Summary: Asymptotic behavior of solutions of some parabolic equation associated with the \(p\)-Laplacian as \(p\to+\infty\) 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, \(\partial\varphi_p(u)=-\Delta_pu\), where \(\varphi_p:L^2(\Omega)\to[0, +\infty]\). 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 \(p \to+\infty\); 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 \(m \to+\infty\), is also given.