Regularizing effect in a Keller–Segel system with density-dependent sensitivity for Lp-initial data of cell density

Abstract

The fully parabolic Keller–Segel system{ut=Δu−χ∇⋅(u(u+1)α−1∇v)inΩ×(0,∞),vt=Δv−v+uinΩ×(0,∞) is considered in a bounded domain Ω⊂RN (N∈N) with smooth boundary, where χ,α∈R. It is shown that an associated Neumann initial-boundary value problem admits a global classical solution for initial data with low regularity, in particular, for Lp-initial data (p>max⁡{1,α,NN+1(α+1)}) of u when α<2N and when α=2N and the initial mass is small. This extends a well-known result on global solvability for reasonably regular initial data.