Blow up and global existence in a nonlinear viscoelastic wave equation

In this paper the nonlinear viscoelastic wave equation u(tt) - Deltau + integral(0)(t) g(t - tau)Deltau(tau) dtau + au(t) \u(t)\(m-2) = bu \u\(p-2) associated with initial and Dirichlet boundary conditions is considered. Under suitable conditions on g, it is proved that any weak solution with negative initial energy blows up in finite time if p > m. Also the case of a stronger damping is considered and it is showed that solutions exist globally for any initial data, in the appropriate space, provided that m > p. (C) 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.