Parabolic–elliptic chemotaxis model with space–time dependent logistic sources on [formula omitted]. II. Existence, uniqueness, and stability of strictly positive entire solutions

Abstract

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≤μχ<infx,t⁡b(x,t), then (0.1) has a strictly positive entire solution, which is time-periodic (respectively time homogeneous) when the logistic source function is time-periodic (respectively time homogeneous). Next, we show that there is positive constant χ0, depending on N, λ, μ, a and b such that for every 0≤χ<χ0, (0.1) has a unique positive entire solution which is uniform and exponentially stable with respect to strictly positive perturbations. In particular, we prove that χ0 can be taken to be infx,t⁡b(x,t)2μ when the logistic source function is either space homogeneous or the function (x,t)↦b(x,t)a(x,t) is constant. We also investigate the disturbances to Fisher-KKP dynamics caused by chemotactic effects, and prove thatsup0<χ≤χ1⁡supt0∈R,t≥0⁡1χ‖uχ(⋅,t+t0;t0,u0)−u0(⋅,t+t0;t0,u0)‖∞<∞ for every 0<χ1<binfμ and every uniformly continuous initial function u0, with infx⁡u0(x)>0, 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<binf.