Propagation of regularity for solutions to a Kawahara equation in two spatial dimensions

Abstract

We prove a propagation of regularity result for solutions of an IVP for a Kawahara equation in two spatial dimensions. We show that if the initial data has sufficient smoothness and belongs to an appropriate Sobolev space when restricted to a half-plane {(x,y)|ax+by≥c0} for some a>0, b,c0∈R, then the solution u(t) lies in the same Sobolev space when restricted to the half-plane {(x,y)|ax+by≥c} for any real number c and any t such that 0<t≤T where T is the existence time of the solution. In addition, we prove a propagation of regularity result for initial data belonging to an anisotropic Sobolev space on a half-plane. These propagation of regularity results show that regularity of the initial data is transferred to the solution with infinite speed.