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Life span of solutions for a semilinear heat equation with initial data having positive limit inferior at infinity. (English) Zbl 1221.35076
Summary: We present a new upper bound of the life span of positive solutions of a semilinear heat equation for initial data having positive limit inferior at space infinity. The upper bound is expressed by the data in limit inferior, not in every direction, but around a specific direction. It is also shown that the minimal time blow-up occurs when initial data attains its maximum at space infinity.

35B44Blow-up (PDE)
35K57Reaction-diffusion equations
35K58Semilinear parabolic equations
35B09Positive solutions of PDE
Full Text: DOI
[1] Deng, K.; Levine, H. A.: The role of critical exponents in blow-up theorems: the sequel, J. math. Anal. appl. 243, 85-126 (2000) · Zbl 0942.35025 · doi:10.1006/jmaa.1999.6663
[2] Fujita, H.: On the blowing up of solutions of the Cauchy problem for $ut={\Delta}u+u1+{\alpha}$, J. fac. Sci. univ. Tokyo sect. IA math. 13, 109-124 (1966) · Zbl 0163.34002
[3] Hayakawa, K.: On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan acad. 49, 503-505 (1973) · Zbl 0281.35039 · doi:10.3792/pja/1195519254
[4] Kobayashi, K.; Sirao, T.; Tanaka, H.: On the growing up problem for semilinear heat equations, J. math. Soc. Japan 29, 407-424 (1977) · Zbl 0353.35057 · doi:10.2969/jmsj/02930407
[5] Levine, H. A.: The role of critical exponents in blow-up theorems, SIAM rev. 32, 262-288 (1990) · Zbl 0706.35008 · doi:10.1137/1032046
[6] Weissler, F. B.: Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math. 38, 29-40 (1981) · Zbl 0476.35043 · doi:10.1007/BF02761845
[7] Lee, T. Y.; Ni, W. M.: Global existence, large time behavior and life span of solutions on a semilinear parabolic Cauchy problem, Trans. amer. Math. soc. 333, 365-378 (1992) · Zbl 0785.35011 · doi:10.2307/2154114
[8] Mizoguchi, N.; Yanagida, E.: Blowup and life span of solutions for a semilinear parabolic equation, SIAM J. Math. anal. 29, 1434-1446 (1998) · Zbl 0909.35056 · doi:10.1137/S0036141097324934
[9] 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 · doi:10.1016/j.jmaa.2005.05.007
[10] Giga, Y.; Umeda, N.: Blow-up directions at space infinity for semilinear heat equations, Bol. soc. Parana. mat. 23, 9-28 (2005) · Zbl 1173.35531
[11] Gui, C.; Wang, X.: Life span of solutions on the Cauchy problem for a semilinear heat equation, J. differential equations 115, 166-172 (1995) · Zbl 0813.35034 · doi:10.1006/jdeq.1995.1010
[12] Mizoguchi, N.; Yanagida, E.: Life span of solutions with large initial data in a semilinear parabolic equation, Indiana univ. Math. J. 50, 591-610 (2001) · Zbl 0996.35006
[13] Mochizuki, K.; Suzuki, R.: Blow-up sets and asymptotic behavior of interfaces for quasilinear degenerate parabolic equations in RN, J. math. Soc. Japan 44, 485-504 (1992) · Zbl 0805.35065 · doi:10.2969/jmsj/04430485
[14] Ozawa, T.; Yamauchi, Y.: Life span of positive solutions for a semilinear heat equation with general non-decaying initial data, J. math. Anal. appl. 379, 518-523 (2011) · Zbl 1215.35091 · doi:10.1016/j.jmaa.2011.01.050
[15] Seki, Y.: On directional blow-up for quasilinear parabolic equations with fast diffusion, J. math. Anal. appl. 338, 572-587 (2008) · Zbl 1144.35030 · doi:10.1016/j.jmaa.2007.05.033
[16] Seki, Y.; Umeda, N.; Suzuki, R.: Blow-up directions for quasilinear parabolic equations, Proc. R. Soc. Edinburgh A 138, 379-405 (2008) · Zbl 1167.35393 · doi:10.1017/S0308210506000801
[17] Yamaguchi, M.; Yamauchi, Y.: Life span of positive solutions for a semilinear heat equation with non-decaying initial data, Differential integral equations 23, 1151-1157 (2010) · Zbl 1240.35218