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Blow-up results for a quasilinear von Karman equation of memory type

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Abstract

In this paper, we consider the blow-up result of solution for a quasilinear von Karman equation of memory type with nonpositive initial energy as well as positive initial energy. For nonincreasing function \(g>0\) and nondecreasing function f, we prove a finite time blow-up result under suitable condition on the initial data.

Introduction

Let \(\rho >0, \alpha >0\), and \(p>2\). Moreover, let us denote by Ω an open bounded set of \(\mathbb{R}^{2} \) with sufficiently smooth boundary Γ. We assume that \(\varGamma _{0} \cup \varGamma _{1}=\varGamma \), \(\varGamma _{0} \cap \varGamma _{1} = \emptyset \), \(\varGamma _{0} \neq \emptyset \), and \(\varGamma _{0} \) and \(\varGamma _{1} \) have positive measure. In this paper we investigate a blow-up result for the following quasilinear von Karman equation of memory type:

$$\begin{aligned} & \vert y_{t} \vert ^{\rho }y_{tt} -\alpha \Delta y_{tt} +\Delta ^{2} y - \int _{0}^{t} g(t-s) \Delta ^{2} y(s) \,ds =[y,z]\quad \text{in } \varOmega \times (0, \infty ), \end{aligned}$$
(1.1)
$$\begin{aligned} & \Delta ^{2} z=-[y,y] \quad\text{in } \varOmega \times (0, \infty ), \end{aligned}$$
(1.2)
$$\begin{aligned} & z=\frac{\partial z}{\partial \nu } =0 \quad\text{on } \varGamma \times (0, \infty ), \end{aligned}$$
(1.3)
$$\begin{aligned} & y=\frac{\partial y}{\partial \nu } =0 \quad\text{on } \varGamma _{0} \times (0, \infty ), \end{aligned}$$
(1.4)
$$\begin{aligned} & \mathcal{B}_{1} y -\mathcal{B}_{1} \biggl( \int _{0}^{t} g(t-s) y(s) \,ds \biggr) =0 \quad\text{on } \varGamma _{1} \times (0, \infty ), \end{aligned}$$
(1.5)
$$\begin{aligned} & \alpha \frac{\partial y_{tt}}{\partial \nu }-\mathcal{B}_{2} y+ \mathcal{B}_{2} \biggl( \int _{0}^{t} g(t-s) y(s) \,ds \biggr) + f(y_{t}) = \vert y \vert ^{p-2} y \quad\text{on } \varGamma _{1} \times (0, \infty ), \end{aligned}$$
(1.6)
$$\begin{aligned} & y(x,0) =y_{0} (x), \qquad y_{t} (x,0) =y_{1} (x) \quad\text{in } \varOmega, \end{aligned}$$
(1.7)

where \(\nu =(\nu _{1}, \nu _{2} )\) is the outward unit normal vector on Γ. The relaxation function g is a positive nonincreasing function and f is a nondecreasing function. Here

$$ \mathcal{B}_{1} \varpi =\Delta \varpi +(1-\mu )B_{1} \varpi,\qquad \mathcal{B}_{2} \varpi =\frac{\partial \Delta \varpi }{\partial \nu } +(1-\mu ) \frac{\partial B_{2} \varpi }{\partial \tau }, $$

where

$$\begin{aligned} & B_{1} \varpi = 2 \nu _{1} \nu _{2} \frac{\partial ^{2} \varpi }{ \partial x_{1} \partial x_{2}} -\nu _{1}^{2} \frac{\partial ^{2} \varpi }{\partial x_{2}^{2}} -\nu _{2}^{2}\frac{\partial ^{2} \varpi }{\partial x_{1}^{2} }, \\ & B_{2} \varpi =\bigl(\nu _{1}^{2} -\nu _{2}^{2} \bigr)\frac{\partial ^{2} \varpi }{\partial x_{1} \partial x_{2}} +\nu _{1} \nu _{2} \biggl(\frac{ \partial ^{2} \varpi }{\partial x_{2}^{2}}- \frac{\partial ^{2} \varpi }{\partial x_{1}^{2}} \biggr) \end{aligned}$$

and the constant \(\mu \in (0, \frac{1}{2} )\) represents Poisson’s ratio. The von Karman bracket \([\varpi, \phi ] \) is given by

$$ [\varpi,\phi ] = \frac{\partial ^{2} \varpi }{\partial x_{1}^{2}} \frac{ \partial ^{2} \phi }{\partial x_{2}^{2}} + \frac{\partial ^{2} \varpi }{ \partial x_{2}^{2}} \frac{\partial ^{2} \phi }{\partial x_{1}^{2}} -2 \frac{ \partial ^{2} \varpi }{\partial x_{1} \partial x_{2}} \frac{\partial ^{2} \phi }{\partial x_{1} \partial x_{2}}. $$

The authors in [1,2,3,4,5] studied the asymptotic behavior of the solutions to a von Karman system with dissipative effects. The uniform decay rate for the von Karman system with frictional dissipative effect in the boundary has been proved by several authors [6,7,8]. For a von Karman equation with rotational inertia and memory of the form

$$\begin{aligned} & y_{tt} -\alpha \Delta y_{tt} +\Delta ^{2} y - \int _{0}^{t} g(t-s) \Delta ^{2} y(s) \,ds =[y,z]\quad \text{in } \varOmega \times (0, \infty ), \\ & \Delta ^{2} z=-[y,y]\quad \text{in } \varOmega \times (0, \infty ), \end{aligned}$$

many authors [9,10,11,12] showed the existence and stability of solutions. Several authors [13,14,15] investigated the general stability for a von Karman system with memory boundary conditions. The stability for a von Karman system with acoustic boundary conditions was treated by [16, 17]. Some authors discussed the energy decay for a von Karman equation with time-varying delay (see [18, 19] and the reference therein).

On the other hand, many authors have considered the global existence, uniform decay rates, and blow-up of solutions for the wave equation with nonlinear damping and source terms:

$$\begin{aligned} y_{tt} -\Delta y + a \vert y_{t} \vert ^{m-2} y_{t} = b \vert y \vert ^{p-2}y \quad\text{in } \varOmega \times (0, \infty ), \end{aligned}$$

where \(a, b>0\) and \(p, m>2\). When \(a=0\), Ball [20] showed that the source term \(|u|^{p-2}u\) causes blow-up of solutions with negative initial energy in finite time. For \(m=2\), Levine [21, 22] proved that solutions with negative initial energy blow up in finite time. Georgiev and Todorova [23] extended Levin’s result to the nonlinear damping case. Messaoudi [24] improved the blow-up result of [23] to the solutions with negative initial energy. Messaoudi [25] studied the blow-up property of solutions with negative initial energy for the following viscoelastic wave equation with \(p>m\):

$$\begin{aligned} y_{tt}- \Delta y + \int ^{t}_{0} g(t-s) \Delta y(s) \,ds + \vert y_{t} \vert ^{m-2} y_{t} = \vert y \vert ^{p-2}y \quad\text{in } \varOmega \times (0,\infty ). \end{aligned}$$
(1.8)

Messaoudi [26] extended the blow-up result of [25] to the solution with positive initial energy. Song [27] proved the finite time blow-up of some solutions whose initial data have arbitrarily positive initial energy for (1.8). Recently, Park et al. [28] showed the blow-up of the solutions for a viscoelastic wave equation with weak damping. Liu and Yu [29] investigated the blow-up of the solutions for the following viscoelastic wave equation with boundary damping and source terms:

$$\begin{aligned} & y_{tt}- \Delta y + \int ^{t}_{0} g(t-s) \Delta y(s) \,ds =0 \quad\text{in } \varOmega \times (0,\infty ), \\ &y=0\quad \text{in } \varGamma _{0} \times [0,\infty ), \\ &\frac{\partial y}{\partial \nu }- \int _{0}^{t} g(t-s) \frac{\partial y}{\partial \nu }(s) \,ds+ \vert y_{t} \vert ^{m-2} y_{t} = \vert y \vert ^{p-2}y \quad\text{in } \varGamma _{1} \times [0,\infty ). \end{aligned}$$

For more related works, we refer to [30,31,32,33,34,35,36,37,38] and the references therein.

