Boundedness criteria for a chemotaxis consumption model with gradient nonlinearities

This work deals with the consumption chemotaxis problem equation* cases* uₜ = u - u v + u - u² - c u ^, & in (0, ), vₜ = v - uv, & in (0, ), cases* equation* in a bounded and smooth domain ⁿ, n 3, under Neumann boundary conditions, for, , , 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 (2nn+1, 2]. Moreover, we have the same result for =2nn+1 and a condition that involves the parameters c, , n, and the initial data.