Global classical solutions for a class of reaction-diffusion system with density-suppressed motility

Abstract

This paper is concerned with a class of reaction-diffusion system with density-suppressed motility (Formula Presented) under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ Rn (n ≤ 2), where α > 0 and D > 0 are constants. The random motility function γ satisfies (Formula Presented) The intake rate function F satisfies F ε C1([0, +∞)), F(0) = 0 and F > 0 on (0, +∞). We show that the above system admits a unique global classical solution for all non-negative initial data u0 ∈ W1,∞(Ω), v0 ∈ W1,∞(Ω), w0 ∈ W1,∞(Ω). Moreover, if there exist k > 0 and (Formula Presented) such that (Formula Presented), then the global solution is bounded uniformly in time