Abstract
In this paper, we consider a homogeneous Neumann initial-boundary value problem (IBVP) for the following two-species and two-stimuli chemotaxis model with both paracrine and autocrine loops:
u(t) = del . (D-1(u)del u - S-1(u)del v); x is an element of Omega, t > 0;
tau(1)v(t) = Delta v - v + w; x is an element of Omega, t > 0;
w(t) = del . (D-2(w)del w - S-2(w)del z - S-3(w)del v); x is an element of Omega, t > 0
tau(2)z(t) = Delta z - z + u; x is an element of Omega, t > 0,
where u(t; x) and w(t; x) denote the density of macrophages and tumor cells at time t and location x is an element of Omega; respectively, v(t; x) and z(t; x) represent the concentration of colony stimulating factor 1 (CSF-1) secreted by the tumor cells and epidermal growth factor (EGF) secreted by macrophages at time t and location x is an element of Omega; respectively. boolean AND is an element of R-n is a bounded region with smooth boundary, tau(i) >= 0 (i = 1; 2), D-i(s) >= d(i)(s + 1)m(i)-1 with parameters m(i) >= 1 (i = 1; 2) and S-j (s) less than or similar to (s + 1)(qj) with parameters q(j) > 0 (j = 1; 2; 3). For the case without autocrine loop (i.e., S-3(w) = 0), it is shown that when q(j) <= 1 (j = 1; 2), if one of q(j) is smaller than one or one of mi is larger than one, then the IBVP has a global classical solution which is uniformly bounded. Moreover, when m(1) = m(2) = q(1) = q(2) = 1, an inequality involving the product d(1)d(2) and the product of the two species' initial mass is obtained which guarantees the existence of global bounded classical solutions. More specifically, it allows one of d(i) to be small or one of the species initial mass to be large. For the case with autocrine loop (i.e S-3(w) not equal 0), similar results hold only if q(3) < 1. If q(3) = 1, solutions to the IBVP exist globally only when d(2) is suitably large or the mass of species w is suitably small.
