Parabolic–elliptic chemotaxis model with space–time-dependent logistic sources on ℝN. I. Persistence and asymptotic spreading

Abstract

The current work is the third of a 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,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 [45], we studied the phenomena of pointwise and uniform persistence for solutions with strictly positive initials, and the asymptotic spreading for solutions with compactly supported or front like initials. In the second of the series [46], we investigate the existence, uniqueness and stability of strictly positive entire solutions of (0.1). In particular, in the case of space homogeneous logistic source (i.e. a(x, t) ≡ a(t) and b(x, t) ≡ b(t)), we proved in [46] that the unique spatially homogeneous strictly positive entire solution (u * (t), v * (t)) of (0.1) is uniformly and exponentially stable with respect to strictly positive perturbations when 0 < 2χµ < inf t∈R b(t).In the current part of the series, we discuss the existence of transition front solutions of (0.1) connecting (0, 0) and (u * (t), v * (t)) in the case of space homogeneous logistic source. We show that for every χ > 0 with χµ 1 + sup t∈R a(t) inf t∈R a(t) < inf t∈R b(t), there is a positive constant c * χ such that for every c > c * χ and every unit vector ξ, (0.1) has a transition front solution of the form (u(x, t), v(x, t)) = (U (x · ξ − C(t), t), V (x · ξ − C(t), t)) satisfying that C ′ (t) = a(t)+κ 2 κ for some positive number κ, lim inf t−s→∞ C(t)−C(s) t−s = c, and lim x→−∞ sup t∈R |U (x, t) − u * (t)| = 0 and lim x→∞ sup t∈R | U (x, t) e −κx − 1| = 0.Furthermore, we prove that there is no transition front solution (u(x, t), v(x, t)) = (U (x · ξ − C(t), t), V (x · ξ − C(t), t)) of (0.1) connecting (0, 0) and (u * (t), v * (t)) with least mean speed less than 2 √ a, where a = lim inf t−s→∞ 1 t−s t s a(τ )dτ .

Can chemotaxis speed up or slow down the spatial spreading in parabolic–elliptic Keller–Segel systems with logistic source?

“…But by [35,Lemma 3.2], we have that (u(t + t n , x; u 0n ), v(t + t n , x; u 0n )) → (u(t + t * , x;ũ 0 ), v(t + t * , x;ũ 0 )) as n → ∞ locally uniformly. In particular…”

Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain

“…Proof. It can be proved by applying properly modified arguments in [20,Lemma 3.5]. But the modification is not trivial.…”

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

“…As it is mentioned in the above, in the first part of the series ( [25]), we studied the phenomena of pointwise and uniform persistence, and asymptotic spreading in (1.1) for solutions with compactly supported or front like initials. Among others, the following theorems are proved in [25].…”

“…While we referred to [25] for the proof of Lemma 2.4, for the sake of clarity in the arguments in the proof of our main result in next section, it is convenient to point out some fundamental results developed in its proof.…”