Abstract: In this paper, we consider the fully parabolic chemotaxis system with the general logistic source
$\begin{eqnarray*}\left\{\begin{array}{llll}u_t= \Delta(\gamma(v) u )+\lambda u-\mu u^{\kappa},~~~ &x \in \Omega, ~t>0,\\ v_t= \Delta v+wz, &x \in \Omega, ~t>0,\\w_t= -wz, &x \in \Omega, ~t>0,\\z_t= \Delta z - z+ u, &x\in \Omega, ~t>0,\\\end{array}\right.\end{eqnarray*}$
where $\Omega\subset \mathbb{R}^n (n\geq 1)$ is a smooth and bounded domain, $\lambda\geq 0, \mu\geq 0, \kappa>1$, and the motility function satisfies that $\gamma(v)\in C^3([0, \infty))$, $\gamma(v)>0$, $\gamma{'}(v)\leq0$ for all $v\geq 0$. Considering the Neumann boundary condition, we obtain the global boundedness of solutions if one of the following conditions holds: (i) $ \lambda=\mu=0, 1\leq n\leq 3; $(ii) $ \lambda> 0, \mu>0, ~\text{combined with}~ \kappa>1, 1\leq n\leq 3 ~~\text{or}~~\kappa>\frac{n+2}{4}, n>3. $ Moreover,we prove that the solution $(u, v, w, z)$ exponentially converges to the constant steady state $\left(\left(\frac{\lambda}{\mu}\right)^{\frac{1}{\kappa-1}}, \frac{\int_{\Omega}v_0{\rm d}x+\int_{\Omega}w_0{\rm d}x}{|\Omega|}, 0, \left(\frac{\lambda}{\mu}\right)^{\frac{1}{\kappa-1}}\right)$.

Key words: chemotaxis, signal-dependent motility, logistic source, boundedness, asymptotic behavior