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Abstract: We prove some regularity results for solutions of some flux limited diffusion equations of porous media type for Lipschitz initial data (or assuming a uniform gradient bound on some power of the data), including the fast diffusion case in which the results are global in time. We also develop the existence and uniqueness theory for solutions of the fast diffusion case, which was not covered in the current literature. [Copyright &y& Elsevier]

Abstract: In this paper we give a characterization of the notion of entropy solutions of some flux limited diffusion equations for which we can prove that the solution is a function of bounded variation in space and time. This includes the case of the so-called relativistic heat equation and some generalizations. For them we prove that the jump set consists of fronts that propagate at the speed given by Rankine–Hugoniot condition and we give on it a geometric characterization of the entropy conditions. Since entropy solutions are functions of bounded variation in space once the initial condition is, to complete the program we study the time regularity of solutions of the relativistic heat equation under some conditions on the initial datum. An analogous result holds for some other related equations without additional assumptions on the initial condition. [Copyright &y& Elsevier]

Abstract: In this paper we study a Fisher–Kolmogorov type equation with a flux limited diffusion term and we prove the existence and uniqueness of finite speed moving fronts and the existence of some explicit solutions in a particular regime of the equation. [Copyright &y& Elsevier]

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]

Abstract: We prove existence and uniqueness of entropy solutions for the quasilinear elliptic equation , where , , and f is a convex function of with linear growth as , satisfying other additional assumptions. In particular, this class of equations includes the elliptic problems associated to a relativistic heat equation and a flux limited diffusion equation used in the theory of radiation hydrodynamics, respectively. In a second part of this work, using Crandall–Liggett''s iteration scheme, this result will permit us to prove existence and uniqueness of entropy solutions for the corresponding parabolic Cauchy problem. [Copyright &y& Elsevier]