To our best knowledge, there are no blow-up results of solution for the von Karman equation with memory. Motivated by the previous results, we consider the quasilinear von Karman equation with memory and boundary weak damping. We study a finite time blow-up result under suitable condition on the initial data.

The outline of the paper is the following. In Sect. 2, we give some notations and hypotheses for our work. In Sect. 3, we prove our main result.

Preliminary

In this section, we present some material needed in the proof of our result. Throughout this paper we denote

$$\begin{aligned} & V=\bigl\{ y\in H^{1} (\varOmega ): y=0 \text{ on } \varGamma _{0}\bigr\} , \\ & W=\biggl\{ y\in H^{2} (\varOmega ): y=\frac{\partial y}{\partial \nu }=0 \text{ on } \varGamma _{0} \biggr\} , \\ & (y,z) = \int _{\varOmega }y(x)z(x) \,dx,\qquad (y,z)_{\varGamma _{1}} = \int _{\varGamma _{1}} y(x) z(x) \,d\varGamma. \end{aligned}$$

For a Banach space X, \(\Vert \cdot \Vert _{X} \) denotes the norm of X. For simplicity, we denote \(\Vert \cdot \Vert _{L^{2} (\varOmega )} \) by the norm \(\Vert \cdot \Vert \) and \(\Vert \cdot \Vert _{L^{2} (\varGamma _{1})}\) by \(\Vert \cdot \Vert _{ \varGamma _{1}}\), respectively. We define, for all \(1\leq p<\infty \),

$$ \Vert y \Vert _{p,\varGamma _{1}}^{p} = \int _{\varGamma _{1}} \bigl\vert y(x) \bigr\vert ^{p} \,d\varGamma. $$

Let \(0< \mu <\frac{1}{2}\), we define the bilinear form \(a(\cdot, \cdot )\) as follows:

$$\begin{aligned} a(y,\kappa )={}& \int _{\varOmega } \biggl\{ \frac{\partial ^{2} y}{\partial x _{1}^{2} } \frac{\partial ^{2} \kappa }{\partial x_{1}^{2} } + \frac{ \partial ^{2} y}{\partial x_{2}^{2} } \frac{\partial ^{2} \kappa }{ \partial x_{2}^{2} } +\mu \biggl( \frac{\partial ^{2} y}{\partial x_{1} ^{2} }\frac{\partial ^{2} \kappa }{\partial x_{2}^{2} } + \frac{ \partial ^{2} y}{\partial x_{2}^{2} } \frac{\partial ^{2} \kappa }{ \partial x_{1}^{2} } \biggr) \\ &{}+2 (1-\mu ) \frac{\partial ^{2} y}{\partial x_{1} \partial x_{2} } \frac{\partial ^{2} \kappa }{\partial x_{1} \partial x_{2} } \biggr\} \,dx. \end{aligned}$$
(2.1)

A simple calculation, based on the integration by parts formula, yields

$$ \int _{\varOmega }\bigl( \Delta ^{2} y\bigr) \kappa \,dx = a(y,\kappa ) - \biggl( \mathcal{B}_{1} y, \frac{\partial \kappa }{\partial \nu } \biggr)_{ \varGamma } +(\mathcal{B}_{2} y, \kappa )_{\varGamma }. $$

Thus, for \((y, \kappa ) \in (H^{4} (\varOmega )\cap W)\times W\), it holds

$$ \int _{\varOmega }\bigl(\Delta ^{2} y \bigr)\kappa \,dx = a(y,\kappa ) - \biggl( \mathcal{B}_{1} y, \frac{\partial \kappa }{\partial \nu } \biggr)_{ \varGamma _{1}} +(\mathcal{B}_{2} y, \kappa )_{\varGamma _{1}}. $$
(2.2)

Since \(\varGamma _{0} \neq \emptyset \), we have (see [39]) that \(\sqrt{a(y,y)} \) is equivalent to the \(H^{2} (\varOmega )\) norm on W, that is,

$$ C_{1} \Vert \Delta y \Vert ^{2} \leq a(y,y) \leq C_{2} \Vert \Delta y \Vert ^{2} \quad\text{for some } C_{1}, C_{2} >0. $$
(2.3)

Now we state the assumptions for problem (1.1)–(1.7). We will need the following assumptions.

(H1) Hypotheses on g.

Let \(g: \mathbb{R}^{+} \to \mathbb{R}^{+} \) be a nonincreasing \(C^{1} \) function satisfying

$$ g(0)>0,\quad 1- \int _{0}^{\infty }g(s) \,ds:= l >0. $$
(2.4)

(H2) Hypotheses on f.

Let \(f : {\mathbb{R}} \to {\mathbb{R}} \) be a nondecreasing \(C^{1}\) function with \(f(0)=0 \). There exists an odd and strictly increasing function \(\xi : [-1, 1] \to {\mathbb{R}}\) such that

$$\begin{aligned} &\bigl\vert \xi (s) \bigr\vert \leq \bigl\vert f(s) \bigr\vert \leq \bigl\vert \xi ^{-1} (s) \bigr\vert \quad\text{for } \vert s \vert \leq 1, \end{aligned}$$
(2.5)
$$\begin{aligned} &c_{1} \vert s \vert ^{m-1} \leq \bigl\vert f(s) \bigr\vert \leq c_{2} \vert s \vert ^{m-1} \quad\text{for } \vert s \vert >1, \end{aligned}$$
(2.6)

where \(c_{1}\) and \(c_{2} \) are positive constants, \(m>2\), and \(\xi ^{-1}\) denotes the inverse function of ξ.

We state the well-posedness which can be established by the arguments of [11,12,13, 29, 40].

