Boundedness criteria for a chemotaxis consumption model with gradient nonlinearities

This work deals with the consumption chemotaxis problem { u t = Δ u − χ ∇ ⋅ u ∇ v + λ u − μ u 2 − c | ∇ u | γ , in Ω × ( 0 , T max ) , v t = Δ v − u v , in Ω × ( 0 , T max ) , in a bounded and smooth domain Ω ⊂ R n , n ≥ 3 , under homogeneous Neumann boundary conditions, for χ , λ , μ , c > 0 , T max ∈ ( 0 , ∞ ] and for u 0 , v 0 positive initial data with a certain regularity. We will show that the problem has a unique and uniformly bounded classical solution for γ ∈ ( 2 n n + 1 , 2 ] . Moreover, we have the same result for γ = 2 n n + 1 and a condition that involves the parameters c , μ , n , χ and the initial data.