Degenerated scalar balance laws: stability of the critical wave

We consider scalar balance laws whith degenerated growth function at one infinity. It is the transition point between monostable and bistable waves. We find it admits smooth profiles arising from both settings, and show stability for the slowest of these waves, under a sign assumption on the flux. The presence of a characteristic point at infinity weaken the temporal decay: an exponential spatial localization loss is necessary to get algebraic temporal decay.