How to check if a system is uniformly globally asymptotically stable We have the system:
\begin{align}
\dot x_1 &= -x_1 - e^{-2t} x_2\\
\dot x_2 &= x_1 - x_2
\end{align}
We are interested in whether it is stable or not and its type of stability. 
Assume the Lyapunov candidate:
$$
V = \frac{x_1^2}{2} + \frac{x_2^2}{4} e^{-2t}
$$
We get:
$$
\dot V = -2~ x_2^2 ~e^{-2t}-x_1^2 < 0,~ ~\forall~ (x_1, x_2) \in R^2 / (0,0)
$$
Does that prove that the system is GUAS?
I am a little concerned with my Lyapunov candidate.
 A: The condition on Lyapunov function is that it should to be positive definite and decresent. Your Lyapunov candidate meets that.
Condition for global uniform asymptotic stability is that $-\dot{V}(\cdot)$ should be positive definite. That is also met for the given system except at $t=\infty$. At $t=\infty$,  $-\dot{V}(\cdot)$  is positive semidefinite.
If the term, $e^{-2t}$ is the concern, that term is just a function of time and as $t\to\infty$, the magnitude of that term fades to zero. Hence the conclusion that the system is GUAS seems sensible to me.
However, you may need to check (and show) whether the equilibrium point does not loose stability to ensure uniform stability.
Lasalle's theorem can be used to check stability when $-\dot{V}(\cdot)$  is positive semidefinite. You need to show that $-\dot{V}(\cdot)$ is positive semi-definite only when $x_2$ is zero and is Positive definite elsewhere. However, as this lecture note says, Lasalle's theorem requires system to be time invariant. But this system is time dependent. Barbalat's lemma can be considered here, which requires $\dot{V}(\cdot)$ to be uniformly continuous. In your case we can say $\dot{V}(\cdot)$ is uniformly continuous and bounded with which we can say that the system is GUAS. Please mention here if you later figure out this is not he case.
