Asymptotic Behavior of a Nonlocal Diffusive Logistic Equation

Abstract

Abstract. The long time behavior of a logistic-type equation modeling the motion of cells is investigated. The equation we consider takes into account birth and death processes using a simple logistic effect as well as a nonlocal motion of cells using a nonlocal Darcy's law with regular kernel. Using the periodic framework we first investigate the well-posedness of the problem before deriving some information about its long time behavior. The lack of asymptotic compactness of the system is overcome by making use of the Young measure theory. This allows us to conclude that the semiflow converges for the Young measure topology.

Existence and uniqueness of solutions for a hyperbolic Keller�CSegel equation

“…Next we estimate (16) and (17) (remark that (15) is a particular case of (17), for s = t), starting with (17). We have…”

“…For Turing and Turing-Hopf bifurcation due to the non-local effect, we refer to Ducrot et al [16] and Song et al [37]. We also refer to Mogliner et al [30], Eftimie et al [20], Ducrot and Magal [17], Ducrot and Manceau [18] for more topics on non-local advection equations. For the derivation of such models, we refer to the work of Bellomo et al [5] and Morale, Capasso and Oelschläger [31].…”

“…Pattern formation for a model similar to (7) by Ducrot, Fu and Magal [16]. Let us mention that the inviscid equation (5) has been studied in a periodic cell by Ducrot and Magal [17]. A substantial literature has been produced for conservative systems of interacting particles and their kinetic limit (Balagué et al [4], Carrillo et al [11], Bernoff and Topaz [6], Bertozzi, Laurent and Rosado [7], among others).…”

Asymptotic behavior of a nonlocal advection system with two populations

“…Compared to the work in [13], one of the technical difficulties in this paper is that we do not have a L 2 uniform boundedness of the solution a priori. This is because we allow function h to be of more general type than that in [13] (see Assumption 1.1 and 4.1). This difficulty obliges us to find another way to prove the L ∞ uniform boundedness of the solution (see Lemma 4.9, Remark 4.11 and Theorem 4.10).…”

“…The single species model of equation (1.1) has been studied by Ducrot and Magal in [13] (see the derivation of the model therein). Compared to the work in [13], one of the technical difficulties in this paper is that we do not have a L 2 uniform boundedness of the solution a priori. This is because we allow function h to be of more general type than that in [13] (see Assumption 1.1 and 4.1).…”