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

Lianzhang Bao; Wenxian Shen

doi:10.3934/dcds.2020072

Article Contents

2020, Volume 40, Issue 2: 1107-1130. Doi: 10.3934/dcds.2020072

This issue Previous Article Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity Next Article On the solvability of singular boundary value problems on the real line in the critical growth case

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

Lianzhang Bao^1,2,3, , and
Wenxian Shen^3,

1.
School of Mathematical Science, Zhejiang University, Hangzhou 310027, China
2.
School of Mathematics, Jilin University, Changchun, Jilin 130012, China
3.
Department of Mathematics and Statistics, Auburn University, AL 36849, USA

^* Corresponding author: Lianzhang Bao
^* Corresponding author: Lianzhang Bao

Received: April 30, 2019

Revised: July 31, 2019

Published: November 2019

The first author is supported by China Postdoctoral Science Foundation (183816). The second author is supported by NSF grant DMS–1645673

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Abstract

The current series of research papers is to investigate the asymptotic dynamics in logistic type chemotaxis models in one space dimension with a free boundary or an unbounded boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front. In this first part of the series, we investigate the dynamical behaviors of logistic type chemotaxis models on the half line $ \mathbb{R}^+ $, which are formally corresponding limit systems of the free boundary problems. In the second of the series, we will establish the spreading-vanishing dichotomy in chemoattraction-repulsion systems with a free boundary as well as with double free boundaries.

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Mathematics Subject Classification: Primary: 35R35, 35J65, 35K20; Secondary: 92B05.

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