Chemotaxis models with mixed mechanisms: Boundedness in growth-dominated regimes

We investigate a class of zero-flux chemotaxis-growth systems featuring nonlinear local and nonlocal reaction terms, as well as a gradient-dependent damping, given by {ut=Δu−χ∇·(u∇v)+auρ−b(∫Ωuβ)δ−c|∇u|γ,inΩ×(0,Tmax),τvt=Δv−v+u,inΩ×(0,Tmax), where χ, a, b, c > 0, ρ, β, δ, γ ≥ 1, and τ ∈ {0, 1}. Assuming that Ω⊂Rn ( n ≥ 1) is a bounded domain with smooth boundary and that the initial data ( u 0, τv 0) are sufficiently regular, we prove the existence of global (i.e., Tmax=∞), uniformly bounded classical solutions under suitable structural conditions on the exponents defining the reaction term and the dimension n . A central insight of our analysis is that, in contrast to classical scenarios where ρ ≤ β typically ensures automatic control of the total mass ∫javax.xml.bind.JAXBElement@7662c8dau, the regime ρ > β makes that the structure of the source term alone does not suffice to guarantee mass boundedness. Accordingly, the global boundedness of solutions is achieved via a two-tiered strategy: first, we identify a parameter regime in which the total mass remains uniformly bounded in time; second, under further constraints on the model parameters, we establish uniform-in-time boundedness of classical solutions in stronger norms, specifically in L ∞(Ω). These results offer further insight into the interplay between chemotactic aggregation, gradient-driven dissipation, and nonlocal reaction effects, contributing to the analysis of blow-up prevention in structured chemotaxis models.