Boundedness criteria for a chemotaxis consumption model with gradient nonlinearities

This work deals with the consumption chemotaxis problem {equation*} {cases*} u_t = Δ u - χ ∇ · u∇ v + λ u - μ u^2 - c ∇ u ^γ, & in Ω×(0,), v_t = Δ v - uv, & in Ω×(0,), {cases*} {equation*} in a bounded and smooth domain Ω⊂ⁿ, n≥ 3, under Neumann boundary conditions, for χ,λ,μ,c>0, ∈(0,∞] and for u₀,v₀ positive initial data with a certain regularity. We will show that the problem has a unique and uniformly bounded classical solution for γ∈(2n/n+1,2]. Moreover, we have the same result for γ=2n/n+1 and a condition that involves the parameters c,μ,n,χ and the initial data.