Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions

Abstract

We study the qualitative behavior of a system of parabolic conservation laws, derived from a Keller-Segel type chemotaxis model with singular sensitivity, on the unit square or cube subject to various types of boundary conditions. It is shown that for given initial data in $H^3(\Omega)$, under the assumption that only the entropic energy associated with the initial data is small, there exist global-in-time classical solutions to the initial-boundary value problems of the model subject to the Neumann-Stress-free and Dirichlet-Stress-free type boundary conditions; these solutions converge to equilibrium states, determined from initial and/or boundary data, exponentially rapidly as time goes to infinity. In addition, it is shown that the solutions of the fully dissipative model converge to those of the corresponding partially dissipative model as the chemical diffusion rate tends to zero under the Neumann-Stress-free type boundary conditions. Numerical analysis is performed for a discretization of the model with the Dirichlet-Stress-free type boundary conditions, and a monotonic exponential decay to the equilibrium solution (analogous to the continuous case) is proven. Numerical simulations are supplemented to illustrate the exponential decay, test the assumptions of the exponential decay theorem, and to predict boundary layer formation under the Dirichlet-Stress-free type boundary conditions.