Zusammenfassung
We consider the spatially inhomogeneous and anisotropic reaction-diffusion equation u(t) = m(s)(-1) div[m(x)a(p)(x, del u)] + epsilon(-2) f(u), involving a small parameter epsilon > 0 and a bistable nonlinear term whose stable equilibria are 0 and 1. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an ...
Zusammenfassung
We consider the spatially inhomogeneous and anisotropic reaction-diffusion equation u(t) = m(s)(-1) div[m(x)a(p)(x, del u)] + epsilon(-2) f(u), involving a small parameter epsilon > 0 and a bistable nonlinear term whose stable equilibria are 0 and 1. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces. More precisely, we show that, within the time-scale of order epsilon(2) vertical bar ln epsilon vertical bar the unique weak solution u(epsilon) develops a steep transition layer that separates the regions {u(epsilon) approximate to 0} and {u(epsilon) approximate to 1}. Then, on a much slower time-scale, the layer starts to propagate. Consequently, as epsilon -> 0, the solution u(epsilon) converges almost everywhere (a.e.) to 0 in Omega(-)(t) and 1 in Omega(+)(t), where Omega(-)(t) and Omega(+)(t) are sub-domains of Omega separated by an interface P(t), whose motion is driven by its anisotropic mean curvature. We also prove that the thickness of the transition layer is of order epsilon.
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