Wave patterns, stability, and slow motions in inviscid and viscous hyperbolic equations with stiff reaction terms

We study the behavior of solutions to the inviscid (A=0) and the viscous (A>0) hyperbolic conservation laws with stiff source terms with W(u) being the double-well potential. The initial-value problem of this equation gives, to the leading order, piecewise constant solutions connected by shock layers and rarefaction layers. In this paper, we establish the layer motion for the inviscid case at the next order, which moves exponentially slowly. In the viscous case we study the patterns of the traveling wave solutions and structures of the internal layers.