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

Authors : Ducival PEREİRA, Carlos RAPOSO

Pages : 137-149

Doi:10.33434/cams.1012330

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Publication Date : 2022-09-30

Article Type : Research Paper

Abstract :This paper deals with existence, uniqueness and energy decay of solutions to a degenerate hyperbolic equations given by \begin{align*} Kinsert ignore into journalissuearticles values(x,t);u`` - M\leftinsert ignore into journalissuearticles values(\int_\Omega |\nabla u|^2\,dx \right); \Delta u - \Delta u` = 0, \end{align*} with operator coefficient $Kinsert ignore into journalissuearticles values(x,t);$ satisfying suitable properties and $Minsert ignore into journalissuearticles values(\,\cdot \,); \in C^1insert ignore into journalissuearticles values([0, \infty););$ is a function which greatest lower bound for $ M insert ignore into journalissuearticles values(\,\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