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

Abstract

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 t t ( x , t ) − Δ u ( x , t ) + ∫ 0 t g ( t − s ) Δ u ( x , s ) d s − Δ u t ( x , t ) = | u ( x , t ) | p − 2 u ( x , t ) ln | u ( x , t ) | $$ u_{tt}(x,t) - \Delta u (x,t) + \int ^{t}_{0} g(t-s) \Delta u(x,s)\,ds - \Delta u_{t} (x,t) = \bigl\vert u(x,t) \bigr\vert ^{p-2} u(x,t) \ln \bigl\vert u(x,t) \bigr\vert $$ in a bounded domain Ω ⊂ R n $\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.

Keywords

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