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

Abstract

We consider the quasilinear wave equation \t\t\tutt−△ut−div(|∇u|α−2∇u)−div(|∇ut|β−2∇ut)+a|ut|m−2ut=b|u|p−2u\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$u_{tt} -\\triangle u_{t} -\\operatorname{div}\\bigl(\\vert \\nabla u\\vert ^{\\alpha-2} \\nabla u\\bigr) - \\operatorname{div}\\bigl(\\vert \\nabla u_{t}\\vert ^{\\beta-2} \\nabla u_{t} \\bigr) +a \\vert u_{t}\\vert ^{m-2} u_{t} =b|u|^{p-2} u $$\\end{document}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.