Does the following hold?

Let $x=0$ be an equilibrium point for the system $\dot x(t)=f(x(t))$ and suppose the existence and uniqueness conditions of solutions on $[t_0, +\infty)$ are satisfied. If $x=0$ is locally asymptotically stable in $\mathcal{D}\subset \mathbb{R}^n$, can one conclude that, for any given signal $\tilde x(t)$ satisfying $\lim_{t\rightarrow \infty}\tilde x(t)=0$, the trajectory of the differential system $\dot x=f(x)+\tilde x(t)$ with initial state in $\mathcal{D}$ satisfies $\lim_{t\rightarrow \infty} x(t)=0$?

If not, what further conditions should be imposed?

Any help would be greatly appreciated.