Electron. J. Diff. Equ., Vol. 2013 (2013), No. 92, pp. 1-25.

Nonautonomous ill-posed evolution problems with strongly elliptic differential operators

Matthew A. Fury

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 $\theta \in (0,\pi/2]$ 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.

Show me the PDF file (402 KB), TEX file, and other files for this article.

Matthew Fury
Division of Science & Engineering, Penn State Abington
1600 Woodland Road
Abington, PA 19001, USA
Tel: 215-881-7553 Fax: 215-881-7333
email: maf44@psu.edu

Return to the EJDE web page