Global existence and blow‐up of solutions for nonlinear viscoelastic wave equation with degenerate damping and source

In this paper we investigate the global existence and finite time blow‐up of solutions to the nonlinear viscoelastic equation \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$$ u_{tt}-\triangle u+\int _0^t g(t-s)\triangle u(s)\,\mathrm{d} s+|u|^k\partial j(u_t)=|u|^{p-1}u \quad {\rm in}\quad \Omega \times (0, T) $$\end{document} associated with initial and Dirichlet boundary conditions. Here ∂j denote the sub‐differential of j. Under suitable assumptions on g(·), j(·) and the parameters in the equation, we obtain the global existence of generalized solutions, weak solutions for the equation. The finite time blow‐up of weak solutions for the equation is also established provided the initial energy is negative and the exponent p is greater than the critical value k + m. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim