In this paper we study the initial boundary value problem for the system $u_t-\Delta u^m=-\mbox{div}(u^{q}\nabla v),\ v_t-\Delta v+v=u$. 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.

中文翻译：

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抛物线-抛物线型 Keller-Segel 模型中的非线性扩散

在本文中我们研究系统的初始边值问题$u_t-\Delta u^m=-\mbox{div}(u^{q}\nabla v),\ v_t-\Delta v+v=u$ . 这个问题就是所谓的具有非线性扩散的 Keller-Segel 模型。我们的调查表明，非线性扩散可以防止过度拥挤。准确地说，我们证明了只要 $m>q>0$ 解决方案是有界的，从而在很大程度上推广了该领域的已知结果。此外，我们的结果似乎暗示 Keller-Segel 模型可以同时具有有界解和爆炸解。