MR2867018 Masahiko SHIMOJO and Noriaki UMEDA Masahiko SHIMOJO and Noriaki UMEDA Blow-Up at Space Infinity for Solutions of Cooperative Reaction-Diffusion Systems Funkcialaj Ekvacioj. Serio Internacia 54 2011 315--334 http://fe.math.kobe-u.ac.jp/FE/FullPapers/54-2/54_315.pdf http://www.ams.org/mathscinet-getitem?mr=MR2867018 We consider the Cauchy problem of cooperative reaction-diffusion systems with nonnegative initial data. Here we discuss the blow-up of a solution that occurs only at space infinity. We give sufficient conditions for such phenomena, and study an asymptotic behaviour at space infinity of the solutions at the blow-up time. In general, relatively little is proved on the locations of blow-up point for semilinear parabolic systems. However, our results can be applied to a large class of nonlinearity for some class of initial value. The reason of this is that, when the blow-up occurs only at space infinity, the effect of reaction is much stronger than that of diffusion, and the behaviour at space infinity is well approximated by the flat solution. Blow-up at space infinity, Cooperative reaction-diffusion systems. 35K45, 35K57. 54-315 2011 Blow-Up at Space Infinity for Solutions of Cooperative Reaction-Diffusion Systems Masahiko SHIMOJO and Noriaki UMEDA Masahiko SHIMOJO and Noriaki UMEDA
1 Cortazar, C.; Elgueta, M.; Rossi, J. D. The blow-up problem for a semilinear parabolic equation with a potential J. Math. Anal. Appl. 335 2007 418--427 MR2340331
2 Cheng, T.; Zheng, G.-F. Some blow-up problem for a semilinear parabolic equation with a potential J. Differential Equations 244 2008 716--802 MR2391344
3 Deng, K. Blow-up rates for parabolic systems Z. Angew. Math. Phys. 47 1996 132--143 MR1408675
4 Deng, K.; Levine, H. A. The role of critical exponents in blow-up theorems J. Math. Anal. Appl. 243 2000 85--126 MR1742850
5 Eidel'man, S. D. Parabolic systems North-Holland Pub. Co. 1969 MR0252806
6 Escobedo, M.; Herrero, M. A. Boundedness and blow up for a semilinear reaction-diffusion system J. Differential Equations 89 1991 176--202 MR1088342
7 Friedman, A. Partial differential equations of parabolic type Prentice-Hall, Inc. Englewood Cliffs, New Jersey 1964 MR0181836
8 Friedman, A.; Giga, Y. A single point blow-up for solutions of semilinear parabolic systems J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 34 1987 65--79 MR0882125
9 Friedman, A.; McLeod, B. Blow-up of positive solutions of semilinear heat equations Indiana Univ. Math. J. 34 1985 425--447 MR0783924
10 Giga, Y.; Kohn, R. V. Nondegeneracy of blowup for semilinear heat equations Comm. Pure Appl. Math. 42 1989 845--884 MR1003437
11 Giga, Y.; Umeda, N. On blow-up at space infinity for semilinear heat equations J. Math. Anal. Appl. 316 2006 538--555 MR2206688
12 Giga, Y.; Umeda, N. Blow-up directions at space infinity for solutions of semilinear heat equations Bol. Soc. Parana. Mat. (3) 23 2005 9--28: Correction is available in Bol. Soc. Parana. Mat. (3) 24 (2006), 19--24 MR2242285
13 Gladkov, A. L. Behavior of solutions of semilinear parabolic equations as $x\to\infty $ Math. Notes 51 1992 124--128 MR1165462
14 Gladkov, A. L. Stabilization of solutions for semilinear parabolic systems as $\vert x\vert\to\infty$ Electron. J. Differential Equations 2009 2009 No. 7, 1--5 MR2471116
15 Guo, J.-S.; Sasayama, S.; Wang, C.-J. Blowup rate estimate for a system of semilinear parabolic equations Comm. Pure Appl. Anal. 8 2009 711--718 MR2461571
