On Patlak-Keller-Segel system for several populations: A gradient flow approach

Abstract

We study the global in time existence of solutions to the parabolic-elliptic Patlak-Keller-Segel system of multi-species populations. We prove that if the initial mass satisfies an appropriate notion of sub-criticality, then the system has a solution defined for all time. We explore the gradient flow structure in the Wasserstein space to study the question of existence. Moreover, we show that the obtained solution satisfies energy dissipation inequality.

Critical mass phenomenon for a parabolic–elliptic multispecies chemotaxis system in a two‐dimensional disk

“…the solutions exist globally in the sub-critical or critical condition [27,28], while there exist a blow-up solution in the supercritical case [29]. It should be noted that this paper improves and extends those results to a system of n populations and m sensitivity agents.…”

“…Let $$$ \mid \mathcal{J}\mid =\mid \mathcal{I}\mid =n\ge 2 $$$, and let $$$ \boldsymbol{\theta} =\left\{{\theta}_{i,j}\right\} $$$ be a symmetric matrix with nonnegative entries and $$$ \boldsymbol{\gamma} =\left\{{\gamma}_{i,j}\right\} $$$ be an identity matrix. If the $$$ {v}_j $$$‐th equation in () is replaced by $$$$ -\Delta {v}_j={u}_j, $$$$ the solutions exist globally in the sub‐critical or critical condition [27, 28], while there exist a blow‐up solution in the supercritical case [29]. It should be noted that this paper improves and extends those results to a system of $$$ n $$$ populations and $$$ m $$$ sensitivity agents.…”

The fully parabolic multi-species chemotaxis system in

“…(1.3) https://doi.org/10.1017/S0956792523000372 Published online by Cambridge University Press Karmakar and Wolansky [22] had derived the global well-posedness of weak solutions with respect to time in the sub-critical regime…”

Critical curve for a two-species chemotaxis model with two chemicals in R2 *

“…, n) and the unit sensitivity coefficients matrix (a i j ) n×n , He and Tadmort established some sharp conditions to identify the global existence and the finite time blow-up of solutions in [14], with the critical case left open. Karmakar and Wolansky considered the parabolic-elliptic model (1.2) with Newtonian potential in [16,17], where the chemical generation coefficients matrix (b i j ) n×n is unit and the (a i j ) n×n is a symmetric and nonnegative matrix. By using the gradient flow structure in Wasserstein space, the global existence of solutions in the subcritical case Λ J (M) := i∈J m i (8π − j∈J a i j m j ) > 0 (a i j 0) for nonempty J ⊂ I = {1, 2, .…”