr) , \end{aligned}$$

we get

$$\begin{aligned} E'(t) = - \Vert \nabla w_{t} \Vert _{2}^{2} - \frac{h(t)}{2} \Vert \Delta w \Vert _{2}^{2} + \frac{1}{2} \bigl(h' \circ \Delta w \bigr) \leq - \Vert \nabla w_{t} \Vert _{2}^{2} \leq 0, \quad t\in [0, T_{*}) \end{aligned}$$
(3.1)

and

$$\begin{aligned} E(t) + \int ^{t}_{0} \bigl\Vert \nabla w_{t} (s) \bigr\Vert _{2}^{2} \,ds \leq E(0), \quad t \in [0, T_{*}) , \end{aligned}$$
(3.2)

where

$$\begin{aligned} \begin{aligned}[b] E(t) = {}&\frac{1}{2} \Vert w_{t} \Vert _{2}^{2} +\frac{1}{2} \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} + \frac{1}{2} (h \circ \Delta w) \\&{}+ \frac{1}{4} \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} - \int _{\Omega } \frac{ \vert w \vert ^{q(x)}}{q(x)} \,dx,\end{aligned} \end{aligned}$$
(3.3)

here

$$ (h \circ \Delta w) (t) = \int ^{t}_{0} h(t-s) \bigl\Vert \Delta w(t) - \Delta w(s) \bigr\Vert _{2}^{2} \,ds , $$

and \(T_{*} = \sup \{ T : [0,T] \text{ is the existence interval of the solution to (1.1)--(1.4)} \}\).

3.1 Case of non-positive initial energy

Theorem 3.1

Let \(q_{1} \geq 4\) and the kernel function h satisfy

$$ \int ^{\infty }_{0} h(s) \,ds \leq \frac{ q_{1} ( q_{1} -2 ) }{ q_{1} ( q_{1} -2 ) + 1 }. $$
(3.4)

Let one of the following hold.

  1. (i)

    \(E(0) < 0 \);

  2. (ii)

    \(E( 0) = 0 \) and \((w_{0} , w_{1} ) > \frac{ 2 \Vert \nabla w_{0} \Vert _{2}^{2} }{ q_{1} -2} \).

Then the solution w of problem (1.1)–(1.4) blows up in a finite time \(T_{*}\), that is,

$$ \lim_{t \to T_{*}^{-}} \bigl\Vert \Delta w(t) \bigr\Vert _{2} = \infty . $$
(3.5)

In addition, \(T_{*}\) satisfies

$$ T_{*} \leq \frac{ 2 \Vert w_{0} \Vert _{2}^{2} + 2 \alpha \beta ^{2} }{ ( q_{1} -2) \{ (w_{0}, w_{1}) + \alpha \beta \} -2 \Vert \nabla w_{0} \Vert _{2}^{2} } , $$
(3.6)

where

$$\begin{aligned} \textstyle\begin{cases} 0 < \alpha \leq -2E(0) \quad \textit{and} \quad \beta > \max \{ 0 , - \frac{( w_{0}, w_{1} )}{ \alpha } , \frac{ 2 \Vert \nabla w_{0} \Vert _{2}^{2} - ( q_{1} -2 ) ( w_{0} , w_{1}) }{ q_{1} -2 } \} \quad \textit{if } E( 0) < 0 ; \\ \alpha =0 \quad \textit{if } E( 0) = 0 \quad \textit{and}\quad (w_{0} , w_{1} ) > \frac{ 2 \Vert \nabla w_{0} \Vert _{2}^{2} }{ q_{1} -2} . \end{cases}\displaystyle \end{aligned}$$
(3.7)

Proof

Suppose that w is global. For \(0 < T < T_{*} \), we define a function F on \([0,T]\) by

$$ F(t) = \Vert w \Vert _{2}^{2} + \int ^{t}_{0} \bigl\Vert \nabla w(s) \bigr\Vert _{2}^{2} \,ds + (T-t) \Vert \nabla w_{0} \Vert _{2}^{2} + \alpha (t+ \beta )^{2}, $$
(3.8)

where \(\alpha \geq 0 \) and \(\beta >0 \) are the constants satisfying (3.7), then

$$\begin{aligned}& F(t) > 0 \quad \text{for } t \in [0,T], \end{aligned}$$
(3.9)
$$\begin{aligned}& F'(t) = 2(w, w_{t}) + 2 \int ^{t}_{0} \bigl(\nabla w(s), \nabla w_{t}(s) \bigr) \,ds + 2 \alpha (t+ \beta ), \end{aligned}$$
(3.10)

and

$$\begin{aligned} F''(t) = & 2 \Vert w_{t} \Vert _{2}^{2} - 2 \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} + 2 \bigl( w, \bigl[ w, \chi ( w ) \bigr] \bigr) + 2 \int _{ \Omega } \vert w \vert ^{q(x)} \,dx \\ & {} - 2 \int ^{t}_{0} h(t-s) \bigl(\Delta w(t), \Delta w(t) - \Delta w(s) \bigr) \,ds + 2 \alpha . \end{aligned}$$

