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Blow-up phenomena for a viscoelastic wave equation with strong damping and logarithmic nonlinearity

Abstract

In this paper we consider the initial boundary value problem for a viscoelastic wave equation with strong damping and logarithmic nonlinearity of the form

$$ u_{tt}(x,t) - \Delta u (x,t) + \int ^{t}_{0} g(t-s) \Delta u(x,s)\,ds - \Delta u_{t} (x,t) = \bigl\vert u(x,t) \bigr\vert ^{p-2} u(x,t) \ln \bigl\vert u(x,t) \bigr\vert $$

in a bounded domain \(\varOmega \subset {\mathbb{R}}^{n} \), where g is a nonincreasing positive function. Firstly, we prove the existence and uniqueness of local weak solutions by using Faedo–Galerkin’s method and contraction mapping principle. Then we establish a finite time blow-up result for the solution with positive initial energy as well as nonpositive initial energy.

Introduction

In this paper, we are concerned with the following viscoelastic wave equation with strong damping and logarithmic nonlinearity source:

$$\begin{aligned}& u_{tt} - \Delta u + \int ^{t}_{0} g(t-s) \Delta u(s)\,ds - \Delta u_{t} = \vert u \vert ^{p-2} u \ln \vert u \vert \quad \text{in } \varOmega \times (0, T), \end{aligned}$$
(1.1)
$$\begin{aligned}& u = 0 \quad \text{on } \partial \varOmega \times (0, T), \end{aligned}$$
(1.2)
$$\begin{aligned}& u(0) = u_{0}, \quad\quad u_{t}(0)= u_{1} \quad \text{on } \varOmega , \end{aligned}$$
(1.3)

where \(\varOmega \subset {\mathbb{R}}^{n} \), \(n\geq 1\), is a bounded domain with smooth boundary ∂Ω. This type of equation is related to viscoelastic mechanics, quantum mechanics theory, nuclear physics, optics, geophysics and so on. For instance, the logarithmic nonlinearity arises in the inflation cosmology and super-symmetric fields in the quantum field theory. In the case \(n=1, 2\), Eq. (1.1) describes the transversal vibrations of a homogeneous viscous string and the longitudinal vibrations of a homogeneous bar, respectively. For the physical point of view, we refer to [13] and the references therein.

During the past decades, the strongly damped wave equations with source effect

$$ u_{tt} - \Delta u - \omega \Delta u_{t} + \mu u_{t} = f(u) $$
(1.4)

have been studied extensively on existence, nonexistence, stability, and blow-up of solutions. In the case of power nonlinearity \(f(u) = \vert u \vert ^{p-2}u\), Sattinger [4] firstly considered the existence of local as well as global solutions for equation (1.4) with \(\omega = \mu =0\) by introducing the concepts of stable and unstable sets. Since then, the potential well method has become an important theory to the study of the existence and nonexistence of solutions [515]. Ikehata [8] gave properties of decay estimates and blow-up of solutions to (1.4) with linear damping (\(\omega =0\) and \(\mu >0\)). Gazzola and Squassina [6] proved the global existence and finite time blow-up of solutions to problem (1.4) with weak and strong damping (\(\omega >0\)). Liu [11] considered a viscoelastic version of (1.4). He investigated decay estimates for global solutions when the initial data enter the stable set and showed finite blow-up results when the initial data enter the unstable set.

In the case of logarithmic nonlinearity \(f(u) =u \ln \vert u \vert ^{k} \), Ma and Fang [16] proved the existence of global solutions and infinite time blow-up to problem (1.4) with \(\omega =1\), \(\mu =0\), and \(k=2\). Lian and Xu [17] investigated global existence, energy decay and infinite time blow-up when \(\omega \geq 0 \) and \(\mu > -\omega \lambda _{1} \), where \(\lambda _{1}\) is the first eigenvalue of the operator −Δ under homogeneous Dirichlet boundary conditions. The results of [16, 17] were obtained by use of the potential well method and the logarithmic Sobolev inequality.

By the way, there is not much literature for strongly damped wave equations with the logarithmic nonlinear source \(\vert u \vert ^{p-2} u \ln \vert u \vert \). Recently, Di et al. [18] considered problem (1.1)–(1.3) when the kernel function \(g = 0\). The presence of the Laplacian operator \(- \Delta u\) and the logarithmic nonlinearity \(\vert u \vert ^{p-2} u \ln \vert u \vert \) causes some difficulty so that one cannot apply the logarithmic Sobolev inequality [19]. Thus, they discussed the global existence, uniqueness, energy decay estimates and finite time blow-up of solutions by modifying the potential well method. We also refer to [20, 21] and the references therein for problems with logarithmic nonlinearity.

Motivated by these results, we study the existence and finite time blow-up of weak solutions for problem (1.1)–(1.3) in the present work by applying the ideas in [11, 18]. To the best our knowledge, this is the first work in the literature that takes into account a viscoelastic wave equation with strong damping and logarithmic nonlinearity in a bounded domain \(\varOmega \subset {\mathbb{R}}^{n} \).

The outline of this paper is as follows. In Sect. 2, we give materials needed for our work. In Sect. 3, we prove the local existence of solutions for problem (1.1)–(1.3) using Faedo–Galerkin’s method and contraction mapping principle. In Sect. 4, we establish a finite time blow-up result.

Preliminaries

In this section we give notations, hypotheses, and some lemmas needed in our main results.

For a Banach space X, \(\Vert \cdot \Vert _{X} \) denotes the norm of X. As usual, \((\cdot , \cdot )\) and \(\langle \cdot , \cdot\rangle \) denote the inner product in the space \(L^{2}(\varOmega )\) and the duality pairing between \(H^{-1}(\varOmega )\) and \(H^{1}_{0}(\varOmega )\), respectively. \(\Vert \cdot \Vert _{q}\) denotes the norm of the space \(L^{q}(\varOmega )\). For brevity, we denote \(\Vert \cdot \Vert _{2}\) by \(\Vert \cdot \Vert \). Let \(c_{q}\) be the best constants in the Poincaré type inequality

$$ \Vert v \Vert _{q} \leq c_{q} \Vert \nabla v \Vert ^{2} \quad \text{for } v \in H^{1}_{0} ( \varOmega ), $$

where

$$ 2 \leq q < \infty , \quad \text{if } n=1,2 ; \quad\quad 2 \leq q \leq \frac{2n}{n-2}, \quad \text{if } n \geq 3 . $$

We need the following lemma.

Lemma 2.1

For each\(q >0 \),

$$ \bigl\vert s^{q} \ln s \bigr\vert \leq \frac{1}{e q} \quad \textit{for } 0< s < 1 \quad \textit{and} \quad 0 \leq s^{-q} \ln s \leq \frac{1}{e q} \quad \textit{for } s \geq 1 . $$

Proof

We can easily show this from simple calculation. So, we omit it here. □

Lemma 2.2

([9])

Let\(L(t)\)be a positive, twice differentiable function satisfying the inequality

$$ L(t)L''(t) - ( 1+ \delta ) \bigl(L'(t) \bigr)^{2} \geq 0 \quad \textit{for } t>0, $$

with some\(\delta >0\). If\(L(0) >0\)and\(L'(0) >0\), then there exists a time\({ T_{*} \leq \frac{L(0)}{\delta L'(0)} }\)such that

$$ \lim_{t \to T_{*}^{-} } L(t) = + \infty . $$

With regard to problem (1.1)–(1.3), we impose the following assumptions:

\((H_{1})\):

Hypotheses onp.

The exponent p satisfies

$$ 2< p < \infty , \quad \text{if } n=1,2 ; \quad\quad 2< p < \frac{2(n-1)}{n-2}, \quad \text{if } n \geq 3 . $$
(2.1)
\((H_{2})\):

Hypotheses ong.

