Journal
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
Volume 18, Issue 3, Pages 601-641Publisher
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/dcdsb.2013.18.601
Keywords
Chemotaxis; bacteria; angiogenesis; traveling waves; wave speed; stability; chemical diffusion; transformation; Fisher equation; conservation laws
Categories
Funding
- Hong Kong CRC General Research Fund [502711]
- Center for Partial Differential Equation (CPDE) in the East China Normal University
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This article surveys the mathematical aspects of traveling waves of a class of chemotaxis models with logarithmic sensitivity, which describe a variety of biological or medical phenomena including bacterial chemotactic motion, initiation of angiogenesis and reinforced random walks. The survey is focused on the existence, wave speed, asymptotic decay rates, stability and chemical diffusion limits of traveling wave solutions. The main approaches are reviewed and related analytical results are given with sketchy proofs. We also develop some new results with detailed proofs to fill the gap existing in the literature. The numerical simulations of steadily propagating waves will be presented along the study. Open problems are proposed for interested readers to pursue.
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