Immediate smoothing and global solutions for initial data in L1 × W1, 2 in a Keller–Segel system with logistic terms in 2D. Issue 4 (August 2021) - Lankeit, Johannes
Abstract : This paper deals with the logistic Keller–Segel model \[ \begin{cases} u_t = \Delta u - \chi \nabla\cdot(u\nabla v) + \kappa u - \mu u^2, \\ v_t = \Delta v - v + u \end{cases} \] in bounded two-dimensional domains (with homogeneous Neumann boundary conditions and for parameters χ, κ ∈ ℝ and μ > 0), and shows that any nonnegative initial data ( u 0, v 0 ) ∈ L 1 × W 1, 2 lead to global solutions that are smooth in $\bar {\Omega }\times (0, \infty )$ .