Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources

Abstract

This paper investigates a quasilinear wave equation with Kelvin-Voigt damping, utt − Δpu − Δut = f(u), in a bounded domain Ω ⊂ ℝ3 and subject to Dirichlét boundary conditions. The operator Δp, 2 < p < 3, denotes the classical p-Laplacian. The nonlinear term f(u) is a source feedback that is allowed to have a supercritical exponent, in the sense that the associated Nemytskii operator is not locally Lipschitz from W01,p(Ω) into L2(Ω). Under suitable assumptions on the parameters, we prove existence of local weak solutions, which can be extended globally provided the damping term dominates the source in an appropriate sense. Moreover, a blow-up result is proved for solutions with negative initial total energy.