Mathematische Annalen

, Volume 339, Issue 4, pp 839-877

First online:

Rate of Type II blowup for a semilinear heat equation

  • Noriko MizoguchiAffiliated withDepartment of Mathematics, Tokyo Gakugei University Email author 

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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 tT 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.

Mathematics Subject Classification (2000)

35K20 35K55 58K57