Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line

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.