Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions

We consider the following fully parabolic Keller-Segel system

$\left\{\begin{array}{ll}u_t=\nabla · (D(u) \nabla u-S(u) \nabla v)+u(1-u^γ),&x ∈ Ω,t>0, \\v_t=Δ v-v+u,&x ∈ Ω,t>0, \\\frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=0,&x∈\partial Ω,t>0\end{array}\right.$

over a multi-dimensional bounded domain $Ω \subset \mathbb R^N$, $N≥2$. Here $D(u)$ and $S(u)$ are smooth functions satisfying: $D(0)>0$, $D(u)≥ K_1u^{m_1}$ and $S(u)≤ K_2u^{m_2}$, $\forall u≥0$, for some constants $K_i∈\mathbb R^+$, $m_i∈\mathbb R$, $i=1, 2$. 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 $γ≥1$, and [3,29] when $m_1>γ-\frac{2}{N}$ if $γ∈(0, 1)$ or $m_1>γ-\frac{4}{N+2}$ if $γ∈[1, ∞)$. Moreover, according to our results, the index $\frac{2}{N}$ is, in contrast to the model without cellular growth, no longer critical to the global existence or collapse of this system.