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

Abstract

This paper deals with a two-species chemotaxis model with two chemicals in R 2 . In our previous work (2019 Nonlinearity 32 4762–78), the critical mass was obtained that the solutions exist globally if m 1 m 2 − 4π(m 1 + m 2) < 0, and the finite time blow-up of solutions may occur if m 1 m 2 − 4π(m 1 + m 2) > 0, where m 1 and m 2 describe the initial mass of the two species, respectively. In the present paper we furthermore determine that the critical situation belongs to the global existence case, namely, the system admits global solutions if m 1 m 2 − 4π(m 1 + m 2) = 0. To apply the key logarithmic Hardy–Littlewood–Sobolev inequality of vector form for the critical case, we should establish a positive lower bound for L 1-norms of the two species in exterior domains { x ∈ R 2 : | x | > R } uniformly for t > 0.

The fully parabolic multi-species chemotaxis system in

“…In particular, in the case of parabolic-elliptic system (1.2) with |I| = 2, |J | = 1, Espejo et al [7,12] discovered a threshold curve to help us to determine whether the solutions of system are blow-up or global in time. See related works for parabolic-elliptic system (1.2) with |I| = |J | = 2 in [18,19]. Moreover, [9][10][11]13] could be refereed to characterise the simultaneous or non-simultaneous blow-up results in two-species model.…”