The kernel function \(g: [0, \infty ) \to [0, \infty )\) is a nonincreasing and differentiable function satisfying

$$ 1- \int ^{\infty }_{0} g(s)\,ds: = l >0 . $$
(2.2)

Definition 2.1

Let\(T>0\). We say that a functionuis a weak solution of problem (1.1)(1.3) if

$$ u \in C\bigl([0,T]; H^{1}_{0} (\varOmega ) \bigr) \cap C^{1} \bigl([0,T]; L^{2} ( \varOmega )\bigr)\cap C^{2} \bigl([0,T]; H^{-1} (\varOmega )\bigr), $$

leads to

$$\begin{aligned} \begin{aligned}[b] &\bigl\langle u_{tt}(t), w\bigr\rangle + \bigl( \nabla u(t), \nabla w\bigr) - \int ^{t}_{0} g(t-s) \bigl( \nabla u(s), \nabla w \bigr)\,ds + \bigl(\nabla u_{t}(t), \nabla w\bigr) \\ &\quad = \int _{\varOmega } \bigl\vert u(x,t) \bigr\vert ^{p-2} u(x,t) \ln \bigl\vert u(x,t) \bigr\vert w \,dx \end{aligned} \end{aligned}$$
(2.3)

for any\(w \in H^{1}_{0} (\varOmega )\)and\(t\in (0,T) \), and

$$ u(0)=u_{0} \quad \textit{in } H^{1}_{0} (\varOmega ) , \quad\quad u_{t}(0) = u_{1} \quad \textit{in } L^{2}( \varOmega ). $$

Local existence of solutions

In this section we prove the local existence of solutions making use of the Faedo–Galerkin method and the contraction mapping principle. For a fixed \(T>0\), we consider the space

$$ \mathcal{H} = C\bigl([0,T];H^{1}_{0}(\varOmega )\bigr)\cap C^{1}\bigl([0,T];L^{2}( \varOmega )\bigr) $$

with the norm

$$ \Vert v \Vert ^{2}_{\mathcal{H}} = \max_{0 \leq t \leq T} \bigl( \bigl\Vert v_{t} (t) \bigr\Vert ^{2} + l \bigl\Vert \nabla v(t) \bigr\Vert ^{2} \bigr). $$

To show the existence and uniqueness of local solution to problem (1.1)–(1.3), we firstly establish the following result.

Lemma 3.1

Assume that\((H_{1})\)and\((H_{2})\)hold. Then, for every\(u_{0} \in H^{1}_{0} (\varOmega )\), \(u_{1} \in L^{2}(\varOmega )\), \(v \in \mathcal{H}\), there exists a unique

$$ u \in C\bigl([0,T]; H^{1}_{0} (\varOmega ) \bigr) \cap C^{1} \bigl([0,T]; L^{2} ( \varOmega )\bigr)\cap C^{2} \bigl([0,T]; H^{-1} (\varOmega )\bigr) $$

such that\(u_{t} \in L^{2}([0,T];H^{1}_{0}(\varOmega ))\)and

$$\begin{aligned}& u_{tt} - \Delta u + \int ^{t}_{0} g(t-s) \Delta u(s)\,ds - \Delta u_{t} = \vert v \vert ^{p-2} v \ln \vert v \vert \quad \textit{in } \varOmega \times (0, T), \end{aligned}$$
(3.1)
$$\begin{aligned}& u = 0 \quad \textit{on } \partial \varOmega \times (0, T), \end{aligned}$$
(3.2)
$$\begin{aligned}& u(0) = u_{0}, \quad\quad u_{t}(0)= u_{1} \quad \textit{on } \varOmega . \end{aligned}$$
(3.3)

Proof

Existence. Let \(\{ w_{j} \}_{j\in {\mathbb{N}}}\) be an orthogonal basis of \(H^{1}_{0} (\varOmega )\) which is orthonormal in \(L^{2}(\varOmega ) \) and \(W_{m} = \operatorname{span} \{ w_{1}, w_{2},\ldots, w_{m} \}\), then there exist subsequences \(u_{0}^{m} \in W_{m} \) and \(u_{1}^{m} \in W_{m} \) such that \(u^{m}_{0} \to u_{0}\) in \(H^{1}_{0} (\varOmega )\) and \(u^{m}_{1} \to u_{1}\) in \(L^{2} (\varOmega )\), respectively. We will seek an approximate solution

$$ u^{m} (x,t) = \sum^{m}_{j=1} h^{m}_{j}(t) w_{j}(x) $$

satisfying

$$\begin{aligned}& \bigl( u^{m}_{tt}(t), w\bigr) + \bigl( \nabla u^{m} (t), \nabla w\bigr) - \int ^{t}_{0} g(t-s) \bigl(\nabla u^{m}(s), \nabla w\bigr)\,ds + \bigl(\nabla u^{m}_{t} (t), \nabla w\bigr) \\& \quad = \int _{\varOmega } \bigl\vert v(x,t) \bigr\vert ^{p-2} v(x,t) \ln \bigl\vert v(x,t) \bigr\vert w(x)\,dx \quad \text{for } w \in W_{m} \end{aligned}$$
(3.4)

and the initial conditions

$$\begin{aligned} u^{m}(0) = u_{0}^{m} , \quad \quad u^{m}_{t}(0)= u^{m}_{1} . \end{aligned}$$
(3.5)

Since (3.4)–(3.5) is a normal system of ordinary differential equations, there exists a solution \(u^{m}\) on the interval \([0, t_{m}) \subset [0,T] \). We obtain an a priori estimate for the solution \(u^{m}\) so that it can be extended to the whole interval \([0,T]\) according to the extension theorem.

Step 1. A priori estimate. Replacing w by \(u^{m}_{t}(t) \) in (3.4) and using the relation

$$\begin{aligned} \int ^{t}_{0} g(t-s) \bigl(\nabla u^{m}(s), \nabla u^{m}_{t}(t)\bigr)\,ds = & - \frac{g(t)}{2} \bigl\Vert \nabla u^{m} (t) \bigr\Vert ^{2} + \frac{1}{2} \bigl(g' \circ \nabla u^{m}\bigr) (t) \\ & {}- \frac{1}{2} \frac{d}{dt} \biggl( \bigl(g\circ \nabla u^{m} \bigr) (t) - \int ^{t}_{0} g(s)\,ds \bigl\Vert \nabla u^{m} (t) \bigr\Vert ^{2} \biggr), \end{aligned}$$

where

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

we have

$$\begin{aligned} \begin{aligned}[b] & \frac{d}{dt} \biggl\{ \bigl\Vert u^{m}_{t}(t) \bigr\Vert ^{2} + \biggl( 1 - \int ^{t}_{0} g(s)\,ds \biggr) \bigl\Vert \nabla u^{m} (t) \bigr\Vert ^{2} + \bigl(g\circ \nabla u^{m} \bigr) (t) \biggr\} + 2 \bigl\Vert \nabla u^{m}_{t} (t) \bigr\Vert ^{2} \\ &\quad = \bigl(g' \circ \nabla u^{m}\bigr) (t) - g(t) \bigl\Vert \nabla u^{m} (t) \bigr\Vert ^{2} + 2 \int _{\varOmega } \bigl\vert v(x,t) \bigr\vert ^{p-2}v(x,t) \ln \bigl\vert v(x,t) \bigr\vert u_{t}^{m} (x,t)\,dx . \end{aligned} \end{aligned}$$

Integrating this over \((0,t)\) and making use of \((H_{2})\),

$$\begin{aligned}& \bigl\Vert u^{m}_{t} (t) \bigr\Vert ^{2} + l \bigl\Vert \nabla u^{m} (t) \bigr\Vert ^{2} + \bigl(g \circ \nabla u^{m}\bigr) (t) + 2 \int ^{t}_{0} \bigl\Vert \nabla u^{m}_{t}(s) \bigr\Vert ^{2}\,ds \\& \quad \leq \bigl\Vert u^{m}_{1} \bigr\Vert ^{2} + \bigl\Vert \nabla u^{m}_{0} \bigr\Vert ^{2} + 2 \int ^{t}_{0} \bigl\Vert \bigl\vert v(s) \bigr\vert ^{p-2}v(s) \ln \bigl\vert v(s) \bigr\vert \bigr\Vert _{\frac{p}{p-1}} \bigl\Vert u_{t}^{m} (s) \bigr\Vert _{p}\,ds. \end{aligned}$$
(3.6)

