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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.

MSC:

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

Citations:

Zbl 0755.35011
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References:

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