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Global existence and blow‐up of solutions for nonlinear viscoelastic wave equation with degenerate damping and source
Author(s) -
Han Xiaosen,
Wang Mingxin
Publication year - 2011
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200810168
Subject(s) - mathematics , degenerate energy levels , viscoelasticity , mathematical analysis , dirichlet boundary condition , nonlinear system , exponent , boundary value problem , wave equation , physics , thermodynamics , linguistics , philosophy , quantum mechanics
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