Theorem 2.1

Suppose that (H1)–(H2) hold and \((y_{0}, y_{1}) \in (H^{4} (\varOmega )\cap W )\times (H^{3} (\varOmega ) \cap V))\). Then, for any \(T>0\), there exists a unique solution of problem (1.1)(1.7) such that

$$\begin{aligned} y\in C \bigl([0,T]; H^{4} (\varOmega )\cap W\bigr) \cap C^{1} \bigl([0,T]; H^{3} (\varOmega )\cap V\bigr)\cap C^{2} \bigl([0,T]; L^{2} (\varOmega )\bigr). \end{aligned}$$

A direct calculation gives

$$\begin{aligned} a\bigl((g\ast y) (t), y_{t} (t) \bigr) ={}&{-}\frac{1}{2} \frac{d}{dt} \biggl[ \bigl(g \square \partial ^{2} y\bigr) (t) - \biggl( \int _{0}^{t} g(s) \,ds \biggr) a\bigl(y(t), y(t)\bigr) \biggr] \\ &{} -\frac{1}{2} g(t) a\bigl(y(t),y(t)\bigr) +\frac{1}{2} \bigl(g'\square \partial ^{2} y\bigr) (t), \end{aligned}$$
(2.7)

where

$$\begin{aligned} (g\ast y) (t) = \int _{0}^{t} g(t-s) y(s) \,ds, \bigl(g \square \partial ^{2} y\bigr) (t)= \int _{0}^{t} g(t-s) a\bigl(y(t)-y(s), y(t)-y(s) \bigr) \,ds. \end{aligned}$$

We recall the trace Sobolev embedding

$$ W \hookrightarrow L^{p} (\varGamma _{1}) \quad\text{for } p \geq 2 $$

and the embedding inequality

$$\begin{aligned} \Vert y \Vert _{p,\varGamma _{1}} \leq B \Vert \Delta y \Vert \quad\text{for } y \in W, \end{aligned}$$
(2.8)

where \(B >0\) is the optimal constant. We define the energy associated with problem (1.1)–(1.7) by

$$\begin{aligned} E(t) :={}&E\bigl(y(t),z(t)\bigr) \\ ={}& \frac{1}{\rho +2} \bigl\Vert y_{t}(t) \bigr\Vert _{\rho +2}^{ \rho +2} + \frac{\alpha }{2} \bigl\Vert \nabla y_{t} (t) \bigr\Vert ^{2} + \frac{1}{2} \biggl( 1 - \int ^{t}_{0} g(s) \,ds \biggr) a\bigl(y(t),y(t) \bigr) \\ & {} + \frac{1}{2} \bigl(g \square \partial ^{2} y\bigr) (t)- \frac{1}{p} \bigl\Vert y(t) \bigr\Vert ^{p} _{p, \varGamma _{1}}+\frac{1}{4} \bigl\Vert \Delta z(t) \bigr\Vert ^{2}, \end{aligned}$$
(2.9)

then

$$ E ' (t) = \frac{1}{2} \bigl(g' \square \partial ^{2} y\bigr) (t) - \frac{g(t)}{2} a\bigl(y(t),y(t)\bigr) - \bigl(f\bigl( y_{t} (t) \bigr), y_{t} (t)\bigr)_{\varGamma _{1}} \leq 0. $$
(2.10)

So the energy E is a nonincreasing function. Next, we define the functionals

$$\begin{aligned} I(t) :={}&I\bigl(y(t),z(t)\bigr) \\ ={}& \biggl( 1 - \int ^{t}_{0} g(s) \,ds \biggr) a\bigl(y(t),y(t) \bigr) + \bigl(g \square \partial ^{2} y\bigr) (t)+ \frac{1}{2} \bigl\Vert \Delta z (t) \bigr\Vert ^{2} - \bigl\Vert y(t) \bigr\Vert ^{p} _{p, \varGamma _{1}}, \end{aligned}$$
(2.11)
$$\begin{aligned} H(t):={}&H\bigl(y(t),z(t)\bigr) \\ ={}& \frac{1}{2} \biggl[ \biggl( 1 - \int ^{t}_{0} g(s) \,ds \biggr) a\bigl(y(t),y(t) \bigr) + \bigl(g \square \partial ^{2} y\bigr) (t) + \frac{1}{2} \bigl\Vert \Delta z (t) \bigr\Vert ^{2} \biggr] \\ &{}- \frac{1}{p} \bigl\Vert y(t) \bigr\Vert ^{p}_{p, \varGamma _{1}}. \end{aligned}$$
(2.12)

We define

$$ e(t)={\inf_{(y,z)\in W\times H^{2}_{0}(\varOmega ), y\neq 0, } \sup_{\lambda \geq 0}H(\lambda y, \lambda z)}, \quad t\geq 0. $$

Lemma 2.1

For \(t\geq 0\), we get

$$ 0< e_{0} \leq e(t) \leq \sup_{\lambda \geq 0}H(\lambda y, \lambda z), $$

where \(e_{0} = \frac{p-2}{2p} ( \frac{C_{1} l}{B^{2}} )^{\frac{p}{p-2}}\) and

$$ \sup_{\lambda \geq 0}H(\lambda y,\lambda z) = \frac{p-2}{2p} \biggl( \frac{( 1 - \int ^{t}_{0} g(s) \,ds ) a(y(t),y(t)) + (g \square \partial ^{2} y)(t) +\frac{1}{2} \Vert \Delta z (t) \Vert ^{2}}{ \Vert y(t) \Vert _{p,\varGamma _{1}}^{2} } \biggr)^{\frac{p}{p-2}}. $$

Proof

We find that

$$ H(\lambda y, \lambda z) =\frac{\lambda ^{2}}{2} \biggl[ \biggl( 1- \int _{0}^{t} g(s) \,ds \biggr) a\bigl(y(t),y(t) \bigr) + \bigl(g \square \partial ^{2} y\bigr) (t) + \frac{1}{2} \bigl\Vert \Delta z (t) \bigr\Vert ^{2} \biggr]-\frac{\lambda ^{p}}{p} \bigl\Vert y (t) \bigr\Vert _{p,\varGamma _{1}}^{p}. $$

If \(\frac{d H(\lambda y,\lambda z)}{d\lambda }=0\), then we obtain

$$ \lambda _{1} = \biggl[ \frac{( 1 - \int ^{t}_{0} g(s) \,ds ) a(y(t),y(t) ) + (g \square \partial ^{2} y)(t) +\frac{1}{2} \Vert \Delta z(t) \Vert ^{2} }{ \Vert y(t) \Vert _{p, \varGamma _{1}}^{p}} \biggr]^{\frac{1}{p-2}}. $$

It is easy to verify that \(\frac{d^{2} H}{d\lambda ^{2}} |_{\lambda =\lambda _{1}} <0 \), then from (2.3), (2.4), and (2.8)

$$\begin{aligned} \sup_{\lambda \geq 0} H(\lambda y,\lambda z) &=H(\lambda _{1} y, \lambda _{1} z) \\ & = \biggl(\frac{p-2}{2p} \biggr) \biggl( \frac{( 1 - \int ^{t}_{0} g(s) \,ds ) a(y(t),y(t)) + (g \square \partial ^{2} y)(t) +\frac{1}{2} \Vert \Delta z(t) \Vert ^{2}}{ \Vert y(t) \Vert _{p,\varGamma _{1}}^{2} } \biggr)^{ \frac{p}{p-2}} \\ & \geq \biggl( \frac{p-2}{2p} \biggr) \biggl( \frac{ C_{1} l \Vert \Delta y(t) \Vert ^{2}}{ \Vert y(t) \Vert _{p, \varGamma _{1}}^{2} } \biggr)^{\frac{p}{p-2}} \geq \biggl( \frac{p-2}{2p} \biggr) \biggl( \frac{C_{1} l}{B^{2}} \biggr)^{\frac{p}{p-2}}. \end{aligned}$$

