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Near field asymptotic behavior for the porous medium equation on the half-line

Kamin and Vázquez proved in 1991 that solutions to the Cauchy-Dirichlet problem for the porous medium equation $u_t=(u^m)_{xx}$ 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 $t\to\infty$, in the so called near field, the dipole solution is $o(t^{-1/m})$, and their result does not give neither the right rate of decay of the solution, nor a nontrivial asymptotic profile. In this paper we will show that the error is $o\big(t^{-(2m+1)/(2m^2)}(1+x)^{1/m}\big)$. 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.