The Development of interfaces in a Parabolic p-Laplacian type diffusion equation with weak convection

This work has the objective to analyse the initial growth of interface and structure of nonnegative weak solution for one-dimensional parabolic p-Laplacian type diffusion-convection with non-positive convection coefficient c. In this situation, the interfaces may expand, shrink or remain stationary relying on the competition between these two factors. In this paper, we concentrate on three regions to classify the behavior of local solutions near the asymptotic interface in the irregular domain. In the first and second regions, the slow diffusion dominates over the convection term with expanding interfaces under some restrictions. In the third region, the slow diffusion dominates over the convection, but the interfaces have a waiting time. In our proof, the rescaling method and blow-up techniques are applied.