Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$

Abstract

The current paper is devoted to the study of spreading speeds and traveling wave solutions of the following parabolic-elliptic chemotaxis system,We then consider the extensions of the results in the case N = 1 to the case N ≥ 2.

Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. II. Spreading-vanishing dichotomy in a domain with a free boundary

“…Hence U (·, ·; u) ∈ E. By the similar arguments as those in [14,Lemma 4.3], we can prove that the mapping E ∋ u → U (·, ·; u) ∈ E is continuous and compact. Then there is u * ∈ E such that U (t, x; u * ) = u * (t, x).…”

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

“…Therefore, U (·, ·; u) ∈ E. By the similar arguments as those in [19,Lemma 4.3], the mapping E ∋ u → U (·, ·; u) ∈ E is continuous and compact, and then by Schauder's fixed theorem, it has a fixed point u * . Clearly (u * (·, ·), v 1 (·, ·; u * ), v 2 (·, ·; u * )) is a classical solution of (1.2).…”

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

“…(2) In [38,37], the first two authors of the current paper obtained some constants c * low (χ, µ, a, b, λ, µ) < 2 √ a < c * up (χ, a, b, λ, µ) depending explicitly on the parameter χ, a, b, λ and µ such that…”

“…Some lower and upper bounds for the propagation speeds of solutions with compactly supported initial functions were derived, and some lower bound for the speeds of traveling wave solutions was also derived. It is proved that all these bounds converge to the spreading speed c * 0 = 2 √ a of (1.2) as χ → 0 (see [36], [37], [38]). The reader is also referred to [13] for the lower and upper bounds of propagation speeds of (1.1), and is referred to [1,2,12,16,24,28,31,43], etc., for the studies on traveling wave solutions of various types of chemotaxis models.…”