A new result for global existence and boundedness of solutions to a parabolic--parabolic Keller--Segel system with logistic source

We consider the following fully parabolic Keller–Segel system with logistic source $$\{\begin{array}{l}{u}_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (u\mathrm{\nabla }v)+au-\mu u^2,x\in \mathrm{\Omega },t>0,\text{disp}{v}_t=\mathrm{\Delta }v-v+u,x\in \mathrm{\Omega },t>0,\end{array}\text{eqno}(KS)$$ over a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N(N\ge 1)$, with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$, the parameters $a\in \mathbb{R},\mu >0,\chi >0$. It is proved that if $\mu >0$, then $(KS)$ admits a global weak solution, while if $\mu>\frac{(N-2){+}}{N}\chi C^{\frac{1}{\frac{N}{2}+1}}{\frac{N}{2}+1}$,then$(KS)$ possesses a global classical solution which is bounded, where $C^{\frac{1}{\frac{N} {2}+1}}_{\frac{N}{2}+1}$ is a positive constant which is corresponding to the maximal Sobolev regularity. Apart from this, we also show that if $a=0$ and $\mu >\frac{(N-2{)}_{+}}{N}\chi C_{\frac{N}{2}+1}^{\frac{1}{\frac{N}{2}+1}}$, then both $u(\cdot ,t)$ and $v(\cdot ,t)$ decay to zero with respect to the norm in $L^{\mathrm{\infty }}(\mathrm{\Omega })$ as $t\to \mathrm{\infty }$.