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, $\begin{array}{cc}\left\{\begin{array}{cc}{\partial }_{t}u=\Delta u+{u}^{p},\hfill & x\in \Omega ,\phantom{\rule{0.166667em}{0ex}}t>0,\hfill \\ u\left(x,t\right)=0,\hfill & x\in \partial \Omega ,\phantom{\rule{0.166667em}{0ex}}t>0,\hfill \\ u\left(x,0\right)=\varphi \left(x\right),\hfill & x\in \Omega ,\hfill \end{array}& \text{(P)}\end{array}$ where Ω is a (possibly unbounded) domain in ${𝐑}^{N}$, $N⩾1$, and $p>1$. We prove that, if $\varphi \in {L}^{\infty }\left(\Omega \right)\cap {L}^{q}\left(\Omega \right)$ for some $q\in \left[1,\infty \right)$, 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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