  We consider the following chemotaxis model %fully parabolic Keller-Segel
system with logistic source $$ \left\{\begin{array}{ll}
u_t=\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)+\mu (u-u^2),\quad x\in
\Omega, t>0, \disp{v_t-\Delta v=-uv },\quad x\in \Omega, t>0, %\disp{\tau
w_t+\delta w=u },\quad %x\in \Omega, t>0, \disp{(\nabla D(u)-\chi u\cdot \nabla
v)\cdot \nu=\frac{\partial v}{\partial\nu}=0},\quad x\in \partial\Omega, t>0,
\disp{u(x,0)=u_0(x)},\quad v(x,0)=v_0(x),~~ x\in \Omega \end{array}\right. $$
on a bounded domain $\Omega\subset\mathbb{R}^N(N\geq1)$, with smooth boundary
$\partial\Omega, \chi$ and $\mu$ are positive constants. Besides appropriate
smoothness assumptions, in this paper it is only required that $D(u)\geq
C_{D}(u+1)^{m-1}$ for all $u\geq 0$ with some $C_{D} > 0$ and some $$
m>\left\{\begin{array}{ll}
1-\frac{\mu}{\chi[1+\lambda_{0}\|v_0\|_{L^\infty(\Omega)}2^{3}]}~~\mbox{if}~~
N\leq2, % >1+\frac{(N+2-2r)^+}{N+2}~~~~~~\mbox{if}~~ % \frac{N+2}{2}\geq
r\geq\frac{N+2}{N}, 1~~~~~~\mbox{if}~~ N\geq3, \end{array}\right. $$ then for
any sufficiently smooth initial data there exists a classical solution which is
global in time and bounded, where $\lambda_{0}$ is a positive constant which is
corresponding to the maximal sobolev regularity. The results of this paper
extends the results of Jin (J. Diff. Eqns., 263(9)(2017), 5759-5772), who
proved the possibility of boundness of weak solutions, in the case $m>1$ and
$N=3$.
