论文标题
具有sublinear隔离和超级线性聚集的完全抛物线凯勒 - 塞格模型的界限
Boundedness for a fully parabolic Keller-Segel model with sublinear segregation and superlinear aggregation
论文作者
论文摘要
这项工作涉及具有非线性产生和趋化剂的完全抛物线趋化模型。该问题是在有界域上提出的，具体取决于与此类生产和化学吸收剂相关的系数之间的特定相互作用，我们确定相关的初始有限值问题具有独特的经典解决方案，该解决方案均匀地界定。确切地说，我们研究此零升级问题\ begin {equination} \ label {QUALICE_ABSTRACT} \ tag {$ \ diamond $} \ begin {cases} u_t =Δu-\ nabla \ nabla \ cdot（f（f（nabla v）v） v_t =Δv-v+g（u）＆\ text {in}ω\ times（0，t_ {max}），\\ \ end {cases} \ end {case} \ end {equation}，其中$ω$是$ \ \ mathbb {r}^n $的限制且平稳的域，$}函数分别概括了原型$ f（u）= u^α$和$ g（u）= u^l $，具有适当的$α，l> 0 $。在证明任何足够光滑的$ u（x，0）= u_0（x）\ geq 0，\，\，v（x，0）= v_0（x）\ geq 0 $散发出唯一的经典和非固定解决方案$（u，v）$ compare \ eqref \ eqref {cools_abstract}的$ phime（ $ t_ {max} $表示生存的最长时间，我们确定（0，\ frac {2} {2} {n}）$和$ \ frac {2} {n} \leqα<1+ \ frac \ frac \ frac {1} {1} {n} {n} {n} - \ freac {lac}实际上，$ u $和$ v $的时间均匀地界定。 此外，本文符合Horstmann和Winkler的贡献，此外，Liu和Tao的结果扩展了结果。的确，在第一项工作中，证明对于$ g（u）= u $，值$α= \ frac {2} {n} $代表模型的关键爆炸指数，而第二个则表示$ f（u）= u $，对应于$ $α= 1 $，在假设$ 0 <lac $ 0 <lac}下显示了解决方案的界面。
This work deals with a fully parabolic chemotaxis model with nonlinear production and chemoattractant. The problem is formulated on a bounded domain and, depending on a specific interplay between the coefficients associated to such production and chemoattractant, we establish that the related initial-boundary value problem has a unique classical solution which is uniformly bounded in time. To be precise, we study this zero-flux problem \begin{equation}\label{problem_abstract} \tag{$\Diamond$} \begin{cases} u_t= Δu - \nabla \cdot (f(u) \nabla v) & \text{ in } Ω\times (0,T_{max}),\\ v_t=Δv-v+g(u) & \text{ in } Ω\times (0,T_{max}),\\ \end{cases} \end{equation} where $Ω$ is a bounded and smooth domain of $\mathbb{R}^n$, for $n\geq 2$, and $f(u)$ and $g(u)$ are reasonably regular functions generalizing, respectively, the prototypes $f(u)=u^α$ and $g(u)=u^l$, with proper $α, l>0$. After having shown that any sufficiently smooth $ u(x,0)=u_0(x)\geq 0, \, v(x,0)=v_0(x)\geq 0$ emanate a unique classical and nonnegative solution $(u,v)$ to problem \eqref{problem_abstract}, which is defined on $Ω\times (0,T_{max})$ with $T_{max}$ denoting the maximum time of existence, we establish that for any $l\in (0,\frac{2}{n})$ and $\frac{2}{n}\leq α<1+\frac{1}{n}-\frac{l}{2}$, $T_{max}=\infty$ and $u$ and $v$ are actually uniformly bounded in time. This paper is in line with the contribution by Horstmann and Winkler, moreover, extends the result by Liu and Tao. Indeed, in the first work it is proved that for $g(u)=u$ the value $α=\frac{2}{n}$ represents the critical blow-up exponent to the model, whereas in the second, for $f(u)=u$, corresponding to $α=1$, boundedness of solutions is shown under the assumption $0<l<\frac{2}{n}.$
