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)

• Record Type: Journal Article
• Authors: Lankeit, Johannes
• Is Part Of: Proceedings. Volume 151:Issue 4(2021)
• Page Start: 1204
• Page End: 1224
• Publication Date: 2021-08
• Subjects: Chemotaxis -- Keller–Segel -- logistic -- regularity -- initial data -- classical solutions; 35B65 -- 35K45 -- 35A09 -- 35Q92 -- 92C17; Mathematics -- Periodicals; 510
• DOI: 10.1017/prm.2020.55
• Languages: English
• ISSNs: 0308-2105

Abstract

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 $\bar {\Omega }\times (0, \infty )$ .