Boundedness in a quasilinear fully parabolic Keller–Segel system with logistic source
Boundedness in a quasilinear fully parabolic Keller–Segel system with logistic source
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DOI：
10.1007/s00033-015-0532-z
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发表时间：
2015-04
期刊：
影响因子：
--
通讯作者：
Qingshan Zhang;Yuxiang Li
中科院分区：
文献类型：
--
作者：
Qingshan Zhang;Yuxiang Li
This paper deals with the Neumann boundary value problem for the system $$\left\{\begin{array}{lll}u_t = \nabla \cdot \left(D(u) \nabla u\right) - \nabla \cdot \left(S(u) \nabla v\right) + f(u), &\quad x \in \Omega, \, t > 0,\\ v_t = \Delta v - v + u, &\quad x \in \Omega, \, t > 0\end{array}\right.$$in a smooth bounded domain, where the functionsD(u) andS(u) are supposed to be smooth satisfyingandwithM> 0,andfor all, and the logistic sourcef(u) is smooth fulfillingas well aswith,andfor all. It is shown that if $$\alpha + 2\beta < \left\{\begin{array}{lll}\gamma - 1 + \frac{2}{n}, &\quad {\rm for} \, 1 \leq \gamma < 2,\\ \gamma - 1 + \frac{4}{n + 2}, &\quad {\rm for} \, \gamma \geq 2,\end{array}\right.$$then for sufficiently smooth initial data, the problem possesses a unique global classical solution which is uniformly bounded.
This paper deals with the Neumann boundary value problem for the system $$\left\{\begin{array}{lll}u_t = \nabla \cdot \left(D(u) \nabla u\right) - \nabla \cdot \left(S(u) \nabla v\right) + f(u), &\quad x \in \Omega, \, t > 0,\\ v_t = \Delta v - v + u, &\quad x \in \Omega, \, t > 0\end{array}\right.$$in a smooth bounded domain, where the functionsD(u) andS(u) are supposed to be smooth satisfyingandwithM> 0,andfor all, and the logistic sourcef(u) is smooth fulfillingas well aswith,andfor all. It is shown that if $$\alpha + 2\beta < \left\{\begin{array}{lll}\gamma - 1 + \frac{2}{n}, &\quad {\rm for} \, 1 \leq \gamma < 2,\\ \gamma - 1 + \frac{4}{n + 2}, &\quad {\rm for} \, \gamma \geq 2,\end{array}\right.$$then for sufficiently smooth initial data, the problem possesses a unique global classical solution which is uniformly bounded.