Blow-up set for type I blowing up solutions for a semilinear heat equation
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 2, p. 231-247
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Let u be a type I blowing up solution of the Cauchy–Dirichlet problem for a semilinear heat equation, { t u=Δu+u p ,xΩ,t>0,u(x,t)=0,xΩ,t>0,u(x,0)=ϕ(x),xΩ,(P) where Ω is a (possibly unbounded) domain in 𝐑 N , N1, and p>1. We prove that, if ϕL (Ω)L q (Ω) for some q[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.

@article{AIHPC_2014__31_2_231_0,
     author = {Fujishima, Yohei and Ishige, Kazuhiro},
     title = {Blow-up set for type I blowing up solutions for a semilinear heat equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {31},
     number = {2},
     year = {2014},
     pages = {231-247},
     doi = {10.1016/j.anihpc.2013.03.001},
     zbl = {1297.35052},
     mrnumber = {3181667},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2014__31_2_231_0}
}
Fujishima, Yohei; Ishige, Kazuhiro. Blow-up set for type I blowing up solutions for a semilinear heat equation. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 2, pp. 231-247. doi : 10.1016/j.anihpc.2013.03.001. http://www.numdam.org/item/AIHPC_2014__31_2_231_0/

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