Immediate smoothing and global solutions for initial data inL1×W1,2in a Keller–Segel system with logistic terms in 2D

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 andμ> 0), and shows that any nonnegative initial data (u0,v0) ∈L1×W1,2lead to global solutions that are smooth in$\bar {\Omega }\times (0,\infty )$.