Corners and collapse: Some simple observations concerning critical masses and boundary blow-up in the fully parabolic Keller–Segel system

Abstract

Our main result shows that the mass 2π is critical for the minimal Keller–Segel system (⋆)ut=Δu−∇⋅(u∇v),vt=Δv−v+u,considered in a quarter disc Ω={(x1,x2)∈R2∣x1>0,x2>0,x12+x22<R2}, R>0, in the following sense: For all reasonably smooth nonnegative initial data u0,v0 with ∫Ωu0<2π, there exists a global classical solution to the Neumann initial boundary value problem associated to (⋆), while for all m>2π there exist nonnegative initial data u0,v0 with ∫Ωu0=m so that the corresponding classical solution of this problem blows up in finite time.At the same time, this gives an example of boundary blow-up in (⋆).Up to now, precise values of critical masses had been observed in spaces of radially symmetric functions or for parabolic–elliptic simplifications of (⋆) only.