Boundedness in the Higher-Dimensional Parabolic-Parabolic Chemotaxis System with Logistic Source

We consider nonnegative solutions of the Neumann boundary value problem for the chemotaxis system in a smooth bounded convex domain Ω ⊂ ℝ n , n ≥ 1, where τ > 0, χ ∈ ℝ and f is a smooth function generalizing the logistic source It is shown that if μ is sufficiently large then for all sufficiently smooth initial data the problem possesses a unique global-in-time classical solution that is bounded in Ω × (0, ∞). Known results, asserting boundedness under the additional restriction n ≤ 2, are thereby extended to arbitrary space dimensions.