In order to estimate the last term in the right hand side of (3.6), we let

$$ \varOmega _{1} = \bigl\{ x \in \varOmega: \bigl\vert u^{m}(x,t) \bigr\vert < 1 \bigr\} \quad \text{and} \quad \varOmega _{2} = \bigl\{ x\in \varOmega: \bigl\vert u^{m}(x,t) \bigr\vert \geq 1 \bigr\} . $$

Since \(2< p < \frac{2n}{n-2} \), we can take \(\mu > 0\) such that \(2< p+ \frac{\mu p}{p-1} < \frac{2n}{n-2}\). Applying Lemma 2.1, we infer that

$$\begin{aligned} \bigl\Vert \vert v \vert ^{p-2}v \ln \vert v \vert \bigr\Vert ^{\frac{p}{p-1}}_{\frac{p}{p-1}} = & \int _{ \varOmega _{1}} \bigl( \bigl\vert \vert v \vert ^{p-1} \ln \vert v \vert \bigr\vert \bigr)^{\frac{p}{p-1}}\,dx + \int _{\varOmega _{2}} \bigl( \bigl\vert \vert v \vert ^{-\mu + (p-1+\mu ) } \ln \vert v \vert \bigr\vert \bigr)^{\frac{p}{p-1}}\,dx \\ \leq & \biggl( \frac{1}{e(p-1)} \biggr)^{\frac{p}{p-1}} \vert \varOmega _{1} \vert + \int _{\varOmega _{2}} \bigl( \vert v \vert ^{-\mu } \ln \vert v \vert \bigr)^{\frac{p}{p-1}} \vert v \vert ^{ \frac{p(p-1+\mu )}{p-1}}\,dx \\ \leq & \biggl( \frac{1}{e(p-1)} \biggr)^{\frac{p}{p-1}} \vert \varOmega _{1} \vert + \biggl( \frac{1}{e \mu } \biggr)^{\frac{p}{p-1}} \int _{\varOmega _{2}} \vert v \vert ^{ \frac{p(p-1+\mu )}{p-1}}\,dx \\ \leq & \biggl( \frac{1}{e(p-1)} \biggr)^{\frac{p}{p-1}} \vert \varOmega _{1} \vert + c^{ \frac{p(p-1+\mu )}{p-1}}_{\frac{p(p-1+\mu )}{p-1}} \biggl( \frac{1}{e \mu } \biggr)^{\frac{p}{p-1}} \Vert \nabla v \Vert ^{ \frac{p(p-1+\mu )}{p-1}} \\ \leq & C, \end{aligned}$$
(3.7)

we used the fact that \(v \in \mathcal{H}\) in the last inequality. Here and in the sequel, C denotes a generic positive constant independent of m and t and different from line to line or even in the same line.

From (3.7), we see that

$$\begin{aligned} 2 \int ^{t}_{0} \bigl\Vert \vert v \vert ^{p-2}v \ln \vert v \vert \bigr\Vert _{\frac{p}{p-1}} \bigl\Vert u_{t}^{m} \bigr\Vert _{p}\,ds \leq & C \int ^{t}_{0} \bigl\Vert u_{t}^{m} (s) \bigr\Vert _{p}\,ds \\ \leq & C \int ^{t}_{0} \bigl\Vert \nabla u_{t}^{m} (s) \bigr\Vert \,ds \leq CT + \int ^{t}_{0} \bigl\Vert \nabla u_{t}^{m} (s) \bigr\Vert ^{2}\,ds. \end{aligned}$$

Adapting this to (3.6), we get

$$\begin{aligned} \bigl\Vert u^{m}_{t} (t) \bigr\Vert ^{2} + l \bigl\Vert \nabla u^{m} (t) \bigr\Vert ^{2} + \bigl( g\circ \nabla u^{m}\bigr) (t) + \int ^{t}_{0} \bigl\Vert \nabla u^{m}_{t}(s) \bigr\Vert ^{2}\,ds \leq &\bigl\Vert u^{m}_{1} \bigr\Vert ^{2} + \bigl\Vert \nabla u^{m}_{0} \bigr\Vert ^{2} + CT \\ \leq& C. \end{aligned}$$

Step 2. Passage to the limit. So, there exists a subsequence of \(\{ u^{m} \}\), which we still denote by \(\{ u^{m} \}\), such that

$$\begin{aligned}& u^{m} \to u\quad \text{weakly star in } L^{\infty }\bigl(0,T; H^{1}_{0}( \varOmega )\bigr), \end{aligned}$$
(3.8)
$$\begin{aligned}& u^{m}_{t} \to u_{t}\quad \text{weakly star in } L^{\infty }\bigl(0,T; L^{2}( \varOmega )\bigr), \end{aligned}$$
(3.9)
$$\begin{aligned}& u^{m}_{t} \to u_{t}\quad \text{weakly in } L^{2}\bigl(0,T; H^{1}_{0}( \varOmega )\bigr). \end{aligned}$$
(3.10)

Now, we integrate (3.4) over \((0,t)\) to get

$$\begin{aligned} \bigl( u^{m}_{t}(t), w\bigr) - \bigl(u_{1}^{m}, w\bigr) + \int ^{t}_{0} \bigl( \nabla u^{m} (s), \nabla w\bigr)\,ds - \int ^{t}_{0} \int ^{\tau }_{0} g(\tau - s) \bigl(\nabla u^{m}(s), \nabla w\bigr)\,ds \,d\tau \\ \quad{} + \bigl(\nabla u^{m} (t), \nabla w\bigr) - \bigl(\nabla u^{m}_{0}, \nabla w \bigr) = \int ^{t}_{0} \int _{\varOmega } \bigl\vert v(x,s) \bigr\vert ^{p-2} v(x,s) \ln \bigl\vert v(x,s) \bigr\vert w(x)\,dx \,ds . \end{aligned}$$

Taking the limit \(m\to \infty \) in this, we have from (3.8) and (3.9) that

$$\begin{aligned}& \bigl( u_{t}(t), w\bigr) - (u_{1}, w) + \int ^{t}_{0} \bigl( \nabla u (s), \nabla w\bigr) \,ds - \int ^{t}_{0} \int ^{\tau }_{0} g(\tau - s) \bigl(\nabla u(s), \nabla w\bigr)\,ds \,d\tau \\& \quad{} + \bigl(\nabla u (t), \nabla w\bigr) - (\nabla u_{0}, \nabla w ) = \int ^{t}_{0} \int _{\varOmega } \bigl\vert v(x,s) \bigr\vert ^{p-2} v(x,s) \ln \bigl\vert v(x,s) \bigr\vert w(x)\,dx \,ds . \end{aligned}$$
(3.11)

This remains valid for all \(w\in H^{1}_{0}(\varOmega ) \). Differentiating (3.11) with respect to t, we have

$$\begin{aligned}& \bigl\langle u_{tt}(t), w\bigr\rangle + \bigl( \nabla u (t), \nabla w \bigr) - \int ^{t}_{0} g(t-s) \bigl(\nabla u(s), \nabla w \bigr)\,ds + \bigl(\nabla u_{t} (t), \nabla w\bigr) \\& \quad = \int _{\varOmega } \bigl\vert v(x,t) \bigr\vert ^{p-2} v(x,t) \ln \bigl\vert v(x,t) \bigr\vert w(x)\,dx \quad \text{for } w \in H^{1}_{0}(\varOmega ). \end{aligned}$$
(3.12)

Now, we are left with verifying that the limit function u satisfies the initial conditions, that is,

$$ u(0)=u_{0} \quad \text{in } H^{1}_{0}(\varOmega ), \quad\quad u_{t}(0)=u_{1} \quad \text{in } L^{2}(\varOmega ). $$

From (3.8), (3.9), and Lion’s lemma [22], we get

$$\begin{aligned} u^{m} \to u \quad \text{in } C\bigl([0,T]; L^{2}(\varOmega )\bigr). \end{aligned}$$
(3.13)

