BOUNDEDNESS IN LOGISTIC KELLER-SEGEL MODELS WITH NONLINEAR DIFFUSION AND SENSITIVITY FUNCTIONS

We consider the following fully parabolic Keller-Segel system {u(t) = del center dot(D(u)del u - S(u)del v) - u(1-u(gamma)), x is an element of Omega, t>0, v(t) = Delta v - v+u, x is an element of Omega, t>0, partial derivative u/partial derivative nu = partial derivative u/partial derivative nu = 0, x is an element of partial derivative Omega, t>0 over a multi dimensional bounded domain Q C R-N, N >= 2. Here D(u) and S(u) are smooth functions satisfying: D(0) > 0, D(u) >= K1u(m1), and S(u) <= K(2)u(m2), for all u >= 0, for some constants K-i is an element of R+ , m(i) is an element of R, i = 1, 2. It is proved that, when the parameter pair (mi, m2) lies in some specific regions, the system admits global classical solutions and they are uniformly bounded in time. We cover and extend [m(1), m(2)], in particular when N >= 3 and gamma >= 1, and [i, 29] when m(1)> gamma- 2/N if gamma is an element of(0,1) or m(1) > gamma-4/N+2 if gamma is an element of[1,infinity). Moreover, according to our results, the index 2/N is, in contrast to the model without cellular growth, no longer critical to the global existence or collapse of this system.