Global stability of a system of viscous balance laws arising from chemotaxis with dynamic boundary flux

Abstract

This paper considers the global dynamics of classical solutions to an initial-boundary value problem of the system of viscous balance laws arising from chemotaxis in one space dimension:$u_t-{\left(uv\right)}_x=u_{xx}+u(1-u),x\in (a,b),t>0,v_t-{(u+v^2)}_x=v_{xx},x\in (a,b),t>0.$ The system of equations is supplemented with time-dependent influx boundary condition for u and homogeneous Dirichlet boundary condition for v. Under suitable assumptions on the dynamic boundary data, it is shown that classical solutions with generic initial data exist globally in time. Moreover, the solutions are shown to converge to the constant equilibrium $(1,0)$, as $t\to \infty$. There is no smallness assumption on the initial data. This is the first rigorous mathematical study of the model subject to dynamic Neumann boundary condition, and generalizes previous works in content and technicality.