Special Session 8: Emergence and dynamics of patterns in nonlinear partial differential equations from mathematical science
Contents
We are concerned with a two-component reaction-diffusion system with conservation of mass, which is proposed as a simple cell polarity model. We first provide a Lyapunov function of the system. Then we reduce the system of stationary problem to a scalar equation with a nonlocal term and bring an energy functional whose critical points in a function space are given by solutions to the reduced scalar equation. We further show that the dimension of the unstable manifold of each equilibrium solution to the system coincides with the Morse index of the corresponding critical point of the energy functional. The proof for the last part is done by applying the spectral comparison argument to the linearized eigenvalue problems. This talk is based on a joint work with S. Jimbo (2013) and a recent joint work with E. Latos and T. Suzuki.