Fujita型反应扩散方程组整体解的存在性、非存在性与渐近性质

Existence, Nonexistence and Asymptotic Behavior of Global Solutions of Reaction-Diffusion System of Fujita Type

本文研究 Ｆｕｊｉｔａ型反应扩散方程组的初值问题：ｕｔ－△ｕ＝ａ１ｕ~α１－１ｕ＋ｂ１ｖ~β１－１ｖ，ｖｔ－△ｖ＝ａ２ｕ~α２－１ｕ＋ｂ２ｖ~β２－１ｖ，ｕ（Ｘ，０）＝ｕ０（Ｘ），Ｖ（Ｘ，０）＝Ｖ０（Ｘ），（Ｘ，ｔ）Ｒ~Ｎ ｘ Ｒ~＋，其中 ａｉ，ｂｉ≥ ０， αｉ，βｉ≥ １（ｉ＝ １，２），给出了非负整体 Ｌ~ｐ解与古典解存在性与非存在性的一系列充分条件，并讨论了解的渐近性质．本文所用方法和所得结果与已有的工作［１－４］，有很大的不同，不但在某些方面推广了［１－５］，而且从某些方面改进了［１］的结果。

This paper studies the initial value problem of Reaction-Diffusion system of Fujita type: ut - △u = a1u~α1-12u + b1v~β1-1v, vt - △v = a2u~α2-1u + b2v~β2-1v, u(x,0) = uo(x), v(x,0) = vo(x), (x,t) E R^N x R^+, where ai,bi≥ 0, on,αiβi≥ 1 (i = 1, 2), gives a series of sufficient conditions of the existence and nonexistence of the nonnegative global L^P solutions and classical solutions, and discusses the asymptotic behavior of solutions. The method used in this paper and obtained results are completly different from previous works [1-4], this paper not only generalies the results of [1-5], but also improves the results of [1] on some respect.