From Lemma 2.4, (1.2), and (1.3), we have

$$\begin{aligned} 2 \bigl( w, \bigl[ w, \chi ( w ) \bigr] \bigr) = 2 \bigl( [ w, w ] , \chi ( w ) \bigr) = - 2 \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} . \end{aligned}$$

Thus, we get

$$\begin{aligned} F''(t) = & 2 \Vert w_{t} \Vert _{2}^{2} - 2 \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} - 2 \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} + 2 \int _{\Omega } \vert w \vert ^{q(x)} \,dx \\ & {} - 2 \int ^{t}_{0} h(t-s) \bigl(\Delta w(t), \Delta w(t) - \Delta w(s) \bigr) \,ds + 2 \alpha . \end{aligned}$$
(3.11)

Using the inequality \(( a \xi _{1} + b \xi _{2} + c \xi _{3} )^{2} \leq ( a^{2} + b^{2} + c^{2} ) ( \xi _{1}^{2} + \xi _{2}^{2} + \xi _{3}^{2} ) \) and (3.8), we get

$$\begin{aligned} \bigl(F'(t) \bigr)^{2} \leq 4 F(t) \biggl( \Vert w_{t} \Vert _{2}^{2} + \int ^{t}_{0} \bigl\Vert \nabla w_{t}(s) \bigr\Vert _{2}^{2} \,ds + \alpha \biggr) . \end{aligned}$$
(3.12)

Using (3.11), (3.12), (3.3), (3.2), and Young’s inequality, one finds

$$\begin{aligned}& F(t) F''(t) - \frac{q_{1} + 2}{4} \bigl(F'(t) \bigr)^{2} \\& \quad \geq F(t) \biggl\{ - q_{1} \Vert w_{t} \Vert _{2}^{2} - 2 \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} - 2 \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} + 2 \int _{\Omega } \vert w \vert ^{q(x)} \,dx \\& \quad \quad {} - 2 \int ^{t}_{0} h(t-s) \bigl(\Delta w(t), \Delta w(t) - \Delta w(s) \bigr) \,ds - ( q_{1} +2 ) \int ^{t}_{0} \bigl\Vert \nabla w_{t}(s) \bigr\Vert _{2}^{2} \,ds - q_{1} \alpha \biggr\} \\& \quad = F(t) \biggl\{ - 2 q_{1} E(t)+ ( q_{1} - 2 ) \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} + q_{1} (h \circ \Delta w) \\& \quad \quad {} - 2 \int ^{t}_{0} h(t-s) \bigl(\Delta w(t), \Delta w(t) - \Delta w(s) \bigr) \,ds + \biggl( \frac{q_{1}}{2} -2 \biggr) \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} \\& \quad \quad {} - ( q_{1} +2 ) \int ^{t}_{0} \bigl\Vert \nabla w_{t}(s) \bigr\Vert _{2}^{2} \,ds - 2 q_{1} \int _{\Omega } \frac{ \vert w \vert ^{q(x)} }{q(x)} \,dx + 2 \int _{ \Omega } \vert w \vert ^{q(x)} \,dx - q_{1} \alpha \biggr\} \\& \quad \geq F(t) \biggl\{ - 2 q_{1} E(0)+ ( q_{1} - 2 ) \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} + q_{1} (h \circ \Delta w) \\& \quad \quad {} - 2 \int ^{t}_{0} h(t-s) \bigl(\Delta w(t), \Delta w(t) - \Delta w(s) \bigr) \,ds + \biggl( \frac{q_{1}}{2} -2 \biggr) \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} \\& \quad \quad {} + ( q_{1} - 2 ) \int ^{t}_{0} \bigl\Vert \nabla w_{t}(s) \bigr\Vert _{2}^{2} \,ds - 2 q_{1} \int _{\Omega } \frac{ \vert u \vert ^{q(x)} }{q(x)} \,dx + 2 \int _{ \Omega } \vert w \vert ^{q(x)} \,dx - q_{1} \alpha \biggr\} . \end{aligned}$$
(3.13)

