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

摘要

We consider the following fully parabolic Keller-Segel system (u_t = ▽ · (D(u)▽u - S(u)▽u) + u(1 - u~γ), x E ∈ Ω, t > 0, v_t = △v - v + u, x ∈ Ω,t > 0, (∂u)/(∂v) = (∂u)/(∂v) = 0, x ∈ ∂Ω,t > 0 over a multi-dimensional bounded domain Ω ⊂ R~N, N ≥ 2. Here D(u) and S(u) are smooth functions satisfying: D(0) > 0, D(u) > K_1u~(m_1) and S(u) ≤ K_2u~(m_2), ∀u ≥ 0, for some constants K_i ∈ R~+, m_i ∈ R, i = 1,2. It is proved that, when the parameter pair (m_1, m_2) lies in some specific regions, the system admits global classical solutions and they are uniformly bounded in time. We cover and extend [22, 28], in particular when N > 3 and 7 > 1, and [3, 29] when m_1 > γ - 2/N if γ ∈ (0,1) or m_1 > γ - 4/(N+2) if γ ∈ [1,∞). Moreover, according to our results, the index is, in contrast to the model without cellular growth, no longer critical to the global existence or collapse of this system.