Preprint No. A-02-13

Jörg Härterich, Corrado Mascia

Front formation and motion in quasilinear parabolic equations

Abstract: This paper deals with the singular limit for
$$
{\mathcal L}^\eps u:= u_t - F(u,\eps u_x)_x - \eps^{-1}g(u)=0.
$$
where the function $F$ is assumed to be smooth and uniformly elliptic, and $g$ is a ``bistable'' nonlinearity. Denoting with $u_m$ the unstable zero of $g$, for any smooth initial datum $u_0$ for which $u_0-u_m$ has a finite number of simple zeroes, we show the existence of solutions and describe the structure of the limiting function $u^0=\lim_{\eps\to 0^+} u^\eps$, where $u^\eps$ is the solution of a corresponding Cauchy problem.
The analysis is based on the construction of travelling waves connecting the stable zeros of $g$ and on the use of a comparison principle.

Keywords: front motion, balance law, traveling waves, singular perturbation, comparison priniciple

Mathematics Subject Classification (MSC2000): 35K55, 35B40, 35B25

Language: ENG

Available: Pr-A-02-13.ps

Contact: Jörg Härterich, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (haerter@math.fu-berlin.de)

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