Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 92, pp. 1-25.
Title: Nonautonomous ill-posed evolution problems
with strongly elliptic differential operators
Author: Matthew A. Fury (Penn State Abington, PA, USA)
Abstract:
In this article, we consider the nonautonomous evolution problem
$du/dt=a(t)Au(t), 0\leq s\leq t< T$ with initial condition $u(s)=\chi$
where -A generates a holomorphic semigroup of angle $\theta \in (0,\pi/2]$
on a Banach space X and $a\in C([0,T]:\mathbb{R}^+)$.
The problem is generally ill-posed under such conditions, and so we employ
methods to approximate known solutions of the problem. In particular,
we prove the existence of a family of regularizing operators for the
problem which stems from the solution of an approximate well-posed problem.
In fact, depending on whether $\theta \in (0,\pi/4]$ or
$\theta \in (\pi/4,\pi/2]$, we provide two separate approximations each
yielding a regularizing family. The theory has applications to ill-posed
partial differential equations in $L^p(\Omega)$, $1