By the definition of \(e_{0} \), we conclude that \(e_{0}>0\). □

Lemma 2.2

Assume that (H1)–(H2) hold. Suppose that \((y_{0}, y_{1}) \in W\times L^{2}(\varOmega )\) and satisfy

$$\begin{aligned} I(0)< 0, \qquad E(0) < \epsilon e_{0} \quad\textit{for any } \epsilon < 1. \end{aligned}$$
(2.13)

Then, for some \(T>0\), we get \(I(t)<0 \) and

$$\begin{aligned} e_{0} &< \frac{p-2}{2p} \biggl[ \biggl( 1 - \int ^{t}_{0} g(s) \,ds \biggr) a\bigl(y(t),y(t) \bigr) + \bigl(g \square \partial ^{2} y\bigr) (t) + \frac{1}{2} \bigl\Vert \Delta z (t) \bigr\Vert ^{2} \biggr] \\ &< \frac{p-2}{2p} \bigl\Vert y(t) \bigr\Vert ^{p}_{p, \varGamma _{1}} \end{aligned}$$
(2.14)

for all \(t\in [0, T)\).

Proof

Using (2.10) and (2.13), we obtain \(E(t)< \epsilon e_{0}\) for all \(t\in [0,T)\). We can also have \(I(t)<0\) for all \(t\in [0,T)\). It can be showed by contradiction. Suppose that there exists some \(t_{0}>0\) such that \(I(t_{0}) =0\) and \(I(t)<0\) for \(0\leq t < t_{0}\). Then

$$ \biggl( 1- \int _{0}^{t} g(s) \,ds \biggr) a\bigl(y(t),y(t) \bigr) + \bigl(g \square \partial ^{2} y\bigr) (t)+ \frac{1}{2} \bigl\Vert \Delta z(t) \bigr\Vert ^{2} < \bigl\Vert y(t) \bigr\Vert _{p,\varGamma _{1}} ^{p},\quad 0\leq t< t_{0}. $$
(2.15)

Using Lemma 2.1 and (2.15), we see that

$$\begin{aligned} e_{0}< {}& \frac{p-2}{2p} \biggl\{ \frac{ ( 1-\int _{0}^{t} g(s) \,ds ) a(y(t),y(t) ) + (g \square \partial ^{2} y)(t)+\frac{1}{2} \Vert \Delta z (t) \Vert ^{2}}{ [ ( 1-\int _{0}^{t} g(s) \,ds ) a(y(t) ,y(t) ) + (g \square \partial ^{2} y)(t)+\frac{1}{2} \Vert \Delta z (t) \Vert ^{2} ]^{\frac{2}{p}} } \biggr\} ^{\frac{p}{p-2}} \\ ={}&\frac{p-2}{2p} \biggl[ \biggl( 1- \int _{0}^{t} g(s) \,ds \biggr) a\bigl(y(t) ,y(t) \bigr) + \bigl(g \square \partial ^{2} y\bigr) (t)+ \frac{1}{2} \bigl\Vert \Delta z (t) \bigr\Vert ^{2} \biggr], \\ & 0\leq t< t_{0}. \end{aligned}$$
(2.16)

Applying (2.15) and (2.16), we obtain

$$ \bigl\Vert y (t) \bigr\Vert _{p,\varGamma _{1}}^{p} > \frac{2p e_{0}}{p-2} >0, \quad 0\leq t< t_{0} . $$

From \(t \rightarrow \Vert y(t) \Vert _{p,\varGamma _{1}}^{p} >0 \) is continuous, we have \(y(t_{0})|_{\varGamma _{1}} \neq 0\). By (2.12) and \(I(t_{0})=0\), we find that

$$ e_{0} \leq \frac{p-2}{2p} \bigl\Vert y(t_{0} ) \bigr\Vert _{p, \varGamma _{1}}^{p} =H(t_{0}). $$

This is contradiction to \(H(t_{0}) \leq E(t_{0} ) < e_{0} \). From Lemma 2.1, we get (2.14). □

We set

$$\begin{aligned} G(t) = \hat{ \epsilon } e_{0} - E(t), \end{aligned}$$
(2.17)

where \(\hat{\epsilon } =\max \{ 0, \epsilon \}\). By (2.10), G is an increasing function. Using (2.9), (2.13), (2.14), and (2.17), we obtain

0<G(0)G(t) ϵ ˆ e 0 + 1 p y ( t ) p , Γ 1 p p 0 y ( t ) p , Γ 1 p ,t[0,T),
(2.18)

where \(p_{0}=\frac{\hat{ \epsilon } }{2} + (1-\hat{\epsilon }) \frac{1}{p}\).

Lemma 2.3

Let the conditions of Lemma 2.2 hold. Then the solution y of problem (1.1)(1.7) satisfies

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert ^{s}_{p, \varGamma _{1}} \leq C_{3} \bigl\Vert y(t) \bigr\Vert ^{p}_{p, \varGamma _{1}},\quad t\in [0,T), \textit{ for any } 2 \leq s \leq p, \end{aligned}$$
(2.19)

where \(C_{3}>0\).

Proof

If \(\Vert y(t) \Vert _{p,\varGamma _{1}} \geq 1\), then \(\Vert y(t) \Vert _{p,\varGamma _{1}} ^{s} \leq \Vert y(t) \Vert _{p, \varGamma _{1}}^{p}\).

If \(\Vert y(t) \Vert _{p,\varGamma _{1}} \leq 1\), then

$$ \bigl\Vert y(t) \bigr\Vert _{p,\varGamma _{1}}^{s} \leq \bigl\Vert y(t) \bigr\Vert _{p, \varGamma _{1}}^{2} \leq B ^{2} \bigl\Vert \Delta y(t) \bigr\Vert ^{2} \leq \frac{B^{2}}{C_{1}} a\bigl(y(t), y(t)\bigr), $$

where we used (2.3) and (2.8). Then there exists a positive constant \(C_{4}=\max \{1, \frac{B^{2}}{C_{1}}\} \) such that

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert ^{s}_{p, \varGamma _{1}} \leq C_{4} \bigl( \bigl\Vert y(t) \bigr\Vert ^{p}_{p, \varGamma _{1}} +a\bigl(y(t), y(t)\bigr) \bigr) \quad\text{for any } 2 \leq s \leq p. \end{aligned}$$
(2.20)

By (2.4), (2.9), (2.17), and (2.18),

$$\begin{aligned} & \frac{l}{2} a\bigl(y(t),y(t)\bigr) \\ & \quad \leq \hat{\epsilon } e_{0} -G(t)- \frac{1}{\rho +2} \bigl\Vert y_{t}(t) \bigr\Vert _{ \rho +2}^{\rho +2} - \frac{\alpha }{2} \bigl\Vert \nabla y_{t} (t) \bigr\Vert ^{2} \\ &\qquad{}- \frac{1}{2} \bigl(g \square \partial ^{2} y\bigr) (t) + \frac{1}{p} \bigl\Vert y(t) \bigr\Vert ^{p} _{p, \varGamma _{1}}-\frac{1}{4} \bigl\Vert \Delta z(t) \bigr\Vert ^{2} \\ & \quad \leq \hat{\epsilon } e_{0} +\frac{1}{p} \bigl\Vert y(t) \bigr\Vert _{p,\varGamma _{1}} ^{p} \leq p_{0} \bigl\Vert y(t) \bigr\Vert ^{p}_{p, \varGamma _{1}}. \end{aligned}$$
(2.21)

Using (2.20) and (2.21), we get the desired result (2.19). □

A blow-up of solution

To obtain the blow-up result for solutions with nonpositive initial energy as well as positive initial energy, we use a similar method of [26, 29].

