The Remark about Fujita Exponent for a
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 , there is something wrong in the proof for belong to the blow-up case, while  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
 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.
 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.
 K. Hayakawa. On nonexistence of global solutions of some semilinear parabolic equations. Japan Academy Proceedings, 1973, 49(7): 503-505.
 F. B. Weissler. Existence and nonexistence of global solutions in exterior domains. Israel Journal of Mathematics, 1981, 38(1-2): 29-40.
 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.
 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.
 K. Mochizuki, R. Suzuki. Critical Exponent and critical blow-up for quasilinear parabolic equations, Israel Journal of Mathematics, 1997, 98(1): 141-156.