Finite-time blowup in attraction–repulsion systems with nonlinear signal production. (October 2021) - Liu, Meng
Abstract: This paper investigates a multi-dimensional attraction–repulsion system u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ), x ∈ Ω, t > 0, 0 = Δ v − μ 1 ( t ) + f 1 ( u ), x ∈ Ω, t > 0, 0 = Δ w − μ 2 ( t ) + f 2 ( u ), x ∈ Ω, t > 0, where μ 1 ( t ) = 1 | Ω | ∫ Ω f 1 ( u ) d x, μ 2 ( t ) = 1 | Ω | ∫ Ω f 2 ( u ) d x, Ω = B R ( 0 ) ⊂ R n ( n ≥ 2 ) and f 1 and f 2 are suitably regular functions generalizing the prototype determined by f 1 ( s ) = s γ 1 and f 2 ( s ) = s γ 2, s ≥ 0, with γ 1, γ 2 > 0 . Under homogeneous boundary conditions of Neumann type for u, v and w, it is proved that, among other things, if γ 1 > γ 2 and γ 1 > 2 n, the solution with initial mass concentrating enough in a small ball centered at origin will blow up in finite time, for any γ 1, γ 2, if γ 1 < 2 n, then for suitable smooth initial data ( u 0, v 0, w 0 ), the system possesses a unique global bounded classical solution. We point out that there exists a gap in the parameter regime: if γ 1 < γ 2, does the solution exist globally?