We study the limit behaviour of solutions of a class of solutions of
nonlinear parabolic equations with a degenerate strong absorption. We prove
that two types of phenomena can occur: the pointwise singularity or the
formation of razor blades (or persistent singularities).Comment: 29 page

Shishkov, Andrey

Veron, Laurent

English

arXiv

International audienceWe 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)

Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

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