Simultaneous finite time blow-up in a two-species model for chemotaxis

Abstract

A system of two classical chemotaxis equations, coupled with an elliptic equation for an attractive chemical, is analyzed. Depending on the parameter values for the three respective diffusion coefficients and the two chemotactic sensitivities in the radial symmetric setting, conditions are given for global existence of solutions and finite time blow-up. A question of interest is, whether there exist parameter regimes, where the two chemotactic species differ in their long time behavior. This questions arises in the context of differential chemotactic behavior in early aggregates of Dictyostelium discoideum.

Remarks on the blowup and global existence for a two species chemotactic Keller–Segel system in2

“…As pointed out by this author, it would be interesting to know which are the optimal conditions on the parameters θ 1 , θ 2 that allows us to conclude the existence of blowup or global existence in time. In the radial symmetrical case, it was shown that it has to be simultaneous (see Espejo et al [6]), in the general, non-radial case the question remains open up to our knowledge. Another interesting question is: In case of blowup, should this be simultaneous or not?…”

“…Making a dimensional analysis like in Espejo et al [6], Section 2 , it reduces to ∂ t u 1 = µ∆u 1 − χ 1 ∇ · (u 1 ∇v), ∂ t u 2 = ∆u 2 − χ 2 ∇ · (u 2 ∇v),…”

Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant

“…More recently, systems of two species with one chemoattractant have been studied by different research groups, the flnite-time blow-up in bounded domains for the ParabolicParabolic-Elliptic issue has been analyzed by Espejo, Stevens and Velázquez [7] and [8] for simultaneous and non-simultaneous blow-up. See also the results in Biler, Espejo and Guerra [3], Biler and Guerra for bounded domains and Conca and Espejo [5,6] for the two-dimensional case in the whole space.…”

Two-species flocking particles immersed in a fluid

“…Two-species models have many applications such as pedestrian flows [29], opinion formation between two groups with different leanings [18,19], and so on. A mathematical study of existence, stability, finite-time blow up, and the large-time behavior for two competitive populations of biological species which are attracted by random diffusion and chemotaxis is another recent active research area [12,20,23,31]. We also refer to [21,25] for nonlocal interaction PDEs with two-species.…”