Advances in Differential Equations

Type-II blowup for a semilinear heat equation

Noriko Mizoguchi

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Abstract

The present paper is concerned with a Cauchy problem for a semilinear heat equation, \[ \left\{ \begin{array}{ll} u_t = \Delta u + u^p & \quad \mbox{ in } {{\bf R}}^N \times (0,\infty), \hspace{3.5cm}\\ u(x,0) = u_0(x) \geq 0 & \quad \mbox{ in } {{\bf R}}^N. \end{array} \right. \tag*{(P)} \] A solution $ u $ of (P) is said to exhibit type-II blowup at $ t = T < + \infty $ if \[ \limsup_{ t \nearrow T } (T-t)^{ 1/(p-1) } |u(t)|_\infty = + \infty \] with the supremum norm $ | \cdot |_\infty $ in $ {{\bf R}}^N $. We show the existence of type-II-blowup solutions. Though our proof is based on the argument due to [9, 10], it is simpler than theirs by taking account of the number of intersections with the singular steady state of (P).

Article information

Source
Adv. Differential Equations, Volume 9, Number 11-12 (2004), 1279-1316.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867903

Mathematical Reviews number (MathSciNet)
MR2099557

Zentralblatt MATH identifier
1122.35050

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions 35K15: Initial value problems for second-order parabolic equations 35K57: Reaction-diffusion equations

Citation

Mizoguchi, Noriko. Type-II blowup for a semilinear heat equation. Adv. Differential Equations 9 (2004), no. 11-12, 1279--1316. https://projecteuclid.org/euclid.ade/1355867903


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