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

• * Corresponding author. QW acknowledges the support from National Natural Science Foundation of China (Grant No. 11501460). All authors thank the two anonymous referees for their helpful suggestions, which improve the presentation of this paper

Currently at Department of Mathematics, University of Central Florida, Orlando, USA. yfeng@knights.ucf.edu.

• 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.

Mathematics Subject Classification: Primary: 92C17, 35K55; Secondary: 35K51.

 Citation: • • Figure 1.  An illustration of parameter regions for the global existence of (1.1) in the $m_1$-$m_2$ plane as $\gamma$ varies. For each pair of $(m_1,m_2)$ in the shaded region, (1.1) over bounded domain $\Omega\subset \mathbb R^N$, $N\geq2$, admits global bounded classical solutions. The slopes of $L_1(L^*_1)$ and $L_2(L^*_2)$ are 1 and the slope of $L_3(L^*_3)$ is $\frac{1}{2}$. Note that in the lower right plot, we use $\frac{1}{N}+\frac{2}{N+2}$, instead of $\frac{3N+2}{N(N+2)}$, for better illustration

Figure 2.  An illustration of up-to-date summary results on global existence of the nonlinear diffusion system (1.1). For each pair of $(m_1,m_2)$ in the shaded region, (1.1) over bounded domain $\Omega\subset \mathbb R^N$, $N\geq2$, admits global bounded classical solutions. The slope of $L_1(L^*_1)$ is 1 and of $L^*_2$ is $\frac{1}{2}$; $L_4$ is the horizontal line $m_2=\gamma$

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