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

Fujishima, Yohei; Ishige, Kazuhiro

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 231-247 / Harvested from Numdam

Résumé

Let u be a type I blowing up solution of the Cauchy–Dirichlet problem for a semilinear heat equation, $\begin{matrix} {\begin{matrix} \partial_{t} u = Δ u + u^{p}, & x \in Ω, t > 0, \\ u (x, t) = 0, & x \in \partial Ω, t > 0, \\ u (x, 0) = ϕ (x), & x \in Ω, \end{matrix} & (P) \end{matrix}$ where Ω is a (possibly unbounded) domain in $𝐑^{N}$ , $N ⩾ 1$ , and $p > 1$ . We prove that, if $ϕ \in L^{\infty} (Ω) \cap L^{q} (Ω)$ for some $q \in [1, \infty)$ , 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.

Publié le : 2014-01-01

MR 3181667 | Zbl 1297.35052

DOI : https://doi.org/10.1016/j.anihpc.2013.03.001

@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},
     volume = {31},
     year = {2014},
     pages = {231-247},
     doi = {10.1016/j.anihpc.2013.03.001},
     mrnumber = {3181667},
     zbl = {1297.35052},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 31 (2014) pp. 231-247. doi : 10.1016/j.anihpc.2013.03.001. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_2_231_0/

Bibliographie

[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 986159 | Zbl 0692.35013

[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 2340331 | Zbl 1131.35039

[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 2214937 | Zbl 1100.35044

[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 | MR 2349898 | Zbl 1190.35038

[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 1867324 | Zbl 0993.35046

[6] A. Friedman, B. Mcleod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425-447 | MR 783924 | Zbl 0576.35068

[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 2975694 | Zbl 1265.35039

[8] Y. Fujishima, K. Ishige, Blow-up set for a semilinear heat equation with small diffusion, J. Differential Equations 249 (2010), 1056-1077 | MR 2652163 | Zbl 1204.35054

[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 2756074 | Zbl 1225.35034

[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 2853563 | Zbl 1255.35055

[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 3043591

[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 742818 | Zbl 0556.35075

[13] V.A. Galaktionov, S.A. Posashkov, The equation $u_{t} = u_{x x} + u^{β}$ . Localization, asymptotic behavior of unbounded solutions, Keldysh Inst. Appl. Math. Acad. Sci., USSR, preprint No. 97, 1985. | MR 832277

[14] Y. Giga, R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297-319 | MR 784476 | Zbl 0585.35051

[15] Y. Giga, R.V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), 1-40 | MR 876989 | Zbl 0601.35052

[16] Y. Giga, R.V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989), 845-884 | MR 1003437 | Zbl 0703.35020

[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 2060042 | Zbl 1058.35096

[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 1895115 | Zbl 1036.35096

[19] K. Ishige, N. Mizoguchi, Blow-up behavior for semilinear heat equations with boundary conditions, Differential Integral Equations 16 (2003), 663-690 | MR 1973274 | Zbl 1035.35052

[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 2021027 | Zbl 1049.35022

[21] K. Ishige, H. Yagisita, Blow-up problems for a semilinear heat equation with large diffusion, J. Differential Equations 212 (2005), 114-128 | MR 2130549 | Zbl 1072.35096

[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 2488333 | Zbl 1178.35084

[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 2799578 | Zbl 1223.35088

[24] F. Merle, Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math. 45 (1992), 263-300 | MR 1151268 | Zbl 0785.35012

[25] F. Merle, H. Zaag, Stability of the blow-up profile for equations of the type $u_{t} = Δ u + {| u |}^{p - 1} u$ , Duke Math. J. 86 (1997), 143-195 | MR 1427848 | Zbl 0872.35049

[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 1488298 | Zbl 0926.35024

[27] F. Merle, H. Zaag, A Liouville theorem for vector-valued nonlinear heat equations and applications, Math. Ann. 316 (2000), 103-137 | MR 1735081 | Zbl 0939.35086

[28] N. Mizoguchi, Blowup rate of solutions for a semilinear heat equation with the Dirichlet boundary condition, Asymptot. Anal. 35 (2003), 91-112 | MR 2007733 | Zbl 1061.35038

[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 2317549 | Zbl 1122.35051

[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 2346798 | Zbl 1128.35003

[31] J.J.L. Velázquez, Estimates on the $(n - 1)$ -dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, Indiana Univ. Math. J. 42 (1993), 445-476 | MR 1237055 | Zbl 0802.35073

[32] F.B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differential Equations 55 (1984), 204-224 | MR 764124 | Zbl 0555.35061

[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 2091413 | Zbl 1065.35154

[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 | Numdam | MR 1922468 | Zbl 1012.35039

[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 2228461 | Zbl 1096.35062