Existence and Gevrey regularity for a two-species chemotaxis system in homogeneous Besov spaces

We study the Cauchy problem of a two-species chemotactic model. Using the Fourier frequency localization and the Bony paraproduct decomposition, we establish a unique local solution and blow-up criterion of the solution, when the initial data(u0, v0, w0) belongs to homogeneous Besov spaces˙B^(-2+3/p)_(p,1)(R^3) ×˙B^(-2+3/r)_(r,1)(R^3) ×˙B^(3/q)_(q,1)(R^3) for p, q and r satisfying some technical assumptions. Furthermore, we prove that if the initial data is sufficiently small, then the solution is global. Meanwhile, based on the so-called Gevrey estimates, we particularly prove that the solution is analytic in the spatial variable. In addition, we analyze the long time behavior of the solution and obtain some decay estimates for higher derivatives in Besov and Lebesgue spaces.

We study the Cauchy problem of a two-species chemotactic model. Using the Fourier frequency localization and the Bony paraproduct decomposition, we establish a unique local solution and blow-up criterion of the solution, when the initial data（u0, v0, w0） belongs to homogeneous Besov spaces B^˙p,1^-2＋3/p（R^3） ×B^˙r,1^-2＋3/r（R^3） ×B^˙q,1^3/q（R^3） for p, q and r satisfying some technical assumptions. Furthermore, we prove that if the initial data is sufficiently small, then the solution is global. Meanwhile, based on the so-called Gevrey estimates, we particularly prove that the solution is analytic in the spatial variable. In addition, we analyze the long time behavior of the solution and obtain some decay estimates for higher derivatives in Besov and Lebesgue spaces.