## 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

Zbl 0755.35011
Full Text:

### References:

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