Global Solution and Stability of a Haptotaxis Mathematical Model for Complex MAP

A critical function of polymeric matrices in biological systems is to exert selective control over the transport of thousands of nanoparticulate species. Utilizing “third-party” molecular anchors to crosslink nanoparticulates to the matrix is an effective strategy, and a trapped nanoparticulate formed a desired complex MAP that is necessary to keep the nanoparticulate immobilized at any given time. In this paper, the global solution and stability of a parabolic–ordinary-parabolic haptotaxis system to complex MAP are studied. First, the existence of a local classical solution to system (4) has been observed using fixed point argument and parabolic Schauder estimates. Furthermore, some a priori estimates that can raise the regularity estimate of the solution for the relatively complicated first equation of system (3) from L ρ to L 2 ρ ( ρ ≥ 1 ) are given; then, the local classic solution can thus extend to the global classic solution when the space dimension N ≤ 3 . Lastly, by using various analytical methods, a threshold value ξ 00 ( ξ 00 < 0 ) is found, such that positive constant steady state ( u ∗ , v ∗ , w ∗ ) becomes unstable when ξ < ξ 00 . Our results show that the haptotaxis plays a crucial role in determining the stability to the model (3), that is, it can have a destabilizing effect.