Relativistic heat equation in bounded domain with Dirichlet boundary condition

Abstract

We study a one-dimensional relativistic heat equation with Dirichlet boundary conditions. We prove the equivalence between the entropy formulations introduced by Benilan-Toure [8] and  the one introduced by Mazon-Caselles-Moll [7]. The analysis relies on tools from nonlinear semigroup theory and functions of bounded variation. The existence of entropy solutions is obtained via an implicit time discretization scheme (Rothe's method), combined with compactness arguments. Uniqueness follows from a Kruzhkov-type doubling of variables technique, yielding an L¹-contraction principle.