摘要
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 ⊂ ℝ <sup>n</sup> by u <sub>t</sub> = ∈∇ · (k(x)∇u) + ∈ <sup>-1</sup> Vu(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 ∈ <sup>-1</sup> Vu (u) is large, while the diffusion coefficient is small. If the initial condition u(x, 0) = φ(x) is positive in the open domain Ω <sub>1</sub> , negative in the open domain Ω <sub>2</sub> (with Ω <sub>1</sub> ∩ Ω <sub>2</sub> = ∅), and zero on a surface Γ <sub>∈</sub> ⊂ Ω, with Ω <sub>1</sub> ∪ Ω <sub>2</sub> ∪ Γ <sub>∈</sub> = Ω, then u rapidly tends to the positive stable state on Ω <sub>1</sub> , and to the negative stable state on Ω <sub>2</sub> ; an interface of width O(∈) develops at Γ <sub>∈</sub> . Then each interface moves on a longer O(1/∈) timescale, either towards a stable equilibrium position Γ <sub>∈</sub> near Γ <sub>0</sub> for ∈ small, or away from unstable equilibrium positions. Here Γ <sub>0</sub> is the limit curve that arises from {Γ <sub>∈</sub> } as ∈ → 0. The equilibrium locations for Γ <sub>0</sub> 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.