Díaz Díaz, Jesús Ildefonso and Badii, M (2010) On the Time Periodic Free Boundary Associated to Some Nonlinear Parabolic Equations. Boundary value problems . ISSN 16872762

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Abstract
We give sufficient conditions, being also necessary in many cases, for the existence of a periodic free boundary generated as the boundary of the support of the periodic solution of a general class of nonlinear parabolic equations. We show some qualitative properties of this free boundary. In some cases it may have some vertical shape (linking the free boundaries of two stationary solutions), and, under the assumption of a strong absorption, it may have several periodic connected components.
Item Type:  Article 

Uncontrolled Keywords:  diffusion; systems; operators; evolution; obstacle; space 
Subjects:  Sciences > Mathematics > Differential equations 
ID Code:  15067 
References:  J. I. Diaz, Nonlinear Partial Differential Equations and Free Boundaries, Pitman, London, UK, 1985. S. N. Antontsev, J. I. Diaz, and S. Shmarev, Energy Methods for Free Boundary Problems. Applications to Nonlinear PDEs and Fluid Mechanics, vol. 48 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Mass, USA, 2002. C. Bandle, R. P. Sperb, and I. Stakgold, "Diffusion and reaction with monotone kinetics," Nonlinear Analysis: Theory, Methods & Applications, vol. 8, no. 4, pp. 321–333, 1984. C. Bandle and I. Stakgold, "The formation of the dead core in parabolic reactiondiffusion problems," Transactions of the American Mathematical Society, vol. 286, no. 1, pp. 275–293, 1984. A. S. Kalašnikov, "The nature of the propagation of perturbations in problems of nonlinear heat conduction with absorption," Žurnal Vyčislitel' noĭ Matematiki i Matematičeskoĭ Fiziki, vol. 14, pp. 891–905, 1974. M. Badii and J. I. Diaz, "Periodic solutions of a quasilinear parabolic boundary value problem arising in unsaturated flow through a porous medium," Applicable Analysis, vol. 56, no. 3–4, pp. 279–301, 1995. M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, New York, NY, USA, 1964. G. M. Lieberman, "Timeperiodic solutions of quasilinear parabolic differential equations. I. Dirichlet boundary conditions," Journal of Mathematical Analysis and Applications, vol. 264, no. 2, pp. 617–638, 2001. R. Caşcaval and I. I. Vrabie, "Existence of periodic solutions for a class of nonlinear evolution equations," Revista Matemática de la Universidad Complutense de Madrid, vol. 7, no. 2, pp. 325–338, 1994. M. Badii and J. I. Diaz, "Time periodic solutions for a diffusive energy balance model in climatology," Journal of Mathematical Analysis and Applications, vol. 233, no. 2, pp. 713–729, 1999. J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites NonLinéaires, Dunod, Paris, France, 1969. T. I. Seidman, "Periodic solutions of a nonlinear parabolic equation," Journal of Differential Equations, vol. 19, no. 2, pp. 242–257, 1975. J. Mawhin, "Periodic solutions of systems with pLaplacianlike operators," in Nonlinear Analysis and Its Applications to Differential Equations (Lisbon, 1998), vol. 43 of Progress in Nonlinear Differential Equations and Their Applications., pp. 3763, Birkhäuser, Boston, Mass, USA, 2001. W. Yifu, Y. Jingxue, and W. Zhuoqun, "Periodic solutions of evolution pLaplacian equations with nonlinear sources," Journal of Mathematical Analysis and Applications, vol. 219, pp. 76–96, 1988. N. Mizoguchi, "Periodic solutions for degenerate diffusion equations," Indiana University Mathematics Journal, vol. 44, no. 2, pp. 413–432, 1995. N. Kenmochi, D. Kröner, and M. Kubo, "Periodic solutions to porous media equations of parabolicelliptic type," Journal of Partial Differential Equations, vol. 3, no. 3, pp. 63–77, 1990. M. Kubo and N. Yamazaki, "Periodic stability of ellipticparabolic variational inequalities with timedependent boundary double obstacles," Discrete and Continuous Dynamical Systems A, pp. 614–623, 2007. W. Liu, "Periodic solutions of evolution mLaplacian equations with a nonlinear convection term," International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 27368, 10 pages, 2007. F. E. Browder, "Nonlinear maximal monotone operators in Banach space," Mathematische Annalen, vol. 175, pp. 89–113, 1968. F. E. Browder, "Periodic solutions of nonlinear equations of evolution in infinite dimensional spaces," in Lectures in Differential Equations, Vol. I, A. K. Aziz, Ed., pp. 71–96, Van Nostrand, New York, NY, USA, 1969. Ph. Benilan and H. Brezis, "Solutions faibles d'équations d'évolution dans les espaces de Hilbert," Université de Grenoble. Annales de l'Institut Fourier, vol. 22, no. 2, pp. 311–329, 1972. H. Brezis, Operateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert, NorthHolland, Amsterdam, The Netherlands, 1973. Ph. Bénilan, Équation d'Évolution dans un Espace de Banach Quelconque et Applications, thesis, Orsay, France, 1972. