Abstract and Applied Analysis
Volume 2004 (2004), Issue 11, Pages 907-933

Convergence of functionals and its applications to parabolic equations

Goro Akagi

Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1 Okubo Shinjuku-ku, Tokyo 169-8555, Japan

Received 11 December 2003

Copyright © 2004 Goro Akagi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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|m2u as m+, is also given.