Asymptotic behaviour for a parabolic p-Laplacian equation with an advection force

Abstract

The rescaling method is presented to allow us establishing the interface functions and the nonnegative local solutions to the evolution of nonlinear degenerate parabolic p-Laplacian process with conservation laws that are posed in one-dimensional space. This equation has a self-similar solution which represents the main feature of this work. The Cauchy problem (CP) for this equation with specific restrictions in the range of parameters and the negative advection coefficient is considered. In this work, there are several regions to discuss the qualitative analysis for the local weak solutions and the asymptotic interfaces in the irregular domains. The dominating of the advection force over the p-Laplacian type diffusion will appear clearly in these regions. Moreover, the solutions of the CP for the degenerate parabolic p-Laplacian advection equations are asymptotically equal to the solutions of advection equations under some restrictions.