### Front Motion in Muti-dimensional conservation Laws with Stiff Source Terms Driven by Mean Curvature and Variation of Front Thickness

Haitao Fan and Shi Jin

Abstract: The bistable reaction-diffusion-convection equation
\del_t u + \nabla \cdot \bff(u) = -{1\over\e}g(u) +\e\Delta u, \quad x\in\RR^n, \ u\in\RR\leqno(1)
is considered. Stationary traveling waves of above equation are proved to exist when $f(u)$ is symmetric and $g(u)$ is antisymmetric about $u=0$. Solutions of initial value problems tends to almost piecewise constant functions within $O(1)\e$ time. The almost constant pieces are separated by sharp interior layers, called fronts. The motion of these fronts are studied by asymptotic expansion. The equation for the motion of the front is obtained. In the case of $\bff=\bb u^2$ and $g(u)=au(1-u^2)$, where $\bb\in\RR^n$ and \$0

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Author(s):
Haitao Fan, <fan@math.georgetown.edu>
Shi Jin, <jin@math.wisc.edu>
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