Global Weak Solution, Uniqueness and Exponential Decay for a Class of Degenerate Hyperbolic Equation

Year 2022, Volume: 5 Issue: 3, 137 - 149, 30.09.2022

Ducival Pereira Carlos Raposo

https://doi.org/10.33434/cams.1012330

Abstract

This paper deals with existence, uniqueness and energy decay of solutions to a degenerate hyperbolic equations given by
\begin{align*}
K(x,t)u'' - M\left(\int_\Omega |\nabla u|^2\,dx \right) \Delta u - \Delta u' = 0,
\end{align*}
with operator coefficient $K(x,t)$ satisfying suitable properties and $M(\,\cdot \,) \in C^1([0, \infty))$ is a function which greatest lower bound for $ M (\,\cdot\,) $ is zero. For global weak solution and uniqueness we use the Faedo-Galerkin method. Exponential decay is proven by using a theorem due to M. Nakao.

Keywords

Degenerate hyperbolic equations , global weak solution , exponential decay.

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