Using the relations

$$\begin{aligned} - 2 q_{1} \int _{\Omega } \frac{ \vert w \vert ^{q(x)} }{q(x)} \,dx \geq - 2 \int _{\Omega } \vert w \vert ^{q(x)} \,dx \end{aligned}$$

and

$$\begin{aligned} - 2 \int ^{t}_{0} h(t-s) \bigl(\Delta w (t), \Delta w(t) - \Delta w(s) \bigr) \,ds \geq - \epsilon (h \circ \Delta w ) - \frac{1}{\epsilon } \biggl( \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} \end{aligned}$$

for \(\epsilon >0 \), we infer

$$\begin{aligned} F(t) F''(t) - \frac{q_{1} + 2}{4} \bigl(F'(t) \bigr)^{2} \geq F(t)G(t) , \end{aligned}$$
(3.14)

where

$$\begin{aligned} G(t) = & {-} 2 q_{1} E(0) - q_{1} \alpha + ( q_{1} - \epsilon ) (h \circ \Delta w) \\ & {} + \biggl\{ ( q_{1} - 2 ) - \biggl( q_{1} - 2 + \frac{1}{\epsilon } \biggr) \int ^{t}_{0} h(s) \,ds \biggr\} \Vert \Delta w \Vert _{2}^{2} . \end{aligned}$$
(3.15)

Taking \(\epsilon = q_{1}\) in (3.15), and using (3.4) and (3.7), we find

$$\begin{aligned} G(t) \geq - 2 q_{1} E(0) + \biggl\{ ( q_{1} - 2 ) - \biggl( q_{1} - 2 + \frac{1}{ q_{1} } \biggr) \int ^{t}_{0} h(s) \,ds \biggr\} \Vert \Delta w \Vert ^{2} - q_{1} \alpha \geq 0. \end{aligned}$$

From the condition (3.7), it is clear that

$$ F'(0) = 2 (w_{0}, w_{1}) + 2 \alpha \beta >0 . $$

Thus, applying Lemma 3.1, we get the existence of \(T_{*}\) satisfying

$$ T_{*} \leq \frac{ 4 F(0)}{(q_{1} -2) F '(0)} = \frac{ 2 ( \Vert w_{0} \Vert _{2}^{2} + T \Vert \nabla w_{0} \Vert _{2}^{2} + \alpha \beta ^{2} )}{ (q_{1} -2) \{ ( w_{0}, w_{1}) + \alpha \beta \} } $$
(3.16)

and

$$ \lim_{t \to T_{*}^{-} } F(t) = \infty , $$

which gives

$$ \lim_{t \to T_{*}^{-}} \bigl\Vert \Delta w(t) \bigr\Vert _{2} = \infty . $$

Moreover, using (3.16) and the relation \(0 < T < T_{*} \), we see

$$ T_{*} \leq \frac{ 2 ( \Vert w_{0} \Vert _{2}^{2} + T_{*} \Vert \nabla w_{0} \Vert _{2}^{2} + \alpha \beta ^{2} )}{ (q_{1} -2) \{ ( w_{0}, w_{1}) + \alpha \beta \} } . $$

This gives (3.6) under the condition β given in (3.7). □

3.2 Case of certain positive initial energy

We set

$$ {\overline{B}} = \max \biggl\{ 1, \frac{B}{\sqrt{l}} \biggr\} , \quad\quad \eta _{1} = \biggl( \frac{1}{ {\overline{B}} } \biggr)^{\frac{ q_{1} }{ q_{1} -2}}, \quad\quad E_{1} = \frac{( q_{1} -2 ) \eta _{1}^{2}}{2 q_{1} }, $$
(3.17)

where B is the embedding constant given in (2.2), and define a function g by

$$ g(\eta ) = \frac{1}{2} \eta ^{2} - \frac{ {\overline{B}}^{ q_{1} }}{ q_{1} } \eta ^{ q_{1} } . $$
(3.18)

Then one knows

  1. (i)

    \(g(0)=0 \) and \({ \lim_{\eta \to + \infty } g(\eta ) = -\infty }\),

  2. (ii)

    g is increasing on \((0, \eta _{1})\) and decreasing on \((\eta _{1}, \infty ) \),

  3. (iii)

    g has the maximum value \(g(\eta _{1}) = E_{1} \).

Lemma 3.2

Let w be the solution of problem (1.1)–(1.4). Assume that

$$ E(0) < E_{1} \quad \textit{and}\quad \eta _{1} < \sqrt{l} \Vert \Delta w_{0} \Vert _{2} \leq \frac{1}{ {\overline{B}} }. $$
(3.19)

Then there exists a constant \(\eta _{*} > \eta _{1}\) such that

$$ l \Vert \Delta w \Vert _{2}^{2} \geq \eta _{*}^{2} \quad \textit{for } 0 \leq t < T_{*} . $$
(3.20)

