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Blow-up for a semilinear parabolic equation with large diffusion on \(\mathbf R^N\). II. (English) Zbl 1255.35055

This is a continuation of a previous paper of the authors [J. Differ. Equations 250, No. 5, 2508–2543 (2011; Zbl 1225.35034)], where a different class of initial functions was considered. It is shown here that the behavior of the blow-up time and the blow-up set (as the diffusion coefficient tends to infinity) is completely different from the case studied before.

MSC:

35B44 Blow-up in context of PDEs
35K58 Semilinear parabolic equations

Citations:

Zbl 1225.35034
Full Text: DOI

References:

[1] Chavel, I.; Karp, L., Movement of hot spots in Riemannian manifolds, J. Anal. Math., 55, 271-286 (1990) · Zbl 0718.53036
[2] 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
[3] 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
[4] 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
[5] 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
[6] Friedman, A.; McLeod, B., Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34, 425-447 (1985) · Zbl 0576.35068
[7] 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
[8] Fujishima, Y.; Ishige, K., Blow-up for a semilinear parabolic equation with large diffusion on \(R^N\), J. Differential Equations, 250, 2508-2543 (2011) · Zbl 1225.35034
[9] Y. Fujishima, K. Ishige, Blow-up set for a semilinear heat equation and pointedness of the initial data, Indiana Univ. Math. J., in press.; Y. Fujishima, K. Ishige, Blow-up set for a semilinear heat equation and pointedness of the initial data, Indiana Univ. Math. J., in press. · Zbl 1277.35072
[10] Giga, Y.; Kohn, R. V., Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 42, 845-884 (1989) · Zbl 0703.35020
[11] Giga, Y.; Umeda, N., On blow-up at space infinity for semilinear heat equations, J. Math. Anal. Appl., 316, 538-555 (2006) · Zbl 1106.35029
[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
[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
[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
[17] Ladyženskaja, O. A.; Solonnikov, V. A.; Uralʼceva, N. N., Linear and Quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Transl., vol. 23 (1968), American Mathematical Society: American Mathematical Society Providence, RI · 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
[19] Merle, F.; Zaag, H., Stability of the blow-up profile for equations of the type \(u_t = \Delta u + | u |^{p - 1} u\), Duke Math. J., 86, 143-195 (1997) · Zbl 0872.35049
[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
[21] P. Quittner, P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007.; P. Quittner, P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 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
[23] Weissler, F. B., Single point blow-up for a semilinear initial value problem, J. Differential Equations, 55, 204-224 (1984) · Zbl 0555.35061
[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
[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
[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
[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
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