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.

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},
pages = {231--247},
publisher = {Elsevier},
volume = {31},
number = {2},
year = {2014},
doi = {10.1016/j.anihpc.2013.03.001},
mrnumber = {3181667},
zbl = {1297.35052},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2013.03.001/}
}
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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://archive.numdam.org/articles/10.1016/j.anihpc.2013.03.001/

[1] X.-Y. Chen, H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations 78 (1989), 160-190 | MR | Zbl

[2] C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, J. Math. Anal. Appl. 335 (2007), 418-427 | MR | Zbl

[3] F. Dickstein, Blowup stability of solutions of the nonlinear heat equation with a large life span, J. Differential Equations 223 (2006), 303-328 | MR | Zbl

[4] M. Fila, M. Winkler, Single-point blow-up on the boundary where the zero Dirichlet boundary condition is imposed, J. Eur. Math. Soc. 10 (2008), 105-132 | EuDML | MR | Zbl

[5] M. Fila, P. Souplet, The blow-up rate for semilinear parabolic problems on general domains, NoDEA Nonlinear Differential Equations Appl. 8 (2001), 473-480 | MR | Zbl

[6] A. Friedman, B. Mcleod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425-447 | MR | Zbl

[7] Y. Fujishima, Location of the blow-up set for a superlinear heat equation with small diffusion, Differential Integral Equations 25 (2012), 759-786 | MR | Zbl

[8] Y. Fujishima, K. Ishige, Blow-up set for a semilinear heat equation with small diffusion, J. Differential Equations 249 (2010), 1056-1077 | MR | Zbl

[9] Y. Fujishima, K. Ishige, Blow-up for a semilinear parabolic equation with large diffusion on ${𝐑}^{N}$, J. Differential Equations 250 (2011), 2508-2543 | MR | Zbl

[10] Y. Fujishima, K. Ishige, Blow-up for a semilinear parabolic equation with large diffusion on ${𝐑}^{N}$. II, J. Differential Equations 252 (2012), 1835-1861 | MR | Zbl

[11] Y. Fujishima, K. Ishige, Blow-up set for a semilinear heat equation and pointedness of the initial data, Indiana Univ. Math. J., in press. | MR

[12] V.A. Galaktionov, S.P. Kurdyumov, A.A. Samarskii, Asymptotic stability of invariant solutions of nonlinear equations of heat conduction with a source, Differential Equations 20 (1984), 461-476 | MR | Zbl

[13] V.A. Galaktionov, S.A. Posashkov, The equation ${u}_{t}={u}_{xx}+{u}^{\beta }$. Localization, asymptotic behavior of unbounded solutions, Keldysh Inst. Appl. Math. Acad. Sci., USSR, preprint No. 97, 1985. | MR

[14] Y. Giga, R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297-319 | MR | Zbl

[15] Y. Giga, R.V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), 1-40 | MR | Zbl

[16] Y. Giga, R.V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989), 845-884 | MR | Zbl

[17] Y. Giga, S. Matsui, S. Sasayama, Blow up rate for semilinear heat equations with subcritical nonlinearity, Indiana Univ. Math. J. 53 (2004), 483-514 | MR | Zbl

[18] K. Ishige, Blow-up time and blow-up set of the solutions for semilinear heat equations with large diffusion, Adv. Difference Equ. 8 (2002), 1003-1024 | MR | Zbl

[19] K. Ishige, N. Mizoguchi, Blow-up behavior for semilinear heat equations with boundary conditions, Differential Integral Equations 16 (2003), 663-690 | MR | Zbl

[20] K. Ishige, N. Mizoguchi, Location of blow-up set for a semilinear parabolic equation with large diffusion, Math. Ann. 327 (2003), 487-511 | MR | Zbl

[21] K. Ishige, H. Yagisita, Blow-up problems for a semilinear heat equation with large diffusion, J. Differential Equations 212 (2005), 114-128 | MR | Zbl

[22] H. Matano, F. Merle, Classification of type I and type II behaviors for a supercritical nonlinear heat equation, J. Funct. Anal. 256 (2009), 992-1064 | MR | Zbl

[23] H. Matano, F. Merle, Threshold and generic type I behaviors for a supercritical nonlinear heat equation, J. Funct. Anal. 261 (2011), 716-748 | MR | Zbl

[24] F. Merle, Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math. 45 (1992), 263-300 | MR | Zbl

[25] F. Merle, H. Zaag, Stability of the blow-up profile for equations of the type ${u}_{t}=\Delta u+{|u|}^{p-1}u$, Duke Math. J. 86 (1997), 143-195 | MR | Zbl

[26] F. Merle, H. Zaag, Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math. 51 (1998), 139-196 | MR | Zbl

[27] F. Merle, H. Zaag, A Liouville theorem for vector-valued nonlinear heat equations and applications, Math. Ann. 316 (2000), 103-137 | MR | Zbl

[28] N. Mizoguchi, Blowup rate of solutions for a semilinear heat equation with the Dirichlet boundary condition, Asymptot. Anal. 35 (2003), 91-112 | MR | Zbl

[29] P. Polàčik, P. Quittner, P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: parabolic equations, Indiana Univ. Math. J. 56 (2007), 879-908 | MR | Zbl

[30] P. Quittner, P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, Birkhäuser Adv. Texts, Basler Lehrbücher Birkhäuser Verlag, Basel (2007) | MR | Zbl

[31] J.J.L. Velázquez, Estimates on the $\left(n-1\right)$-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, Indiana Univ. Math. J. 42 (1993), 445-476 | MR | Zbl

[32] F.B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differential Equations 55 (1984), 204-224 | MR | Zbl

[33] H. Yagisita, Blow-up profile of a solution for a nonlinear heat equation with small diffusion, J. Math. Soc. Japan 56 (2004), 993-1005 | MR | Zbl

[34] H. Zaag, On the regularity of the blow-up set for semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 505-542 | EuDML | Numdam | MR | Zbl

[35] H. Zaag, Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation, Duke Math. J. 133 (2006), 499-525 | MR | Zbl

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