Abstract and Applied Analysis
Volume 2004 (2004), Issue 11, Pages 907-933
doi:10.1155/S1085337504403030

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→+∞ 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, ∂φp(u)=−Δpu, where φp:L2(Ω)→[0,+∞]. To this end, the notion of Mosco convergence is employed and it is proved that φp converges to the indicator function over some closed convex set on L2(Ω) in the sense of Mosco as p→+∞; 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 ut=Δ|u|m−2u as m→+∞, is also given.