On wave equations of the [formula omitted]-Laplacian type with supercritical nonlinearities

Abstract

This article focuses on a quasilinear wave equation of p-Laplacian type: utt−Δpu−Δut=f(u)in a bounded domain Ω⊂R3 with a sufficiently smooth boundary Γ=∂Ω subject to a generalized Robin boundary condition featuring boundary damping and a nonlinear source term. The operator Δp, 2<p<3, denotes the classical p-Laplacian. The interior and boundary terms f(u), h(u) are sources that are allowed to have a supercritical exponent, in the sense that their associated Nemytskii operators are not locally Lipschitz from W1,p(Ω) into L2(Ω) or L2(Γ). Under suitable assumptions on the parameters we provide a rigorous proof of existence of a local weak solution which can be extended globally in time, provided the damping terms dominate the corresponding sources in an appropriate sense. Moreover, a blow-up result is proved for solutions with negative initial total energy.