Thus, \(u^{m} (0)\to u(0)\) in \(L^{2}(\varOmega ) \). Since \(u^{m}(0)=u^{m}_{0} \to u_{0}\) in \(H^{1}_{0}(\varOmega ) \), we observe that

$$ u(0) = u_{0} \quad \text{in } H^{1}_{0}( \varOmega ) . $$
(3.14)

Next, multiplying (3.4) by \(\phi \in C^{\infty }_{0} (0,T)\) and integrating it over \((0,T)\), we find

$$\begin{aligned}& - \int ^{T}_{0} \bigl( u^{m}_{t}(t), w \phi '(t)\bigr)\,dt + \int ^{T}_{0} \bigl( \nabla u^{m}(t), \nabla w \phi (t)\bigr)\,dt \\& \quad{} - \int ^{T}_{0} \int ^{t}_{0} g(t-s) \bigl( \nabla u^{m}( \tau ), \nabla w \phi (t)\bigr) \,d\tau \,dt \\& \quad{} - \int ^{T}_{0} \bigl( \nabla u^{m}(t) , \nabla w \phi '(t) \bigr)\,dt = \int ^{T}_{0} \bigl( \bigl\vert v(t) \bigr\vert ^{p-2} v(t) \ln \bigl\vert v(t) \bigr\vert , w \phi (t) \bigr)\,dt \quad \text{for } w \in W_{m} . \end{aligned}$$

Letting \(m\to \infty \), we get

$$\begin{aligned}& - \int ^{T}_{0} \bigl( u_{t}(t), w \phi '(t)\bigr)\,dt + \int ^{T}_{0} \bigl( \nabla u(t), \nabla w \phi (t) \bigr)\,dt \\& \quad{} - \int ^{T}_{0} \int ^{t}_{0} g(t-s) \bigl( \nabla u(\tau ), \nabla w \phi (t)\bigr) \,d\tau \,dt \\& \quad{} - \int ^{T}_{0} \bigl( \nabla u(t) , \nabla w \phi '(t) \bigr)\,dt = \int ^{T}_{0} \bigl( \bigl\vert v(t) \bigr\vert ^{p-2} v(t) \ln \bigl\vert v(t) \bigr\vert , w \phi (t) \bigr)\,dt \quad \text{for } w \in H^{1}_{0}(\varOmega ) . \end{aligned}$$

This yields \(u_{tt} \in L^{2}(0,T ; H^{-1}(\varOmega ) ) \). This and the fact that \(u_{t} \in L^{2}(0,T; H^{1}_{0}(\varOmega ))\) imply that

$$ u_{t} \in C\bigl([0,T] ; H^{-1}(\varOmega ) \bigr). $$

Thus, \(u^{m}_{t}(0) \to u_{t}(0)\) in \(H^{-1}(\varOmega ) \). Owing to \(u^{m}_{t}(0)= u^{m}_{1} \to u_{1}\) in \(L^{2}(\varOmega ) \), we conclude

$$ u_{t}(0) = u_{1} \quad \text{in } L^{2}(\varOmega ). $$

Uniqueness. Let u and ũ be the solutions of the linearized problem (3.1)–(3.3) and \(w=u - \tilde{u}\). Then w satisfies

$$\begin{aligned}& w_{tt} - \Delta w + \int ^{t}_{0} g(t-s) \Delta w(s)\,ds - \Delta w_{t} = 0 \quad \text{in } \varOmega \times (0, T), \\& w = 0 \quad \text{on } \partial \varOmega \times (0, T), \\& w(0) = 0, \quad\quad w_{t}(0)= 0 \quad \text{on } \varOmega . \end{aligned}$$

By the same arguments of (3.6), we observe

$$\begin{aligned} \bigl\Vert w_{t} (t) \bigr\Vert ^{2} + l \bigl\Vert \nabla w (t) \bigr\Vert ^{2} + (g\circ \nabla w) (t) + 2 \int ^{t}_{0} \bigl\Vert \nabla w_{t}(s) \bigr\Vert ^{2}\,ds \leq 0, \end{aligned}$$

and hence \(w \equiv 0 \). This completes the proof. □

Now, we are ready to prove the local existence of problem (1.1)–(1.3).

Theorem 3.1

Assume that\((H_{1})\)and\((H_{2})\)hold. Then, for the initial data\(u_{0} \in H^{1}_{0} (\varOmega )\), \(u_{1} \in L^{2}(\varOmega ) \), there exists a unique solutionuof problem (1.1)(1.3).

Proof

Existence. For \(M>0\) large enough and \(T>0\), we let

$$ \mathcal{M}_{T} = \bigl\{ u \in \mathcal{H}: \Vert u \Vert _{\mathcal{H}} \leq M \bigr\} . $$

For a given \(v \in \mathcal{H}\), there exists a unique solution u of problem (3.1)–(3.3). So, we can define a map \(S: \mathcal{M}_{T} \to \mathcal{H} \) by \(S(v) =u \). We will show that the map S is a contraction mapping on \(\mathcal{M}_{T}\). By a similar computation to that of (3.6), we find

$$\begin{aligned}& \bigl\Vert u_{t} (t) \bigr\Vert ^{2} + l \bigl\Vert \nabla u (t) \bigr\Vert ^{2} + (g\circ \nabla u) (t) + 2 \int ^{t}_{0} \bigl\Vert \nabla u_{t}(s) \bigr\Vert ^{2}\,ds \\& \quad \leq \Vert u_{1} \Vert ^{2} + \Vert \nabla u_{0} \Vert ^{2} + 2 \int ^{t}_{0} \bigl\Vert \bigl\vert v(t) \bigr\vert ^{p-2}v(t) \ln \bigl\vert v(t) \bigr\vert \bigr\Vert _{\frac{p}{p-1}} \bigl\Vert u_{t} (t) \bigr\Vert _{p}\,ds \\& \quad \leq \Vert u_{1} \Vert ^{2} + \Vert \nabla u_{0} \Vert ^{2} + 2 \int ^{t}_{0} \bigl\Vert \nabla u_{t} (s) \bigr\Vert \,ds \\& \quad\quad{} + 2 \int ^{t}_{0} \biggl\{ \biggl( \frac{1}{e(p-1)} \biggr)^{ \frac{p}{p-1}} \vert \varOmega _{1} \vert + c_{\frac{p(p-1+\mu )}{p-1}}^{ \frac{p(p-1+\mu )}{p-1}} \biggl( \frac{1}{e \mu } \biggr)^{\frac{p}{p-1}} \bigl\Vert \nabla v(s) \bigr\Vert ^{\frac{p(p-1+\mu )}{p-1}} \biggr\} ^{\frac{2(p-1)}{p}}\,ds \\& \quad \leq \Vert u_{1} \Vert ^{2} + \Vert \nabla u_{0} \Vert ^{2} + 2 \int ^{t}_{0} \bigl\Vert \nabla u_{t} (s) \bigr\Vert \,ds + CT\bigl(1+M^{2(p-1 + \mu )}\bigr) , \end{aligned}$$

we used \(v \in \mathcal{M}_{T}\) in the last inequality. Thus, we see

$$\begin{aligned} \bigl\Vert u_{t} (t) \bigr\Vert ^{2} + l \bigl\Vert \nabla u (t) \bigr\Vert ^{2} + (g\circ \nabla u) (t) \leq \Vert u_{1} \Vert ^{2} + \Vert \nabla u_{0} \Vert ^{2} + CT\bigl(1+M^{2(p-1 + \mu )}\bigr). \end{aligned}$$
(3.15)

