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

Authors

Pei Pei, Earlham CollegeFollow
Mohammad A. Rammaha, University of Nebraska-LincolnFollow
Daniel Toundykov, University of Nebraska-LincolnFollow

Document Type

Article

Date of this Version

2015

Citation

JOURNAL OF MATHEMATICAL PHYSICS 56, 081503 (2015)
doi: 10.1063/1.4927688]

Comments

Copyright 2015 AIP Publishing LLC. Used by permission.

Abstract

This paper investigates a quasilinear wave equation with Kelvin-Voigt damping, u_tt − Δ_pu − Δu_t = f (u), in a bounded domain Ω ⊂ R³ and subject to Dirichlét boundary conditions. The operator Δ_p, 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 W₀^{1, 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.