Asymptotic stability in a quasilinear chemotaxis-haptotaxis model with general logistic source and nonlinear signal production

This paper deals with the quasilinear chemotaxis-haptotaxis model of cancer invasion {u(t) = del.(D(u)del u) + del.(S-1(u)del v) + del. S-2(u)del w) + f(u,w), x is an element of Omega, t > 0, tau v(t) = Delta v - v + g(1)(w)g(2)(u), x is an element of Omega, t > 0, w(t) = -vw, x is an element of Omega, t > 0 in a bounded smooth domain Omega subset of R-N (N >= 1) with zero-flux boundary conditions, where tau is an element of {0, 1}, the functions D(u), S-1(u), S-2(u) is an element of C-2 ([0, infinity)), f(u, w) is an element of C-1 ([0, infinity)(2)), g(1)(w), g(2)(u) is an element of C-1 ([0, infinity)) ful-fill D(u) >= C-D(u + 1)(-alpha), S-1(u) < chi u(u + 1)(beta-1), S-2(u) <= xi u(u + 1)(gamma-1), f(u, w) <= u(a - mu ur(-1)-lambda w), f (0, w) >= 0, g(1)(w) >= 0, 0 <= g(2)(u) <= Ku(kappa) with C-D, chi, xi, mu, kappa > 0, lambda >= 0, r > 1 and alpha, beta, gamma, a is an element of R. Under specific parameters conditions, it is shown that for any appropriately regular initial data, the associated initial-boundary value problem admits a globally bounded classical solution. Moreover, when f = u(a - mu ur(-1) - lambda w), g(1)(w) 1 and g(2)(u) u(kappa), the asymptotic stability of solutions is also investigated. Specifically, for some a(0), mu(0) > 0 independent of (u(0), v(0)), the bounded classical solution (u, v, w) exponentially converges to ((a/mu)(1/r-1), (a/mu)(kappa/r-1), 0) in L-p (Omega) x L-infinity (Omega) x W-1,W-infinity (Omega) for any p >= 2 if a > a(0) and mu > mu(0). These results improve or extend previously known ones, and partial results are new. (C) 2020 Elsevier Inc. All rights reserved.