16 Igarashi, T.; Umeda, N. Existence and nonexistence of global solutions in time for a reaction-diffusion system with inhomogeneous terms Funkcial. Ekvac. 51 2008 17--37 MR2427541 2427541
17 Igarashi, T.; Umeda, N. Nonexistence of global solutions in time for reaction-diffusion systems with inhomogeneous terms in cones Tsukuba J. Math. 33 2009 131--145 MR2553841
18 Igarashi, T.; Umeda, N., Existence of global solutions in time for reaction-diffusion systems with inhomogeneous terms in cones, submitted
19 Ishige, K.; Mizoguchi, N. Location of blow-up set for a semilinear parabolic equation with large diffusion Math. Ann. 327 2003 487--511 MR2021027
20 Ishige, K.; Yagisita, H. Blow-up problem for a semilinear heat equation with large diffusion J. Differential Equations 212 2005 114--128 MR2130549
21 Lacey, A. A. The form of blow-up for nonlinear parabolic equations Proc. Roy. Soc. Edinburgh Sect. A 98 1984 183--202 MR0765494
22 Matano, H.; Merle, F. On nonexistence of Type II blowup for a supercritical nonlinear heat equation Comm. Pure. Appl. Math. 57 2004 1494--1541 MR2077706
23 Matano, H.; Merle, F. Classification of type I and type II behaviors for a supercritical nonlinear heat equation J. Funct. Anal. 256 2009 992--1064 MR2488333
24 Merle, F. Solution of a nonlinear heat equation with arbitrarily given blow-up points Comm. Pure. Appl. Math. 45 1992 263--300 MR1151268
25 Merle, F.; Zaag, H. Optimal estimates for blowup rate and behavior for nonlinear heat equations Comm. Pure. Appl. Math. 51 1998 139--196 MR1488298
26 Mochizuki, K.; Huang, Q. Existence and behavior of solutions for a weakly coupled system of reaction-diffusion equations Methods Appl. Anal. 5 1998 109--124 MR1636542
27 Protter, M. H.; Weinberger, H. F. Maximum principles in differential equations Englewood Cliffs, N. J., Prentice-Hail 1967 MR0219861
28 Rossi, J. D.; Souplet, P. Coexistence of simultaneous and nonsimultaneous blow-up in a semilinear parabolic system Differential Integral Equations 18 2005 405--418 MR2122706
29 Seki, Y. On directional blow-up for quasilinear parabolic equations with fast diffusion J. Math. Anal. Appl. 338 2008 572--587 MR2386440
30 Seki, Y.; Suzuki, R.; Umeda, N. Blow-up directions for quasilinear parabolic equations Proc. Roy. Soc. Edinburgh Sect. A 138 2008 379--405 MR2406697
31 Souplet, Ph. Single point blow-up for a semilinear parabolic system J. Eur. Math. Soc. (JEMS) 11 2009 169--188 MR2471135
32 Shimojō, M. The global profile of blow-up at space infinity in semilinear heat equations J. Math. Kyoto Univ. 48 2008 339--361 MR2436740
33 Shimojō, M., On blow-up phenomenon at space infinity and its locality for semilinear heat equations (in Japanese), Master's Thesis, The University of Tokyo (2005)
34 Umeda, N. Blow-up and large time behavior of solutions of a weakly coupled system of reaction-diffusion equations Tsukuba J. Math. 27 2003 31--46 MR1999233
35 Umeda, N. Existence and nonexistence of global solutions of a weakly coupled system of reaction-diffusion equations Commun. Appl. Anal. 10 2006 57--78 MR2203590
36 Weissler, F. B. Single point blow-up for a semilinear initial value problem J. Differential Equations 55 1984 204--224 MR0764124
37 Wang, M. Blow-up rate estimates for semilinear parabolic systems J. Differential Equations 170 2001 317--324 MR1815186
38 Zaag, H. On the regularity of the blow-up set for semilinear heat equations Ann. Inst. H. Poincaré Anal. Non Linéaire 19 2002 505--542 MR1922468