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Blow-up rate of type II and the braid group theory. (English) Zbl 1228.35069
The author considers positive radially symmetric solutions of the semilinear heat equation $u_t=\Delta u+u^p$, where the spatial variable belongs to $\Bbb R^N$ or to a ball in $\Bbb R^N$, $N\geq11$, and $p$ is supercritical in the sense of Joseph and Lundgren ($p>1+4/(N-4-2\sqrt{N-1})$). In the case of a ball, the equation is complemented by the homogeneous Dirichlet boundary condition. It is known that solutions of these problems can blow-up in finite time $T$ and the blow-up rate can be of type II (that is, the estimate $u(x,t)\leq C(T-t)^{-1/(p-1)}$ is not true for any constant $C$). Considering such solutions and assuming additional intersection properties for their initial data, the author establishes upper and lower estimates for their blow-up rates. The proofs are based on intersection-comparison arguments, including the use of the braid group theory.

35B44Blow-up (PDE)
35K58Semilinear parabolic equations
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