Finite-time blow-up in a two-dimensional Keller–Segel system with an environmental dependent logistic source. (April 2020) - Fuest, Mario
Abstract: The Neumann initial–boundary problem for the chemotaxis system ( ⋆ ) u t = Δ u − ∇ ⋅ ( u ∇ v ) + κ ( | x | ) u − μ ( | x | ) u p, 0 = Δ v − m ( t ) | Ω | + u, m ( t ) ≔ ∫ Ω u ( ⋅, t ) is studied in a ball Ω = B R ( 0 ) ⊂ R 2, R > 0 for p ≥ 1 and sufficiently smooth functions κ, μ : [ 0, R ] → [ 0, ∞ ) . We prove that whenever μ ′, − κ ′ ≥ 0 as well as μ ( s ) ≤ μ 1 s 2 p − 2 for all s ∈ [ 0, R ] and some μ 1 > 0 then for all m 0 > 8 π there exists u 0 ∈ C 0 ( Ω ¯ ) with ∫ Ω u 0 = m 0 and a solution ( u, v ) to ( ⋆ ) with initial datum u 0 blowing up in finite time. If in addition κ ≡ 0 then all solutions with initial mass smaller than 8 π are global in time, displaying a certain critical mass phenomenon. On the other hand, if p > 2, we show that for all μ satisfying μ ( s ) ≥ μ 1 s p − 2 − ε for all s ∈ [ 0, R ] and some μ 1, ε > 0 the system ( ⋆ ) admits a global classical solution for each initial datum 0 ≤ u 0 ∈ C 0 ( Ω ¯ ) .