Global nonexistence of solutions for a quasilinear wave equation with acoustic boundary conditions

Abstract

Abstract We consider the quasilinear wave equation u t t − △ u t − div ( | ∇ u | α − 2 ∇ u ) − div ( | ∇ u t | β − 2 ∇ u t ) + a | u t | m − 2 u t = b | u | p − 2 u $$u_{tt} -\triangle u_{t} -\operatorname{div}\bigl(\vert \nabla u\vert ^{\alpha-2} \nabla u\bigr) - \operatorname{div}\bigl(\vert \nabla u_{t}\vert ^{\beta-2} \nabla u_{t} \bigr) +a \vert u_{t}\vert ^{m-2} u_{t} =b|u|^{p-2} u $$ a , b > 0 $a,b>0$ , associated with initial and Dirichlet boundary conditions at one part and acoustic boundary conditions at another part, respectively. We prove, under suitable conditions on α, β, m, p and for negative initial energy, a global nonexistence of solutions.

Keywords

• quasilinear wave equation
• blow-up
• acoustic boundary