Proof

From (3.3), Lemma 2.1, (2.2) and (3.17), we have

$$\begin{aligned} E(t) \geq & \frac{1}{2} \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} - \int _{\Omega } \frac{ \vert w \vert ^{q(x)} }{q(x)} \,dx \\ \geq & \frac{l}{2} \Vert \Delta w \Vert _{2}^{2} - \frac{1}{ q_{1} } \max \bigl\{ \Vert w \Vert ^{ q_{1} }_{q(\cdot )}, \Vert w \Vert ^{ q_{2} }_{q(\cdot )} \bigr\} \\ \geq & \frac{l}{2} \Vert \Delta w \Vert _{2}^{2} - \frac{1}{ q_{1} } \max \bigl\{ B^{ q_{1} } \Vert \Delta w \Vert _{2}^{ q_{1} }, B^{ q_{2} } \Vert \Delta w \Vert _{2}^{ q_{2} } \bigr\} \} \\ \geq & \frac{1}{2} \bigl( \sqrt{l} \Vert \Delta w \Vert _{2} \bigr) ^{2} - \frac{1}{ q_{1} } \max \bigl\{ { \overline{B}}^{ q_{1} } \bigl( \sqrt{l} \Vert \Delta w \Vert _{2} \bigr)^{q_{1} }, {\overline{B}}^{ q_{2} } \bigl( \sqrt{l} \Vert \Delta w \Vert _{2} \bigr)^{ q_{2} } \bigr\} \\ = & f \bigl( \sqrt{l} \Vert \Delta w \Vert _{2} \bigr), \end{aligned}$$
(3.21)

where

$$ f(\eta ) = \frac{1}{2} \eta ^{2} - \frac{1}{ q_{1} } \max \bigl\{ { \overline{B}}^{ q_{1} } \eta ^{ q_{1} }, {\overline{B}}^{ q_{2} } \eta ^{q_{2} } \bigr\} . $$

It is easily seen that

$$ f(\eta ) = g(\eta ) \quad \text{for } 0 \leq \eta \leq \frac{1}{ {\overline{B}} }. $$
(3.22)

Since \(E(0) < E_{1}\), there exists \(\eta _{*} > \eta _{1}\) such that

$$ E(0) = g(\eta _{*}). $$
(3.23)

From (3.23), (3.21), and (3.19), we observe

$$ g(\eta _{*}) = E(0) \geq f \bigl( \sqrt{l} \Vert \Delta w_{0} \Vert _{2} \bigr) = g \bigl( \sqrt{l} \Vert \Delta w_{0} \Vert _{2} \bigr) . $$

Since g is decreasing on \((\eta _{1}, \infty )\), we see that

$$ \eta _{*} \leq \sqrt{l} \Vert \Delta w_{0} \Vert _{2}. $$

From (3.19), we also know

$$ \eta _{1} < \eta _{*} \leq \frac{1}{{\overline{B}}} . $$
(3.24)

We will show (3.20) by contradiction. Suppose that there exists \(t_{0} \in [0, T_{*} ) \) such that

$$ \sqrt{l} \bigl\Vert \Delta w ( t_{0}) \bigr\Vert _{2} < \eta _{*} . $$

Because the solution w is continuous in t, there exists \(t_{1} >0\) such that

$$ \eta _{1} < \sqrt{l} \bigl\Vert \Delta w(t_{1}) \bigr\Vert _{2} < \eta _{*} . $$
(3.25)

Noting that g is decreasing on \((\eta _{1}, \infty )\) and using (3.23), (3.25), (3.22), (3.24), (3.21), (3.1), we get

$$ E(0) = g( \eta _{*} ) < g \bigl( \sqrt{l} \bigl\Vert \Delta w (t_{1}) \bigr\Vert _{2} \bigr) = f \bigl( \sqrt{l} \bigl\Vert \Delta w (t_{1}) \bigr\Vert _{2} \bigr) \leq E(t_{1}) \leq E(0). $$

This is a contradiction. □

Theorem 3.2

Let the conditions of Lemma 3.2are valid. If \(E(0) = \gamma E_{1} \), where \(0< \gamma <1 \), and

$$ \int ^{\infty }_{0} h(s) \,ds \leq \frac{ q_{1} -2}{ q_{1} -2 + \frac{1}{(1- \gamma )^{2} q_{1} + 2 \gamma (1- \gamma ) } }, $$
(3.26)

the solution w to problem (1.1)–(1.4) blows up in a finite time \(T_{*} \). Moreover, \(T_{*}\) satisfies

$$ T_{*} \leq \frac{ 2 \Vert w_{0} \Vert _{2}^{2} + 2 \alpha \beta ^{2} }{ ( q_{1} -2) \{ (w_{0}, w_{1}) + \alpha \beta \} -2 \Vert \nabla w_{0} \Vert _{2}^{2} } , $$

where

$$ \begin{gathered} 0 < \alpha \leq \frac{ \gamma \lambda ( q_{1} -2 ) }{ q_{1} }, \quad \textit{and} \\ \beta > \max \biggl\{ 0 , - \frac{( w_{0}, w_{1} )}{ \alpha } , \frac{ 2 \Vert w_{0} \Vert _{2}^{2} - ( q_{1} -2 ) ( w_{0} , w_{1}) }{ q_{1} -2 } \biggr\} , \end{gathered} $$
(3.27)

here \(0 < \lambda < \eta _{*}^{2} - \eta _{1}^{2} \).

