增长区域上斑图形成的自由边界问题

A free boundary problem of modeling pattern formation on growing domains

研究一个在增长区域上斑图形成的自由边界问题,首先利用不动点定理证明解的局部存在性,然后通过扰动分析方法得到斑图形成的参数条件.区域指一块生物组织,该组织由活的细胞组成.细胞的再生和死亡引起细胞间局部压力的变化,从而产生了一个细胞速度场,推动区域的增长.细胞的再生和死亡由2种化学物质决定,区域的增长影响这2种化学物质的空间分布.反过来,这2种化学物质的空间不均匀分布也会影响区域增长.生物斑图的形成由这2种化学物质的空间不均匀分布引起.

A free boundary problem of modeling pattern formation with domain growth is studied. A fixed point argument is used to prove the existence of local solution, and a perturbation analysis is used to find explicit parameter ranges for pattern formation. The generic tissue which consists of live ceils is represented by a domain. A velocity field that drives domain growth is generated by the local pressure changing associated with cell birth and death. The spatial distribution of the two chemicals which regulate the rates of cell birth and death are affected by the growth of domain. On the other hand, the domain growth is also affected by the two chemicals＇ heterogeneous spatial distribution. The spatial pattern is established by the heterogeneous spatial distribution.