Singular limit of solutions of the porous medium equation with absorption

Abstract:

We prove that as $m\to \infty$ the solutions $u^{(m)}$ of $u_{t}=\Delta u^{m}-u^{p}$, $(x,t)\in R^{n}\times (0,T)$, $T>0$, $m>1$, $p>1$, $u\ge 0$, $u(x,0)=f(x)\in L^{1}(R^{n})\cap L^{\infty }(R^{n})$, converges in $L^{1}_{loc}(R^{n}\times (0,T))$ to the solution of the ODE $v_{t}=-v^{p}$, $v(x,0)=g(x)$, where $g\in L^{1}(R^{n})$, $0\le g\le 1$, satisfies $g-\Delta \widetilde {g}=f$ in $\mathcal {D}’(R^{n})$ for some function $\widetilde {g}\in L^{\infty }_{loc}(R^{n})$, $\widetilde {g}\ge 0$, satisfying $\widetilde {g}(x)=0$ whenever $g(x)<1$ for a.e. $x\in R^{n}$, $\int _{E}\widetilde {g}dx\le C|E|^{2/n}$ for $n\ge 3$ and $\int _{E}|\nabla \widetilde {g}|dx\le C|E|^{1/2}$ for $n=2$, where $C>0$ is a constant and $E$ is any measurable subset of $R^{n}$.