We take \(M>0\) large enough so that

$$ \Vert u_{1} \Vert ^{2} + \Vert \nabla u_{0} \Vert ^{2} \leq \frac{M^{2}}{2}, $$

then we choose \(T>0\) sufficiently small so that

$$ CT\bigl(1+M^{2(p-1 + \mu )}\bigr) \leq \frac{M^{2}}{2}. $$

From (3.15), we have \(\Vert u \Vert _{\mathcal{H}} \leq M \), that is,

$$ S(\mathcal{M}_{T}) \subset \mathcal{M}_{T}. $$

It remains to show that S is a contraction mapping. Let \(v_{1}, v_{2} \in \mathcal{M}_{T}\), \(u=S(v_{1})\), \(w=S(v_{2})\) and \(z=u-w\). Then z satisfies

$$\begin{aligned}& z_{tt} - \Delta z + \int ^{t}_{0} g(t-s) \Delta z(s)\,ds - \Delta z_{t} \\& \quad = \vert v_{1} \vert ^{p-2} v_{1} \ln \vert v_{1} \vert - \vert v_{2} \vert ^{p-2} v_{2} \ln \vert v_{2} \vert \quad \text{in } \varOmega \times (0, T), \end{aligned}$$
(3.16)
$$\begin{aligned}& z = 0 \quad \text{on } \partial \varOmega \times (0, T), \end{aligned}$$
(3.17)
$$\begin{aligned}& z(0) = 0, \quad\quad z_{t}(0)= 0 \quad \text{on } \varOmega . \end{aligned}$$
(3.18)

Multiplying \(z_{t}\) in (3.16) and integrating it over \((0,t)\),

$$\begin{aligned}& \Vert z_{t} \Vert ^{2} + l \Vert \nabla z \Vert ^{2} + (g\circ \nabla z) + 2 \int ^{t}_{0} \bigl\Vert \nabla z_{t}(s) \bigr\Vert ^{2}\,ds \\& \quad \leq 2 \int ^{t}_{0} \int _{\varOmega } \bigl( \vert v_{1} \vert ^{p-2} v_{1} \ln \vert v_{1} \vert - \vert v_{2} \vert ^{p-2} v_{2} \ln \vert v_{2} \vert \bigr) z_{t}\,dx \,ds \\& \quad = 2 \int ^{t}_{0} \int _{\varOmega } \bigl( (p-1) \vert \zeta \vert ^{p-2} \ln \vert \zeta \vert + \vert \zeta \vert ^{p-2} \bigr) (v_{1} - v_{2}) z_{t}\,dx \,ds \\& \quad = 2 \int ^{t}_{0} \int _{\varOmega } \vert \zeta \vert ^{p-2} (v_{1} - v_{2}) z_{t}\,dx \,ds + 2(p-1) \int ^{t}_{0} \int _{\varOmega } \vert \zeta \vert ^{p-2} \ln \vert \zeta \vert (v_{1} - v_{2}) z_{t}\,dx \,ds \\& \quad : = \varXi _{1} + \varXi _{2} , \end{aligned}$$
(3.19)

where \(\zeta = \theta v_{1} + (1-\theta ) v_{2}\), here \(0 < \theta <1\). Young’s inequality yields

$$\begin{aligned} \varXi _{1} \leq & 2 \int ^{t}_{0} \Vert \zeta \Vert _{n(p-2)}^{p-2} \Vert v_{1} - v_{2} \Vert _{ \frac{2n}{n-2}} \Vert z_{t} \Vert \,ds \\ \leq & 2 c_{\frac{2n}{n-2}} c_{n(p-2)}^{p-2} \int ^{t}_{0} \Vert \nabla \zeta \Vert ^{p-2} \Vert \nabla v_{1} - \nabla v_{2} \Vert \Vert z_{t} \Vert \,ds \\ \leq & C \int ^{t}_{0} \Vert \nabla \zeta \Vert ^{2(p-2)} \Vert \nabla v_{1} - \nabla v_{2} \Vert ^{2}\,ds + \int ^{t}_{0} \Vert \nabla z_{t} \Vert ^{2}\,ds \\ \leq & C M^{2(p-2)}T \Vert v_{1} -v_{2} \Vert ^{2}_{\mathcal{H}} + \int ^{t}_{0} \Vert \nabla z_{t} \Vert ^{2}\,ds \end{aligned}$$
(3.20)

and

$$\begin{aligned} \varXi _{2} \leq & 2(p-1) \int ^{t}_{0} \bigl\Vert \vert \zeta \vert ^{p-2} \ln \vert \zeta \vert \bigr\Vert _{n} \Vert v_{1} - v_{2} \Vert _{\frac{2n}{n-2}} \Vert z_{t} \Vert \,ds \\ \leq & 2(p-1)c_{\frac{2n}{n-2}} \int ^{t}_{0} \bigl\Vert \vert \zeta \vert ^{p-2} \ln \vert \zeta \vert \bigr\Vert _{n} \Vert \nabla v_{1} - \nabla v_{2} \Vert \Vert z_{t} \Vert \,ds . \end{aligned}$$
(3.21)

Since \(p-2 < \frac{2}{n-2}\), there exists \(\eta >0\) such that \(n(p-2+ \eta ) < \frac{2n}{n-2}\). By similar arguments to (3.7), we deduce

$$\begin{aligned} \bigl\Vert \vert \zeta \vert ^{p-2} \ln \vert \zeta \vert \bigr\Vert _{n}^{n} \leq & \biggl( \frac{1}{e(p-1)} \biggr)^{n} \vert \varOmega _{1} \vert + \int _{\varOmega } \bigl( \vert \zeta \vert ^{-\eta } \ln \vert \zeta \vert \bigr)^{n} \vert \zeta \vert ^{n(p-2 + \eta )} \,dx \\ \leq & \biggl( \frac{1}{e(p-1)} \biggr)^{n} \vert \varOmega _{1} \vert + c_{n(p-2 + \eta )}^{n(p-2 + \eta )} \biggl( \frac{1}{e \eta } \biggr)^{n} \Vert \nabla \zeta \Vert ^{n(p-2 + \eta )} \\ \leq &C \bigl(1+ M^{n(p-2 + \eta )} \bigr). \end{aligned}$$

Applying this to (3.21), we get

$$\begin{aligned} \varXi _{2} \leq & C \bigl( 1+ M^{(p-2 + \eta )} \bigr) \int ^{t}_{0} \Vert v_{1} - v_{2} \Vert _{ \mathcal{H}} \Vert \nabla z_{t} \Vert \,ds \\ \leq & C T \bigl( 1+ M^{(p-2 + \eta )} \bigr) \Vert v_{1} - v_{2} \Vert _{\mathcal{H}}^{2} + \int ^{t}_{0} \Vert \nabla z_{t} \Vert ^{2}\,ds. \end{aligned}$$
(3.22)

Collecting (3.19), (3.20), (3.22), we arrive at

$$\begin{aligned} \bigl\Vert z_{t} (t) \bigr\Vert ^{2} + l \bigl\Vert \nabla z (t) \bigr\Vert ^{2} + ( g\circ \nabla z) (t) \leq C T \bigl( 1+ M^{2(p-2)} + M^{(p-2 + \eta )} \bigr) \Vert v_{1} - v_{2} \Vert _{ \mathcal{H}}^{2} . \end{aligned}$$
(3.23)

Taking \(T>0\) sufficiently small so that \(C T ( 1+ M^{2(p-2)} + M^{(p-2 + \eta )} ) <1 \), we conclude

$$ \bigl\Vert S(v_{1}) -S(v_{2}) \bigr\Vert _{\mathcal{H}} < \Vert v_{1} - v_{2} \Vert _{ \mathcal{H}} . $$

Thus, the contraction mapping principle ensures the existence of weak solutions.

