A new result for boundedness in the quasilinear parabolic-parabolic Keller-Segel model (with logistic source)

Abstract

The current paper considers the boundedness of solutions to the following quasilinear Keller-Segel model (with logistic source) $$\{\begin{array}{l}{u}_t=\nabla \cdot (D(u)\nabla u)-\chi \nabla \cdot (u\nabla v)+\mu (u-u^2),x\in \Omega ,t>0,\\ v_t-\Delta v=u-v,x\in \Omega ,t>0,\\ (D(u)\nabla u-\chi u\cdot \nabla v)\cdot \nu =\frac{\partial v}{\partial \nu }=0,x\in \partial \Omega ,t>0,\\ u(x,0)=u_0(x),v(x,0)=v_0(x),x\in \Omega ,\end{array}$$ where $\Omega \subset {\mathbb{R}}^N(N\ge 1)$ is a bounded domain with smooth boundary $\partial \Omega ,$ $\chi >0$ and $\mu \ge 0$. We prove that for nonnegative and suitably smooth initial data $(u_0,v_0)$, if $D(u)\ge C_D(u+1{)}^{m-1}$ for all $u\ge 0$ with some $C_D>0$ and some $m>2-\frac{2}{N}\frac{\chi \max \{1,{\lambda }_0\}}{[\chi \max \{1,{\lambda }_0\}-\mu {]}_{+}}$ or $m=2-\frac{2}{N}$ and $C_D>\frac{C_{GN}(1+\Vert u_0{\Vert }_{L^1(\Omega )})}{3}(2-\frac{2}{N}{)}^2\max \{1,{\lambda }_0\}\chi$, the $(KS)$ possesses a global classical solution which is bounded in $\Omega \times (0,\infty )$, where $C_{GN}$ and ${\lambda }_0$ are the constants which are corresponding to the Gagliardo--Nirenberg inequality and the maximal Sobolev regularity. To our best knowledge, this seems to be the first rigorous mathematical result which (precisely) gives the relationship between $m$ and $\frac{\mu }{\chi }$ that yields to the boundedness of the solutions.

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