Pattern formation for a nonlinear diffusion chemotaxis model with logistic source

摘要

This paper deals with a Neumann boundary value problem in a d-dimensional box (mathbb{T}^{d}=(0,pi)^{d}) ((d=1, 2, 3)) for a nonlinear diffusion chemotaxis model with logistic source. By using the embedding theorem, the higher-order energy estimates and bootstrap arguments, the condition of chemotaxis-driven instability and the nonlinear evolution near an unstable positive constant equilibrium for this chemotaxis model are proved. Our result provides a quantitative characterization for early spatial pattern formation on the positive constant equilibrium. Finally, numerical simulations are carried out to support our theoretical nonlinear instability results.