Proof

Let F be the function given in (3.8) with (3.27). Then (3.9), (3.10), (3.11), (3.14), and (3.15) are valid. Taking \(\epsilon = (1- \gamma ) q_{1} + 2 \gamma \) in (3.15), we have

$$\begin{aligned} G(t) \geq & {-} 2 q_{1} E(0) - q_{1} \alpha + \gamma ( q_{1} -2) (h \circ \Delta w ) \\ & {} + \biggl\{ ( q_{1} -2) - \biggl( q_{1} - 2 + \frac{1}{(1- \gamma ) q_{1} + 2 \gamma } \biggr) \int ^{t}_{0} h(s)\,ds \biggr\} \Vert \Delta w \Vert _{2}^{2} . \end{aligned}$$
(3.28)

The condition (3.26) implies

$$\begin{aligned} ( q_{1} -2) - \biggl( q_{1} -2 + \frac{1}{ (1- \gamma ) q_{1} + 2 \gamma } \biggr) \int ^{t}_{0} h(s)\,ds \geq \gamma ( q_{1} -2) \biggl( 1 - \int ^{\infty }_{0} h(s)\,ds \biggr). \end{aligned}$$

Since w is continuous in t, Lemma 3.2 guarantees the existence of \(\lambda >0\) with

$$\begin{aligned} \eta _{1}^{2} + \lambda < \eta _{*}^{2} \leq l \Vert \Delta w \Vert _{2}^{2} \quad \text{for all } t\in [0,T] . \end{aligned}$$

Adapting these too, noting the definition of \(E_{1}\) given in (3.17), and using (3.27), we have

$$\begin{aligned} G(t) \geq & {-} 2 q_{1} E(0) - q_{1} \alpha + \gamma ( q_{1} -2) l \Vert \Delta w \Vert _{2}^{2} \\ \geq & {-} 2 q_{1} \gamma E_{1} - q_{1} \alpha + \gamma ( q_{1} -2 ) \bigl( \eta _{1}^{2} + \lambda \bigr) \\ = & \gamma ( q_{1} - 2 ) \lambda - q_{1} \alpha \geq 0. \end{aligned}$$

By the same argument of Theorem 3.1, we complete the proof. □

3.3 Case of high initial energy

Lemma 3.3

If \(q_{1} \geq 4\) and h satisfies (3.4), it fulfills

$$\begin{aligned} \biggl( (w, w_{t}) - \frac{2 q_{1} + k B_{2}^{2}}{ 2k Q } E(t) \biggr) \geq e^{rt} \biggl( (w_{0} , w_{1}) - \frac{2 q_{1} + k B_{2}^{2}}{ 2k Q } E(0) \biggr) \quad \textit{for } t\in [0,T_{*}), \end{aligned}$$
(3.29)

here

$$ \begin{gathered} Q = ( q_{1} - 2 ) - \biggl( q_{1} - 2 + \frac{1}{q_{1} } \biggr) (1- l ) , \\ k = \min \biggl\{ \frac{ q_{1} +2}{ q_{1} -2}, \frac{1}{B_{1}^{2} B_{2}^{2}} \biggr\} , \quad\quad r = \frac{2 k q_{1} Q }{2 q_{1} + k B_{2}^{2}} .\end{gathered} $$
(3.30)

Proof

Using (1.1)–(1.4) and Young’s inequality, we get

$$\begin{aligned} \frac{d}{dt}(w, w_{t} ) = & \Vert w_{t} \Vert _{2}^{2} - \biggl( 1- \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} - (\nabla w, \nabla w_{t} ) - \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} \\ & {} + \int ^{t}_{0} h(t-s) \bigl( \Delta w(s) - \Delta w(t), \Delta w(t) \bigr) \,ds + \int _{\Omega } \vert w \vert ^{q(x)} \,dx \\ \geq & \Vert w_{t} \Vert _{2}^{2} - \biggl\{ \biggl( 1- \int ^{t}_{0} h(s) \,ds \biggr) + \frac{1}{2 q_{1}} \int ^{t}_{0} h(s) \,ds \biggr\} \Vert \Delta w \Vert _{2}^{2} - \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} \\ & {} - \frac{ \delta }{2} \Vert \nabla w \Vert _{2}^{2} - \frac{ 1 }{2 \delta } \Vert \nabla w_{t} \Vert _{2}^{2} - \frac{q_{1}}{2} ( h \circ \Delta w) + \int _{\Omega } \vert w \vert ^{q(x)} \,dx \end{aligned}$$
(3.31)

