Blow-up of a hyperbolic equation of viscoelasticity with supercritical nonlinearities

We investigate a hyperbolic PDE, modeling wave propagation in viscoelastic media, under the influence of a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as an energy-amplifying supercritical nonlinear source: {u(tt) - k(0)Delta u - integral(infinity)(0) k'(s)Delta u(t-s)ds + [u(t)](m-1) u(t) = vertical bar u vertical bar(P-1)u, in Omega x (0, T), u(x, t) = u(0)(x, t), in Omega x (-infinity, 0], where Omega is a bounded domain in R-3 with a Dirichlet boundary condition. The relaxation kernel k is mono-tone decreasing and k(infinity) = 1. We study blow-up of solutions when the source is stronger than dissipations, i.e., p > max{m, root K(0)}, under two different scenarios: first, the total energy is negative, and the second, the total energy is positive with sufficiently large quadratic energy. This manuscript is a follow-up work of the paper [30] in which Hadamard well-posedness of this equation has been established in the finite energy space. The model under consideration features a supercritical source and a linear memory that accounts for the full past history as time goes to -infinity, which is distinct from other relevant models studied in the literature which usually involve subcritical sources and a finite-time memory. (C) 2016 Elsevier Inc. All rights reserved.