A Free Boundary Problem Modeling Pattern Formation On A Growing Domain

In this paper we study a free boundary problem modeling pattern formation with domain growth. The domain represents a generic tissue, which consists of live cells. The local pressure changes associated with cell birth and death generate a velocity field that drives domain growth. The rates of cell birth and death are regulated by two diffusible chemicals. The growth of domain affects the two chemicals’ spatial distribution. On the other hand, the two chemicals’ heterogeneous spatial distribution can also affect the domain growth, and this heterogeneous spatial distribution establishes the spatial pattern. We first reduce this free boundary problem into an initial-boundary value problem for a nonlinear parabolic system on a fixed domain. Then, we use a fixed point argument and the sup- and sub-solution method together with Schauder theory and L~p estimates to prove the existence of a global solution for an appropriate parameter condition. Furthermore, we find explicit parameter ranges for pattern formation by performing a perturbation analysis.