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

This paper is concerned with a class of reaction-diffusion system with density-suppressed motility document equation* cases uₓ = ( (v) u) + u F (w), x, t0, \\ vₓ = D v+u-v, x, t0, \\ wₓ = w-u F (w), x, t0, cases equation* document under homogeneous Neumann boundary conditions in a smooth bounded domain Rⁿ\; (n 2), where 0 and D 0 are constants. The random motility function satisfies document equation* C³ ( (0, +) ), \ 0, \ '0\, \ on\, \ (0, +) \ \ and\ \ ₕ + (v) = 0. equation* document The intake rate function F satisfies F 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 W^1, (), \, v₀ W^1, (), \, w₀ W^1, (). Moreover, if there exist k 0 and v 0 such that document equation* ₕₕvᵏ (v) 0, equation* document then the global solution is bounded uniformly in time.