# Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

Abstract : We study the limit behaviour of solutions of $\prt_tu-\Gd u+h(\abs x)\abs u^{p-1}u=0\quad\text { in }\BBR^N\ti (0,T)$ with initial data $k\gd _{0}$ when $k\to\infty$, where $h$ is a positive nondecreasing function and $p>1$. If $h(r)=r^{\gb}$, $\gb>N(p-1)-2$, we prove that the limit function $u_{\infty}$ is an explicit very singular solution, while such a solution does not exist if $\gb\leq N(p-1)-2$. If $\liminf_{r\to 0}r^2\ln (1/h(r))>0$, $u_{\infty}$ has a persistent singularity at $(0,t)$ ($t\geq 0$). If $\int_{0}^{r_{0}}r\ln (1/h(r))\,dr<\infty$, $u_{\infty}$ has a pointwise singularity localized at $(0,0)$
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Cited literature [17 references]

https://hal.archives-ouvertes.fr/hal-00282501
Contributor : Laurent Veron <>
Submitted on : Tuesday, May 27, 2008 - 4:16:14 PM
Last modification on : Tuesday, August 13, 2019 - 2:00:02 PM
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• HAL Id : hal-00282501, version 1

### Citation

Andrey Shishkov, Laurent Veron. Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption. Calculus of Variations and Partial Differential Equations, Springer Verlag, 2008, 33, pp.343-375. ⟨hal-00282501⟩

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