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Quenching rate of solutions for a semilinear parabolic equation. (English) Zbl 1227.35168
In this article, the author investigates the behavior of solutions of the Cauchy problem
\begin{aligned} &u_t=\Delta u-u^{-q},\quad x\in \mathbb R^n,\quad t>0,\\ &u(x, 0)=u_0(x),\quad x\in \mathbb R^n, \end{aligned}\tag{1} where $$n\geq 3$$,
$0<q<q_c={{n-2\sqrt{n-1}}\over {2\sqrt{n-1}-(n-4)}}={{(n-2)^2-4n+8\sqrt{n-1}}\over {(n-2)(n-10)}} \quad\text{for }n\in [3, 10)$ and $$q_c=\infty$$ for $$n\geq 10$$, $$u_0(x)$$ is a given positive continuous function on $$\mathbb R^n$$ and grows to infinity as $$|x|\to \infty$$ at most of polynomial order.
The author gives conditions for the initial data $$u_0(x)$$ so that the solution of the problem $$(1)$$ is an infinite time quenching solution and $$u(0, t)\leq C_1(1+t)^{-A_1}$$ or $$u(0, t)\geq C_2(1+t)^{-A_2}$$ for all $$t>0$$ and for some positive constants $$C_1$$ and $$C_2$$, the positive constants $$A_1$$ and $$A_2$$ are suitably chosen.

##### MSC:
 35K58 Semilinear parabolic equations 35K15 Initial value problems for second-order parabolic equations 35B35 Stability in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35K67 Singular parabolic equations