ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

# Differential Equations and Control Processes (Differencialnie Uravnenia i Protsesy Upravlenia)

## The Global Attractor of a Multivalued Dynamical System Generated by a Two-phase Heating System

### Author(s):

Dmitry Zyryanov

Saint-Petersburg State University,
Faculty of Mathematics and Mechanics
Chair of Applied Cybernetics
Klychevaya str., 7
195221, St. Petersburg, Russia
Bachelor degree

dmitry.zyr@gmail.com

Volker Reitmann

Saint-Petersburg State University,
Faculty of Mathematics and Mechanics
Chair of Applied Cybernetics
Botanicheskaya ul. 70/2,
198504, Petergof, St. Petersburg, Russia
Professor of the Chair of Applied Cybernetics

vreitmann@aol.com

### Abstract:

In this work we study the asymptotic behavior of solutions of Maxwell's equations coupled with the heat equation for the Stefan problem in 3-dimensional space describing a microwave heating process. These results generalize some well-known properties of the one-dimensional heating system. We use in the 3-dimensional case weak solutions which are generated by a couple of integral identities. Based on this solutions a multi-valued dynamical system is constructed and the existence of a global attractor is proved. The resulting multi-valued dynamical system is considered only at a subspace of the solutions space. A numerical experiment for the temperature approximation is carried out.

### Keywords

• dynamical system
• global attractor
• multi-valued
• Stefan problem
• two-phased system with heating

### References:

1. Budak B. M., Vasil'ev F. P., Yspenskiy A. B. [Numerical methods in aerodynamics]. Raznostnye metodi resheniya nekotorih kraevih zadach tipa Stefana [Finite element methods solving some boundary value Stefan problems]. Lomonosov Moskow State University. 1965. 139-182. (In Russ. )
2. Dautray R., Lions J. -L. Mathematical Analysis and Numerical Methods for Science and Technology. Spectral Theory and Applications. Berlin: Springer-Verlag. 1990. P. 515
3. Dyvo G., Lions J-L. Neravenstva v mehanike i fizike [inequalities in mechanics and physics]. Moskow, Nauka Publ., 1982. 602 p
4. Girault V., Raviart P. A., Finite Element Methods for Navier-Stokes Equations. Berlin: Springer-Verlag. 1986. 376 p
5. Grundas S. Advances in induction and microwave heating of mineral and organic materials. Rijeca: InTech. 2011. 752 p
6. Kalinin Y., Reitmann V., Yumaguzin N., Asymptotic behavior of Maxwell's equation in one-space dimension with thermal effect. Discrete and Cont. Dyn. Sys. 2011. Vol. 2. 754-762
7. Kamennomostskaya S. L. [About Stefan problem]. Matematicheskiy sbornik, 1961; (4):489-514. (In Russ. )
8. Kumar S., Katiyar V. K., Numerical study on phase change heat transfer during combined hyperthermia and cryosurgical treatment of lung cancer. Int. J. of Appl. Math and Mech. 2007. Vol. 3. 1-17 p
9. Ladizhenskaya O. A., Solonnikov V. A., Yral'ceva N. N. Lineynye i kvazilineynie yravneniya parabolicheskogo tipa [Linear and quasilinear parabolic equations]. Moskow, Nauka Publ., 1967. 736 p
10. Manoranjan V. S., Yin H. -M. On two-phase Stefan problem arising from a microwave heating process. Discrete and Cont. Dyn. Sys. - Series A. 2006. Vol. 4, 1155-1168 p
11. Melnik V. S., Valero J. On attractors of multivalued semi-flows and differential inclusions. Set-Valued Analysis. 1998. Vol. 6. 83-111 p
12. Phung K. D. Controle et Stabilization D'Ondes Electromagnetiques. ESAIM Control Optim. Calc. Var. 2000. 87-137 p
13. Reitmann V., Yumaguzin N., Stability analysis for Maxwell's equations with a thermal effect in one-space dimension. Journal of Mathematical Sciences. 2012. Vol 46. 1-12 p
14. Yin H. -M. On Maxwells equations in an electromagnetic field with the temperature effect. SIAM J. of Mathematical Analysis. 1998. Vol. 29, 637-651 p
15. Yumaguzin N. Y., Asimptoticheskoe povedenie resheniy dvyhfazovoy problemi microvolnovogo nagreva v odnomernom slychae. Kand. Diss. [Asymptotic behavior of solutions for two-phased microwave heating problem in one-space dimension], Saint-Petersburg. 2012. 96 p