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

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.
Citation: Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209
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K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations,, Proc. Japan Acad., 49 (1973), 503. doi: 10.3792/pja/1195519254.

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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. doi: 10.2969/jmsj/02930407.

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

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

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F. Spitzer, "Principles of Random Walk,", Second edition, 34 (1976).

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F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation,, Israel J. Math., 38 (1981), 29. doi: 10.1007/BF02761845.

show all references

References:
[1]

J. Bebernes and D. Eberly, "Mathematical Problems from Combustion Theory,", Appl. Math. Sci., 83 (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.

[3]

K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations,, Proc. Japan Acad., 49 (1973), 503. 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. doi: 10.2969/jmsj/02930407.

[5]

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

[6]

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

[7]

F. Spitzer, "Principles of Random Walk,", Second edition, 34 (1976).

[8]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation,, Israel J. Math., 38 (1981), 29. doi: 10.1007/BF02761845.

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