Boundedness and stabilization of solutions to a chemotaxis May-Nowak model

The chemotaxis May-Nowak model {u(t)=D-u Delta(u)- del center dot(uf (u) del u) - g (u) w + r - u, x is an element of Omega, t > 0, v(t) = D-u Delta(v) + g(u) w- v, x is an element of Omega, t > 0, w(t) =D-w Delta(w) + v - w, x is an element of Omega, t > 0, is considered in a bounded domain Omega subset of R-n(n >= 1) with homogeneous Neumann boundary conditions and the parameters D-u,D-v,D-w,r>0. The chemotactic sensitivity function and the conversion function are given by f(s)-alpha K-f (1+s)(-alpha) and g(s)=Kg(s beta)for all s>0, respectively, where K-f is an element of R, K-g, alpha,beta>0. The global boundedness of solution is shown if the following case holds: alpha>max {n beta/4,beta/2,n(n+2)/6n+8 beta+1/2}. Moreover, for the large time behavior of the global smooth bounded solution, the basic reproduction number R-0:=K(g)r(beta) has an important effect (Lai and Zou in Bull Math Biol 76:2806-2833, 2014; Wang et al. in Nonlinear Anal RWA 33:253-283, 2017), and system has the infection-free steady state if 0<R-0<1 0 and the coexistence equilibrium steady state if R0>1 (Korobeinikov in Bull Math Biol 66:879-883, 2004). By constructing an appropriate energy function, under the conditions that K-f and K-g are appropriately mild, it is shown that: . If R-0 is an element of (0, 1), then any global bounded solution converges to (r, 0, 0) as t ->infinity; . If R-0 is an element of (1,8), beta = 1, then any global bounded solution converges to (1/K-g, r-1/K-g, r-1/K-g) as t ->infinity.