Global existence, boundedness and asymptotic behavior of classical solutions to a fully parabolic two-species chemotaxis-competition model with singular sensitivity

This paper deals with the following parabolic-parabolic-parabolic chemotaxis system with singular sensitivity and Lotka-Volterra competition kinetics { ut=Delta u-chi 1 del<middle dot>(uw del w)+mu 1u(1-u-a1v),t >0,x is an element of Omega vt=Delta v-chi 2 del<middle dot>(vw del w)+mu 2v(1-v-a2u),t >0,x is an element of Omega wt=Delta w-w+u+v,t >0,x is an element of Omega partial derivative u partial derivative nu=partial derivative v partial derivative nu=partial derivative w partial derivative nu=0,t>0,x is an element of partial derivative Omega, u(0,x)=u0(x), v(0,x)=v0(x), w(0,x)=w0(x), x is an element of Omega where Omega subset of R-N (N >= 2) is a bounded smooth convex domain, and the parameters X1 , X2 , mu 1 , mu 2 , a1 and a2 are positive constants. It is shown that the system possesses globally bounded classical solutions under the following conditions X1 , X2 is an element of (0, 1/N-2 2 ) for N = 2, 3 or X1 , X2 is an element of (0, N-1 ) for N >= 4. Moreover, if min{mu 1 , mu 2} > max{X1 ,X2}2 , we obtain the uniformly lower bound for w. Finally, when 4 X1 , X2 are suitably small, it is proved that if 0 < a1 , a2 < 1, then the solution (u, v, w) converges to ( 1-a1 /1-a1a2 , 1-a2 /1-a1a2 , 2-a1-a2 in L(infinity )norm as t -> infinity; if 0 < a2 < 1 <= a1 , then the solution (u, v, w) con1-a1a2 verges to (0, 1, 1) in L-infinity norm as t ->infinity.