Existence and Asymptotic Behavior of Solutions of Boundary Value Problems for Tiknohov-Type Reaction–Diffusion Systems in the Case of Stability Exchange

Singularly perturbed systems of reaction-diffusion equations in the case of fast and slow equations, known as Tikhonov-type systems, are studied. The case where the roots of the degenerate equation intersect and hence are not isolated is considered. Effective conditions for the existence of a solution whose approximation is the so-called composite stable solution of the degenerate system are obtained. The existence of a solution is proved and the accuracy of the asymptotic approximation is estimated by using the asymptotic method of differential inequalities extended to this class of problems. Conditions for the Lyapunov stability of solutions such as those of the corresponding initial boundary value problems are given.