Boundedness in a fully parabolic chemotaxis-consumption system with nonlinear diffusion and sensitivity, and logistic source

In this paper we study the zero-flux chemotaxis-system (Formula presented.) Ω being a convex smooth and bounded domain of ℝn, n≥1, and where m,k∈ℝ, μ > 0 and α<m+1/2. For any v ≥ 0 the chemotactic sensitivity function is assumed to behave as the prototype x(v)=x0/(1+av)2, with a ≥ 0 and x0> 0. We prove that for nonnegative and sufficiently regular initial data u(x, 0) and v(x, 0), the corresponding initial-boundary value problem admits a unique globally bounded classical solution provided μ is large enough.