Sharp condition for blow-up and global existence in a two species chemotactic Keller–Segel system in ²

Abstract

Link to this article: http://journals.cambridge.org/abstract_S0956792512000411 How to cite this article: ELIO ESPEJO, KARINA VILCHES and CARLOS CONCA (2013). Sharp condition for blow-up and global existence in a two species chemotactic Keller-Segel system in 2 .For the parabolic-elliptic Keller-Segel system in R 2 it has been proved that if the initial mass is less than 8π/χ, a global solution exists, and in case the initial mass is larger than 8π/χ, blow-up happens. The case of several chemotactic species introduces an additional question: What is the analog for the critical mass obtained for the single species system? We find a threshold curve in the two species case that allows us to determine if the system is a blow-up or a global in time solution. No radial symmetry is assumed.

Numerical study of two-species chemotaxis models

“…We show that different scenarios are possible: depending on the initial masses, either one or both cell densities may blow up, or a global solution may exist. In particular, our numerical results indicate answers on some open questions of possible blow up stated in [4,7]. We then introduce two regularizations of the studied models and demonstrate that their solutions are capable of developing spiky structure without blowing up.…”

“…In this section, we present the results of our numerical experiments that clarify the behavior of the solutions of the studied two-species chemotaxis systems in two space dimensions. We restrict our consideration to the two-dimensional (2-D) case since the theoretical results/open questions in [4,5,7] were obtained/formulated for the 2-D version of the system (2).…”

“…Following [4,7], we split the (θ 1 , θ 2 )-plane into the following four regions, outlined in Figure 1:…”

“…We first numerically investigate two models proposed and analytically studied in [4,5,6,7,8,9,20,34]. The first system reads    (ρ 1 ) t + χ 1 ∇ · (ρ 1 ∇c) = µ 1 ∆ρ 1 , (ρ 2 ) t + χ 2 ∇ · (ρ 2 ∇c) = µ 2 ∆ρ 2 , c t = D∆c + α 1 ρ 1 + α 2 ρ 2 − βc,…”

“…In the latter case, we would also like to determine whether the cell densities of both species blow up simultaneously or the cell density of the species with a larger chemosensitivity constant blows up first. These questions were first raised in [34] and further studied in [4,5,7]. More precisely, it has been proven analytically that depending on a particular set of parameter values and initial cell densities the solution may be globally regular or it may blow up within a finite time.…”