Theorem 3.1

Let (H1)–(H2) and the conditions of Lemma 2.2 hold, \(\epsilon < \frac{p-4}{p-2}\) and \(p> \max \{ 4, m \}\). Moreover, we assume that g satisfies

$$\begin{aligned} \int ^{\infty }_{0} g(s) \,ds < \frac{ p-2 }{ p-2+\frac{1}{[(1-\hat{\epsilon })^{2} (p-2) +2(1-\hat{\epsilon })] } }, \end{aligned}$$
(3.1)

where \(\hat{\epsilon }=\max \{ 0, \epsilon \}\) and

$$\begin{aligned} \xi ^{-1}(1) < \biggl( \frac{ p\beta \eta \hat{\epsilon } e_{0} }{(p-1) \vert \varGamma _{1} \vert } \biggr)^{ \frac{p-1}{p} }, \end{aligned}$$
(3.2)

where \(0 < \eta < \min \{2 \theta _{0},2 \theta _{1}, 4 \theta _{2}\}\), \(0<\beta <\eta ^{\frac{1}{p-1}}\), for some \(\delta >0\),

$$\begin{aligned} & \theta _{0} = \biggl( \frac{p}{2} -1 \biggr) (1-\hat{ \epsilon }) - \biggl\{ \biggl( \frac{p}{2} -1 \biggr) (1-\hat{\epsilon }) + \frac{1}{4\delta } \biggr\} \int ^{t}_{0} g(s) \,ds>0, \end{aligned}$$
(3.3)
$$\begin{aligned} & \theta _{1} = \biggl(\frac{p}{2}-1 \biggr) (1-\hat{ \epsilon })+(1-\delta ) >0, \end{aligned}$$
(3.4)
$$\begin{aligned} & \theta _{2} = \biggl(\frac{p}{4}-1 \biggr)-\hat{ \epsilon } \biggl(\frac{p}{4}-\frac{1}{2} \biggr)>0. \end{aligned}$$
(3.5)

Then the solution of system (1.1)(1.7) blows up in finite time.

Proof

We suppose that there exists some positive constant \(B_{0}\) such that, for \(t>0\), the solution \(y(t)\) of (1.1)–(1.7) satisfies

$$ \bigl\Vert y_{t}(t) \bigr\Vert _{\rho +2}^{\rho +2} + \bigl\Vert \nabla y_{t} (t) \bigr\Vert ^{2} + \bigl\Vert \Delta y(t) \bigr\Vert ^{2} + \bigl\Vert y(t) \bigr\Vert _{p,\varGamma _{1}}^{p} \leq B_{0}. $$
(3.6)

Let us define

$$\begin{aligned} F(t) = G^{1-\sigma } (t) + \frac{\varepsilon }{\rho +1} \int _{\varOmega } \bigl\vert y_{t} (t) \bigr\vert ^{\rho }y_{t}(t) y(t) \,dx +\alpha \varepsilon \int _{\varOmega } \nabla y_{t} (t) \nabla y(t) \,dx, \end{aligned}$$
(3.7)

where \(\varepsilon >0\) shall be taken later and

$$\begin{aligned} 0 < \sigma < \min \biggl\{ \frac{1}{\rho +2}, \frac{p-m}{p(m-1)} \biggr\} . \end{aligned}$$
(3.8)

Using (1.1)–(1.6), (2.2), (2.9), and (2.17), we get

$$\begin{aligned} F '(t) ={}&(1- \sigma ) G^{-\sigma } (t) G' (t) + \frac{\varepsilon }{ \rho +1} \bigl\Vert y_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2}+\alpha \varepsilon \bigl\Vert \nabla y_{t} (t) \bigr\Vert ^{2} - \varepsilon \bigl\Vert \Delta z (t) \bigr\Vert ^{2} \\ &{} -\varepsilon a\bigl(y(t),y(t) \bigr) \\ & {} +\varepsilon a \bigl((g\ast y) (t),y(t) \bigr) -\varepsilon \bigl( f\bigl(y _{t} (t)\bigr), y(t) \bigr)_{\varGamma _{1}} +\varepsilon \bigl\Vert y(t) \bigr\Vert ^{p}_{p,\varGamma _{1}} +\varepsilon p E(t) - \varepsilon p E(t) \\ ={}&(1- \sigma ) G^{-\sigma } (t) G' (t) + \frac{\varepsilon }{\rho +1} \bigl\Vert y_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2}+\alpha \varepsilon \bigl\Vert \nabla y_{t} (t) \bigr\Vert ^{2} - \varepsilon \bigl\Vert \Delta z(t) \bigr\Vert ^{2} \\ &{} -\varepsilon a\bigl(y(t),y(t)\bigr) \\ & {} +\varepsilon a \bigl((g\ast y) (t),y(t) \bigr) -\varepsilon \bigl( f\bigl(y _{t}(t)\bigr), y(t) \bigr)_{\varGamma _{1}} +\varepsilon p \bigl( G(t)- \hat{\epsilon } e_{0} \bigr)+ \frac{\varepsilon p }{\rho +2} \bigl\Vert y_{t} (t) \bigr\Vert ^{\rho +2}_{\rho +2} \\ & {} +\varepsilon p \biggl(\frac{\alpha }{2} \bigl\Vert \nabla y_{t} (t) \bigr\Vert ^{2} + \frac{1}{2} \biggl( 1- \int _{0}^{t} g(s) \,ds \biggr) a\bigl(y(t),y(t) \bigr) \\ &{}+ \frac{1}{2} \bigl( g \square \partial ^{2} y\bigr) (t) +\frac{1}{4} \bigl\Vert \Delta z(t) \bigr\Vert ^{2} \biggr). \end{aligned}$$
(3.9)

From (2.14), we find that

$$\begin{aligned} -\hat{\epsilon } e_{0} > \hat{ \epsilon } \biggl( \frac{1}{p}- \frac{1}{2} \biggr) \biggl( \biggl( 1- \int _{0}^{t} g(s) \,ds \biggr) a\bigl(y(t),y(t) \bigr)+\bigl(g \square \partial ^{2} y\bigr) (t) + \frac{1}{2} \bigl\Vert \Delta z(t) \bigr\Vert ^{2} \biggr). \end{aligned}$$
(3.10)

