DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

SUBDIFFERENTIAL OPERATOR APPROACH TO THE
DIRICHLET PROBLEM OF NONLINEAR DEGENERATE
PARABOLIC EQUATIONS
A. ITO, M. KUBO, AND Q. LU
We define a convex function on H −1 (Ω) whose subdifferential generates an evolution
equation for the Dirichlet problem of a nonlinear parabolic equation associated with an
arbitrary maximal monotone graph. Some applications to the Penrose-Fife phase transition
model are given.
Copyright © 2006 A. Ito et al. 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.
1. Introduction
This paper is concerned with the following initial-boundary value problem for a nonlinear
parabolic partial differential equation.
Problem 1.1.
u t − Δv = f (t,x), v ∈ α(u)in(0,T) × Ω,
v = h(x) on(0,T) × ∂Ω,
u(0,x) = u 0 (x) inΩ,
(1.1)
where Ω ⊂ R N (N ≥ 1) is a bounded domain, f (t,x) is a given function in (0,T) × Ω,
h(x) is a given boundary value on ∂Ω, u 0 is a given initial value in Ω,andα is a maximal
monotone graph in R × R.
The present paper aims to formulate Problem 1.1 in a form of the Cauchy problem of
an evolution equation as follows.
Problem 1.2.
u ′ (t)+∂ϕ ( u(t) ) ∋ f ∗ (t),
u(0) = u 0 .
0