伪抛物方程Fujita指标的注记
The Remark about Fujita Exponent for a Pseudo-Parabolic Equation

作者: 解斌强 * , 曾有栋 :福州大学,数学与计算机科学学院;

关键词: 伪抛物柯西问题爆破Fujita指标Pseudo-Parabolic Cauchy Problem Blow-Up Fujita Exponent

摘要:
本文考虑伪抛物方程 的柯西问题的非负解。对于柯西问题,已经知道 是爆破的临界指标;即当 ,所有的非负非平凡解在有限时刻爆破(爆破情况),当 时,存在着非平凡的全局解(全局存在情况)。由于文献[6]对于 是属于爆破的情况的证明有错,而文献[5]对于 是属于经典解的爆破的情况的证明较繁。本文是对于临界指标 属于爆破的情况给出了一个新且简洁的证明方法且经典解推广到更为一般的弱解。

Abstract:
In this paper we consider nonnegative solutions to the Cauchy problem for the pseudo-parabolic equation . It is well known that is the critical exponent of blow up. Namely, if , then all the nontrivial solutions blow up in finite time (blow-up case), and if , then there are nontrivial global solutions (global existence case). In this paper we show for the Cauchy problem, belongs to the blow-up case. Because [6], there is something wrong in the proof for belong to the blow-up case, while [5] the method is too complicate. In the paper we give a new simpler proof for the critical exponent which belongs to the blow-up case, moreover we generalize classical solution to the general weak solution case.

文章引用: 解斌强 , 曾有栋 (2013) 伪抛物方程Fujita指标的注记。 理论数学, 3, 300-304. doi: 10.12677/PM.2013.35046

参考文献

[1] H. Fujita. On the blowing up of solutions of the Cauchy problem for . Journal of the Faculty of Science of the University of Toky, 1966, 13(2): 109-124.

[2] V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov and A. A. Samarskii. Unbounded solutions of the Cauchy problem for the parabolic equation . Soviet Physics Doklady, 1980, 25(5): 458-459.

[3] K. Hayakawa. On nonexistence of global solutions of some semilinear parabolic equations. Japan Academy Proceedings, 1973, 49(7): 503-505.

[4] F. B. Weissler. Existence and nonexistence of global solutions in exterior domains. Israel Journal of Mathematics, 1981, 38(1-2): 29-40.

[5] Y. Cao, J. X. Yin and C. P. Wang. Cauchy problem of semilinear pseudo-parabolic equations. Journal of Differential Equations, 2009, 246(12): 4568-4590.

[6] El. Kaikina, P. I. Naumkin and I. A. Shishmarev. The Cauchy problem for an equation of Sobolev type with power non-linearity. Izvestiya Mathematics, 2005, 69(1): 61-114.

[7] K. Mochizuki, R. Suzuki. Critical Exponent and critical blow-up for quasilinear parabolic equations, Israel Journal of Mathematics, 1997, 98(1): 141-156.

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