Advances in Differential Equations

Hadamard well-posedness for wave equations with p-Laplacian damping and supercritical sources

Mohammad A. Rammaha and Zahava Wilstein

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Abstract

We study the global well-posedness of the nonlinear wave equation $$ u_{tt} - \Delta u - \Delta _p u_t = f(u) $$ in a bounded domain ${\Omega} \subset \mathbb{R}^n$ with Dirichlét boundary conditions. The nonlinearity $f(u)$ represents a strong source which is allowed to have a supercritical exponent; i.e., the Nemytski operator $f(u)$ is not locally Lipschitz from $H^1_0({\Omega})$ into $L^2({\Omega})$. The nonlinear term $- \Delta _p u_t $ is a strong damping where the $-\Delta _p$ denotes the p-Laplacian (defined below). Under suitable assumptions on the parameters and with careful analysis involving the theory of monotone operators, we prove the existence and uniqueness of a local weak solution. Also, such a unique solution depends continuously on the initial data from the finite energy space. In addition, we prove that weak solutions are global, provided the exponent of the damping term dominates the exponent of the source.

Article information

Source
Adv. Differential Equations Volume 17, Number 1/2 (2012), 105-150.

Dates
First available in Project Euclid: 17 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355703099

Mathematical Reviews number (MathSciNet)
MR2906731

Zentralblatt MATH identifier
1262.35151

Subjects
Primary: 35L05: Wave equation 35L20: Initial-boundary value problems for second-order hyperbolic equations 58J45: Hyperbolic equations [See also 35Lxx]

Citation

Rammaha, Mohammad A.; Wilstein, Zahava. Hadamard well-posedness for wave equations with p-Laplacian damping and supercritical sources. Adv. Differential Equations 17 (2012), no. 1/2, 105--150. https://projecteuclid.org/euclid.ade/1355703099.


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