Stationary equation of the relativistic heat diffusion in transparent media having $L^1$--data

Abstract

Our objective is to prove existence of a solution to the Dirichlet problem for an equation arising in the theory of radiation hydrodynamics to deal with the radiating energy in transparent media. We study its stationary equation with $L^1$--datum in a bounded domain. This problem was addressed in [11] for regular data (data belonging to $L^N(\Omega)$) and a bounded solution was obtained. In our framework, the proof of existence is far from trivial since the solution sought cannot be bounded. Consequently, the Anzellotti theory of pairings does not apply and we have to use new developments to introduce the meaning of solution. We also study the regularity of solutions when data belong to $L^p(\Omega)$, with $1<p<N$. Our result is coherent with the regularity found in [11].