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.