Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 92, pp. 1-30.
Title: Time-dependent domains for nonlinear evolution operators and
partial differential equations
Author: Chin-Yuan Lin (National Central Univ., Chung-Li, Taiwan)
Abstract:
This article concerns the nonlinear evolution equation
$$\displaylines{
\frac{du(t)}{dt} \in A(t)u(t), \quad 0 \leq s < t < T, \cr
u(s) = u_0
}$$
in a real Banach space X, where the nonlinear, time-dependent,
and multi-valued operator
$ A(t) : D(A(t)) \subset X \to X$
has a time-dependent domain D(A(t)).
It will be shown that, under certain assumptions on A(t),
the equation has a strong solution. Illustrations are
given of solving quasi-linear partial differential equations
of parabolic type with time-dependent boundary conditions.
Those partial differential equations are studied to a
large extent.
Submitted June 15, 2011. Published July 18, 2011.
Math Subject Classifications: 47B44, 47H20, 35J25, 35K20.
Key Words: Dissipative operators; evolution equations;
parabolic and elliptic equations.