On ε-Regularity Theorem and Asymptotic Behaviors of Solutions for Keller-Segel Systems
On ε-Regularity Theorem and Asymptotic Behaviors of Solutions for Keller-Segel Systems
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DOI：
10.1137/080721078
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发表时间：
2009-10
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通讯作者：
Y. Sugiyama
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作者：
Y. Sugiyama
我们处理方程（KS）$_m$对于$q = m + 2/N$的临界情形，其中$N \ge 3$，$m > 1$，$q \ge 2$：$\partial_t u = \Delta u^m - \nabla \cdot（u^{q-1} \nabla v）$，$x \in \mathbb{R}^N$，$t>0$; $0 = \Delta v - \gamma v + u$，$x \in \mathbb{R}^N$，$t>0$;$u（x，0）= u_0（x）$，$\tau v（x，0）= \tau v_0（x）$，$x \in \mathbb{R}^N$.基于[Y. Sugiyama，Partial regularity and blow-up asymptomatics of weak solutions to degenerate parabolic solutions of porous media type，submitted]，我们首先证明了弱解u的爆破点集S_u$至多具有零Hausdorff维数，如果u \in C_w（[0，T]; L^1（\mathbb{R}^N））$.其次，我们给出了弱解u的各种条件，使得集合S_u$由100个点组成。此外，我们得到了一个明确的常数$\vareps $在这样一种方式，如果当地的质量浓度周围的某个点$x \在S_u$是小于$\vareps $，那么u实际上是当地有界的x，这可能是.
We deal with the equation (KS)$_m$ for the critical case of $q = m + 2/N$ with $N \ge 3$, $m > 1$, $q \ge 2$: $\partial_t u = \Delta u^m - \nabla \cdot (u^{q-1} \nabla v)$, $x \in \mathbb{R}^N$, $t>0$; $0 = \Delta v - \gamma v + u$, $x \in \mathbb{R}^N$, $t>0$; $u(x,0) = u_0(x)$, $\tau v(x,0) = \tau v_0(x)$, $x \in \mathbb{R}^N$. Based on a $\varepsilon$-regularity theorem in [Y. Sugiyama, Partial regularity and blow-up asymptotics of weak solutions to degenerate parabolic systems of porous medium type, submitted], we first show that the set $S_u$ of blow-up points of the weak solution u has at most the zero-Hausdorff dimension if $u \in C_w([0,T]; L^1(\mathbb{R}^N))$. Next, we give various conditions on the weak solution u so that the set $S_u$ consists of finitely many points. Furthermore, we obtain an explicit constant for $\varepsilon$ in such a way that if the local concentration of mass around some point $x \in S_u$ is less than $\varepsilon$, then u is in fact locally bounded around x, which may be...