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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.

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