Mathematische Annalen

, Volume 339, Issue 4, pp 839–877

Rate of Type II blowup for a semilinear heat equation

Authors

    • Department of MathematicsTokyo Gakugei University
Article

DOI: 10.1007/s00208-007-0133-z

Cite this article as:
Mizoguchi, N. Math. Ann. (2007) 339: 839. doi:10.1007/s00208-007-0133-z

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 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 (u0)t change sign at most finitely many times. We determine the exact blowup rate of Type II blowup solution with initial data u0 in the case of p > pL, where pL is the Lepin exponent.

Mathematics Subject Classification (2000)

35K2035K5558K57

Copyright information

© Springer-Verlag 2007