Region of attraction of a periodic solution to a singularly perturbed parabolic problem

Abstract

We consider the singularly perturbed parabolic differential equation under the assumption that f is T-periodic in t and that the degenerate equation f(u, x, t, 0) = 0 has two intersecting roots. In a previous paper [V.F. Butuzov, N.N. Nefedov, L. Recke, and K.R. Schneider, On a class of periodic solutions of a singularly perturbed parabolic problem, J. Math. Anal. Appl. 348 (2008), pp. 508--515] we presented conditions under which there exists an asymptotically stable T-periodic solution u _p (x, t, ϵ) satisfying no-flux boundary conditions. In this note we characterize a set of initial functions belonging to the region of attraction of u _p (x, t, ϵ).

Keywords:

• singularly perturbed reaction diffusion equation
• periodic boundary value problem
• boundary layers
• region of attraction

AMS Subject Classifications::

• 35B25
• 35B10
• 35K20
• 35K57

Acknowledgements

This work was partially supported by RFBR-DFG grant, the program of cooperation of the Moscow State University and the Humboldt University of Berlin, and RFBR grant N08-01-00413.