Nonautonomous ill-posed evolution problems with strongly elliptic differential operators
Matthew A. Fury · DOAJ (DOAJ: Directory of Open Access Journals) · 2013
In this article, we consider the nonautonomous evolution problem $du/dt=a(t)Au(t), 0leq sleq t< T$ with initial condition $u(s)=chi$ where -A generates a holomorphic semigroup of angle $heta in (0,pi/2]$ on a Banach space X and $ain C([0,T]:mathbb{R}^+)$. 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 $heta in (0,pi/4]$ or $heta in (pi/4,pi/2]$, 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