Qualitative behavior of solutions for a chemotaxis-haptotaxis system with gradient-dependent flux-limitation

This paper deals with the coupled chemotaxis-haptotaxis model with gradient-dependent flux-limitation {u(t) = Delta u - chi del center dot (u|del v|(p-2)del v) - xi del center dot (u del w) +mu u(1 - u - w), x is an element of Omega, t > 0, 0 = Delta v - v + u, x is an element of Omega, t > 0 w(t) = -vw, x is an element of Omega, t > 0, where Omega subset of R-n(n >= 1) is a bounded domain with smooth boundary, chi, xi and mu are positive parameters. center dot Under the condition that 1 < p < n/n -1, then for any chi, xi, mu > 0 and all reasonably smooth initial data (u(0), w(0)), the above system possesses a globally classical solution, which is uniformly bounded in time. center dot Under the conditions that 1 < p < n/n -1 and mu > chi(2)/8, it asserts exponential decay of w in the large time limit, whereas both u and v persist in some certain sense.