Moment Estimates and Higher-Order Asymptotic Expansions of Solutions to a Parabolic System of Chemotaxis in the Whole Space

Abstract

We study the asymptotic profiles of bounded solutions to the initial value problem for a parabolic system of chemotaxis. Very recently, it was shown in Yamada [18] that under suitable initial conditions, Theorem 2.1 of Yamada [17] also holds true for bounded solutions by deriving the moment estimate of n-th order for the solution. In this paper, we prove that if the l-th moment of initial data is finite for some l ≥ 1, then the bounded solution admits the moment estimate of l-th order. Moreover, as an application, we give the asymptotic profiles of bounded solutions by space-time higher-order asymptotic expansions under suitable moment conditions on the initial data.