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The current paper considers the boundedness of solutions to the followingquasilinear Keller-Segel model (with logistic source)$$\left\{\begin{array}{ll} u_t = \nabla\cdot(D(u)\nablau)-\chi\nabla\cdot(u\nabla v)+\mu (u-u^2),\quad x\in \Omega, t>0,\\ v_t-\Deltav = u-v,\quad x\in \Omega, t>0,\\ (D(u)\nabla u-\chi u\cdot \nabla v)\cdot \nu= \frac{\partial v}{\partial\nu}=0,\quad x\in \partial\Omega, t>0,\\ u(x,0) =u_0(x),\quad v(x,0) = v_0(x),\ \ x\in \Omega, \end{array}\right.$$ where$\Omega\subset\mathbb{R}^N(N\geq1)$ is a bounded domain with smooth boundary$\partial\Omega,$ $\chi>0$ and $\mu\geq0$. We prove that for nonnegative andsuitably smooth initial data $(u_0, v_0)$, if $D(u)\geq C_{D}(u+1)^{m-1}$ forall $u\geq 0$ 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$C_D>\frac{C_{GN}(1+\|u_0\|_{L^1(\Omega)})}{3}(2-\frac{2}{N})^2\max\{1,\lambda_0\}\chi$,the $(KS)$ possesses a global classical solution which is bounded in$\Omega\times (0,\infty)$, where $C_{GN}$ and $\lambda_0$ are the constantswhich are corresponding to the Gagliardo--Nirenberg inequality and the maximalSobolev regularity. To our best knowledge, this seems to be the first rigorousmathematical result which (precisely) gives the relationship between $m$ and$\frac{\mu}{\chi}$ that yields to the boundedness of the solutions.

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