Sets of admissible initial data for porous-medium equations with absorption

Abstract

In this article, we study a porous-medium equation with absorption in $mathbb{R}^{N}imes (0,T)$ or in $Omega imes (0,T)$: $$ u_{t}-Delta u^{m}+u^{p}=0,. $$ We give a rather complete qualitative picture of the initial trace problem in all the range $m>1$, $pgeqslant 0$. We consider nonnegative Borel measures as initial data (not necessarily locally bounded) and discuss whether or not the Cauchy problem admits a solution. In the case of non-admissible data we prove the existence of some projection operators which map any Borel measure to an admissible measure for this equation.

Keywords

• Initial trace
• Borel measures
• initial projection
• singular solution.