Moreover, we give

$$\begin{aligned} a\bigl((g \ast y) (t),y(t) \bigr) &= \int _{0}^{t} g(t-s) a\bigl( y(s)-y(t), y(t) \bigr) \,ds + \biggl( \int _{0}^{t} g(s) \,ds \biggr) a\bigl(y(t),y(t) \bigr) \\ & \geq \biggl( 1-\frac{1}{4\delta } \biggr) \biggl( \int _{0}^{t} g(s) \,ds \biggr) a\bigl(y(t),y(t) \bigr)- \delta \bigl(g \square \partial ^{2} y\bigr) (t), \end{aligned}$$
(3.11)

for some \(\delta >0\). Combining (3.9), (3.10), and (3.11), we deduce that

$$\begin{aligned} F'(t) \geq{}& (1-\sigma ) G^{-\sigma }(t) G' (t) +\varepsilon \biggl( \frac{1}{ \rho +1} + \frac{p}{\rho +2} \biggr) \bigl\Vert y_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2}+ \varepsilon \alpha \biggl( 1+ \frac{p}{2} \biggr) \bigl\Vert \nabla y_{t} (t) \bigr\Vert ^{2} \\ & {} +\varepsilon \biggl[ \biggl(\frac{p}{2}-1 \biggr) (1-\hat{\epsilon }) - \biggl\{ \biggl(\frac{p}{2} -1 \biggr) (1-\hat{\epsilon } ) + \frac{1}{4 \delta } \biggr\} \int _{0}^{t} g(s) \,ds \biggr]a\bigl(y(t),y(t) \bigr) \\ & {} +\varepsilon \biggl[ \biggl( \frac{p}{2} -1 \biggr) (1-\hat{ \epsilon }) +(1- \delta ) \biggr] \bigl(g\square \partial ^{2} y\bigr) (t) \\ &{}+\varepsilon \biggl[ \biggl( \frac{p}{4}-1 \biggr)-\hat{\epsilon } \biggl( \frac{p}{4}-\frac{1}{2} \biggr) \biggr] \bigl\Vert \Delta z(t) \bigr\Vert ^{2} \\ & {} +\varepsilon p G(t) -\varepsilon \bigl(f\bigl(y_{t} (t)\bigr), y(t)\bigr)_{\varGamma _{1}} \end{aligned}$$
(3.12)

for some δ with \(0<\delta <1+ (\frac{p}{2}-1 )(1- \hat{\epsilon })\). By (3.1), (3.3)–(3.5), estimate (3.12) can be rewritten by

$$\begin{aligned} F'(t) \geq{}& (1-\sigma ) G^{-\sigma }(t) G' (t) +\varepsilon \biggl( \frac{1}{ \rho +1} + \frac{p}{\rho +2} \biggr) \bigl\Vert y_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} + \varepsilon \alpha \biggl( 1+ \frac{p}{2} \biggr) \bigl\Vert \nabla y_{t} (t) \bigr\Vert ^{2} \\ &{} + \varepsilon \theta _{0} a\bigl(y(t),y(t)\bigr) + \varepsilon \theta _{1} \bigl(g \square \partial ^{2} y\bigr) (t)+ \varepsilon \theta _{2} \bigl\Vert \Delta z(t) \bigr\Vert ^{2} \\ &{} + \varepsilon p G(t) - \varepsilon \bigl(f\bigl(y_{t} (t)\bigr), y(t)\bigr)_{\varGamma _{1}}. \end{aligned}$$
(3.13)

Using a method similar to [30], we now estimate the last term of the right-hand side of (3.13). Setting \(\varGamma _{11} = \{ x\in \varGamma _{1} : |y_{t} (x,t)| \leq 1 \}\) and \(\varGamma _{12} = \{ x\in \varGamma _{1} : |y_{t} (x,t)| > 1 \} \), we obtain

$$\begin{aligned} \bigl(f\bigl(y_{t} (t)\bigr), y(t) \bigr)_{\varGamma _{1}} \leq \int _{\varGamma _{11}} \bigl\vert f\bigl(y_{t} (x, t)\bigr) \bigr\vert \bigl\vert y(x,t) \bigr\vert \,d\varGamma + \int _{\varGamma _{12}} \bigl\vert f\bigl(y_{t} (x, t)\bigr) \bigr\vert \bigl\vert y(x,t) \bigr\vert \,d\varGamma. \end{aligned}$$
(3.14)

From (2.5) and Young’s inequality, we get

$$\begin{aligned} &\int _{\varGamma _{11}} \bigl\vert f\bigl(y_{t} (x, t)\bigr) \bigr\vert \bigl\vert y(x,t) \bigr\vert \,d\varGamma\\ &\quad\leq \biggl( \int _{\varGamma _{11}} \bigl\vert \xi ^{-1}(1) \bigr\vert ^{\frac{p}{p-1}} \,d \varGamma \biggr)^{ \frac{p-1}{p}} \biggl( \int _{\varGamma _{11}} \bigl\vert y(x,t) \bigr\vert ^{p} \,d\varGamma \biggr)^{ \frac{1}{p}} \\ &\quad \leq \frac{\beta ^{p-1}}{p} \bigl\Vert y(t) \bigr\Vert ^{p}_{p, \varGamma _{1}} + \frac{(p-1) \vert \varGamma _{1} \vert }{p \beta } \bigl(\xi ^{-1}(1) \bigr)^{\frac{p}{p-1}},\quad \beta >0. \end{aligned}$$
(3.15)

On the other hand, by using (2.6), (2.10), (2.17), and Young’s inequality, we have

$$\begin{aligned} &\int _{\varGamma _{12}} \bigl\vert f\bigl(y_{t} (x, t)\bigr) \bigr\vert \bigl\vert y(x,t) \bigr\vert \,d\varGamma\\ &\quad \leq c_{2} \biggl( \int _{\varGamma _{12}} \bigl\vert y_{t}(x,t) \bigr\vert ^{m}\,d\varGamma \biggr)^{ \frac{m-1}{m}} \biggl( \int _{\varGamma _{12}} \bigl\vert y(x,t) \bigr\vert ^{m} \,d \varGamma \biggr)^{ \frac{1}{m}} \\ &\quad \leq c_{2} \biggl( \frac{1}{c_{1}} \int _{\varGamma _{12}} f\bigl(y_{t}(x,t)\bigr)y _{t}(x,t) \,d\varGamma \biggr)^{\frac{m-1}{m}} \biggl( \int _{\varGamma _{12}} \bigl\vert y(x,t) \bigr\vert ^{m} \,d \varGamma \biggr)^{\frac{1}{m}} \\ &\quad\leq \frac{c_{2}^{m}\gamma ^{m}}{m} \bigl\Vert y(t) \bigr\Vert ^{m}_{p, \varGamma _{1}} + \frac{m-1 }{c_{1}m \gamma ^{\frac{m}{m-1}}} G'(t), \quad\gamma >0. \end{aligned}$$
(3.16)

