Boundedness and Finite-Time Stability for Multivalued Doubly-Nonlinear Evolution Systems Generated by a Microwave Heating Problem

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Abstract

Doubly-nonlinear evolutionary systems are considered. Sufficient conditions of the boundedness of solutions of such systems are derived. Analogical results for a one-dimensional microwave heating problem are proved. The notions of global process and of a local multivalued process are introduced. Sufficient conditions for the finite-time stability of a global process and of a local multivalued process are shown. For local multivalued processes sufficient conditions for the finite-time instability are derived. For the one-dimensional microwave heating problem conditions of the finite-time stability are shown.

About the authors

S Popov

St. Petersburg State University

Email: psa.87@mail.ru
St. Petersburg, Russia

V Reitmann

St. Petersburg State University

Email: vreitmann@aol.com
St. Petersburg, Russia

S Skopinov

St. Petersburg State University

Email: serg_vologda@mail.ru
St. Petersburg, Russia

References

  1. Березанский Ю. М. Разложение по собственным функциям самосопряженных операторов. - Киев: Наукова думка, 1965.
  2. Ермаков И. В., Калинин Ю. Н., Райтманн В. Определяющие моды и почти периодические интегралы для коциклов// Дифф. уравн. - 2011. - 47, № 13. - С. 1-16.
  3. Лихтарников А. Л. Критерии абсолютной устойчивости нелинейных операторных уравнений// Изв. АН СССР. Сер. Мат. - 1977. - 41, № 5. - C. 1064-1083.
  4. Лихтарников А. Л., Якубович В. Частотная теорема для уравнений эволюционного типа// Сиб. мат. ж. - 1976. - 17, №5. - C. 790-803.
  5. Райтманн Ф., Скопинов С. Н. Устойчивость на конечном промежутке времени в одномерной задаче микроволнового нагрева// Вестн. СПб. ун-та. Сер. 1. - 2015. - 2(60), № 1. - С. 54-59.
  6. Райтманн Ф., Юмагузин Н. Ю. Ассимптотическое поведение решений двухфазовой проблемы микроволнового нагрева в одномерном случае// Вестн. СПб. ун-та. Сер. 1. - 2012. - № 3. - С. 59-62.
  7. Четаев Н. Г. О некоторых вопросах, относящихся к задаче об устойчивости неустановившихся движений// Прикл. мат. мех. - 1960. - 34.- С. 6-18.
  8. Dafermos C. M. An invariance principle for compact process// J. Differ. Equ. - 1971. - 9. - С. 239-252.
  9. Datko R. Extending a theorem of A. M. Liapunov to Hilbert spaces// Rend. Mat. Acc. Lincei. - 1994. - 5. - С. 297-302.
  10. DiBenedetto E., Vespri V. Exponential attractors for a doubly nonlinear equation// J. Math. Anal. Appl. - 1994. - 185. - С. 321-339.
  11. Eden A., Rakotoson J. M. Continuity for bounded solutions of multiphase Stefan problem// J. Math. Anal. Appl. - 1974. - 32. - С. 610-616.
  12. Glassey К., Yin H.-M. On Maxwell’s equations with a temperature effect. II// Commun. Math. Phys. - 1998. - 194. - С. 343-358.
  13. Kalinichenko D. Yu., Reitmann V., Skopinov S. N. Asymptotic behavior of solutions to a coupled system of Maxwell’s equations and a controlled differential inclusion// Discrete Contin. Dyn. Syst. - 2013. - Supplement. - С. 407-414.
  14. Kapustyan A. V., Melnik V. S., Valero J. Attractors of multivalued dynamical processes generated by phase-field equations// Internat. J. Bifur. Chaos Appl. Sci. Engrg. - 2003. - 13, № 7. - С. 1969-1983.
  15. Lions J. L., Magenes E. Non-homogeneous boundary value problems and applications. - Berlin-Heidelberg-N.-Y.: Springer-Verlag, 1972.
  16. Manoranjan V. S., Showalter R., Yin H.-M. On two-phase Stefan problem arising from a microwave heating process// Discrete Contin. Dyn. Syst. Ser. A. - 2006. - 15, № 4. - С. 1155-1168.
  17. Matas A., Merker J. Strong solutions of doubly nonlinear parabolic equations// Z. Anal. Anwend. - 2012. - 31, № 2. - С. 217-235.
  18. Merker J. Strong solutions of doubly nonlinear Navier-Stokes equations// Discrete Contin. Dyn. Syst. - 2011. - Supplement. - С. 1052-1060.
  19. Pankov A. Bounded and almost periodic solutions of nonlinear operator differential equations. - Dordrecht: Kluwer Academic Publishers, 1990.
  20. Popov S. A., Reitmann V. Frequency domain conditions for finite dimensional projectors and determining observations for the set of amenable solutions// Discrete Contin. Dyn. Syst. - 2014. - 34, № 1. - С. 249- 267.
  21. Weiss L., Infante E. F. On the stability of systems defined over a finite time interval// Proc. Natl. Acad. Sci. USA. - 1965. - 54.- С. 44-48.
  22. Yin H.-M. On Maxwell’s equations in an electromagnetic field with the temperature effect// SIAM J. Math. Anal.- 1998.- 29, № 3. - С. 637-651.
  23. Zyryanov D. A., Reitmann V. Attractors in multivalued dynamical systems for the two-phase heating problem// Electron. J. Differ. Equ. Control Processes. - 2017. - 4. - С. 118-138.

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