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

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.