Stationary Points of the 'Reaction-Diffusion'

Of concern is a stationary 'reaction-diffusion' equation with cubic non-linearity is Neumann boundary conditions and fixed average value of the desired bifurcating solutions. A method of approximate calculation of bifurca-ting solutions for small and finite values of supercritical parameter increment are presented. Computing is based on the Lyapunov-Schmidt reducing procedure and is leaning on key functions Ritz' approximation of the set of eigenfunctions (modes) of main linear part of gradient energy functional. A technique of evaluating of a functional space size, where Lyapunov - Schmidt reduction can be applied is performed. In case of local reduction the main part of the key function has been found and asymptotic presentation of bifurcating solutions for small supercritical increment of bifurcation parameter is calculated. The relation between solutions search procedures for 'reaction-diffusion' equations and Cahn - Hilliard equation (with extended Neumann boundary conditions) is also performed. Graphs are presented.

Keywords

continuously differentiable functional; extremal; bifurcation; Lyapunov - Shmidt method.

References

1. Mikhlin S.G. Variational Methods in Mathematical Physics. Oxford, London, Edinburgh, N.Y., Paris, Frankfurt, Pergamon Press, 1964. 584 p. DOI: 10.1002/zamm.19650450439
2. Darinskii B.M., Sapronov Y.I., Tsarev S.L. Bifurcations of Extremals Fredholm Functional. Journal of Mathematical Science, 2007, vol. 145, issue 6, pp. 5311-5453. DOI: 10.1007/s10958-007-0356-2
3. Murray J.D. Lectures on Nonlinear Differential-Equation. Models in Biology About the Models. Oxford, Clarendon Press, 1977.
4. Hassard B.D., Kazarinoff N.D., Wan Y.-H. Theory and Applications of Hopf Bifurcation. Cambridge, Cambridge University Press, 1981. 320 p.
5. Kazarnikov A.V., Revina S.V. The Onset of Auto-Oscillations in Raileigh Systems with Diffusion. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2016, vol. 9, no. 2, pp. 16-28. (in Russian) DOI: 10.14529/mmp160202
6. Sviridyuk G.A. Quasistationary Trajectories of Semilinear Dynamical Equations of Sobolev Type. Russian Academy of Sciences. Izvestiya Mathematics, 1994, vol. 42, no. 3, pp. 601-614. (in Russian) DOI: 10.1070/IM1994v042n03ABEH001547
7. Sviridyuk G.A., Zagrebina S.A. On the Verigin Problem for the Generalized Boussinesq Filtration Equation. Russian Mathematics (Izvestiya VUZ. Matematika), 2003, vol. 47, no. 7, pp. 55-59. (in Russian)
8. Zagrebina S.A. On the Showalter - Sidorov Problem. Russian Mathematics (Izvestiya VUZ. Matematika), 2007, vol. 51, no. 3, pp. 19-24. (in Russian) DOI: 10.3103/S1066369X07030036
9. Kostina T.I. [Nonlocal Calculation of Key Functions in the Problem of Periodic Solutions of Variational Equations]. Proceeding of Voronezh State University. Series: Physics. Mathematics, 2011, no. 1, pp. 181-186. (in Russian)
10. Sapronov Yu.I. Modelling Liquid Flows in Diffusers by Reduced. Bulletin of South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 2, pp. 74-86. (in Russian) DOI: 10.14529/mmp140207
11. Krasnosel'skij M.A., Krejn S.G. [The Iterative Process Minimal Residuals]. Sbornik: Mathematics, 1952, vol. 31 (73), no. 2, pp. 315-334. (in Russian)
12. Lemeshko A.A. [Uniform Convergence with Derivatives Galerkin Approximations to the Solutions of Equations with Parameters]. Matematicheskij modeli i operatornye uravnenija [Mathematical Models and Operator Equations], 2003, vol. 2, pp. 94-103. (in Russian)
13. Lemeshko A.A. [Uniform Convergence of Newton's Approximations to Solutions of Equations with Parameters]. Sbornik trudov molodyh uchenyh matematicheskogo fakul'teta Voronezhskogo gosudarstvennogo Universiteta [Proceedings of Young Scientists Mathematical Faculty of Voronezh State University], 2003, pp. 74-83. (in Russian)
14. Kovaleva M.I., Kostina T.I., Sapronov Y.I. Ogibayushchie krivye, tochki vozvrata i bifurkatsionnyy analiz nelineynykh zadach [The Envelope Curve, the Point of Return and Bifurcation Analysis of Nonlinear Problems]. Voronezh, Military Training and Research Center of the Air Force 'Air Force Academy named after Professor N.E. Zhukovsky and Y.A. Gagarin', 2015. 242 p.