Uniqueness. Let w and z be the solutions of problem (1.1)–(1.3) and \(U= w-z\). Then U satisfies

$$\begin{aligned}& U_{tt} - \Delta U + \int ^{t}_{0} g(t-s) \Delta U(s)\,ds - \Delta U_{t} = \vert w \vert ^{p-2} w \ln \vert w \vert - \vert z \vert ^{p-2} z \ln \vert z \vert \quad \text{in } \varOmega \times (0, T), \\& U = 0 \quad \text{on } \partial \varOmega \times (0, T), \\& U(0) = 0, \quad\quad U_{t}(0)= 0 \quad \text{on } \varOmega . \end{aligned}$$

By the same arguments as of (3.19), (3.20) and (3.21), we observe

$$\begin{aligned} \bigl\Vert U_{t} (t) \bigr\Vert ^{2} + \bigl\Vert \nabla U (t) \bigr\Vert ^{2} \leq C \int ^{t}_{0} \bigl( \bigl\Vert U_{t} (s) \bigr\Vert ^{2} + \bigl\Vert \nabla U (s) \bigr\Vert ^{2} \bigr)\,ds . \end{aligned}$$

Gronwall’s inequality gives \(U \equiv 0 \). This completes the proof. □

Finite time blow-up of solutions

In this section we establish the blow-up of the weak solution for problem (1.1)–(1.3). For this purpose, we set the following functionals:

$$\begin{aligned}& J(v) = \frac{1}{2} \biggl( 1 - \int ^{\infty }_{0} g(s)\,ds \biggr) \Vert \nabla v \Vert ^{2} - \frac{1}{p} \int _{\varOmega } \bigl\vert v(x) \bigr\vert ^{p} \ln \bigl\vert v(x) \bigr\vert \,dx + \frac{1}{p^{2}} \Vert v \Vert ^{p}_{p}, \end{aligned}$$
(4.1)
$$\begin{aligned}& I(v) = \biggl( 1 - \int ^{\infty }_{0} g(s)\,ds \biggr) \Vert \nabla v \Vert ^{2} - \int _{\varOmega } \bigl\vert v(x) \bigr\vert ^{p} \ln \bigl\vert v(x) \bigr\vert \,dx , \end{aligned}$$
(4.2)

then

$$\begin{aligned} J(v) = \biggl( \frac{1}{2} - \frac{1}{p} \biggr) \biggl( 1 - \int ^{\infty }_{0} g(s)\,ds \biggr) \Vert \nabla v \Vert ^{2} + \frac{1}{p} I(v) + \frac{1}{p^{2}} \Vert v \Vert ^{p}_{p}. \end{aligned}$$
(4.3)

Define the potential depth as

$$ d = \inf_{v \in H^{1}_{0}(\varOmega ) \setminus \{0\} } \sup_{\lambda > 0} J( \lambda v), $$
(4.4)

then, see e.g. [2325],

$$ 0 < d = \inf_{v \in \mathcal{N}} J(v), $$
(4.5)

where \(\mathcal{N}\) is the well-known Nehari manifold given by

$$ \mathcal{N} = \bigl\{ v \in H^{1}_{0}(\varOmega ) \setminus \{0\} \mid I(v)=0\bigr\} . $$

Lemma 4.1

For any\(v \in H^{1}_{0}(\varOmega ) \setminus \{0\} \), there exists a unique\(\lambda _{*} >0\)such that

$$\begin{aligned} I(\lambda v) = \lambda \frac{\partial J(\lambda v)}{\partial \lambda } \textstyle\begin{cases} >0, &0 < \lambda < \lambda _{*}, \\ =0, &\lambda = \lambda _{*}, \\ < 0, &\lambda > \lambda _{*}. \end{cases}\displaystyle \end{aligned}$$
(4.6)

Proof

For \(\lambda >0 \), we have

$$\begin{aligned} \frac{\partial }{\partial \lambda } J(\lambda v) = & \lambda \biggl\{ \biggl( 1 - \int ^{\infty }_{0} g(s)\,ds \biggr) \Vert \nabla v \Vert ^{2} - \lambda ^{p-2} \int _{\varOmega } \bigl\vert v(x) \bigr\vert ^{p} \ln \bigl\vert v(x) \bigr\vert \,dx - \lambda ^{p-2} \ln \lambda \Vert v \Vert ^{p}_{p} \biggr\} \\ : = & \lambda K(\lambda v). \end{aligned}$$
(4.7)

By simple calculation, we also get

$$\begin{aligned} \frac{\partial }{\partial \lambda } K(\lambda v) &= - \lambda ^{p-3} \biggl\{ (p-2) \int _{\varOmega } \bigl\vert v(x) \bigr\vert ^{p} \ln \bigl\vert v(x) \bigr\vert \,dx + (p-2) \ln \lambda \Vert v \Vert ^{p}_{p} + \Vert v \Vert ^{p}_{p} \biggr\} \\ & \textstyle\begin{cases} >0, & 0 < \lambda < \lambda _{1}, \\ =0, & \lambda = \lambda _{1}, \\ < 0, & \lambda > \lambda _{1}, \end{cases}\displaystyle \end{aligned}$$

where

$$ \lambda _{1} = \exp \biggl( \frac{(p-2) \int _{\varOmega } \vert v(x) \vert ^{p} \ln \vert v(x) \vert \,dx + \Vert v \Vert ^{p}_{p} }{ (2-p) \Vert v \Vert ^{p}_{p} } \biggr) < 1. $$
(4.8)

Since \(\lim_{\lambda \to 0^{+}} K(\lambda v) = ( 1 - \int ^{\infty }_{0} g(s)\,ds ) \Vert \nabla v \Vert ^{2} \geq 0 \) and \(\lim_{\lambda \to + \infty } K(\lambda v) = -\infty \), there exists a unique \(\lambda _{*} > \lambda _{1}\) such that \(K(\lambda _{*} v) =0 \). From this and (4.7), we have

$$\begin{aligned} \frac{\partial J(\lambda v)}{\partial \lambda } \textstyle\begin{cases} >0, &0 < \lambda < \lambda _{*}, \\ =0, &\lambda = \lambda _{*}, \\ < 0, & \lambda > \lambda _{*}. \end{cases}\displaystyle \end{aligned}$$

Noting that \(I(\lambda v) = \lambda \frac{\partial J(\lambda v)}{\partial \lambda } \), which is verified by a direct computation, we complete the proof. □

Now, we define the energy for problem (1.1)–(1.3) by

$$\begin{aligned} E(t) = & \frac{1}{2} \bigl\Vert u_{t} (t) \bigr\Vert ^{2} +\frac{1}{2} \biggl( 1 - \int ^{t}_{0} g(s)\,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + \frac{1}{2} (g\circ \nabla u) (t) \\ &{} - \frac{1}{p} \int _{\varOmega } \bigl\vert u(x,t) \bigr\vert ^{p} \ln \bigl\vert u(x,t) \bigr\vert \,dx + \frac{1}{p^{2}} \bigl\Vert u(t) \bigr\Vert ^{p}_{p}, \end{aligned}$$
(4.9)

then

$$\begin{aligned} E(t) \geq J\bigl(u(t)\bigr) + \frac{1}{2} \bigl\Vert u_{t} (t) \bigr\Vert ^{2} + \frac{1}{2} (g \circ \nabla u) (t) \geq J\bigl(u(t)\bigr). \end{aligned}$$
(4.10)

Replacing w in (2.3) by \(u_{t} (t)\) and using \((H_{2})\), one sees

$$\begin{aligned} \frac{d}{dt}E(t) + \bigl\Vert \nabla u_{t} (t) \bigr\Vert ^{2} = \frac{1}{2} \bigl(g' \circ \nabla u(t)\bigr) (t) - \frac{g(t)}{2} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \leq 0 \end{aligned}$$

and hence

$$\begin{aligned} E(t) + \int ^{t}_{0} \bigl\Vert \nabla u_{t} (s) \bigr\Vert ^{2}\,ds \leq E(0) \quad \text{for } 0 \leq t < T_{\max }, \end{aligned}$$
(4.11)

where \(T_{\max }\) is the maximal existence time of the solution u of problem (1.1)–(1.3).