Inserting (3.14)–(3.16) into (3.13), we obtain

$$\begin{aligned} F'(t) \geq{}& \biggl[ (1-\sigma ) G^{-\sigma }(t)- \frac{\varepsilon (m-1) }{c_{1} m \gamma ^{\frac{m}{m-1}}} \biggr] G' (t) + \varepsilon \biggl( \frac{1}{\rho +1} +\frac{p}{\rho +2} \biggr) \bigl\Vert y_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} \\ & {} +\varepsilon \alpha \biggl( 1+\frac{p}{2} \biggr) \bigl\Vert \nabla y_{t} (t) \bigr\Vert ^{2} + \varepsilon \theta _{0} a\bigl(y(t),y(t)\bigr) \\ &{} + \varepsilon \theta _{1} \bigl(g \square \partial ^{2} y\bigr) (t)+\varepsilon \theta _{2} \bigl\Vert \Delta z(t) \bigr\Vert ^{2} + \varepsilon p G(t) \\ & {} - \frac{\varepsilon \beta ^{p-1}}{p} \bigl\Vert y(t) \bigr\Vert ^{p}_{p, \varGamma _{1}}- \frac{ \varepsilon c_{2}^{m}\gamma ^{m}}{m} \bigl\Vert y(t) \bigr\Vert ^{m}_{p, \varGamma _{1}}- \frac{ \varepsilon (p-1) \vert \varGamma _{1} \vert }{p \beta } \bigl(\xi ^{-1}(1) \bigr)^{ \frac{p}{p-1}}. \end{aligned}$$
(3.17)

We choose \(\gamma = ( \tau G^{-\sigma }(t) )^{- \frac{m-1}{m}}\), \(\tau >0\) will be specified later. Using (2.18), (2.19), and (3.8), we see that

$$\begin{aligned} - \frac{\varepsilon c_{2}^{m}\gamma ^{m}}{m} \bigl\Vert y(t) \bigr\Vert ^{m}_{p, \varGamma _{1}} &= - \frac{\varepsilon c_{2}^{m} \tau ^{1-m}}{m} G^{\sigma (m-1)}(t) \bigl\Vert y(t) \bigr\Vert ^{m}_{p, \varGamma _{1}} \\ & \geq - \frac{\varepsilon c_{2}^{m} \tau ^{1-m}}{m} p_{0}^{\sigma (m-1)} \bigl\Vert y(t) \bigr\Vert _{p, \varGamma _{1}}^{\sigma p(m-1) +m} \geq -\varepsilon C_{5} \tau ^{1-m} \bigl\Vert y(t) \bigr\Vert _{p, \varGamma _{1}}^{p}, \end{aligned}$$
(3.18)

where \(C_{5}= \frac{c_{2}^{m} p_{0}^{\sigma (m-1)} C_{3} }{m} \). Substituting (3.18) into (3.17), we have

$$\begin{aligned} F'(t) \geq{}& \biggl[ (1- \sigma ) - \frac{ \varepsilon \tau (m-1) }{c _{1}m} \biggr] G^{-\sigma } (t) G' (t) + \varepsilon \biggl( \frac{1}{ \rho +1}+ \frac{p}{\rho +2} \biggr) \bigl\Vert y_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} \\ &{} + \varepsilon \alpha \biggl( 1+\frac{p}{2} \biggr) \bigl\Vert \nabla y_{t} (t) \bigr\Vert ^{2} + \varepsilon \theta _{0} a\bigl(y(t),y(t)\bigr) \\ &{} + \varepsilon \theta _{1} \bigl(g \square \partial ^{2} y\bigr) (t)+\varepsilon \theta _{2} \bigl\Vert \Delta z(t) \bigr\Vert ^{2} + \varepsilon p G(t) \\ &{} - \varepsilon \biggl( \frac{\beta ^{p-1}}{p} + C_{5}\tau ^{1-m} \biggr) \bigl\Vert y(t) \bigr\Vert ^{p} _{p, \varGamma _{1}} - \frac{\varepsilon (p-1) \vert \varGamma _{1} \vert }{p \beta } \bigl(\xi ^{-1}(1) \bigr)^{\frac{p}{p-1}}. \end{aligned}$$
(3.19)

Adding and subtracting \(\varepsilon \eta G(t) \) on the right-hand side of (3.19) and applying (2.9) and (2.17), we obtain

$$\begin{aligned} F'(t) \geq{}& \biggl[ (1- \sigma ) - \frac{ \varepsilon \tau (m-1) }{c _{1}m} \biggr] G^{-\sigma } (t) G' (t) + \varepsilon \biggl( \frac{1}{ \rho +1}+ \frac{p}{\rho +2} - \frac{\eta }{\rho +2} \biggr) \bigl\Vert y_{t} (t) \bigr\Vert _{ \rho +2}^{\rho +2} \\ &{} + \varepsilon \alpha \biggl( 1+\frac{p}{2}- \frac{\eta }{2} \biggr) \bigl\Vert \nabla y_{t} (t) \bigr\Vert ^{2} + \varepsilon ( p - \eta ) G(t) \\ &{}+ \varepsilon \biggl\{ \theta _{0} - \frac{ \eta }{2} \biggl( 1 - \int ^{t}_{0} g(s) \,ds \biggr) \biggr\} a \bigl(y(t),y(t)\bigr) \\ &{} + \varepsilon \biggl( \theta _{1} - \frac{\eta }{2} \biggr) \bigl(g\square \partial ^{2} y\bigr) (t) +\varepsilon \biggl( \theta _{2} -\frac{\eta }{4} \biggr) \bigl\Vert \Delta z (t) \bigr\Vert ^{2} \\ &{}+ \varepsilon \biggl( \frac{ \eta }{p} - \frac{ \beta ^{p-1}}{p} - C_{5} \tau ^{1-m} \biggr) \bigl\Vert y(t) \bigr\Vert ^{p}_{p, \varGamma _{1}} \\ &{} +\varepsilon \eta e_{0} \hat{\epsilon }- \frac{\varepsilon (p-1) \vert \varGamma _{1} \vert }{p \beta } \bigl(\xi ^{-1}(1) \bigr)^{\frac{p}{p-1}}. \end{aligned}$$
(3.20)

We fix η such that

$$ 0 < \eta < \min \{2 \theta _{0}, 2 \theta _{1}, 4 \theta _{2}\}, $$
(3.21)

then we can choose \(\beta >0\) sufficiently small so that \({ \eta } - {\beta ^{p-1}} >0 \). And then, we select \(\tau >0\) large enough such that \(\frac{ \eta }{p} - \frac{\beta ^{p-1}}{p} - C_{5} \tau ^{1-m} >0 \). Finally, we take \(\varepsilon >0\) with

$$ (1- \sigma ) - \frac{ \varepsilon \tau (m-1) }{c_{1}m} >0 ,\qquad G^{1-\sigma }(0)+ \frac{\varepsilon }{\rho +1} \int _{\varOmega } \vert y_{1} \vert ^{ \rho }y_{1} y_{0} \,dx +\alpha \varepsilon \int _{\varOmega }\nabla y_{1} \nabla y_{0} \,dx >0. $$

Condition (3.2) yields

$$ \eta e_{0}\hat{\epsilon } - \frac{ (p-1) \vert \varGamma _{1} \vert }{p \beta } \bigl( \xi ^{-1}(1) \bigr)^{\frac{p}{p-1}} >0. $$

