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

In this paper we consider the initial boundary value problem for a viscoelastic wave equation with strong damping and logarithmic nonlinearity of the form utt(x,t)−Δu(x,t)+∫0tg(t−s)Δu(x,s)ds−Δut(x,t)=|u(x,t)|p−2u(x,t)ln|u(x,t)|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u_{tt}(x,t) - \Delta u (x,t) + \int ^{t}_{0} g(t-s) \Delta u(x,s)\,ds - \Delta u_{t} (x,t) = \bigl\vert u(x,t) \bigr\vert ^{p-2} u(x,t) \ln \bigl\vert u(x,t) \bigr\vert $$\end{document} in a bounded domain Ω⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varOmega \subset {\mathbb{R}}^{n} $\end{document}, 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: where Ω ⊂ R n , n ≥ 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 [1][2][3] and the references therein. During the past decades, the strongly damped wave equations with source effect u ttuω u t + μu 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) = |u| p-2 u, Sattinger [4] firstly considered the existence of local as well as global solutions for equation (1.4) with ω = μ = 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 [5][6][7][8][9][10][11][12][13][14][15]. Ikehata [8] gave properties of decay estimates and blow-up of solutions to (1.4) with linear damping (ω = 0 and μ > 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 (ω > 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 |u| k , Ma and Fang [16] proved the existence of global solutions and infinite time blow-up to problem (1.4) with ω = 1, μ = 0, and k = 2. Lian and Xu [17] investigated global existence, energy decay and infinite time blow-up when ω ≥ 0 and μ > -ωλ 1 , where λ 1 is the first eigenvalue of the operatorunder 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 |u| p-2 u ln |u|. Recently, Di et al. [18] considered problem (1.1)-(1.3) when the kernel function g = 0. The presence of the Laplacian operatoru and the logarithmic nonlinearity |u| p-2 u ln |u| 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 Ω ⊂ 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, · X denotes the norm of X. As usual, (·, ·) and ·, · denote the inner product in the space L 2 (Ω) and the duality pairing between H -1 (Ω) and H 1 0 (Ω), respectively. · q denotes the norm of the space L q (Ω). For brevity, we denote · 2 by · . Let c q be the best constants in the Poincaré type inequality We need the following lemma.
Proof We can easily show this from simple calculation. So, we omit it here. With regard to problem (1.1)-(1.3), we impose the following assumptions: (H 1 ) Hypotheses on p. The exponent p satisfies for any w ∈ H 1 0 (Ω) and t ∈ (0, T), and

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 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
such that u t ∈ L 2 ([0, T]; H 1 0 (Ω)) and Proof Existence. Let {w j } j∈N be an orthogonal basis of H 1 0 (Ω) which is orthonormal in L 2 (Ω) and W m = span{w 1 , w 2 , . . . , w m }, then there exist subsequences u m 0 ∈ W m and u m 1 ∈ W m such that u m 0 → u 0 in H 1 0 (Ω) and u m 1 → u 1 in L 2 (Ω), respectively. We will seek an approximate solution (3.4) and the initial conditions Since (3.4)-(3.5) is a normal system of ordinary differential equations, there exists a solu- 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 Integrating this over (0, t) and making use of (H 2 ), In order to estimate the last term in the right hand side of (3.6), we let Since 2 < p < 2n n-2 , we can take μ > 0 such that 2 < p + μp p-1 < 2n n-2 . Applying Lemma 2.1, we infer that we used the fact that v ∈ 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 Adapting this to (3.6), we get Step 2.

Uniqueness. Let u andũ be the solutions of the linearized problem (3.1)-(3.3) and w = u -ũ. Then w satisfies
By the same arguments of (3.6), we observe We will show that the map S is a contraction mapping on M T . By a similar computation to that of (3.6), we find we used v ∈ M T in the last inequality. Thus, we see u t (t) 2 + l ∇u(t) 2 + (g • ∇u)(t) ≤ u 1 2 + ∇u 0 2 + CT 1 + M 2(p-1+μ) . (3.15) We take M > 0 large enough so that then we choose T > 0 sufficiently small so that From (3.15), we have u H ≤ M, that is, It remains to show that S is a contraction mapping.
Thus, we conclude from Lemma 2.2 that 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.