Initial–boundary value problems for a system of hyperbolic balance laws arising from chemotaxis

• Published in: Journal of Differential Equations Vol. 258; no. 2; pp. 302 - 338
• Main Authors: Li, Huicong, Zhao, Kun
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.01.2015
• Subjects: Chemotaxis; Classical solution; Global well-posedness; Hyperbolic balance laws; Initial–boundary value problem; Long-time behavior; Initial–boundary value problem; Classical solution; 35K50; 92C15; Global well-posedness; 92C17; 35K57; 35K45; Chemotaxis; 35K55; Hyperbolic balance laws; Long-time behavior
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2014.09.014

This paper is devoted to the analytical study of initial–boundary value problems for a system of hyperbolic balance laws derived from a repulsive chemotaxis model with logarithmic sensitivity. In the first part of the paper we show that, subject to the Dirichlet boundary conditions, classical solutions exist globally in time for large initial data. Asymptotically in time, the solutions are shown to converge to their boundary data at an exponential rate as time goes to infinity. Numerical simulations are supplied to corroborate the analytical results. The analytic approach developed herein can be utilized to handle a family of initial and initial–boundary value problems of the model and related models with similar mathematical structure. As a demonstration of the effectiveness of our approach, in the second part of the paper we show that, subject to the Neumann–Dirichlet boundary conditions, classical solutions exist and converge to constant equilibrium states for large initial data and for arbitrary values of the chemical diffusion coefficient. This improves a previous result obtained in [30] where the smallness of the chemical diffusion coefficient was required.