Therefore, we get from (2.3) and (3.20)

$$\begin{aligned} F'(t) \geq C \bigl( \bigl\Vert y_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} + \bigl\Vert \nabla y_{t}(t) \bigr\Vert ^{2}+ \bigl\Vert \Delta y(t) \bigr\Vert ^{2} + \bigl\Vert y(t) \bigr\Vert ^{p}_{p,\varGamma _{1}} + G(t) \bigr), \end{aligned}$$
(3.22)

where \(C>0\) is a generic constant. Hence we have

$$ F(t) \geq F(0) >0, \quad\forall t \geq 0. $$

By the similar arguments in [31, 32], we see that

$$\begin{aligned} F^{\frac{1}{1-\sigma }}(t) \leq C \bigl( \bigl\Vert y_{t} (t) \bigr\Vert _{\rho +2}^{ \rho +2} + \bigl\Vert \nabla y_{t}(t) \bigr\Vert ^{2} + \bigl\Vert \Delta y(t) \bigr\Vert ^{2}+ \bigl\Vert y(t) \bigr\Vert ^{p} _{p,\varGamma _{1}} \bigr). \end{aligned}$$
(3.23)

Indeed, using Young’s inequality and

$$\begin{aligned} \biggl\vert \int _{\varOmega } \bigl\vert y_{t}(t) \bigr\vert ^{\rho }y_{t} (t)y(t) \,dx \biggr\vert \leq \bigl\Vert y _{t}(t) \bigr\Vert _{\rho +2}^{\rho +1} \bigl\Vert y(t) \bigr\Vert _{\rho +2}, \end{aligned}$$

we obtain

$$\begin{aligned} \biggl\vert \int _{\varOmega } \bigl\vert y_{t}(t) \bigr\vert ^{\rho }y_{t} (t)y(t) \,dx \biggr\vert ^{\frac{1}{1- \sigma }} &\leq \bigl( \bigl\Vert y_{t}(t) \bigr\Vert _{\rho +2}^{\rho +1} \bigl\Vert y(t) \bigr\Vert _{ \rho +2} \bigr)^{\frac{1}{1-\sigma }} \\ &\leq C \bigl( \bigl\Vert y_{t}(t) \bigr\Vert _{ \rho +2}^{\frac{(\rho +1)\kappa }{1-\sigma }}+ \bigl\Vert y(t) \bigr\Vert _{\rho +2}^{\frac{ \mu }{1-\sigma }} \bigr), \end{aligned}$$
(3.24)

where \(\frac{1}{\kappa }+\frac{1}{\mu }=1\). By taking \(\kappa =\frac{(1- \sigma )(\rho +2)}{\rho +1}\) and using (3.8), we get \(\kappa >1\) and \(\frac{\mu }{1-\sigma } = \frac{\rho +2}{(1-\sigma )(\rho +2)-( \rho +1)}\). Since G is an increasing function, (2.18) and (3.6), we arrive at

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert _{\rho +2}^{\frac{\mu }{1-\sigma }} \leq C_{0}^{\frac{\mu }{1- \sigma }} \bigl\Vert \Delta y(t) \bigr\Vert ^{\frac{\mu }{1-\sigma }} \leq \frac{(C_{0} ^{2} B_{0})^{\frac{\mu }{2(1-\sigma )}}}{G(0)}G(t) \leq C \bigl\Vert y(t) \bigr\Vert ^{p} _{p, \varGamma _{1}}, \end{aligned}$$
(3.25)

where \(C_{0}\) is the embedding constant. Similarly, by Young’s inequality, we obtain

$$\begin{aligned} \biggl\vert \int _{\varOmega }\nabla y_{t}(t)\nabla y(t) \,dx \biggr\vert ^{\frac{1}{1- \sigma }} &\leq \bigl\Vert \nabla y_{t}(t) \bigr\Vert ^{\frac{1}{1-\sigma }} \bigl\Vert \nabla y(t) \bigr\Vert ^{\frac{1}{1- \sigma }} \\ &\leq C \bigl( \bigl\Vert \nabla y_{t}(t) \bigr\Vert ^{2} + \bigl\Vert \nabla y(t) \bigr\Vert ^{\frac{2}{1-2 \sigma }} \bigr). \end{aligned}$$
(3.26)

Like (3.25), we find that

$$\begin{aligned} \bigl\Vert \nabla y(t) \bigr\Vert ^{\frac{2}{1-2\sigma }} \leq C_{*}^{\frac{2}{1-2\sigma }} \bigl\Vert \Delta y(t) \bigr\Vert ^{\frac{2}{1-2\sigma }} \leq \frac{(C_{*}^{2} B_{0})^{\frac{1}{1-2 \sigma }}}{G(0)}G(t)\leq C \bigl\Vert y(t) \bigr\Vert ^{p}_{p, \varGamma _{1}}, \end{aligned}$$
(3.27)

where \(C_{*}\) is the embedding constant. From (2.18), (3.7), (3.24)–(3.27), we see that (3.23) holds. Combining (3.22) and (3.23), we deduce that

$$\begin{aligned} F'(t) \geq C F^{\frac{1}{1-\sigma }}(t) \quad\text{for } t \geq 0. \end{aligned}$$
(3.28)

By a simple integration of (3.28) over \((0,t)\), we get

$$\begin{aligned} F^{\frac{\sigma }{1-\sigma }}(t) \geq \frac{1}{F^{-\frac{\sigma }{1- \sigma }}(0) -\frac{C\sigma t}{1-\sigma }} \quad\text{for } t \geq 0. \end{aligned}$$

Consequently, \(F(t)\) blows up in time \(T^{*} \leq \frac{1-\sigma }{C \sigma F^{\frac{\sigma }{1-\sigma }} (0)} \). Furthermore, we have from (3.23)

$$ \lim_{t\to T^{*-}} \bigl( \bigl\Vert y_{t}(t) \bigr\Vert _{\rho +2}^{\rho +2} + \bigl\Vert \nabla y _{t}(t) \bigr\Vert ^{2} + \bigl\Vert \Delta y(t) \bigr\Vert ^{2} + \bigl\Vert y(t) \bigr\Vert _{p,\varGamma _{1}}^{p} \bigr)=\infty. $$

This leads to a contradiction with (3.6). Therefore the solution of (1.1)–(1.7) blows up in finite time. □

Conclusions

In this paper, we consider the blow-up of solutions for the quasilinear von Karman equation of memory type. In recent years, there has been published much work concerning the wave equation with nonlinear boundary damping. But as far as we know, there was no blow-up result of solutions to the viscoelastic von Karman equation with nonlinear boundary damping and source terms. Therefore, we will prove a finite time blow-up result of solution with positive initial energy as well as non-positive initial energy. Moreover, we generalize the earlier result under a weaker assumption on the damping term.

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Acknowledgements

The authors are thankful to the honorable reviewers and editors for their valuable comments and suggestions, which improved the paper.

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Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Funding

Research of MJL was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2017R1E1A1A03070738). Research of J-RK was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03028291).

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The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Correspondence to Jum-Ran Kang.

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Lee, M.J., Kang, J. Blow-up results for a quasilinear von Karman equation of memory type. Bound Value Probl 2019, 174 (2019) doi:10.1186/s13661-019-1277-y

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MSC

  • 35B44
  • 37B25
  • 35L70
  • 74D10

Keywords

  • Blow-up result
  • Von Karman equation
  • Memory dissipation