Asymptotically self-similar solutions for the parabolic system modelling chemotaxis

Abstract

Abstract. We consider a nonlinear parabolic system modelling chemotaxisWe first prove the existence of time-global solutions, including self-similar solutions, for small initial data, and then show the asymptotically self-similar behavior for a class of general solutions.

On asymptotic behaviors of solutions to parabolic systems modelling chemotaxis

“…For (1.1) the existence of self-similar solutions was proved by Biler [1], and Naito [29] showed rigorously that if the initial data is small enough then the solution to (1.1)-(1.2) behaves like a self-similar solution as t → ∞. More precisely, the following theorem is proved in [29]. [29].)…”

“…Self-similar solutions are recognized as an important class of solutions, for they are heuristically candidates of the asymptotic profiles of general solutions at large time. For (1.1) the existence of self-similar solutions was proved by Biler [1], and Naito [29] showed rigorously that if the initial data is small enough then the solution to (1.1)-(1.2) behaves like a self-similar solution as t → ∞. More precisely, the following theorem is proved in [29].…”

“…More precisely, the following theorem is proved in [29]. [29].) If Ω (1) 0 L 1 and ∇Ω (2) 0 L 2 are sufficiently small, then there is a unique solution Ω to (1.1)-(1.2) such that sup t>0 t 1− 1 p Ω (1) (2) (t) L q < ∞, 1 p ∞, 2 q ∞.…”

Asymptotic behaviors of solutions to evolution equations in the presence of translation and scaling invariance

“…On the other hand, there is a two-dimensional Keller-Segel system which is classified in the critical case. For such a system it is known [2,23] that self-similar solutions describe large time behavior as in (1), while an estimate like (2) seems to have not yet been achieved for this case.…”

On the parabolic-elliptic limit of the doubly parabolic Keller–Segel system modelling chemotaxis

“…[4]. For (PP) with small M such special solutions are also important in the study of space-time decay of general solutions, see [24]. The analysis if any M > 8π may correspond to a self-similar solution is under way, see [7].…”