for \(\delta >0 \). From (3.3), we observe

$$\begin{aligned} \int _{\Omega } \vert w \vert ^{q(x)} \,dx \geq& - q_{1} E(t) + \frac{ q_{1} }{2} \Vert w_{t} \Vert _{2}^{2} + \frac{ q_{1} }{2} \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} \\ &{}+ \frac{ q_{1} }{2} (h \circ \Delta w) + \frac{ q_{1} }{4} \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} . \end{aligned}$$

Applying this to (3.31) and using (3.1), we have

$$\begin{aligned} \frac{d}{dt}(w, w_{t} ) \geq & \frac{ q_{1} +2}{2} \Vert w_{t} \Vert _{2}^{2} + \frac{1}{2 \delta } E'(t) - q_{1} E(t) \\ & {} + \biggl[ \biggl\{ ( q_{1} - 2 ) - \biggl( q_{1} - 2 + \frac{1}{q_{1}} \biggr) \int ^{t}_{0} h(s) \,ds \biggr\} B_{2}^{-2} - \delta \biggr] \frac{ \Vert \nabla w \Vert _{2}^{2} }{2} \\ \geq & \frac{ q_{1} +2}{2} \Vert w_{t} \Vert _{2}^{2} + \frac{1}{2 \delta } E'(t) - q_{1} E(t) \\ & {} + \frac{1}{2 B_{1}^{2} B_{2}^{2} } \biggl[ \biggl\{ ( q_{1} - 2 ) - \biggl( q_{1} - 2 + \frac{1}{q_{1}} \biggr) \int ^{t}_{0} h(s) \,ds \biggr\} - B_{2}^{2} \delta \biggr] \Vert w \Vert _{2}^{2} . \end{aligned}$$

Recalling (3.30), we have

$$\begin{aligned}& \frac{d}{dt} \biggl( (w, w_{t} ) - \frac{1}{2 \delta } E(t) \biggr) \\& \quad \geq \frac{ q_{1} +2}{2} \Vert w_{t} \Vert _{2}^{2} + \frac{ Q - B_{2}^{2} \delta }{2 B_{1}^{2} B_{2}^{2}} \Vert w \Vert _{2}^{2} - q_{1} E(t) \\& \quad \geq \frac{ k ( q_{1} - 2 ) }{2} \Vert w_{t} \Vert _{2}^{2} + \frac{ k ( Q - B_{2}^{2} \delta ) }{2} \Vert w \Vert _{2}^{2} - q_{1} E(t) \\& \quad \geq \frac{ k ( Q - B_{2}^{2} \delta )}{2} \bigl( \Vert w_{t} \Vert _{2}^{2} + \Vert w \Vert _{2}^{2} \bigr) - q_{1} E(t) \\& \quad \geq k \bigl( Q - B_{2}^{2} \delta \bigr) \biggl( ( w, w_{t} ) - \frac{ q_{1} }{ k ( Q - B_{2}^{2} \delta ) } E(t) \biggr) \quad \text{for } 0 < \delta < \frac{ Q }{B_{2}^{2}} . \end{aligned}$$

Taking

$$ \delta = \frac{ k Q }{ 2 q_{1} + k B_{2}^{2} } , $$

we find

$$\begin{aligned} \frac{d}{dt} \biggl( (w, w_{t} ) - \frac{2 q_{1} + k B_{2}^{2}}{ 2k Q } E(t) \biggr) \geq \frac{2 k q_{1} Q }{2 q_{1} + k B_{2}^{2}} \biggl( ( w, w_{t} ) - \frac{2 q_{1} + k B_{2}^{2}}{ 2k Q } E(t) \biggr) . \end{aligned}$$

