Existence and Stability of Periodic Solutions for Reaction-Diffusion Equations in the Two-Dimensional Case

Parabolic singularly perturbed problems have been actively studied in recent years in connection with a large number of practical applications: chemical kinetics, synergetics, astrophysics, biology, and so on. In this work a singularly perturbed periodic problem for a parabolic reaction-diffusion equation is studied in the two-dimensional case. The case when there is an internal transition layer under unbalanced nonlinearity is considered. The internal layer is localised near the so called transitional curve. An asymptotic expansion of the solution is constructed and an asymptotics for the transitional curve is determined. The asymptotical expansion consists of a regular part, an interior layer part and a boundary part. In this work we focus on the interior layer part. In order to describe it in the neighborhood of the transition curve the local coordinate system is introduced and the stretched variables are used. To substantiate the asymptotics thus constructed, the asymptotic method of differential inequalities is used. The upper and lower solutions are constructed by sufficiently complicated modification of the asymptotic expansion of the solution. The Lyapunov asymptotical stability of the solution was proved by using the method of contracting barriers. This method is based on the asymptotic comparison principle and uses the upper and lower solutions which are exponentially tending to the solution to the problem. As a result, the solution is locally unique.
The article is published in the authors’ wording.

1. N. N. Nefedov, L. Recke, K. R. Schnieder, “Existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations.”, Journal of Mathematical Analysis and Applications, 405 (2013), 90–103.

2. N. N. Nefedov, M. A. Davydova, “Contrast structures in singularly perturbed quasilinear reaction-diffusion-advection equations.”, Differentsial’nye Uravneniya, 49 (2013), 715– 733.

3. N.N. Nefedov, L. Recke, K.R. Schneider, “Asymptotic stability via the Krein-Rutman theorem for singularly perturbed parabolic periodic-Dirichlet problems.”, Regular and Chaotic Dynamics, 15 (2010), 382–389.

4. N.N. Nefedov, “The method of differential inequalities for some classes of nonlinear singularly perturbed problems with internal layers.”, Differ. Uravn., 31 (1995), 1142– 1149.

5. V.T. Volkov and N.N. Nefedov, “Development of the asymptotic method of differential inequalities for investigation of periodic contrast structures in reacton-diffusion-advection equations.”, Differ. Uravn., 46 (2006), 585–593.

6. A.B. Vasil’eva and V. F. Butuzov, Asymptotic Expansions of the Solutions of Singularly Perturbed Equations (in Russian), Nauka, Moscow, 1973.

7. P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Math. Series 247, Longman Scientific&Technical, 1991.

8. D.H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Math. J. V. 21, N11, (1972), 979–1001.

9. P. Fife, M. Tang, “Comparison principles for reaction-diffusion systems: Irregular comparison functions and applications to question of stability and speed propagation of front.”, J. Diff. Equations., 40 (1981), 168–175.

10. V.T. Volkov and N.N. Nefedov, “O periodicheskikh resheniyakh s pogranichnymi sloyami odnoy singulyarno vozmushchennoy modeli reaktsiya-diffuziya.”, Computational Mathematics and Mathematical Physics, 34 (1994), 1307–1315.

11. N. N. Nefedov, “Development of the Asymptotic Method of Differential Inequalities for Investigation of Periodic Contrast Structures in Reaction–Diffusion Equations.”, Computational Mathematics and Mathematical Physics, 46 (2006), 614–622.

12. N. N. Nefedov, “Comparison Principle for Reaction-Diffusion-Advection Problems with Boundary and Internal Layers.”, Lecture Notes in Computer Science, 8236 (2013), 62–72.

13. N. N. Nefedov, O. E. Omel’chenko, “Periodic Step-Like Contrast Structures for a Singularly Perturbed Parabolic Equation.”, Differ. Uravn., 36 (2000), 209–218.

14. N. N. Nefedov, “An Asymptotic Method of Differential Inequalities for the Investigation of Periodic Contrast Structures: Existence, Asymptotics, and Stability.”, Differ. Uravn., 36 (2000), 262–269.

15. A. B. Vasil’eva, V. F. Butuzov, N. N. Nefedov, “Kontrastnye struktury v singulyarno vozmushchennykh zadachakh.”, Fundamental’naya i prikladnaya matematika, 4 (1998), 799–851.

16. C.V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York and London, 1992.

17. Amman H. Periodic solutions of semilinear parabolic equations in nonlinear analysis New York: Acad. Press, 1978.