On wave equations of the p-Laplacian type with supercritical nonlinearities. (June 2019) - Kass, Nicholas J.; Rammaha, Mohammad A.
Abstract: This article focuses on a quasilinear wave equation of p -Laplacian type: u t t − Δ p u − Δ u t = f ( u ) in a bounded domain Ω ⊂ R 3 with a sufficiently smooth boundary Γ = ∂ Ω subject to a generalized Robin boundary condition featuring boundary damping and a nonlinear source term. The operator Δ p, 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 W 1, p ( Ω ) into L 2 ( Ω ) or L 2 ( 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.