Hyperbolic conservation laws with stiff reaction terms of monostable type (Q2723468)

The paper deals with two questions related to a hyperbolic conservation law with stiff source term of the form NEWLINE\[NEWLINE u_t + f(u)_x = \frac{u(1/u)}{\epsilon}. NEWLINE\]NEWLINE First, the pointwise limit of solutions of the Cauchy problem is determined in the zero reaction time limit \(\epsilon\to 0\) for initial conditions \(u_0\) which satisfy \(0\leq u_0(x) \leq 1\) and for which the set \(\{x; u_0(x)=0\}\) consists of a finite collection of disjoint intervals. It turns out that the limiting function is piecewise constant and that constant states are separated either by shock discontinuities or by non-shock discontinuities that travel with speed \(f'(0)\). The proof of this result relies on the use of generalized characteristics and follows very closely the previous work of \textit{H. Fan, S. Jin} and \textit{Z. Teng} [J. Differ. Equations 168, No. 2, 270-294 (2000; Zbl 0966.35075)]. The second part of the paper deals with the existence of traveling waves for the viscous regularization NEWLINE\[NEWLINE u_t + f(u)_x = \frac{u(1/u)}{\epsilon} + A\epsilon u_{xx}. NEWLINE\]NEWLINE Here, some necessary conditions for the existence and non-existence of traveling waves are derived.