Matthew A. Fury
Abstract:
In this article, we consider the nonautonomous evolution problem
with initial condition
where -A generates a holomorphic semigroup of angle
on a Banach space X and
.
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
or
,
we provide two separate approximations each
yielding a regularizing family. The theory has applications to ill-posed
partial differential equations in
,
where A is a strongly elliptic differential operator and
is a fixed
domain in
.
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.
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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 |
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