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 {u(t) = del. ((u + 1)(m-1) del u - u(u +1 )(alpha-1) chi(v)del v) + ku - mu u(2), X is an element of Omega, T > 0, v(t) = Delta v - vu, X is an element of Omega, t > 0, Omega being a convex smooth and bounded domain of R-n, n >= 1, and where m, k is an element of R, mu > 0 and alpha < m+1/2. For any v >= 0 the chemotactic sensitivity function is assumed to behave as the prototype chi(v) = chi(0)/(1+av)(2), with a >= 0 and chi(0) > 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 mu is large enough.