In this paper, we investigate weighted asymptotic behavior of solutions to the Sobolev-type differential equation \[ \frac{d}{dt}[u(t)+f(t,u(t))]=A(t)u(t)+g(t,u(t)),\quad tı \cal R, \] where $A(t):D\subseteq\Bbb X\to \Bbb X$ for $t\in\Bbb R$ is a family of densely defined closed linear operator on a domain $D$, independent of $t$, and $f:\Bbb R\times\Bbb X\to \Bbb X$ is a weighted pseudo almost automorphic function and $g:\Bbb R\times\Bbb X\to \Bbb X$ is an $S^p$-weighted pseudo almost automorphic function and satisfying suitable conditions. Some sufficient conditions are established by properties of $S^p$-weighted pseudo almost automorphic functions combined with theories of asymptotically stable of operators.