Wave patterns, stability, and slow motions in inviscid and viscous hyperbolic equations with stiff reaction terms
The qualitative behavior of solutions of a scalar conservation law with a stiff source term of the form \[ u_t + f(u)_x = \varepsilon A U_{xx} - \varepsilon^{-1}W'(u) \] is studied in the case of a convex flux \(f\) and a bistable source \(W'(u)\). The main results concern the inviscid case \(A=0\). Here it had been shown previously that in the zero reaction time limit \(\varepsilon\to 0\) the solution tends almost everywhere to a piecewise constant function \(u(x,t)\) where constant parts are separated by straight lines in the \((x,t)\)-plane. Now the authors study the asymptotic for \(\varepsilon>0\) small and show that shock layers move exponentially slowly while rarefaction layers are stationary. Another result shows that in the viscous case \(A\neq 0\) there exist traveling wave solutions correponding to either a single shock or a rarefaction layer and that these waves are asymptotically stable in \(W^{1,2}\).
scientific article; zbMATH DE number 1901949