Existence and non-existence of global solutions for a discrete semilinear heat equation

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March 2011, 31(1): 209-220. doi: 10.3934/dcds.2011.31.209

Existence and non-existence of global solutions for a discrete semilinear heat equation

Keisuke Matsuya ^1, and Tetsuji Tokihiro ^1,

1.	University of Tokyo, Komaba 3-8-1, Meguro, Tokyo 153-8914, Japan, Japan

Received April 2010 Revised October 2010 Published June 2011

Abstract
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Existence of global solutions to initial value problems for a discrete analogue of a $d$-dimensional semilinear heat equation is investigated. We prove that a parameter $\alpha$ in the partial difference equation plays exactly the same role as the parameter of nonlinearity does in the semilinear heat equation. That is, we prove non-existence of a non-trivial global solution for $0<\alpha \le 2/d$, and, for $\alpha > 2/d$, existence of non-trivial global solutions for sufficiently small initial data.

Keywords: global solution., semilinear heat equation, Discretization.

Mathematics Subject Classification: Primary: 39A14, 74G25; Secondary:35K58, 39A1.

Citation: Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209

References:

[1]	J. Bebernes and D. Eberly, "Mathematical Problems from Combustion Theory," Appl. Math. Sci., 83, Springer-Verlag, New York, 1989.
[2]	H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. A Math., 16 (1966), 109-124.
[3]	K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503-505. doi: 10.3792/pja/1195519254.
[4]	K. Kobayashi, T. Sirao and H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407-424. doi: 10.2969/jmsj/02930407.
[5]	Howard A. Levine, The role of critical exponents in blowup theorems, SIAM Review, 32 (1990), 262-288. doi: 10.1137/1032046.
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[7]	F. Spitzer, "Principles of Random Walk," Second edition, Graduate Texts in Mathematics, Vol. 34, Springer-Verlag, New York-Heidelberg, 1976.
[8]	F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845.

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References:

[1]	J. Bebernes and D. Eberly, "Mathematical Problems from Combustion Theory," Appl. Math. Sci., 83, Springer-Verlag, New York, 1989.
[2]	H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. A Math., 16 (1966), 109-124.
[3]	K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503-505. doi: 10.3792/pja/1195519254.
[4]	K. Kobayashi, T. Sirao and H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407-424. doi: 10.2969/jmsj/02930407.
[5]	Howard A. Levine, The role of critical exponents in blowup theorems, SIAM Review, 32 (1990), 262-288. doi: 10.1137/1032046.
[6]	P. Meier, On the critical exponent for reaction-diffusion equations, Arch. Rational Mech. Anal., 109 (1990), 63-71. doi: 10.1007/BF00377979.
[7]	F. Spitzer, "Principles of Random Walk," Second edition, Graduate Texts in Mathematics, Vol. 34, Springer-Verlag, New York-Heidelberg, 1976.
[8]	F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845.

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