Lemma 4.2

Let\((H_{1})\)and\((H_{2})\)hold. If\(I(u_{0}) <0\)and\(E(0) < d \), then the solutionuof problem (1.1)(1.3) satisfies

$$ I\bigl(u(t)\bigr) < 0 \quad \textit{and}\quad E(t) < d \quad \textit{for } t \in [0, T_{\max }). $$
(4.12)

Proof

From (4.11), it is clear that \(E(t) < d\). Since \(I(u_{0}) <0\) and u is continuous on \([0, T_{\max })\),

$$ I\bigl(u(t)\bigr) < 0 \quad \text{for some interval } [0, t_{1}) \subset [0,T_{\max }). $$
(4.13)

Let \(t_{0}\) be the maximal time satisfying (4.13). Suppose \(t_{0} < T_{\max } \), then \(I(u(t_{0})) =0 \), that is,

$$ u(t_{0}) \in \mathcal{N}. $$

Thus, we have from (4.5)

$$ J\bigl(u(t_{0})\bigr) \geq \inf_{v \in \mathcal{N}} J(v)= d. $$

But this is a contradiction for

$$\begin{aligned} J\bigl(u(t_{0})\bigr) \leq E(t_{0}) \leq E(0) < d. \end{aligned}$$

 □

Theorem 4.1

Let the conditions\((H_{1})\)and\((H_{2})\)hold. Assume that\(I(u_{0}) <0 \), \(E(0) = \alpha d \), where\(\alpha <1 \), and the kernel functiongsatisfies

$$ \int ^{\infty }_{0} g(s)\,ds \leq \frac{ p-2}{ p-2 + \frac{1}{(1- \hat{\alpha } )^{2} p + 2 \hat{\alpha } (1- \hat{\alpha } ) } }, $$
(4.14)

where\(\hat{\alpha }= \max \{ 0, \alpha \}\). Moreover, suppose that\((u_{0}, u_{1}) >0 \)when\(E(0) =0 \). Then the solutionuof problem (1.1)(1.3) blows up in finite time.

Proof

By contradiction, suppose that the solution u is global. For any \(T>0\), we consider \(L: [0,T] \to \mathbb{R}^{+}\) defined by

$$ L(t) = \bigl\Vert u(t) \bigr\Vert ^{2} + \int ^{t}_{0} \bigl\Vert \nabla u(s) \bigr\Vert ^{2}\,ds + (T-t) \Vert \nabla u_{0} \Vert ^{2} + b (t+ T_{0})^{2}, $$
(4.15)

where \(T_{0}> 0\) and \(b \geq 0 \), which are specified later. Then

$$\begin{aligned}& L(t) > 0 \quad \text{for } t \in [0,T], \end{aligned}$$
(4.16)
$$\begin{aligned}& \begin{aligned}[b] L'(t) &= 2\bigl(u(t), u_{t}(t)\bigr) + \bigl\Vert \nabla u(t) \bigr\Vert ^{2} - \Vert \nabla u_{0} \Vert ^{2} + 2b(t+T_{0}) \\ &= 2\bigl(u(t), u_{t}(t)\bigr) + 2 \int ^{t}_{0} \bigl(\nabla u(s), \nabla u_{t}(s)\bigr)\,ds + 2b(t+T_{0}) , \end{aligned} \end{aligned}$$
(4.17)

and, from (1.1),

$$\begin{aligned} L''(t) = & 2 \bigl\Vert u_{t}(t) \bigr\Vert ^{2} - 2 \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + 2 \int ^{t}_{0} g(t-s) \bigl(\nabla u(t), \nabla u(s) \bigr)\,ds \\ &{} + 2 \int _{\varOmega } \bigl\vert u(x,t) \bigr\vert ^{p} \ln \bigl\vert u(x,t) \bigr\vert \,dx + 2b \\ = & 2 \bigl\Vert u_{t}(t) \bigr\Vert ^{2} - 2 \biggl( 1- \int ^{t}_{0} g(s)\,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &{} - 2 \int ^{t}_{0} g(t-s) \bigl(\nabla u(t), \nabla u(t) - \nabla u(s)\bigr)\,ds + 2 \int _{\varOmega } \bigl\vert u(x,t) \bigr\vert ^{p} \ln \bigl\vert u(x,t) \bigr\vert \,dx \\ &{}+ 2b , \end{aligned}$$
(4.18)

for almost every \(t \in [0,T]\). By the Cauchy–Schwartz inequality and (4.15), we see that

$$\begin{aligned} \frac{(L'(t))^{2}}{4} = & \biggl( \bigl(u(t), u_{t}(t)\bigr) + \int ^{t}_{0} \bigl( \nabla u(s), \nabla u_{t}(s)\bigr)\,ds + b(t+T_{0}) \biggr)^{2} \\ \leq & \biggl( \bigl\Vert u(t) \bigr\Vert ^{2} + \int ^{t}_{0} \bigl\Vert \nabla u(s) \bigr\Vert ^{2}\,ds + b(t+T_{0})^{2} \biggr) \biggl( \bigl\Vert u_{t}(t) \bigr\Vert ^{2} + \int ^{t}_{0} \bigl\Vert \nabla u_{t}(s) \bigr\Vert ^{2}\,ds + b \biggr) \\ = & \bigl( L(t) -(T-t) \Vert \nabla u_{0} \Vert ^{2} \bigr) \biggl( \bigl\Vert u_{t}(t) \bigr\Vert ^{2} + \int ^{t}_{0} \bigl\Vert \nabla u_{t}(s) \bigr\Vert ^{2}\,ds + b \biggr) \\ \leq & L(t) \biggl( \bigl\Vert u_{t}(t) \bigr\Vert ^{2} + \int ^{t}_{0} \bigl\Vert \nabla u_{t}(s) \bigr\Vert ^{2}\,ds + b \biggr) . \end{aligned}$$
(4.19)

Thus, we have from (4.18) and (4.19) that

$$\begin{aligned} L(t) L''(t) - \frac{p+2}{4} \bigl(L'(t)\bigr)^{2} \geq L(t) F(t), \end{aligned}$$
(4.20)

where

$$\begin{aligned} F(t) = & - p \bigl\Vert u_{t} (t) \bigr\Vert ^{2} - 2 \biggl( 1- \int ^{t}_{0} g(s)\,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + 2 \int _{\varOmega } \bigl\vert u(x,t) \bigr\vert ^{p} \ln \bigl\vert u(x,t) \bigr\vert \,dx \\ &{} - 2 \int ^{t}_{0} g(t-s) \bigl(\nabla u(t), \nabla u(t) - \nabla u(s)\bigr)\,ds - (p+2) \int ^{t}_{0} \bigl\Vert \nabla u_{t}(s) \bigr\Vert ^{2}\,ds -pb . \end{aligned}$$
(4.21)

Applying (4.9) to this and using (4.11) and Young’s inequality, we get

$$\begin{aligned} F(t) = & - 2p E(t) + (p-2) \biggl( 1- \int ^{t}_{0} g(s)\,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + p (g\circ \nabla u) (t) + \frac{2}{p} \bigl\Vert u(t) \bigr\Vert ^{p}_{p} \\ &{} - (p+2) \int ^{t}_{0} \bigl\Vert \nabla u_{t}(s) \bigr\Vert ^{2}\,ds - 2 \int ^{t}_{0} g(t-s) \bigl(\nabla u(t), \nabla u(t) - \nabla u(s)\bigr)\,ds - pb \\ \geq & - 2pE(0) + (p-2) \biggl( 1- \int ^{t}_{0} g(s)\,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + p (g\circ \nabla u) (t) + \frac{2}{p} \bigl\Vert u(t) \bigr\Vert ^{p}_{p} \\ & {}+ (p-2) \int ^{t}_{0} \bigl\Vert \nabla u_{t}(s) \bigr\Vert ^{2}\,ds - 2 \int ^{t}_{0} g(t-s) \bigl(\nabla u(t), \nabla u(t) - \nabla u(s)\bigr)\,ds - pb \\ \geq & - 2pE(0) + \biggl\{ (p-2) - \biggl(p-2 + \frac{1}{\epsilon } \biggr) \int ^{t}_{0} g(s)\,ds \biggr\} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + (p-\epsilon ) (g\circ \nabla u) (t) \\ & {}+ \frac{2}{p} \bigl\Vert u(t) \bigr\Vert ^{p}_{p} + (p-2) \int ^{t}_{0} \bigl\Vert \nabla u_{t}(s) \bigr\Vert ^{2}\,ds - pb, \end{aligned}$$
(4.22)

where \(\epsilon >0\). We now consider the initial energy \(E(0)\) divided into three cases: \(E(0)<0\), \(E(0)=0\), and \(0< E(0) < d\).

