Asymptotic behavior of solutions of some parabolic equation associated with the $p$ -Laplacian as $p \rightarrow +\infty$ is studied for the periodic problem as well as the initial-boundary value problem by pointing out the variational structure of the $p$ -Laplacian, that is, $\partial \varphi_p(u)= - \Delta_p u$ , where $\varphi_p:L^2(\Omega) \rightarrow [0,+\infty]$ . To this end, the notion of Mosco convergence is employed and it is proved that $\varphi_p$ converges to the indicator function over some closed convex set on $L^2(\Omega)$ in the sense of Mosco as $p \rightarrow +\infty$ ; moreover, an abstract theory relative to Mosco convergence and evolution equations governed by time-dependent subdifferentials is developed until the periodic problem falls within its scope. Further application of this approach to the limiting problem of porous-medium-type equations, such as $u_t = \Delta |u|^{m-2}u$ as $m \rightarrow +\infty$ , is also given.