Convergence of functionals and its applications to parabolic equations

Abstract

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, $\partial {\varphi }_p\left(u\right)=-{\Delta }_{p}u$, where ${\varphi }_p:L^2\left(\Omega \right)\to \left[0,+\infty \right]$. 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 $L^2\left(\Omega \right)$ 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 $u_t=\Delta {\left|u\right|}^{m-2}u$ as $m\to +\infty$, is also given.