This completes the proof. □

Theorem 3.3

Let \(q_{1} \geq 4 \) and h satisfy (3.4). If \(0 < E(0) < \frac{2k Q }{2 q_{1} + k B_{2}^{2}} (w_{0}, w_{1}) \), then the solution w blows up in a finite time \(T_{*}\). Moreover, if \(E(0) < \frac{ Q \Vert u_{0} \Vert _{2}^{2} }{ 2 q_{1} B_{1}^{2} B_{2}^{2}} \), then \(T_{*}\) satisfies

$$ T_{*} \leq \frac{ 2 \Vert w_{0} \Vert _{2}^{2} + 2 \alpha \beta ^{2} }{ ( q_{1} -2) \{ (w_{0}, w_{1}) + \alpha \beta \} -2 \Vert \nabla w_{0} \Vert _{2}^{2} }, $$

where

$$\begin{aligned} \begin{gathered} 0 < \alpha \leq \frac{ - 2 q_{1} B_{1}^{2} B_{2}^{2} E(0) + Q \Vert w_{0} \Vert _{2}^{2} }{ q_{1} B_{1}^{2} B_{2}^{2} } \quad \textit{and} \\ \beta > \max \biggl\{ 0, \frac{ - ( q_{1} -2)(w_{0}, w_{1}) + 2 \Vert \nabla w_{0} \Vert _{2}^{2} }{ ( q_{1} -2) \alpha } \biggr\} . \end{gathered} \end{aligned}$$
(3.32)

Proof

Suppose that w is global. Then, using (3.2), we get

$$\begin{aligned} \Vert w \Vert _{2} \leq & \Vert w_{0} \Vert _{2} + \int ^{t}_{0} \bigl\Vert w_{t} (s) \bigr\Vert _{2} \,ds \leq \Vert w_{0} \Vert _{2} + B_{1} t^{\frac{1}{2}} \biggl( \int ^{t}_{0} \bigl\Vert \nabla w_{t} (s) \bigr\Vert _{2}^{2} \,ds \biggr)^{\frac{1}{2}} \\ \leq & \Vert w_{0} \Vert _{2} + B_{2} \bigl( t \bigl( E(0) -E(t) \bigr) \bigr)^{\frac{1}{2}}, \quad t \geq 0 . \end{aligned}$$
(3.33)

In the case \(E(t) \geq 0\) for all \(t\geq 0 \), from (3.33), we see

$$\begin{aligned} \Vert w \Vert _{2}^{2} \leq 2 \Vert w_{0} \Vert _{2}^{2} + 2 B_{2}^{2} E(0)t, \quad t\geq 0. \end{aligned}$$
(3.34)

Applying Lemma 3.3, we also have

$$\begin{aligned} \Vert w \Vert _{2}^{2} = & \Vert w_{0} \Vert _{2}^{2} + 2 \int ^{t}_{0} \bigl(w(s), w_{t} (s) \bigr) \,ds \\ \geq & \Vert w_{0} \Vert _{2}^{2} + 2 \biggl\{ \int ^{t}_{0} e^{rs} \biggl( (w_{0} , w_{1}) - \frac{2 q_{1} + k B_{2}^{2}}{ 2k Q } E(0) \biggr) \,ds + \int ^{t}_{0} \frac{2 q_{1} + k B_{2}^{2}}{ 2k Q } E(s) \,ds \biggr\} \\ \geq & \Vert w_{0} \Vert _{2}^{2} + 2 \biggl( (w_{0} , w_{1}) - \frac{2 q_{1} + k B_{2}^{2}}{ 2k Q } E(0) \biggr) \frac{ e^{rt} -1 }{r} \quad \text{for all } t\geq 0. \end{aligned}$$
(3.35)

But this contradicts (3.34) for t appropriately large. In the case \(E( t_{1} ) < 0\) for some \(t_{1} > 0 \), there exists the first \(t_{2} >0\) with \(0 < t_{2} < t_{1} \) satisfying \(E(t_{2}) =0 \), \(E(t) >0\) for \(0 \leq t < t_{2}\), and \(E(t_{0}) < 0\) for some \(t_{0} > t_{2}\). Taking \(w(t_{0})\) as a new initial datum, by Theorem 3.1, the solution w blows up after the time \(t_{0}\). This also is a contradiction. Consequently, \(T_{*} < \infty \).

Let F be the function given in (3.8) with (3.32). Then (3.9), (3.10), (3.11), (3.14), and (3.15) are also valid. Taking \(\epsilon = q_{1}\) in (3.15) and using (3.35), we have

$$\begin{aligned} F(t) F''(t) - \frac{q_{1} + 2}{4} \bigl(F'(t) \bigr)^{2} \geq & F(t) \bigl( - 2 q_{1} E(0) - q_{1} \alpha + Q \Vert \Delta w \Vert _{2}^{2} \bigr) \\ \geq & F(t) \biggl( - 2 q_{1} E(0) - q_{1} \alpha + \frac{ Q }{ B_{1}^{2} B_{2}^{2}} \Vert w_{0} \Vert _{2}^{2} \biggr). \end{aligned}$$

From (3.32), we observe

$$ F(t) F''(t) - \frac{ q_{1} +2}{4} \bigl( F'(t) \bigr)^{2} \geq 0 . $$

By the same argument of Theorem 3.1, we complete the proof. □

Theorem 3.4

Let the conditions of one of Theorem 3.1–Theorem 3.3are satisfied. Then the blow-up time \(T_{*}\) verifies

$$ \int ^{\infty }_{D(0)} \frac{1}{2 y + d_{1} y^{3} + d_{2} y^{ q_{1} -1 } + d_{3} y^{ q_{2} -1 } } \,dy \leq T_{*}, $$
(3.36)

where \(D(0) = \Vert w_{1} \Vert _{2}^{2} + \Vert \Delta w_{0} \Vert _{2}^{2} \) and \(d_{i} >0\) (\(i=1,2,3\)) are certain constants.

