Global boundedness of weak solution in an attraction-repulsion chemotaxis system with p-Laplacian diffusion

In this paper, an attraction-repulsion chemotaxis system with p-Laplacian diffusion {u(t) = del . (vertical bar del u vertical bar(p-2)del u) - chi del . (u del v) + xi del . (u del w), x is an element of Omega, t > 0, 0 = Delta v + alpha u -beta v, x is an element of Omega, t > 0, 0 = Delta w + gamma u - delta w, x is an element of Omega, t > 0, u(x,0) = u(0)(x), x is an element of Omega is considered associated with homogeneous Neumann boundary conditions in a smooth bounded domain Omega subset of R-n, n >= 2. chi, xi, alpha, beta, gamma, delta are positive parameters. Global bounded weak solution is constructed for any p > 1 under the following cases: Case I: xi gamma - chi alpha > 0 with p > 1; Case II: xi gamma - chi alpha <= 0 with p > 3n/n+1; Case III: xi gamma - chi alpha <= 0 with 1 < p <= 3n/n+1 and parallel to u0 parallel to(L (3-p)n/p (Omega)) is small. (C) 2019 Elsevier Ltd. All rights reserved.