Inhomogeneous fast reaction, slow diffusion and weighted curve shortening - Norbury, J; Yeh, L (2001)
In this paper, we extend the work of Rubinstein, Sternberg and Keller (1989 SIAM J. Appl. Math. 49 116-33). We consider chemical reactions, phase transitions or other processes governed by a semilinear reaction-diffusion equation (with Neumann boundary conditions) for u(x, t, ∈) defined for t > 0 and x ∈ Ω̄ ⊂ ℝn by ut = ∈∇ · (k(x)∇u) + ∈-1Vu(x, u) x ∈ Ω where ∈ is a small parameter and V is a bistable potential for u; here V and k depend on x ∈ Ω and V is even in u. Here one of the stable minimizers is pointwise positive, and the fact that V is even in u then gives that the other stable minimizer is negative. The reaction rate ∈-1Vu (u) is large, while the diffusion coefficient is small. If the initial condition u(x, 0) = φ(x) is positive in the open domain Ω1, negative in the open domain Ω2 (with Ω1 ∩ Ω2 = ∅), and zero on a surface Γ∈ ⊂ Ω, with Ω1 ∪ Ω2 ∪ Γ∈ = Ω, then u rapidly tends to the positive stable state on Ω1, and to the negative stable state on Ω2; an interface of width O(∈) develops at Γ∈. Then each interface moves on a longer O(1/∈) timescale, either towards a stable equilibrium position Γ∈ near Γ0 for ∈ small, or away from unstable equilibrium positions. Here Γ0 is the limit curve that arises from {Γ∈} as ∈ → 0. The equilibrium locations for Γ0 are calculated from a geometric geodesic condition, together with their local stability. Simple formulae for these are derived, which depend only on the x variation in V and k for ∈ small.