Critical Mass For A Two-species Chemotaxis Model With Rotational Flux Terms

Chemotaxis is a biological phenomenon which describes the directed movement of cells toward or away from the chemical substances,and it plays an important role in various situations,such as tumor invasion,embryo development,diseases prevention and so on.In mathematics,all above biological phenomena can be modeled by chemotaxis models which are formed by nonlinear PDE systems.In this paper we study the optimal critical mass for a two-species chemotaxis model with rotational flux terms in two dimensional whole space.The chemotactic sensitivity in this model is a tensor function,which describes the chemotactic phenomenon that cells or microorganisms are not precisely oriented along the concentration gradient but involves certain rotational flux component.The key observation is that for any angle of rotation α∈(-π,π],the resulting PDE system preserves a dissipative energy structure.Firstly,we obtain the global existence of the regularization system of this problem by the Schauder fixed point theorem and the parabolic regularity theory.Secondly,by prior estimation and Aubin-Lions compactness method,we obtain the local existence and continuation criterion of the free energy solution to the system.Thirdly,on the basis of constructing the energy free energy functional,we show that the system admits a unique global bounded solution provided that the free energy functionals have upper and lower bounds with the Moser-Trudinger inequality of vector form.Finally,we show that the solutions blow up at finite time by the second moment method.In summary,we show that the critical mass can be represented exactly by the curve 4π(m1+m2)-χ(cos α)m1m2=0,where m1 and m2 represent the initial mass of the two species respectively.