Rate of Type II blowup for a semilinear heat equation

Abstract

A solution u of a Cauchy problem for a semilinear heat equation

$$\left\{ \begin{array}{ll}u_{t} = \Delta u + u^{p} & \quad {\rm in}\, {\bf R}^N \times (0,\,T),\\u(x,0) = u_{0}(x) \geq 0 & \quad {\rm in}\, {\bf R}^N \end{array} \right.$$

is said to undergo Type II blowup at t =  T if lim sup $_{t \nearrow T} \; (T-t)^{1/(p-1)} |u(t)|_\infty = \infty .$ Let $\varphi_\infty$ be the radially symmetric singular steady state. Suppose that $u_0 \in L^\infty$ is a radially symmetric function such that $u_0 - \varphi_\infty$ and (u ₀) _t change sign at most finitely many times. We determine the exact blowup rate of Type II blowup solution with initial data u ₀ in the case of p > p _L , where p _L is the Lepin exponent.