Convergence of functionals and its applications to parabolic equations

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