Boundedness, blowup and critical mass phenomenon in competing chemotaxis
Abstract We consider the following attraction repulsion Keller Segel system: u t D u kh u v x u w x O t 0 v t D v a u b v x O t 0 0 D w g u d w x O t 0 u x 0 u 0 x v x 0 v 0 x x O with homogeneous Neumann boundary conditions in a bounded domain O R 2 with smooth boundary. The system models the chemotactic interactions between one species (denoted by u and two competing chemicals (denoted by v and w which has important applications in Alzheimer's disease. Here all parameters kh x a b g and d are positive. By constructing a Lyapunov functional, we establish the global existence of uniformly in time bounded classical solutions with large initial data if the repulsion dominates or cancels attraction (i.e. x g a kh If the attraction dominates (i.e. x g a kh a critical mass phenomenon is found. Specifically speaking, we find a critical mass m 4 p a kh x g such that the solution exists globally with uniform in time bound if M m and blows up if M m and M 4 p m th m N where N denotes the set of positive integers and M O u 0 d x the initial cell mass.