PATTERN FORMATION, WAVE PROPAGATION AND STABILITY IN CONSERVATION LAWS WITH SLOW DIFFUSION AND FAST REACTION

The limiting behavior of the solution of a scalar conservation law with slow diffusion and fast bistable reaction is considered. In a short time the solution develops transition patterns connected by shock layers and rarefaction layers, when the initial data has finitely many monotone pieces. The existence and uniqueness of the front profiles for both shock layers and rarefaction layers are established. A variational characterization of wave speeds of these profiles is derived. These profiles are shown to be stable. Furthermore, it is proved that solutions with monotone initial data approach the shock layer or rarefaction layer waves as time goes to infinity.

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