Unlimited growth in logarithmic Keller-Segel systems

The fully parabolic cross-diffusion system{εut=Δu−χ∇⋅(uv∇v),vt=Δv−v+u, is considered under homogneous Neumann boundary conditions in a ball Ω⊂Rn, n≥1, for ε∈(0,1) and χ>0. Previous literature has asserted global existence of certain generalized solutions whenever χ2nn−2. The present study examines the solution behavior in the singular limit ε↘0, and based on an essentially well-known result on finite-time blow-up in an accordingly obtained nonlocal scalar parabolic problem, under the assumptions that n≥3 and χ>nn−2 a statement on spontaneous emergence of arbitrarily large values of u for appropriately small ε is derived.