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)x⁢x${u_{t}=(u^{m})_{xx}}$, m>1${m>1}$, 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)${o(t^{-1/m})}$. However, on sets of the form 0<x<g⁢(t)${0<x<g(t)}$, with g⁢(t)=o⁢(t1/(2⁢m))${g(t)=o(t^{1/(2m)})}$ as t→∞${t\to\infty}$, in the so-called near field, a scale which includes the particular case of compact sets, the dipole solution is o⁢(t-1/m)${o(t^{-1/m})}$, 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-(2⁢m+1)/(2⁢m2)⁢(1+x)1/m)${o(t^{-(2m+1)/(2m^{2})}(1+x)^{1/m})}$. This allows in particular to obtain a nontrivial asymptotic profile in the near field limit, which is a multiple of x1/m${x^{1/m}}$, thus improving in this scale the results of Kamin and Vázquez.

Keywords

• porous medium equation on the half-line
• asymptotic behavior
• matched asymptotics
• 35b40
• 35k65
• 35r35