Large time behavior of bounded solutions to a parabolic system of chemotaxis in the whole space

Abstract

We consider the Cauchy problem for a parabolic system of chemotaxis in R N (N 1), and give the decay rates and asymptotic profiles of bounded solutions.

Diffusion-dominated asymptotics of solution to chemotaxis model

“…In the parabolic-parabolic case (considered in [20,25,26]), we have parabolic equation describing temporal evolution of v and thus we need to estimate the solutions u and v simultaneously (getting also analogous results for decay of the function v).…”

“…The problem of the asymptotic profile for the problem with parabolic version of (1.2) was studied in the papers [20,25] and [26], under the general assumption that L 1 and L ∞ norms of solutions are bounded. Assuming additionally boundedness of the quantities u∇v and yu 0 (y)dy, the asymptotics of the solutions is given by the heat kernel G(t).…”

Improvement of convergence rates for a parabolic system of chemotaxis in the whole space

“…Then the L q -estimates for rv can be shown by methods similar to those in Nagai and Yamada [14,Section 4].…”

“…1, and asymptotically behaves like the heat kernel. Furthermore, Kato [6] and Nagai and Yamada [14] proved the following: Assume n 1 and 1 Ä p Ä 1, and let .u, v/ be the solution to (1.1) with sup t>0 .ku. , t/k q C kv.…”

Global existence of solutions for a nonlinearly perturbed Keller–Segel system in $${\mathbb{R}}^2$$

“…The asymptotic profile has been obtained for the simpler chemotaxis system by Biler-Dolbeault [4] and Nagai-Syukuinn-Umesako [29] (see also NagaiYamada [30]). In view of Theorem 1.2, we may show the asymptotic profile of the solution u(t), v(t) is the Gauss kernel.…”