Existence of a Periodic Solution in the Form of a Two-Dimensional Front in a System of Parabolic Equations

• Published in: Differential equations Vol. 56; no. 4; pp. 462 - 477
• Main Authors: Melnikova, A. A., Deryugina, N. N.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.04.2020; Springer
• Subjects: Backup software; Chemical reaction, Rate of; Difference and Functional Equations; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations
• ISSN: 0012-2661; 1608-3083
• DOI: 10.1134/S0012266120040060

For a singularly perturbed system consisting of the two parabolic equations and given in the Cartesian product of a two-dimensional simply connected domain with smooth boundary and the time half-line, where is a small parameter and and are sufficiently smooth functions, -periodic in time, of the variables , , we consider the Neumann boundary value problem supplemented with the condition of -periodicity of the solution. Such solutions are called solutions of the periodic front type and can describe a sharp change in the physical characteristics of some spatially inhomogeneous medium. Systems of such type are employed to model transients in ecology, biophysics, chemical kinetics, semiconductor physics, and other fields. An algorithm is presented for constructing an asymptotic approximation to the solution, and a theorem about the existence of a solution of the periodic front type, as well as its local existence and asymptotic stability, is proved.