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 $L^p(\Omega)$, $1<p<\infty$ where A
 is a strongly elliptic differential operator and $\Omega$ is a fixed
 domain in $\mathbb{R}^n$.


Submitted November 3, 2012. Published April 11, 2013.
Math Subject Classifications: 46B99, 47D06.
Key Words: Regularizing family of operators; ill-posed evolution equation;
           holomorphic semigroup; strongly elliptic operator.