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Singular limit of solutions of the porous medium equation with absorption
Authors:Kin Ming Hui
Affiliation:Institute of Mathematics, Academia Sinica, Nankang, Taipei, 11529, Taiwan, R. O. C.
Abstract:
We prove that as $mto infty $ the solutions $u^{(m)}$ of $u_{t}=Delta u^{m}-u^{p}$, $(x,t)in R^{n}times (0,T)$, $T>0$, $m>1$, $p>1$, $uge 0$, $u(x,0)=f(x)in L^{1}(R^{n})cap L^{infty }(R^{n})$, 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 $gin L^{1}(R^{n})$, $0le gle 1$, 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. $xin R^{n}$, $int _{E}widetilde {g}dxle C|E|^{2/n}$ for $nge 3$ and $int _{E}|nabla widetilde {g}|dxle C|E|^{1/2}$ for $n=2$, where $C>0$ is a constant and $E$ is any measurable subset of $R^{n}$.

Keywords:Singular limit   porous medium equation with absorption
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