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Article Contents

# Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping

• This paper presents a study of the nonlinear wave equation with $p$-Laplacian damping: $u_{tt} - \Delta u - \Delta _p u_t = f(u)$ evolving in a bounded domain $\Omega \subset \mathbb{R}^n$ with Dirichlet 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 $H{1\atop 0}(\Omega)$ into $L^2(\Omega)$. The nonlinear term $- \Delta _p u_t$ acts as a strong damping where the $-\Delta _p$ denotes the $p$-Laplacian. Under suitable assumptions on the parameters and with careful analysis involving the Nehari Manifold, we prove the existence of a global solution and estimate the decay rates of the energy.
Mathematics Subject Classification: Primary: 35L05, 35J92; Secondary: 35A01, 35D30, 35B35.

 Citation:

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