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

Abstract

We are interested in the asymptotic behavior of solutions towards a parabolic system of chemotaxis in R n , n 1. It was proved in the previous results that decaying solutions converge to the heat kernel in L p .R n /.1 Ä p Ä 1/ at the rate t n.1 1=p/=2 1=2 as t ! 1. Our aim in this paper is to improve the convergence rates by taking into account the center of mass of such solutions.

On asymptotic behaviors of solutions to parabolic systems modelling chemotaxis

“…Nishihara [32] then proved that a suitably shifted heat kernel gives more precise asymptotics as in (1.5) by introducing a time dependent shift parameter in the spatial variable. An extension to the case of higher space dimensions has been done in Yamada [36] by considering the behavior of the center of mass of solutions.…”

Exponential convergence to equilibrium in a coupled gradient flow system modeling chemotaxis

“…Global existence and blow-up in the classical parabolic-parabolic Keller-Segel model, which is (1)&(2) with φ(v) = −v, W ≡ 0 and linear diffusion, has been thoroughly studied by Calvez and Corrias [10] in space dimension d = 2, and by Corrias and Perthame [16] in higher space dimensions d > 2, see also [3,26,34,37,42,46,49]. Variants with nonlinear diffusion and drift have been studied for instance by Sugiyama [44,45].…”

“…Now apply estimate (49), once with q := 1 and Q := 1/2 to the first term, and once with q := 6/5 and Q := 3/4 to the second term on the right-hand side. Further, since u n τ is of unit mass, one has…”

“…Now assume that (48) holds for some k ∈ N. Using the semigroup property H t1+t2 = H t1 * H t2 of the heat kernel, we find that which is (48) with k + 1 in place of k, using that kΓ(k) = Γ(k + 1). For (49), first observe that r → r k−1 e −r /Γ(k) defines a probability density on R + . We can thus apply Jensen's inequality to obtain…”

Remarks on the local existence of solutions to the Debye system

“…In [18], Ogawa and Shimizu established global well-posedness of (1.1)-(1.4) with small initial data in a two-dimensional Hardy space. For further researches of the problem (1.1)-(1.4) we refer the reader to see [16,19,21] and the references therein.…”