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International, Leyden, The Netherlands, 1976. P. Hess, PeriodicParabolic Boundary Value Problems and Positivity, vol. 247 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1991. D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, vol. 279 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1992. B. A. Ton, "Periodic solutions of nonlinear evolution equations in Banach spaces," Canadian Journal of Mathematics, vol. 23, pp. 189–196, 1971. H. Amann, "Periodic solutions of semilinear parabolic equations," in Nonlinear Analysis, L. Cesari, R. Kannan, and H. Weinberger, Eds., pp. 1–29, Academic Press, New York, NY, USA, 1978. J. Prüss, "Periodic solutions of semilinear evolution equations," Nonlinear Analysis: Theory, Methods & Applications, vol. 3, no. 5, pp. 601–612, 1979. J. Crema and J. L. Boldrini, "Periodic solutions of quasilinear equations with discontinuous perturbations," Southwest Journal of Pure and Applied Mathematics, no. 1, pp. 55–73, 2000. N. Alaa and M. Iguernane, "Weak periodic solutions of some quasilinear parabolic equations with data measures," Journal of Inequalities in Pure and Applied Mathematics, vol. 3, no. 3, article 46, 2002. N. Kenmochi and M. Kubo, "Periodic behavior of solutions to parabolicelliptic free boundary problems," Journal of the Mathematical Society of Japan, vol. 41, no. 4, pp. 625–640, 1989. B. Kawhol and R. Rüll, "Periodic solutions of nonlinear heat equations under discontinuous boundary conditions," in Equadiff 82, vol. 1017 of Lecture Notes in Mathematics, pp. 322–327, Springer, New York, NY, USA, 1990. C. V. Pao, "Periodic solutions of parabolic systems with nonlinear boundary conditions," Journal of Mathematical Analysis and Applications, vol. 234, no. 2, pp. 695–716, 1999. M. Badii, "Periodic solutions for a nonlinear parabolic equation with nonlinear boundary conditions," Rendiconti del Seminario Matematico. Università e Politecnico Torino, vol. 67, no. 3, pp. 341–349, 2009. M. Badii, "Periodic solutions for semilinear parabolic problems with nonlinear dynamical boundary condition," Rivista di Matematica della Università di Parma. Serie 7, vol. 5, pp. 57–67, 2006. Y. Giga and N. Mizoguchi, "On time periodic solutions of the Dirichlet problem for degenerate parabolic equations of nondivergence type," Journal of Mathematical Analysis and Applications, vol. 201, no. 2, pp. 396–416, 1996. E. N. Dancer and P. Hess, "On stable solutions of quasilinear periodicparabolic problems," Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, vol. 14, no. 1, pp. 123–141, 1987. P. Buonocore, "On the principal eigenvalue of a periodicparabolic problem," Ricerche di Matematica, vol. 40, no. 2, pp. 335–343, 1991. J. L. Boldrini and J. Crema, "On forced periodic solutions of superlinear quasiparabolic problems," Electronic Journal of Differential Equations, no. 14, pp. 1–18, 1998. A. El Hachimi and A. Lamrani Alaoui, "Existence of stable periodic solutions for quasilinear parabolic problems in the presence of wellordered lower and uppersolutions," in Proceedings of the Fez Conference on Partial Differential Equations, vol. 9 of Electronic Journal of Differential Equations Conference, pp. 117–126, Southwest Texas State University, San Marcos, Tex, USA. T. Godoy, J. Hernández, U. Kaufmann, and S. Paczka, "On some singular periodic parabolic problems," Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 6, pp. 989–995, 2005. J. J. Morgan and S. L. Hollis, "The existence of periodic solutions to reactiondiffusion systems with periodic data," SIAM Journal on Mathematical Analysis, vol. 26, no. 5, pp. 1225–1232, 1995. M. Badii, "A generalized periodic thermistor model," Rendiconti del Seminario Matematico. Università e Politecnico Torino, vol. 65, no. 3, pp. 353–364, 2007. M. Badii, "Existence of periodic solutions for the thermistor problem with the JouleThomson effect," Annali dell'Universitá di Ferrara. Sezione VII. Scienze Matematiche, vol. 54, no. 1, pp. 1–10, 2008. M. Badii, "Existence of periodic solutions for the quasistatic thermoelastic thermistor problem," Nonlinear Differential Equations and Applications, vol. 16, no. 1, pp. 1–15, 2009. M. Badii, "The obstacle thermistor problem with periodic data," Rendiconti del Seminario Matematico della Università di Padova, vol. 121, pp. 145–159, 2009. J. I. Diaz and J. Hernández, "Global bifurcation and continua of nonnegative solutions for a quasilinear elliptic problem," Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, vol. 329, no. 7, pp. 587–592, 1999. J. I. Diaz, J. Hernández, and F. J. Mancebo, "Branches of positive and free boundary solutions for some singular quasilinear elliptic problems," Journal of Mathematical Analysis and Applications, vol. 352, no. 1, pp. 449–474, 2009. J. I. Diaz, "Estimates of the location of a free boundary for the obstacle and Stefan problems obtained by means of some energy methods," Georgian Mathematical Journal, vol. 15, no. 3, pp. 475–484, 2008. J.M. Coron, Control and Nonlinearity, vol. 136 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 2007. P. Hess, M. A. Pozio, and A. Tesei, "Time periodic solutions for a class of degenerate parabolic problems," Houston Journal of Mathematics, vol. 21, no. 2, pp. 367–394, 1995. 
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