Self-Similar Solutions to a Parabolic System Modeling Chemotaxis

Abstract

We study the forward self-similar solutions to a parabolic system modeling chemotaxisin the whole space R 2 ; where t is a positive constant. Using the Liouville-type result and the method of moving planes, it is proved that self-similar solutions ðu; vÞ must be radially symmetric about the origin. Then the structure of the set of selfsimilar solutions is investigated. As a consequence, it is shown that there exists a threshold in R

Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis

“…and defining S(y) := 4 (ψ(y) − y ψ ′ (y)) ′ = −4 y ψ ′′ (y) = − √ y v ′ ( √ y) as in [2,15], system (27)-(28) becomes, after a differentiation of (28) with respect to y, a first order system in the (φ ′ , S) variables…”

“…Then, using the fact that u, v, and consequently |∇v| are bounded, it has been proved in [15] that there exists a constant σ such that…”

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

“…We remark that a self-similar solution exists for a system replaced the second equation of (P) by v t = v + u (see [13]), but (P) does not have such a self-similar solution. For the large time behavior of a parabolic-elliptic system with degenerate diffusion modeling chemotaxis, we refer to [10].…”

Asymptotically self-similar solutions for the parabolic system modelling chemotaxis

“…(ii) For the properties of self-similar solutions to (1.1), we refer to [3,9]. We also refer to [1,2,10], where the self-similar solutions to the parabolic-elliptic problem have been studied.…”