Proof

We let

$$ D(t) = \Vert w_{t} \Vert _{2}^{2} + \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} + ( h \circ \Delta w ). $$
(3.37)

From (3.5), it is observed

$$ \lim_{t \to T_{*}^{-}} D(t) \geq \lim_{t \to T_{*}^{-}} \bigl( \Vert w_{t} \Vert _{2}^{2} + l \Vert \Delta w \Vert _{2}^{2} \bigr) = \infty . $$
(3.38)

Using (1.1)–(1.4), one gets

$$\begin{aligned} D'(t) = & {-} 2 \Vert \nabla w_{t} \Vert _{2}^{2} - h(t) \Vert \Delta w \Vert _{2}^{2} + \bigl(h' \circ \Delta w \bigr) + 2 \bigl( w_{t}, \bigl[w, \chi ( w ) \bigr] \bigr) + 2 \int _{ \Omega } w_{t} \vert w \vert ^{ q(x) -2} w \,dx \\ \leq & 2 \Vert w_{t} \Vert _{2}^{2} + \bigl\Vert \bigl[w, \chi ( w ) \bigr] \bigr\Vert _{2}^{2} + \int _{ \Omega } \vert w \vert ^{2(q(x) -1 )} \,dx. \end{aligned}$$
(3.39)

From Lemma 2.2 and Lemma 2.3, we see

$$\begin{aligned} \bigl\Vert \bigl[w, \chi ( w ) \bigr] \bigr\Vert _{2}^{2} \leq & \bigl( a_{2} \Vert w \Vert _{H^{2}(\Omega )} \bigl\Vert \chi ( w ) \bigr\Vert _{W^{2, \infty }(\Omega )} \bigr)^{2} \\ \leq & \bigl( a_{1} a_{2} \Vert w \Vert _{H^{2}(\Omega )}^{3} \bigr)^{2} \\ \leq & b_{1} \Vert \Delta w \Vert _{2}^{6} \end{aligned}$$
(3.40)

for some \(b_{1} >0 \). The last term of (3.39) is estimated as

$$\begin{aligned} \int _{\Omega } \vert w \vert ^{2(q(x) -1 )} \,dx \leq & \int _{ \vert w \vert < 1} \vert w \vert ^{2( q_{1} - 1 )} \,dx + \int _{ \vert w \vert \geq 1 } \vert w \vert ^{2( q_{2} - 1 )} \,dx \\ \leq & \Vert w \Vert _{2 ( q_{1} -1 )}^{2 ( q_{1} -1 )} + \Vert w \Vert _{2 ( q_{2} -1 )}^{2 ( q_{2} -1 )} \\ \leq & b_{2} \Vert \Delta w \Vert _{2}^{2 ( q_{1} -1 )} + b_{3} \Vert \Delta w \Vert _{2}^{2 ( q_{2} -1 )} \end{aligned}$$
(3.41)

for some \(b_{2}, b_{3} >0 \). From (3.39), (3.40), (3.41), and (3.37), we arrive at

$$\begin{aligned} D'(t) \leq & 2 \Vert w_{t} \Vert _{2}^{2} + b_{1} \Vert \Delta w \Vert _{2}^{6} + b_{2} \Vert \Delta w \Vert _{2}^{2 ( q_{1} -1 )} + b_{3} \Vert \Delta w \Vert _{2}^{2 ( q_{2} -1 )} \\ \leq & 2 D(t) + d_{1} \bigl(D(t) \bigr)^{3} + d_{2} \bigl(D(t) \bigr)^{ q_{1} -1 } + d_{3} \bigl(D(t) \bigr)^{ q_{2} -1 } \end{aligned}$$

for some \(d_{1}, d_{2}, d_{3} >0\). Using the integration of substitution and (3.38), we get (3.36). □

4 Conclusion

In this paper, the author considered a viscoelastic von Karman equation with strong damping and variable exponent source terms. We showed that the solutions with three levels of initial energy such as non-positive initial energy, certain positive initial energy, and high initial energy blow up in a finite time. Moreover, we estimated not only the upper bound but also the lower bound of the blow-up time.