TY - JOUR

T1 - Blow-up set for type i blowing up solutions for a semilinear heat equation

AU - Fujishima, Yohei

AU - Ishige, Kazuhiro

PY - 2014/1/1

Y1 - 2014/1/1

N2 - Let u be a type I blowing up solution of the Cauchy-Dirichlet problem for a semilinear heat equation, { ∂tu = Δu + up, x ε Ω, t > 0, u(x, t) = 0, x ε ∂Ω, t > 0, u(x, 0) = φ(x), x ε Ω, where Ω is a (possibly unbounded) domain in RN, N≥1, and p>1. We prove that, if φ ε L ∞(Ω)∩Lq(Ω) for some q ε [1, ∞ ), then the blow-up set of the solution u is bounded. Furthermore, we give a sufficient condition for type I blowing up solutions not to blow up on the boundary of the domain Ω. This enables us to prove that, if Ω is an annulus, then the radially symmetric solutions of (P) do not blow up on the boundary ∂Ω.

AB - Let u be a type I blowing up solution of the Cauchy-Dirichlet problem for a semilinear heat equation, { ∂tu = Δu + up, x ε Ω, t > 0, u(x, t) = 0, x ε ∂Ω, t > 0, u(x, 0) = φ(x), x ε Ω, where Ω is a (possibly unbounded) domain in RN, N≥1, and p>1. We prove that, if φ ε L ∞(Ω)∩Lq(Ω) for some q ε [1, ∞ ), then the blow-up set of the solution u is bounded. Furthermore, we give a sufficient condition for type I blowing up solutions not to blow up on the boundary of the domain Ω. This enables us to prove that, if Ω is an annulus, then the radially symmetric solutions of (P) do not blow up on the boundary ∂Ω.

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U2 - 10.1016/j.anihpc.2013.03.001

DO - 10.1016/j.anihpc.2013.03.001

M3 - Article

AN - SCOPUS:84897445598

VL - 31

SP - 231

EP - 247

JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

SN - 0294-1449

IS - 2

ER -