On a parabolic–elliptic system with gradient dependent chemotactic coefficient
On a parabolic–elliptic system with gradient dependent chemotactic coefficient
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DOI：
10.1016/j.jde.2018.01.040
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发表时间：
2018-08
影响因子：
2.4
通讯作者：
M. Negreanu;J. Ignacio Tello
中科院分区：
文献类型：
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作者：
M. Negreanu;J. Ignacio Tello
本文考虑一类具有趋化项的二阶抛物-椭圆型偏微分方程组。该系统描述了生物物种“u”在RN的有界和开放域中向更高浓度的化学刺激“v”移动的进化。在所考虑的系统中，趋化性敏感性取决于v的梯度，即趋化性项具有以下表达式− d i v（χ u|吉夫|p− 2 <$v），其中χ是正常数，且p满足p∈（1，∞），若N= 1，且p∈（1，N N− 1），若N≥ 2。得到了该问题在L∞（Ω）上的一致有界性和整体时间解的存在性.对于一维情形，我们证明了对任意正质量和正χ，p∈（1，2）存在无穷多个非常数平衡态.
We consider a second order PDEs system of Parabolic–Elliptic type with chemotactic terms. The system describes the evolution of a biological species “u” moving towards a higher concentration of a chemical stimuli “v” in a bounded and open domain of R N. In the system considered, the chemotaxis sensitivity depends on the gradient of v, ie, the chemotaxis term has the following expression− d i v (χ u|∇ v| p− 2∇ v), where χ is a positive constant and p satisfies p∈(1,∞), if N= 1 and p∈(1, N N− 1), if N≥ 2. We obtain uniform bounds in L∞(Ω) and the existence of global in time solutions. For the one-dimensional case we prove the existence of infinitely many non-constant steady-states for p∈(1, 2) for any χ positive and a given positive mass.