Wavefront propagation for reaction-diffusion systems of PDE (Q1173825)

The scaled reaction-diffusion system \[ u^ \varepsilon_{k,t}=\varepsilon d_ k\Delta u^ \varepsilon_ k+(1/{\varepsilon})f_ k(u^ \varepsilon)\text{ in } \mathbb{R}^ n\times (0,\infty);\quad u^ \varepsilon_ k=g_ k \text{ on } \mathbb{R}^ n\times\{0\}\quad(k=1,\ldots,m) \] is considered, where \(d_ k>0\) are constants and nonlinear functions \(f_ k\) satisfy several structural assumptions. The asymptotic behavior (as \(\varepsilon\to 0)\) of solutions of this system is characterized in terms of solution \(J\) of the ``simple'' Hamilton-Jacobi equation: it is shown that \(\lim_{\varepsilon\to 0} u^ \varepsilon=0\) uniformly on a compact subset of \(\{J>0\}\), \(\liminf_{\varepsilon\to 0}u^ \varepsilon_ k>0\) uniformly on a compact subset of \(\{J<0\}\). This paper is an extension to systems of an earlier work of the authors on single equations [\textit{L. C. Evans} and \textit{P. E. Soungandis}, Indiana Univ. Math. J. 38, No. 1, 141-172 (1989; Zbl 0692.35014)].