A new (and optimal) result for the boundedness of a solution of a quasilinear chemotaxis-haptotaxis model (with a logistic source)

This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion u(t) = del . (D(u)del u) - chi del . (u del v) - xi del . (u del w) + mu u (1 - u - w), x is an element of Omega, t > 0, tau v(t =) Delta v - v + u, x is an element of Omega, t > 0, w(t) = -vw, x is an element of Omega, t > 0 (0.1) under homogeneous Neumann boundary conditions in a smooth bounded domain Omega subset of R-N (N >= 1), where tau is an element of {0, 1} and chi, xi, and mu are given nonnegative parameters. The diffusivity D(u) is assumed to satisfy D(u) >= C-D(u + 1)(m-1) for all u >= 0 and C-D > 0. In the present work it is shown that if m >= 2 - 2/N lambda with 0 < mu < kappa(0), m > 2 - 2/N lambda with mu >= kappa(0), or m > 2 - 2/N and mu = 0 or m = 2 - 2/N and C-D > C-GN(1 + parallel to u(0)parallel to L-1(Omega))(3)/4 (2 - 2/N)(2)kappa(0), then for all reasonably regular initial data, a corresponding initial-boundary value problem for (0.1) possesses a unique global classical solution that is uniformly bounded in Omega x (0, infinity), where lambda = kappa(0)/(kappa(0) - mu)+ and kappa(0) = {max(s >= 1) lambda(1/s+1)(0)(chi + xi parallel to w0 parallel to L-infinity(Omega)) if tau = 1, chi if tau = 0. Here C-GN and lambda(0) are constants that correspond to the Gagliardo-Nirenberg inequality and the maximal Sobolev regularity, respectively. With use of new L-p-estimate techniques to obtain the a priori estimate of a solution from L-1(Omega) -> L lambda-epsilon(Omega) -> L-lambda(Omega) -> L lambda+epsilon(Omega) -> L-p(Omega) (for all p > 1), these results significantly improve or extend previous results obtained by several authors. (C) 2020 Elsevier Inc. All rights reserved.