带非线性梯度项的渗流方程的梯度爆破

Gradient Blow-up for the Filtration Equations with Nonlinear Gradient Terms

研究了一类带非线性梯度项的Newton渗流方程u_t-△u^m=F(u,▽u)的梯度爆破性质,即▽u在有限时间爆破而u仍然一致有界.首先,在右端反应项只含非线性梯度项时,对任意连续Dirichlet边界条件,得到了当初值充分大时,方程梯度爆破.其次,在右端反应项含有梯度和零阶非线性项时,得到了方程梯度爆破的某些结果,这些结果部分地回答了Souplet于2002年提出的一个相关问题.最后,在右端反应项含有非局部梯度源时,得到了通常不会产生梯度爆破的结果.

A property of gradient blow up for a kind of Newton filtration equation u_t-△um=F（u,▽u） with nonlinear gradient is studied. This property concludes that u is uniformly bounded when ▽u blows up in limited time. We firstly show that if the reaction term in the right side of an equation has only a nonlinear gradition term, then for any continuous boundary condition, this equation blows up gradiently when the initial value is big enough. Secondly, some conclusions about gradient blow-up are reached if the reaction term has a gradient term and a zero-order nonlinear term, which partly solved the problem proposed by Souplet in 2002. Finally, we show an equation does not blow up gradiently when the reaction term of right side has nonlocal gradient source.