Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line

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Kamin and Vázquez [11] proved in 1991 that solutions to the Cauchy-Dirichlet problem for the porous medium equation ut = (um)xx, m > 1, on the half-line with zero boundary data and nonnegative compactly supported integrable initial data behave for large times as a dipole-type solution to the...

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Autor principal: Cortázar, C.
Otros Autores: Quirós, F., Wolanski, N.
Formato: Capítulo de libro
Lenguaje:Inglés
Publicado: Walter de Gruyter GmbH 2017
Acceso en línea:Registro en Scopus
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100 1 |a Cortázar, C. 
245 1 0 |a Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line 
260 |b Walter de Gruyter GmbH  |c 2017 
270 1 0 |m Quirós, F.; Departamento de Matemáticas, Universidad Autónoma de MadridSpain; email: fernando.quiros@uam.es 
506 |2 openaire  |e Política editorial 
504 |a Barenblatt, G.I., On some unsteady motions of a liquid and gas in a porous medium (in Russian) (1952) Akad. Nauk SSSR. Prikl. Mat. Meh., 16 (1), pp. 67-78 
504 |a Barenblatt, G.I., Zel'Dovich, Y.B., On dipole solutions in problems of non-stationary filtration of gas under polytropicregime (in Russian) (1957) Prikl. Mat. Mekh., 21 (5), pp. 718-720 
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 (2), pp. 211-232 
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 (4), pp. 1391-1407 
504 |a Cortázar, C., Quirós, F., Wolanski, N., (2016) Near Field Asymptotics for the Porous Medium Equation in Exterior Domains, , https://arxiv.org/abs/1610.04772, Thecritical two-dimensional case, preprint 
504 |a Esteban, J.R., Vázquez, J.L., Homogeneous diffusion in R with power-like nonlinear diffusivity (1988) Arch. Ration. Mech. Anal., 103 (1), pp. 39-80 
504 |a Gilding, B.H., Goncerzewicz, J., Large-time behaviour of solutions of the exterior-domain Cauchy-Dirichlet problem forthe porous media equation with homogeneous boundary data (2007) Monatsh. Math., 150 (1), pp. 11-39 
504 |a Gilding, B.H., Peletier, L.A., On a class of similarity solutions of the porous media equation (1976) J. Math. Anal. Appl., 55 (2), pp. 351-364 
504 |a Gilding, B.H., Peletier, L.A., On a class of similarity solutions of the porous media equation. II (1977) J. Math. Anal. Appl., 57 (3), pp. 522-538 
504 |a Hulshof, J., Vázquez, J.L., The dipole solution for the porous medium equation in several space dimensions (1993) Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 20 (2), pp. 193-217 
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 (1), pp. 34-45 
504 |a Vázquez, J.L., (2007) The Porous Medium Equation. Mathematical Theory, Oxford Math. Monogr, , Oxford University Press, Oxford 
504 |a Zel'Dovich, Y.B., Kompaneets, A.S., On the theory of propagation of heat with the heat conductivity depending uponthe temperature (in Russian) (1950) Collection in Honor of the Seventieth Birthday of Academician A. F. Ioffe, Izdat. Akad. Nauk SSSR, Moscow, pp. 61-71 
520 3 |a Kamin and Vázquez [11] proved in 1991 that solutions to the Cauchy-Dirichlet problem for the porous medium equation ut = (um)xx, m > 1, on the half-line with zero boundary data and nonnegative compactly supported integrable initial data behave for large times as a dipole-type solution to the equation having the same first moment as the initial data, with an error which is o(t-1/m). However, on sets of the form 0 < x < g(t), with g(t) = o(t1/(2m)) as t →, in the so-called near field, a scale which includes the particular case of compact sets, the dipole solution is o( t-1/m), and their result gives neither the right rate of decay of the solution nor a nontrivial asymptotic profile. In this paper, we will improve the estimate for the error, showing that it is o(t-(2m+1)/(2m2) (1+x)1/m). This allows in particular to obtain a nontrivial asymptotic profile in the near field limit, which is a multiple of x1/m, thus improving in this scale the results of Kamin and Vázquez. © 2017 by De Gruyter.  |l eng 
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, UBA, IMAS, CONICET, Ciudad Universitaria, Pab. I, Buenos Aires, 1428, Argentina 
690 1 0 |a ASYMPTOTIC BEHAVIOR 
690 1 0 |a MATCHED ASYMPTOTICS 
690 1 0 |a POROUS MEDIUM EQUATION ON THE HALF-LINE 
700 1 |a Quirós, F. 
700 1 |a Wolanski, N. 
773 0 |d Walter de Gruyter GmbH, 2017  |g v. 17  |h pp. 245-254  |k n. 2  |p Adv. Nonlinear Stud.  |x 15361365  |t Advanced Nonlinear Studies 
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