Díaz Díaz, Jesús Ildefonso and Alvarez, Luis
(2009)
*On the retention of the interfaces in some elliptic and parabolic nonlinear problems.*
Discrete and Continuous Dynamical Systems. Series A., 25
(1).
pp. 1-17.
ISSN 1078-0947

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## Abstract

We study some retention phenomena on the free boundaries associated to some elliptic and parabolic problems of reaction-diffusion type. This is the case, for instance, of the wait in g time phenomenon for solutions of suitable parabolic equations. We find sufficient conditions in order to have a discrete version of the waiting time property (the so called nondiffusion of the support) for solutions of the associated family of elliptic equations and prove how to pass to the limit in order to get this property for the solutions of the parabolic equation.

Item Type: | Article |
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Uncontrolled Keywords: | strong maximum principle; equations |

Subjects: | Sciences > Mathematics > Differential geometry Sciences > Mathematics > Differential equations |

ID Code: | 15090 |

References: | L. Alvarez, On the behaviour of the free boundary of some nonhomogeneous elliptic problems, Applicable Analysis, 36 (1990), 131–144. L. Alvarez and J. I. Díaz, On the behaviour near the free boundary of solutions of some non homogeneous elliptic problems, in “Actas del IX CEDYA,” Univ. de Valladolid, (1987), 55–59. L. Alvarez and J. I. Díaz, The waiting time property for parabolic problems trough the nondiﬀusion of the support for the stationary problems, Rev. R. Acad. Cien. Serie A Matem, 97 (2003), 83–88. L. Alvarez and J. I. Díaz, On the initial growth of interfaces in reaction-diﬀusion equations with strong absorption, Proceed. Royal Soc. of Edinburgh, 123A (1993), 803–817. L. Alvarez, R. Garc´ıa, J. Perez and A. Suarez, An´alisis num´erico de un problema el´ıptico semilineal. Aspectos computacionales, in “I Congreso Métodos Numéricos en Ingeniería,” Univ. de Las Palmas, (1990), 215–222. F. Andreu, N. Igbida, J. M. Mazón and J. A. Toledo, A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions, Interfaces Free Bound., 8 (2006), 447–479. S. Antontsev, J. I. Díaz and S. Shmarev, “Energy Methods for Free Boundary Problems. Applications to Nonlinear PDEs and Fluid Mechanics,” Series Progress in Nonlinear Diﬀerential Equations and Their Applications No. 48, Birkäuser, Boston, 2002. R. Aris, “The Mathematical Theory of Diﬀusion and Reaction in Permeable Catalysts,” Volumes I and II, Clarendon Press, Oxford, 1975. D. G. Aronson, Regularity properties of ﬂows through porous media: The interface, Arch. Rational Mech. Anal, 37 (1970), 1–10. G. Barles, G. Díaz and J. I. Díaz, Uniqueness and continuum of foliated solutions for a quasilinear elliptic equation with non Lipschitz nonlinearity, Comm. in Partial Diﬀerential Equations, 17 (1992), 1037–1050. J. Bear, “Dynamics of Fluids in Porous Media,” Elsevier, New York, 1972. Ph. Benilan, H. Brezis and M. G. Crandall, A semilinear equation in L 1 , Ann.Scu. Norm. Sup. Pisa, 2 (1975), 532–555. Ph. Benilan and P. Wittbold, On mild and weak solutions of elliptic-parabolic problems, Adv.Diﬀerential Equations, 1 (1996), 1053–1073. I. Bejenaru, J. I. Díaz and I. Vrabie, An abstract approximate controllability results and applications to elliptic and parabolic systems with dynamic boundary conditions, Electron. J. Diﬀ. Eqns., 50 (2001), 1–19. H. Brezis and L. Nirenberg, Removable singularities for nonlinear elliptic equations, Topol. Methods Nonlinear Anal., 9 (1997), 201–219. N. Calvo , J. I. Díaz, J. Durany, E. Schiavi and C. Vázquez, On a doubly nonlinear parabolic obstacle problem modelling ice sheet dynamics, SIAM J. Appl. Math., 63 (2002), 683–707. M. G. Crandall and Th. M. Liggett, A theorem and a counterexample in the theory of semigroups of nonlinear transformations, Trans. Amer. Math. Soc., 160 (1971), 263–278. R. L. Devaney, “An Introduction to Chaotic Dynamical Systems,” Addison-Wesley, Menlo Park California, 2nd edition, 1989. J. I. Díaz, “Nonlinear Partial Diﬀerential Equations and Free Boundaries,” Pitman, Boston, MA, Research Notes in Mathematics 106 London, 1985. J. I. Díaz and Ch. Faghloumi, Analysis of a degenerate obstacle problem on an unbounded set arising in the environment, Applied Math. and Optimization, 45 (2002), 251–267. J.I. Díaz, J. E. Saa and U. Thiel, On the equation of prescribed mean curvature and other elliptic quasilinear equations with locally vanishing solutions (Spanish) , Rev. Un. Mat. Argentina, 35 (1989), 175–206. J. I. Díaz, Qualitative study of nonlinear parabolic equations: An introduction, Extracta Mathematicae, 16 (2001), 303–341. J. I. Díaz, On the formation of the free boundary for the obstacle and Stefan problems via an energy method, in “Actas XVII CEDYA / VII CMA” (L. Ferragut y A. Santos eds.), Servicio de Publicaciones de la Univ. de Salamanca, (2001) J. I. Díaz and J. Henández, Qualitative properties of free boundaries for some nonlinear degenerate parabolic equations, in “Nonlinear Parabolic Equations: Qualitative Properties of Solutions” (L. Boccardo y A.Tesei eds.), Research Notes in Mathematics 149, Pitman, London, (1987), 5–93. J. I. Díaz and J. Henández, Global bifurcation and continua of nonegative solutions for a quasilinear elliptic problem, Comptes Rendus Acad. Sci. Paris, S´erie I, 329 (1999), 587–592. J. I. Díaz, Mathematical analysis of some diﬀusive energy balance models in Climatology, in “Mathematics, Climate and Environment” (J. I. Díaz and J. L. Lions eds., Research Notes in Applied Mathematics 27, Masson, Paris, (1993), 28–56. J. I. Díaz, Energy methods for free boundary problems: New results and some remarks on numerical algorithm, Actes du XXXIV Congres d’Analyse Numerique, May 2002, Canum 2002, Anglet (France), SMAI and GDR 2290 of CNRS Editors, Pau, (2002), 71–86. E. DiBenedetto, J. M. Urbano and V. Vespri, Current issues on singular and degenerate evolution equations. Evolutionary equations, Handb. Diﬀer. Equ., I North-Holland, Amsterdam (2004), 169–286. A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations. (Russian) , Uspekhi Mat. Nauk, 42 (1987), 135–176. S. Kamin and L. Veron, Flat core properties associated to the p–Laplace operator, Proceedings of the Amer. Math. Soc., 118 (1993), 1079–1085. A. M. Meirmanov, “The Stefan Problem,” Walter de Gruyter & Co., Berlin, 1992. R. H. Nochetto and G. Savar´e, Nonlinear evolution governed by accretive operators in Banach spaces: error control and applications, Math. Models Methods Appl. Sci., 16 (2006), 439–477. R. H. Nochetto and C. Verdi, Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal., 25 (1988), 784–814. P. Pucci and J. Serrin, The strong maximum principle revisited, J. Diﬀ. Equations, 196 (2004), 1–66. R. Showalter, “Mechanics of Non-Newtonian Fluids,” Pergamon Press, Oxford, 1978. J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191–202. |

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