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Blow-up for a semilinear parabolic equation with large diffusion on $\Bbb R^N$. (English) Zbl 1225.35034
The blow-up time and the blow-up set of positive solutions of a semilinear heat equation with a large diffusion coefficient is studied. It is shown that under suitable assumptions on initial data the location of the blow-up set depends on the large time behavior of the hot spots of solutions of the linear heat equation.

35B44Blow-up (PDE)
35K91Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian
Full Text: DOI
[1] Chavel, I.; Karp, L.: Movement of hot spots in Riemannian manifolds, J. anal. Math. 55, 271-286 (1990) · Zbl 0718.53036 · doi:10.1007/BF02789205
[2] Cheng, T.; Zheng, G. -F.: Some blow-up problems for a semilinear parabolic equation with a potential, J. differential equations 244, 766-802 (2008) · Zbl 1139.35060 · doi:10.1016/j.jde.2007.11.004
[3] Chen, Y. -G.: Blow-up solutions of a semilinear parabolic equation with the Neumann and Robin boundary conditions, J. fac. Sci. univ. Tokyo 37, 537-574 (1990) · Zbl 0785.35052
[4] Chen, X. -Y.; Matano, H.: Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. differential equations 78, 160-190 (1989) · Zbl 0692.35013 · doi:10.1016/0022-0396(89)90081-8
[5] Cortazar, C.; Elgueta, M.; Rossi, J. D.: The blow-up problem for a semilinear parabolic equation with a potential, J. math. Anal. appl. 335, 418-427 (2007) · Zbl 1131.35039 · doi:10.1016/j.jmaa.2007.01.079
[6] Dickstein, F.: Blowup stability of solutions of the nonlinear heat equation with a large life span, J. differential equations 223, 303-328 (2006) · Zbl 1100.35044 · doi:10.1016/j.jde.2005.08.009
[7] Filippas, S.; Merle, F.: Compactness and single-point blowup of positive solutions on bounded domains, Proc. roy. Soc. Edinburgh sect. A 127, 47-65 (1997) · Zbl 0874.35053 · doi:10.1017/S0308210500023507
[8] Friedman, A.; Mcleod, B.: Blow-up of positive solutions of semilinear heat equations, Indiana univ. Math. J. 34, 425-447 (1985) · Zbl 0576.35068 · doi:10.1512/iumj.1985.34.34025
[9] Fujishima, Y.; Ishige, K.: Blow-up set for a semilinear heat equation with small diffusion, J. differential equations 249, 1056-1077 (2010) · Zbl 1204.35054 · doi:10.1016/j.jde.2010.03.028
[10] Y. Fujishima, K. Ishige, Blow-up set for a semilinear heat equation and pointedness of the initial data, preprint. · Zbl 1277.35072
[11] Giga, Y.; Kohn, R. V.: Nondegeneracy of blowup for semilinear heat equations, Comm. pure appl. Math. 42, 845-884 (1989) · Zbl 0703.35020 · doi:10.1002/cpa.3160420607
[12] Ishige, K.: Blow-up time and blow-up set of the solutions for semilinear heat equations with large diffusion, Adv. differential equations 8, 1003-1024 (2002) · Zbl 1036.35096
[13] Ishige, K.; Ishiwata, M.; Kawakami, T.: The decay of the solutions for the heat equation with a potential, Indiana univ. Math. J. 58, 2673-2707 (2009) · Zbl 1196.35051 · doi:10.1512/iumj.2009.58.3771
[14] Ishige, K.; Mizoguchi, N.: Blow-up behavior for semilinear heat equations with boundary conditions, Differential integral equations 16, 663-690 (2003) · Zbl 1035.35052
[15] Ishige, K.; Mizoguchi, N.: Location of blow-up set for a semilinear parabolic equation with large diffusion, Math. ann. 327, 487-511 (2003) · Zbl 1049.35022 · doi:10.1007/s00208-003-0463-4
[16] Ishige, K.; Yagisita, H.: Blow-up problems for a semilinear heat equation with large diffusion, J. differential equations 212, 114-128 (2005) · Zbl 1072.35096 · doi:10.1016/j.jde.2004.10.021
[17] Ladyženskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N.: Linear and quasi-linear equations of parabolic type, Amer. math. Soc. transl. 23 (1968) · Zbl 0174.15403
[18] Merle, F.: Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. pure appl. Math. 45, 263-300 (1992) · Zbl 0785.35012 · doi:10.1002/cpa.3160450303
[19] Merle, F.; Zaag, H.: Stability of the blow-up profile for equations of the type $ut={\Delta}$u+|u|p-1u, Duke math. J. 86, 143-195 (1997) · Zbl 0872.35049 · doi:10.1215/S0012-7094-97-08605-1
[20] Mizoguchi, N.; Vázquez, J. L.: Multiple blowup for nonlinear heat equations at different places and different times, Indiana univ. Math. J. 56, 2859-2886 (2007) · Zbl 1145.35071 · doi:10.1512/iumj.2007.56.3147
[21] Quittner, P.; Souplet, P.: Superlinear parabolic problems, blow-up, global existence and steady states, Birkhäuser adv. Texts basler lehrbücher (2007) · Zbl 1128.35003
[22] Velázquez, J. J. L.: Estimates on the (n-1)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, Indiana univ. Math. J. 42, 445-476 (1993) · Zbl 0802.35073 · doi:10.1512/iumj.1993.42.42021
[23] Weissler, F. B.: Single point blow-up for a semilinear initial value problem, J. differential equations 55, 204-224 (1984) · Zbl 0555.35061 · doi:10.1016/0022-0396(84)90081-0
[24] Yagisita, H.: Blow-up profile of a solution for a nonlinear heat equation with small diffusion, J. math. Soc. Japan 56, 993-1005 (2004) · Zbl 1065.35154 · doi:10.2969/jmsj/1190905445
[25] Yagisita, H.: Variable instability of a constant blow-up solution in a nonlinear heat equation, J. math. Soc. Japan 56, 1007-1017 (2004) · Zbl 1064.35088 · doi:10.2969/jmsj/1190905446
[26] Zaag, H.: On the regularity of the blow-up set for semilinear heat equations, Ann. inst. H. Poincaré anal. Non linéaire 19, 505-542 (2002) · Zbl 1012.35039 · doi:10.1016/S0294-1449(01)00088-9 · numdam:AIHPC_2002__19_5_505_0
[27] Zaag, H.: Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation, Duke math. J. 133, 499-525 (2006) · Zbl 1096.35062 · doi:10.1215/S0012-7094-06-13333-1 · euclid:dmj/1150201200