TRAVELING WAVES AS LIMITS OF SOLUTION ON BOUNDED DOMAINS

This paper is concerned with the asymptotic behavior as epsilon --> 0 of solutions of the reaction-diffusion equation u(t) = epsilon(2)u(xx) - (u + alpha)(u(2) - 1) defined in (-1, 1) with Neumann boundary cond itions. Far alpha = 0, this equation has a monotone equilibrium soluti on u(epsilon) with the property that u(epsilon)(x) --> -1 (resp. +1) o n [-1, 0) (resp. (0, 1]) as epsilon --> 0; that is, the solution has a sharp transition layer if alpha = 0. Also, it is known that u(epsilon ) has a one-dimensional unstable manifold M(u(epsilon)). Solutions nea r M(u(epsilon)) decrease exponentially to M(u(epsilon)) and move with a speed O(e(-c/epsilon)) along M(u(epsilon)). This paper considers the case where alpha is small and fixed. For each fixed epsilon, alpha no t equal 0, small, there is an equilibrium solution u(epsilon alpha) wi th unstable manifold of dimension one, but u(epsilon alpha) approaches either the function 1 or -1 as epsilon --> 0; that is, there is no mo notone equilibrium solution with a sharp transition layer. If we resca le x to epsilon x and consider the rescaled equation on (-infinity, in finity), then there is a unique (except for translation) monotone trav eling-wave solution on (-infinity, infinity) with wave speed -root 2 a lpha. Using a geometric approach, we prove that there are positive con stants epsilon(0) and alpha(0) such that, for 0 < epsilon < epsilon(0) and \alpha\ < alpha(0), solutions of the rescaled equations on (-1/ep silon, 1/epsilon) in a neighborhood of size C root alpha(0) of a monot one traveling-wave solution decrease exponentially fast before they en ter a neighborhood of size O(epsilon(k)) of such a solution, where k c an be any positive integer. Along the traveling-wave direction, soluti ons move with the traveling-wave speed plus an error term O(epsilon(k) ). It also is proved that, the L(infinity)-norm between the solution a nd a translation of the traveling wave is of order O(epsilon(k)) for C (1)k log 1/epsilon < t < C-2/epsilon.