科研之友
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摘要
This paper deals with the global existence and boundedness of the solutions for the chemotaxis system with logistic source @@@ {ut = del. (phi(u)del u) - del . (phi(u)del v) + f(u), x is an element of Omega, t > 0, vt =Delta v - v +u, x is an element of Omega, t > 0, @@@ under homogeneous Neumann boundary conditions in a convex smooth bounded domain Omega subset of R-n(n >= 2), with non-negative initial data u(0) is an element of C-0 ((Omega) over bar) and v(o) is an element of W-1,W-theta (Omega) (with some theta > n). The nonlinearities phi and phi are assumed to generalize the prototypes @@@ phi(u) = (u +1)(-alpha) , phi(u) = u(u + 1)(beta-1) @@@ with alpha is an element of R and beta is an element of R. f(u) is a smooth function generalizing the logistic function @@@ f(u) = ru - b(u)gamma, U >= 0, with r >= 0, b> 0 and gamma > 1. @@@ It is proved that the corresponding initial-boundary value problem possesses a unique global classical solution that is uniformly bounded provided that some technical conditions are fulfilled.
