A nonlocal 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 L1 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 L1 to the self–similar N–waves of the homogeneous conservation law. Key words: nonlinear heat radiation, entropy solutions, large time asymptotics, wave break-ing threshold. AMS subject classification: 35Q53, 35B40, 35L65