GLOBAL CAUCHY PROBLEM OF A SYSTEM OF PARABOLIC CONSERVATION LAWS ARISING FROM A KELLER-SEGEL TYPE CHEMOTAXIS MODEL

We study the qualitative behavior of solutions to the Cauchy problem of a coupled system in (p, q) of parabolic conservation laws in one space dimension posed on R. This system arises from a Keller-Segel type repulsive model for chemotaxis with singular sensitivity and nonlinear production rate. In particular, we initiate the study of such models that correspond to a nonlinear production rate of g(p) = p(gamma) , where gamma > 1, in the regime when the ratio of chemical-to-cell diffusivity is of order epsilon, where epsilon > 0 denotes the chemical diffusion coefficient. By assuming H-1 initial data and utilizing energy methods, it is shown that regardless of the magnitude of initial data, there exist global-in-time solutions to the Cauchy problem, and the regularity of the solution depends on the specific values of gamma and epsilon. Moreover, the global asymptotic stability of constant ground states and the zero chemical diffusion limit (epsilon -> 0) of solutions are investigated.