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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

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 & 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.  Google Scholar

[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.  Google Scholar

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K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503-505. doi: 10.3792/pja/1195519254.  Google Scholar

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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.  Google Scholar

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Howard A. Levine, The role of critical exponents in blowup theorems, SIAM Review, 32 (1990), 262-288. doi: 10.1137/1032046.  Google Scholar

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P. Meier, On the critical exponent for reaction-diffusion equations, Arch. Rational Mech. Anal., 109 (1990), 63-71. doi: 10.1007/BF00377979.  Google Scholar

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F. Spitzer, "Principles of Random Walk," Second edition, Graduate Texts in Mathematics, Vol. 34, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

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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.  Google Scholar

show all references

References:
[1]

J. Bebernes and D. Eberly, "Mathematical Problems from Combustion Theory," Appl. Math. Sci., 83, Springer-Verlag, New York, 1989.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

Howard A. Levine, The role of critical exponents in blowup theorems, SIAM Review, 32 (1990), 262-288. doi: 10.1137/1032046.  Google Scholar

[6]

P. Meier, On the critical exponent for reaction-diffusion equations, Arch. Rational Mech. Anal., 109 (1990), 63-71. doi: 10.1007/BF00377979.  Google Scholar

[7]

F. Spitzer, "Principles of Random Walk," Second edition, Graduate Texts in Mathematics, Vol. 34, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[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.  Google Scholar

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