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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Singular limit of solutions
of the porous medium equation
with absorption


Author: Kin Ming Hui
Journal: Trans. Amer. Math. Soc. 350 (1998), 4651-4667
MSC (1991): Primary 35B40; Secondary 35B25, 35K55, 35K65
MathSciNet review: 1443877
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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 $R^{n}$.


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Additional Information

Kin Ming Hui
Affiliation: Institute of Mathematics, Academia Sinica, Nankang, Taipei, 11529, Taiwan, R. O. C.
Email: makmhui@ccvax.sinica.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02030-3
PII: S 0002-9947(98)02030-3
Keywords: Singular limit, porous medium equation with absorption
Received by editor(s): April 20, 1996
Received by editor(s) in revised form: December 15, 1996
Article copyright: © Copyright 1998 American Mathematical Society