On wave equations of the p-Laplacian type with supercritical nonlinearities. (June 2019)

• Record Type: Journal Article
• Authors: Kass, Nicholas J.; Rammaha, Mohammad A.
• Is Part Of: Nonlinear analysis. Volume 183(2019)
• Page Start: 70
• Page End: 101
• Publication Date: 2019-06
• Subjects: primary 35L05 35L20 35L72 -- secondary 58J45; Wave equation -- p-Laplacian -- Supercritical sources -- Local existence -- Generalized Robin condition; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.01.005
• Languages: English
• ISSNs: 0362-546X

Abstract

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.