A Note on the Domain of Attraction for the Stationary Solution to a Singularly Perturbed Parabolic Equation

Abstract

We consider a boundary-value problem for a singularly perturbed parabolic equation with an initial function independent of a perturbation parameter in the case where a degenerate stationary equation has smooth possibly intersecting roots. Before, the existence of a stable stationary solution to this problem was proved and the domain of attraction of this solution was investigated — due to exchange of stabilities, the stationary solution approaches the non-smooth (but continuous) composite root of the degenerate equation as the perturbation parameter gets smaller, and its domain of attraction contains all initial functions situated strictly on one side of the other non-smooth (but continuous) composite root of the degenerate equation. We show that if the initial function is out of the boundary of this family of initial functions near some point, the problem cannot have a solution inside the domain of the problem, i.e. this boundary is the true boundary of the attraction domain. The proof uses ideas of the nonlinear capacity method.

Keywords

• small parameter
• singular perturbation
• parabolic equation
• stationary solution
• domain of attraction
• intersecting roots
• exchange of stabilities
• nonexistence
• blow-up
• nonlinear capacity