论文标题
关于上估计的最佳估计值近爆炸keller-segel Systems
On the optimality of upper estimates near blow-up in quasilinear Keller--Segel systems
论文作者
论文摘要
解决趋化系统的解决方案$（u，v）$ \ begin {align*} \ begin {case} u_t = \ nabla \ cdot（（U+1）^{m-1} \ nabla u-u（u+1）^{q-1} \ nabla v），\\ τv_t=ΔV -V + U \end{cases} \end{align*} in a ball $Ω\subset \mathbb R^n$, $n \ge 2$, wherein $m, q \in \mathbb R$ and $τ\in \{0, 1\}$ are given parameters with $m - q > -1$, cannot blow up in finite time provided $u$ is uniformly-in-time bounded in $ l^p（ω）$对于某些$ p> p_0：= \ frac n2（1-（m -q））$。 对于径向对称的解决方案，我们表明，如果$ u $仅以$ l^{p_0}（ω）限制，并且实现了技术条件$ m> \ frac {n-2 p_0} {n} {n} $，那么，对于任何$α> \ frac {n} {p_0} $ u(x, t) \leq C |x|^{-α} \qquad \text{for all $x \in Ω$ and $t \in (0, T_{\max})$}, \end{align*} $T_{\max} \in (0, \infty]$ denoting the maximal existence time. This is essentially optimal in the感觉到，如果对任何$α<\ frac {n} {p_0} $保存的估计值，则$ u $的$ u $将在$ l^{p}（ω）$中以某些$ p> p_0 $限制。 此外，我们还给出了具有非线性信号产生的趋化系统的某些上层估计，即使没有对$ u $的任何额外的有限假设。 该证明主要基于泊松方程解决方案的点梯度估计值，其源术语均匀限制为$ l^{p_0}（ω）$。
Solutions $(u, v)$ to the chemotaxis system \begin{align*} \begin{cases} u_t = \nabla \cdot ( (u+1)^{m-1} \nabla u - u (u+1)^{q-1} \nabla v), \\ τv_t = Δv - v + u \end{cases} \end{align*} in a ball $Ω\subset \mathbb R^n$, $n \ge 2$, wherein $m, q \in \mathbb R$ and $τ\in \{0, 1\}$ are given parameters with $m - q > -1$, cannot blow up in finite time provided $u$ is uniformly-in-time bounded in $L^p(Ω)$ for some $p > p_0 := \frac n2 (1 - (m - q))$. For radially symmetric solutions, we show that, if $u$ is only bounded in $L^{p_0}(Ω)$ and the technical condition $m > \frac{n-2 p_0}{n}$ is fulfilled, then, for any $α> \frac{n}{p_0}$, there is $C > 0$ with \begin{align*} u(x, t) \leq C |x|^{-α} \qquad \text{for all $x \in Ω$ and $t \in (0, T_{\max})$}, \end{align*} $T_{\max} \in (0, \infty]$ denoting the maximal existence time. This is essentially optimal in the sense that, if this estimate held for any $α< \frac{n}{p_0}$, then $u$ would already be bounded in $L^{p}(Ω)$ for some $p > p_0$. Moreover, we also give certain upper estimates for chemotaxis systems with nonlinear signal production, even without any additional boundedness assumptions on $u$. The proof is mainly based on deriving pointwise gradient estimates for solutions of the Poisson or heat equation with a source term uniformly-in-time bounded in $L^{p_0}(Ω)$.
