A non-local conservation law with nonlinear ‘radiation’ inhomogeneity

We study a scalar conservation law with a nonlinear dissipative inhomogeneity, which serves as a simplified model for nonlinear heat radiation effects in high–temperature gases. We establish global existence and uniqueness of weak entropy solutions along with L contraction and monotonicity properties of the solution semigroup. We derive explicit threshold conditions ensuring formation of shocks within finite time. Our main result proves – under further assumptions on the nonlinearity and on the initial datum – large time convergence in L to the self–similar N–waves of the homogeneous conservation law.