GLOBAL EXISTENCE AND BOUNDEDNESS OF SOLUTION OF A PARABOLIC-PARABOLIC-ODE CHEMOTAXIS-HAPTOTAXIS MODEL WITH (GENERALIZED) LOGISTIC SOURCE

摘要

In this paper, we study the following chemotaxis-haptotaxis system with (generalized) logistic source{u(t) = Delta u - chi del . (u del v) - xi V . (u del w) + u(a - mu u(r-1 )- w),v(t) = del v - v + u,w(t )= -vw,partial derivative u/partial derivative v = partial derivative u/partial derivative v = partial derivative w/partial derivative v = 0, x is an element of partial derivative Omega, t 0,u(x, 0) = u(0 )(x), v(x, 0) = v(0 )(x), w(x, 0) = w(0)(x),( )x is an element of Omega, (0.1)( )in a smooth bounded domain R-N (N = 1), with parameter r 1. the parameters a is an element of R, mu 0, chi 0. It is shown that when r 2, ormu mu* = (N-2)(+)/N(chi + C-beta)C-N/2+1(1/N/2+1), if r = 2,the considered problem possesses a global classical solution which is bounded, where C-N/2+1(1/N/2+1) is a positive constant which is corresponding to the maximal sobolev regularity. Here C-beta is a positive constant which depends on xi, parallel to u(0)parallel to(C((Omega) over bar)), parallel to v(0)parallel to(W1,infinity(Omega)) and parallel to w(0)parallel to(L infinity(Omega)). This result improves or extends previous results of several authors.