$ L^1 $ local stability to a nonlinear shallow water wave model

Abstract

<p>A nonlinear shallow water wave equation containing the famous Degasperis$ - $Procesi and Fornberg$ - $Whitham models is investigated. The novel derivation is that we establish the $ L^2 $ bounds of solutions from the equation if its initial value belongs to space $ L^2(\mathbb{R}) $. The $ L^{\infty} $ bound of the solution is derived. The techniques of doubling the space variable are employed to set up the $ L^1 $ local stability of short time solutions.</p>