Search

A non-linear PDE featuring flux limitation effects together with those of the porous media equation (non-linear Fokker-Planck) is presented in this paper. We analyze the balance of such diverse effects through the study of the existence and qualitative behavior of some admissible patterns, namely traveling wave solutions, to this singular reaction-diffusion equation. We show the existence and qualitative behavior of different types of traveling waves: classical profiles for wave speeds high enough, and discontinuous waves that are reminiscent of hyperbolic shock waves when the wave speed lowers below a certain threshold. Some of these solutions are of particular relevance as they provide models by which the whole solution (and not just the bulk of it, as it is the case with classical traveling waves) spreads through the medium with finite speed. [ABSTRACT FROM AUTHOR]

We study the relativistic heat equation in one space dimension. We prove a local regularity result when the initial datum is locally Lipschitz in its support. We propose a numerical scheme that captures the known features of the solutions and allows for analysing further properties of their qualitative behaviour. [ABSTRACT FROM PUBLISHER]

In this paper we provide analytical and/or numerical evidences of the qualitative properties of discontinuity fronts, developed by the solutions of the relativistic heat equation (and some porous media variants). We study the local-in-time existence of radially symmetric smooth solutions (inside the support) for smooth initial conditions whose only discontinuities are at the boundary of its support. Then we show some numerical experiments that permit us to conjecture the regularity and qualitative properties of entropy solutions of the relativistic heat equation and some porous media--type variants. [ABSTRACT FROM AUTHOR]

Abstract: We prove some partial regularity results for the entropy solution u of the so-called relativistic heat equation. In particular, under some assumptions on the initial condition , we prove that is a Radon measure in . Moreover, if is log-concave inside its support Ω, Ω being a convex set, then we show the solution is also log-concave in its support . This implies its smoothness in . In that case we can give a simpler characterization of the notion of entropy solution. [Copyright &y& Elsevier]