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On a singularly perturbed initial value problem in case of a double root of the degenerate equation
We study the initial value problem of a singularly perturbed first order ordinary differential equation in case that the degenerate equation has a double root. We construct the formal asymptotic expansion of the solution such that the boundary layer functions decay exponentially. This requires a modification of the standard procedure. The asymptotic solution will be used to construct lower and upper solutions guaranteeing the existence of a unique solution and justifying its asymptotic expansion.
Global region of attraction of a periodic solution to a singularly perturbed parabolic problem in case of exchange of stability
We consider a singularly perturbed parabolic differential equation in case that the degenerate equation has two intersecting roots. In a previous paper we presented conditions under which there exists an asymptotically stable periodic solution satisfying no-flux boundary conditions. In this note we characterize a set of initial functions belonging to the global region of attraction of that periodic solution.
Existence and asymptotic stability of a periodic solution with boundary layers of reaction-diffusion equations with singularly perturbed Neumann boundary conditions
We consider singularly perturbed reaction-diffusion equations with singularly perturbed Neumann boundary conditions. We establish the existence of a time-periodic solution u(x; t; epsilon) with boundary layers and derive conditions for their asymptotic stability The boundary layer part of u(x; t; ") is of order one, which distinguishes our case from the case of regularly perturbed Neumann boundary conditions, where the boundary layer is of order epsilon.
Asymptotics and stability of a periodic solution to a singularly perturbed parabolic problem in case of a double root of the degenerate equation
For a singularly perturbed parabolic problem with Dirichlet conditions we prove the existence of a solution periodic in time and with boundary layers at both ends of the space interval in the case that the degenerate equation has a double root. We construct the corresponding asymptotic expansion in the small parameter. It turns out that the algorithm of the construction of the boundary layer functions and the behavior of the solution in the boundary layers essentially differ from that ones in case of a simple root. We also investigate the stability of this solution and the corresponding region of attraction.