论文标题
在抛物线抛物线类型的Keller-Segel模型中的非线性扩散
Nonlinear diffusion in the Keller-Segel model of parabolic-parabolic type
论文作者
论文摘要
在本文中，我们研究了系统的初始边界值问题$ u_t-ΔU^m = - \ mbox {div}（u^{q} \ nabla v），\ v_t-t-ΔV+v = u $。这个问题是具有非线性扩散的所谓凯勒 - 塞格模型。我们的调查表明，非线性扩散可以防止拥挤。确切地说，我们表明解决方案的界限长达$ q>> 0 $，从而大大概括了该领域的已知结果。此外，我们的结果似乎暗示了凯勒 - 塞格模型可以同时具有限制的解决方案和爆炸解决方案。
In this paper we study the initial boundary value problem for the system $u_t-Δu^m=-\mbox{div}(u^{q}\nabla v),\ v_t-Δv+v=u$. This problem is the so-called Keller-Segel model with nonlinear diffusion. Our investigation reveals that nonlinear diffusion can prevent overcrowding. To be precise, we show that solutions are bounded as long as $m>q>0$, thereby substantially generalizing the known results in this area. Furthermore, our result seems to imply that the Keller-Segel model can have bounded solutions and blow-up ones simultaneously.
