Cauchy problem of a system of parabolic conservation laws arising from the singular Keller-Segel model in multi-dimensions

In this paper, we study the qualitative behavior of solutions to the Cauchy problem of a system of parabolic conservation laws, derived from a Keller-Segel type chemotaxis model with singular sensitivity, in multiple space dimensions. Assuming H2 initial data, it is shown that under the assumption that only some fractions of the total energy associated with the initial perturbation around a prescribed constant ground state are small, the Cauchy problem admits a unique global-in-time solution, and the solution converges to the prescribed ground state as time goes to infinity. In addition, it is shown that solutions of the fully dissipative model converge to that of the corresponding partially dissipative model with certain convergence rates as a specific system parameter tends to zero.