Near-field asymptotics for the porous medium equation in exterior domains. The critical two-dimensional case

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We consider the porous medium equation in an exterior two-dimensional domain that excludes a hole, with zero Dirichlet data on its boundary. Gilding and Goncerzewicz proved in 2007 that in the far-field scale, which is the adequate one to describe the movement of the free boundary, solutions to this...

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Autor principal: Cortázar, C.
Otros Autores: Quirós, F., Wolanski, N.
Formato: Capítulo de libro
Lenguaje:Inglés
Publicado: Society for Industrial and Applied Mathematics Publications 2018
Acceso en línea:Registro en Scopus
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100 1 |a Cortázar, C. 
245 1 0 |a Near-field asymptotics for the porous medium equation in exterior domains. The critical two-dimensional case 
260 |b Society for Industrial and Applied Mathematics Publications  |c 2018 
506 |2 openaire  |e Política editorial 
504 |a Barenblatt, G.I., On some unsteady motions of a liquid and gas in a porous medium (1952) Akad. Nauk SSSR. Prikl. Mat. Mekh., 16, pp. 67-78. , Russian 
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504 |a Bénilan, P.H., The Laplace operator (1990) Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 1. Physical Origins and Classical Methods, pp. 220-666. , R. Dautray and J.-L. Lions, eds., Springer, Berlin 
504 |a Brändle, C., Quirós, F., Vázquez, J.L., Asymptotic behaviour of the porous media equation in domains with holes (2007) Interfaces Free Bound, 9, pp. 211-232 
504 |a Caffarelli, L.A., Friedman, A., Continuity of the density of a gas flow in a porous medium (1979) Trans. Amer. Math. Soc., 252, pp. 99-113 
504 |a Cortázar, C., Elgueta, M., Quirós, F., Wolanski, N., Asymptotic behavior for a nonlocal diffusion equation on the half line (2015) Discrete Contin. Dyn. Syst., 35, pp. 1391-1407 
504 |a Cortázar, C., Elgueta, M., Quirós, F., Wolanski, N., Asymptotic behavior for a nonlocal diffusion equation in exterior domains: The critical two-dimensional case (2016) J. Math. Anal. Appl., 436, pp. 586-610 
504 |a Cortázar, C., Quirós, F., Wolanski, N., Near field asymptotic behavior for the porous medium equation on the half-line (2017) Adv. Nonlinear Stud., 17, pp. 245-254 
504 |a Friedman, A., Kamin, S., The asymptotic behavior of gas in an n-dimensional porous medium (1980) Trans. Amer. Math. Soc., 262, pp. 551-563 
504 |a Gilding, B.H., Goncerzewicz, J., Large-time behaviour of solutions of the exterior-domain Cauchy-Dirichlet problem for the porous media equation with homogeneous boundary data (2007) Monatsh. Math., 150, pp. 11-39 
504 |a Herraiz, L., Asymptotic behaviour of solutions of some semilinear parabolic problems (1999) Ann. Inst. H. Poincaré Anal. Non Linéaire, 16, pp. 49-105 
504 |a Kamin, S., Vázquez, J.L., Asymptotic behaviour of solutions of the porous medium equation with changing sign (1991) SIAM J. Math. Anal., 22, pp. 34-45 
504 |a King, J.R., Integral results for nonlinear diffusion equations (1991) J. Engrg. Math., 25, pp. 191-205 
504 |a Pattle, R.E., Diffusion from an instantaneous point source with a concentration-dependent coefficient (1959) Quart. J. Mech. Appl. Math., 12, pp. 407-409 
504 |a Vázquez, J.L., Asymptotic beahviour for the porous medium equation posed in the whole space (2003) J. Evol. Equ., 3, pp. 67-118 
504 |a Vázquez, J.L., The Porous Medium Equation (2007) Mathematical Theory, Oxford Math. Monogr., , The Clarendon Press, Oxford 
504 |a Zel’dovič, Y.B., Kompaneets, A.S., On the theory of propagation of heat with the heat conductivity depending upon the temperature (1950) Collection in Honor of The Seventieth Birthday of Academician A. F. Ioffe, pp. 61-71. , Akad. Nauk SSSR, Moscow 
504 |a Ziemer, W.P., Interior and boundary continuity of weak solutions of degenerate parabolic equations (1982) Trans. Amer. Math. Soc., 271, pp. 733-748 
520 3 |a We consider the porous medium equation in an exterior two-dimensional domain that excludes a hole, with zero Dirichlet data on its boundary. Gilding and Goncerzewicz proved in 2007 that in the far-field scale, which is the adequate one to describe the movement of the free boundary, solutions to this problem with integrable and compactly supported initial data behave as an instantaneous point-source solution for the equation with a variable mass that decays to 0 in a precise way, determined by the initial data and the hole. In this paper, starting from their result in the far field, we study the large time behavior in the near field, in scales that evolve more slowly than the free boundary. In this way we get, in particular, the final profile and decay rate on compact sets. Spatial dimension two is critical for this problem, and involves logarithmic corrections. © 2018 Society for Industrial and Applied Mathematics.  |l eng 
536 |a Detalles de la financiación: PIP625 
536 |a Detalles de la financiación: Fondo Nacional de Desarrollo Científico y Tecnológico, MTM2014-53037-P, 1150028 
536 |a Detalles de la financiación: ∗Received by the editors November 3, 2016; accepted for publication (in revised form) December 19, 2017; published electronically May 22, 2018. http://www.siam.org/journals/sima/50-3/M110191.html Funding: The work of the first author was supported by FONDECYT grant 1150028 (Chile). The work of the second author was supported by project MTM2014-53037-P (Spain). The work of the third author was supported by CONICET PIP625, Res. 960/12, ANPCyT PICT-2012-0153, UBACYT X117, and MathAmSud 13MATH03 (Argentina). †Departamento de Matemática, Pontificia Universidad Católica de Chile, Santiago, Chile (ccortaza@mat.puc.cl). ‡Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049-Madrid, Spain (fernando.quiros@uam.es). §Departamento de Matemática, FCEyN, Universidad de Buenos Aires, and IMAS, CONICET, 1428 Buenos Aires, Argentina (wolanski@dm.uba.ar). 
593 |a Departamento de Matemática, Pontificia Universidad Católica de Chile, Santiago, Chile 
593 |a Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, 28049, Spain 
593 |a Departamento de Matemática, FCEyN, Universidad de Buenos Aires, IMAS, CONICET, Buenos Aires, 1428, Argentina 
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690 1 0 |a LOGARITHMIC CORRECTIONS 
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700 1 |a Quirós, F. 
700 1 |a Wolanski, N. 
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