Global existence of solutions to Keller-Segel chemotaxis system with heterogeneous logistic source and nonlinear secretion

Abstract

We study the following Keller-Segel chemotaxis system with logistic source and nonlinear secretion:and Ω ⊂ R n , n ≥ 2. For this system, we prove the global existence of solutions under suitable assumptions on the initial condition and the functions κ(•) and µ(•).

Possible points of blow-up in chemotaxis systems with spatially heterogeneous logistic source

“…In light of these observations, also chemotaxis-growth systems with non-constant κ and µ have been considered: Salako and Shen treated the case that Ω = R n , n ∈ N, and κ, µ are positive functions depending on x ∈ R n and t ∈ R. They showed global existence and uniform persistence, asymptotic spreading of classical solutions [29], established stability of strictly positive entire solutions [30] and found conditions for existence and non-existence of transition front solutions [31]. For bounded domains Ω and κ, µ depending on the space variable only, global solutions have been constructed in [52] (within a generalized solution framework) and [1] (for a related system), while solutions blowing up in finite time have been found in [9,4] (see also [35]). The location of blow-up.…”