Analytical Solutions with Zero Front to the Nonlinear Degenerate Parabolic System

• Published in: Differential equations Vol. 58; no. 11; pp. 1457 - 1467
• Main Authors: Kazakov, A. L., Kuznetsov, P. A.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01-11-2022; Springer; Springer Nature B.V
• Subjects: Boundary conditions; Boundary value problems; Difference and Functional Equations; Differential equations; Exact solutions; Existence theorems; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Population biology; Taylor series; Uniqueness theorems
• ISSN: 0012-2661; 1608-3083
• DOI: 10.1134/S00122661220110039

We consider a special boundary value problem for a nonlinear parabolic system proposed by J. Murray and used to describe population dynamics. The boundary conditions of the problem assume that the possible solutions have a zero front—a line on which the desired functions vanish and the parabolic type of the system degenerates. A particular case of such solutions to single degenerate equations is nonlinear heat (filtration, diffusion) waves considered by Ya.B. Zeldovich, G.I. Barenblatt, and A.A. Samarskii. In the present paper, we prove an existence and uniqueness theorem for a nontrivial analytical solution to the problem under study. In the course of the proof, a solution is constructed in the form of Taylor series, and recurrence coefficient formulas are written that can later be used to verify numerical calculations. Some exact solutions of the system with zero front are presented. Separately, examples are considered that illustrate the behavior of the solution when deviating from the conditions in the theorem. The first example shows the possibility of existence of solutions of the original system with two distinct zero fronts. The second example is an analog of the well-known counterexample by S.V. Kovalevskaya in the considered case.