Case 1:\(\alpha <0\), i.e.\(E(0)<0\).

Taking \(\epsilon = p\) in (4.22) and choosing \(0 < b \leq -2E(0)\), we have from (4.14)

$$\begin{aligned} F(t) \geq & p\bigl( - 2E(0) -b \bigr) + \biggl\{ (p-2) - \biggl(p-2 + \frac{1}{p} \biggr) \int ^{t}_{0} g(s)\,ds \biggr\} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &{} + \frac{2}{p} \bigl\Vert u(t) \bigr\Vert ^{p}_{p} + (p-2) \int ^{t}_{0} \bigl\Vert \nabla u_{t}(s) \bigr\Vert ^{2}\,ds \geq 0. \end{aligned}$$
(4.23)

Case 2:\(\alpha = 0\), i.e.\(E(0)=0\).

Taking \(\epsilon = p\) in (4.22) and \(b=0\), we see from (4.14) that

$$\begin{aligned} F(t) \geq & \biggl\{ (p-2) - \biggl(p-2 + \frac{1}{p} \biggr) \int ^{t}_{0} g(s)\,ds \biggr\} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &{} + \frac{2}{p} \bigl\Vert u(t) \bigr\Vert ^{p}_{p} + (p-2) \int ^{t}_{0} \bigl\Vert \nabla u_{t}(s) \bigr\Vert ^{2}\,ds \geq 0 . \end{aligned}$$
(4.24)

Case 3:\(0< \alpha < 1\), i.e.\(0< E(0)< d\).

Taking \(\epsilon = (1-\alpha ) p + 2 \alpha \) in (4.22), we find

$$\begin{aligned} F(t) \geq & - 2p E(0) + \biggl\{ (p-2) - \biggl( p-2 + \frac{1}{(1-\alpha ) p + 2 \alpha } \biggr) \int ^{t}_{0} g(s)\,ds \biggr\} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &{} + \alpha (p-2) (g\circ \nabla u) (t) + \frac{2}{p} \bigl\Vert u(t) \bigr\Vert ^{p}_{p} + (p-2) \int ^{t}_{0} \bigl\Vert \nabla u_{t}(s) \bigr\Vert ^{2}\,ds - pb. \end{aligned}$$
(4.25)

Due to the condition (4.14), it follows that

$$\begin{aligned} (p-2) - \biggl( p-2 + \frac{1}{ (1-\alpha ) p + 2 \alpha } \biggr) \int ^{t}_{0} g(s)\,ds \geq \alpha (p-2) \biggl( 1 - \int ^{\infty }_{0} g(s)\,ds \biggr), \end{aligned}$$
(4.26)

and hence

$$\begin{aligned} F(t) \geq & - 2p E(0) + \alpha (p-2) \biggl\{ \biggl( 1- \int ^{ \infty }_{0} g(s)\,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + (g\circ \nabla u) (t) \biggr\} \\ & {}+ \frac{2}{p} \bigl\Vert u(t) \bigr\Vert ^{p}_{p} + (p-2) \int ^{t}_{0} \bigl\Vert \nabla u_{t}(s) \bigr\Vert ^{2}\,ds - pb \\ \geq & - 2p E(0) + \alpha (p-2) \biggl( 1- \int ^{\infty }_{0} g(s)\,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + \frac{2 \alpha }{p} \bigl\Vert u(t) \bigr\Vert ^{p}_{p} - pb . \end{aligned}$$
(4.27)

On the other hand, it is noted that \(I(u(t)) <0\) for all \(t \in [0,T]\) from Lemma 4.2. So, Lemma 4.1 ensures that the existence of \(\lambda _{*} \in (0,1)\) satisfying \(I(\lambda _{*} u(t)) =0 \). Hence, from (4.3) and (4.5)

$$\begin{aligned} d \leq & J\bigl(\lambda _{*} u(t)\bigr) = \biggl( \frac{1}{2} - \frac{1}{p} \biggr) \biggl( 1 - \int ^{\infty }_{0} g(s)\,ds \biggr) \lambda _{*}^{2} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + \frac{\lambda _{*}^{p}}{p^{2}} \bigl\Vert u(t) \bigr\Vert ^{p}_{p} \\ < & \frac{p-2}{2p} \biggl( 1 - \int ^{\infty }_{0} g(s)\,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + \frac{1}{p^{2}} \bigl\Vert u(t) \bigr\Vert ^{p}_{p} . \end{aligned}$$
(4.28)

Since u is continuous on \([0,T]\), there exists \(\kappa >0\) such that

$$\begin{aligned} d + \kappa < \frac{p-2}{2p} \biggl( 1 - \int ^{\infty }_{0} g(s)\,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + \frac{1}{p^{2}} \bigl\Vert u(t) \bigr\Vert ^{p}_{p} \quad \text{for all } t\in [0,T] . \end{aligned}$$

From this and (4.27), we get

$$\begin{aligned} F(t) \geq & - 2p \alpha d + 2\alpha p \biggl\{ \frac{p-2}{2p} \biggl( 1- \int ^{\infty }_{0} g(s)\,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + \frac{ 1}{p^{2}} \bigl\Vert u(t) \bigr\Vert ^{p}_{p} \biggr\} - pb \\ > & 2\alpha p \kappa -pb . \end{aligned}$$
(4.29)

Choosing \(b >0\) sufficiently small so that \(2\alpha p \kappa -pb \geq 0 \), we obtain

$$ F(t) \geq 0 . $$
(4.30)

Adapting (4.23), (4.24), (4.30) to (4.20), we infer

$$\begin{aligned} L(t) L''(t) - \frac{p+2}{4} \bigl(L'(t)\bigr)^{2} \geq 0 . \end{aligned}$$
(4.31)

Now it remains to show \(L'(0) >0 \). In the case of \(E(0) =0\), the condition \((u_{0}, u_{1}) >0\) gives

$$ L'(0) = 2 (u_{0}, u_{1}) >0. $$

For the cases of \(E(0)<0\) and \(0 < E(0) <d\), we choose \(T_{0} \) large enough so that

$$ L'(0) = 2 (u_{0}, u_{1}) + 2bT_{0} >0. $$

Thus, we conclude from Lemma 2.2 that

$$ \lim_{t \to T^{-}_{*}} L(t) = + \infty $$
(4.32)

for

$$ T_{*} \leq \frac{4L(0)}{(p-2)L'(0)} = \frac{ 2 \Vert u_{0} \Vert ^{2} + 2T \Vert \nabla u_{0} \Vert ^{2} + 2bT_{0}^{2} }{(p-2) ( (u_{0}, u_{1}) + bT_{0} ) }. $$

Thus, we deduce that

$$ T_{*} \leq \frac{ 2 \Vert u_{0} \Vert ^{2} + 2bT_{0}^{2} }{(p-2) (u_{0}, u_{1}) +(p-2) bT_{0} - 2 \Vert \nabla u_{0} \Vert ^{2} }. $$
(4.33)

From (4.15), (4.32) and (4.33), we have

$$ \lim_{t \to T^{-}_{*}} \biggl( \bigl\Vert u(t) \bigr\Vert + \int ^{t}_{0} \bigl\Vert \nabla u(s) \bigr\Vert ^{2}\,ds \biggr) = + \infty . $$

This contradicts our assumption that the weak solution is global. Thus, we conclude that the weak solution u to problem (1.1)–(1.3) blows up in finite time. □

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Acknowledgements

The authors would like to thank the reviewers for valuable comments and suggestions.

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The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1I1A3A01051714). The second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2020R1I1A3066250).

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Ha, T.G., Park, S. Blow-up phenomena for a viscoelastic wave equation with strong damping and logarithmic nonlinearity. Adv Differ Equ 2020, 235 (2020). https://doi.org/10.1186/s13662-020-02694-x

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MSC

  • 35L05
  • 35L70
  • 35L71
  • 35B44
  • 35B40

Keywords

  • Viscoelastic wave equation
  • Logarithmic nonlinearity
  • Local existence
  • Finite time blow-up