伪抛物方程Fujita指标的注记

The Remark about Fujita Exponent for a Pseudo-Parabolic Equation

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。

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.