Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on RN. II. Existence, uniqueness, and stability of strictly positive entire solutions

The current work is the second of the series of three papers devoted to the study of asymptotic dynamics in the following parabolic-elliptic chemotaxis system with space and time dependent logistic source,(0.1){∂tu=Δu−χ∇⋅(u∇v)+u(a(x,t)−ub(x,t)),x∈RN,0=Δv−λv+μu,x∈RN, where N≥1 is a positive integer, χ,λ and μ are positive constants, and the functions a(x,t) and b(x,t) are positive and bounded. In the first of the series, we studied the phenomena of pointwise and uniform persistence, and asymptotic spreading in (0.1) for solutions with compactly supported or front like initials. In the second of the series, we investigate the existence, uniqueness and stability of strictly positive entire solutions of (0.1). In this direction, we prove that, if 0≤μχ0, where (uχ(x,t+t0;t0,u0),vχ(x,t+t0;t0,u0)) denotes the unique classical solution of (0.1) with uχ(x,t0;t0,u0)=u0(x), for every 0≤χ