Regularity of entropy solutions for the 1D relativistic heat equation

Abstract

This paper investigates the one-dimensional Cauchy problem for the relativistic heat equation with initial data u0 in W1,1(R) intersect BV(R). We prove that the associated entropy solution preserves bounded variation for all positive times, meaning that u(t) belongs to BV(R) for every t > 0, and propagates with finite speed so that its support remains confined within an interval of the form [a - ct, b + ct]. Furthermore, we establish that the time derivative ut is a Radon measure on (0,T) x R, providing a precise characterization of the temporal regularity of entropy solutions. These results contribute to a deeper understanding of the fine regularity properties of solutions to flux-limited diffusion equations.