Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic chemotaxis system with logistic source and nonlinear production

Abstract

This paper deals with the quasilinear parabolic-elliptic chemotaxis system with logistic source and nonlinear production,) dx, and D, S and f are functions generalizing the prototypes D(u) = (u + 1) m−1 , S(u) = u(u + 1) α−1 and f (u) = u ℓ with m ∈ R, α > 0 and ℓ > 0. In the case m = α = ℓ = 1, Fuest (NoDEA Nonlinear Differential Equations Appl.;2021; 28; 16) obtained conditions for κ such that solutions blow up in finite time. However, in the above system boundedness and finite-time blow-up of solutions have been not yet established. This paper gives boundedness and finite-time blow-up under some conditions for m, α, κ and ℓ.