Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type

Abstract

The paper deals with a system of two nonlinear second-order parabolic equations. Similar systems, also known as reaction-diffusion systems, describe different chemical processes. In particular, two unknown functions can represent concentrations of effectors (the activator and the inhibitor respectively), which participate in the reaction. Diffusion waves propagating over zero background with finite velocity form an essential class of solutions of these systems. The existence of such solutions is possible because the parabolic type of equations degenerates if unknown functions are equal to zero. We study the analytic solvability of a boundary value problem with the degeneration for the reaction-diffusion system. The diffusion wave front is known. We prove the theorem of existence of the analytic solution in the general case. We construct a solution in the form of power series and suggest recurrent formulas for coefficients. Since, generally speaking, the solution is not unique, we consider some cases not covered by the proved theorem and present the example similar to the classic example of S.V. Kovalevskaya.

Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type

“…The complication of the models considered as systems of equations can help in more accurately describing complex natural processes [23,49]; 4.…”

“…Further, these results were extended to the cases of circular and spherical symmetry [20] and to the two-dimensional case [21,22], as well as to the case of the simplest nonlinear systems [23]. Since the proved theorems are local, as with all such statements, starting with the Cauchy-Kovalevskaya theorem, the question of the domain of the solution's existence is relevant.…”