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-rho to L-2 rho (rho >= 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 xi(00 )(xi(00)< 0) is found, such that positive constant steady state (u*,v*,w*) becomes unstable when xi < xi(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.