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Generation and propagation of interfaces for reaction-diffusion equations. (English) Zbl 0765.35024

The author considers the reaction-diffusion equation \[ \partial_ t u - \Delta u + (1/\varepsilon^ 2)\Phi (u)=0 \] on \({\mathbf R}^ N \times {\mathbf R}^ +\), where \(\Phi\) is the derivative of a bistable potential. He shows for initial conditions with values in both domains of attraction of the potential that an interface will develop in time \(O(\varepsilon ^ 2 | \text{ln} \varepsilon | )\). Depending on whether the wells of the potential have equal depth or not, he obtains a different propagation behavior of this interface (normal velocity equal to its mean curvature in the first case, constant speed proportional to the difference of the depths of the wells in the latter case). Also, the disappearance and nonextinction of the interface are treated.
Reviewer: G.Hetzer (Auburn)

MSC:

35K57 Reaction-diffusion equations
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