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Abstract:

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.

Registro:

Documento: Artículo
Título:Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line
Autor:Cortázar, C.; Quirós, F.; Wolanski, N.
Filiación:Departamento de Matemática, Pontificia Universidad Católica de Chile, Santiago, Chile
Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, 28049, Spain
Departamento de Matemática, FCEyN, UBA, IMAS, CONICET, Ciudad Universitaria, Pab. I, Buenos Aires, 1428, Argentina
Palabras clave:Asymptotic Behavior; Matched Asymptotics; Porous Medium Equation on the Half-Line
Año:2017
Volumen:17
Número:2
Página de inicio:245
Página de fin:254
DOI: http://dx.doi.org/10.1515/ans-2017-0006
Título revista:Advanced Nonlinear Studies
Título revista abreviado:Adv. Nonlinear Stud.
ISSN:15361365
Registro:http://digital.bl.fcen.uba.ar/collection/paper/document/paper_15361365_v17_n2_p245_Cortazar

Referencias:

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  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
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  • 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
  • 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
  • Vázquez, J.L., (2007) The Porous Medium Equation. Mathematical Theory, Oxford Math. Monogr, , Oxford University Press, Oxford
  • 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

Citas:

---------- APA ----------
Cortázar, C., Quirós, F. & Wolanski, N. (2017) . Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line. Advanced Nonlinear Studies, 17(2), 245-254.
http://dx.doi.org/10.1515/ans-2017-0006
---------- CHICAGO ----------
Cortázar, C., Quirós, F., Wolanski, N. "Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line" . Advanced Nonlinear Studies 17, no. 2 (2017) : 245-254.
http://dx.doi.org/10.1515/ans-2017-0006
---------- MLA ----------
Cortázar, C., Quirós, F., Wolanski, N. "Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line" . Advanced Nonlinear Studies, vol. 17, no. 2, 2017, pp. 245-254.
http://dx.doi.org/10.1515/ans-2017-0006
---------- VANCOUVER ----------
Cortázar, C., Quirós, F., Wolanski, N. Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line. Adv. Nonlinear Stud. 2017;17(2):245-254.
http://dx.doi.org/10.1515/ans-2017-0006