Near field asymptotic behavior for the porous medium equation on the half-line. (English) Zbl 1360.35023

Summary: S. Kamin and J. L. Vazquez proved in [SIAM J. Math. Anal. 22, No. 1, 34–45 (1991; Zbl 0755.35011)] that solutions to the Cauchy-Dirichlet problem for the porous medium equation \({u_{t}=(u^{m})_{xx}}\), \({m>1}\), 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 \({g(t)=o(t^{1/(2m)})}\) as \({t\rightarrow\infty}\), 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 \({o(t^{-(2m+1)/(2m^{2})}(1+x)^{1/m})}\). This allows in particular to obtain a nontrivial asymptotic profile in the near field limit, which is a multiple of \({x^{1/m}}\), thus improving in this scale the results of Kamin and Vázquez.


35B40 Asymptotic behavior of solutions to PDEs
35K65 Degenerate parabolic equations
35R35 Free boundary problems for PDEs


Zbl 0755.35011
Full Text: DOI arXiv


[1] Barenblatt G. I., On some unsteady motions of a liquid and gas in a porous medium (in Russian), Akad. Nauk SSSR. Prikl. Mat. Meh. 16 (1952), no. 1, 67-78. · Zbl 0049.41902
[2] Barenblatt G. I. and Zel’dovich Y. B., On dipole solutions in problems of non-stationary filtration of gas under polytropic regime (in Russian), Prikl. Mat. Mekh. 21 (1957), no. 5, 718-720.
[3] Brändle C., Quirós F. and Vázquez J. L., Asymptotic behaviour of the porous media equation in domains with holes, Interfaces Free Bound. 9 (2007), no. 2, 211-232. · Zbl 1131.35043
[4] Cortázar C., Elgueta M., Quirós F. and Wolanski N., Asymptotic behavior for a nonlocal diffusion equation on the half line, Discrete Contin. Dyn. Syst. 35 (2015), no. 4, 1391-1407. · Zbl 1310.35236
[5] Cortázar C., Quirós F. and Wolanski N., Near field asymptotics for the porous medium equation in exterior domains. The critical two-dimensional case, preprint 2016, . · Zbl 1332.35158
[6] Esteban J. R. and Vázquez J. L., Homogeneous diffusion in \(\mathbb{R}\) with power-like nonlinear diffusivity, Arch. Ration. Mech. Anal. 103 (1988), no. 1, 39-80. · Zbl 0683.76073
[7] Gilding B. H. and Goncerzewicz J., Large-time behaviour of solutions of the exterior-domain Cauchy-Dirichlet problem for the porous media equation with homogeneous boundary data, Monatsh. Math. 150 (2007), no. 1, 11-39. · Zbl 1159.35040
[8] Gilding B. H. and Peletier L. A., On a class of similarity solutions of the porous media equation, J. Math. Anal. Appl. 55 (1976), no. 2, 351-364. · Zbl 0356.35049
[9] Gilding B. H. and Peletier L. A., On a class of similarity solutions of the porous media equation. II, J. Math. Anal. Appl. 57 (1977), no. 3, 522-538. · Zbl 0365.35029
[10] Hulshof J. and Vázquez J. L., The dipole solution for the porous medium equation in several space dimensions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 20 (1993), no. 2, 193-217. · Zbl 0832.35082
[11] Kamin S. and Vázquez J. L., Asymptotic behaviour of solutions of the porous medium equation with changing sign, SIAM J. Math. Anal. 22 (1991), no. 1, 34-45. · Zbl 0755.35011
[12] Vázquez J. L., The Porous Medium Equation. Mathematical Theory, Oxford Math. Monogr., Oxford University Press, Oxford, 2007.
[13] Zel’dovich Y. B. and Kompaneets A. S., On the theory of propagation of heat with the heat conductivity depending upon the temperature (in Russian), Collection in Honor of the Seventieth Birthday of Academician A. F. Ioffe, Izdat. Akad. Nauk SSSR, Moscow (1950), 61-71.
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