  This work is devoted to the study of the asymptotic behavior of nonautonomous
reaction-diffusion equations in Dumbbell domains $\Omega_{\varepsilon} \subset
\mathbb{R}^{N}$. Each $\Omega_{\varepsilon}$ is the union of a fixed open set
$\Omega$ and a channel $R_{\varepsilon}$ that collapses to a line segment $R_0$
as $\varepsilon \rightarrow 0^{+}$. We first establish the global existence of
solution for each problem by using two properties of the parabolic equation
considered, which are the positivity of the solutions and comparison results
for them. We prove the existence of pullback and uniform attractors and we
obtain uniform bounds (in $\varepsilon$) for them.
