Eighth Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 19 (2010), pp. 99-121.

Continuous dependence of solutions for ill-posed evolution problems

Matthew Fury, Rhonda J. Hughes

Abstract:
We prove Holder-continuous dependence results for the difference between certain ill-posed and well-posed evolution problems in a Hilbert space. Specifically, given a positive self-adjoint operator D in a Hilbert space, we consider the ill-posed evolution problem
$$\displaylines{ \frac{du(t)}{dt} = A(t,D)u(t) \quad 0\leq t<T \cr u(0) = \chi. }$$
We determine functions $f:[0,T]\times [0,\infty)\to \mathbb{R}$ for which solutions of the well-posed problem
$$\displaylines{ \frac{dv(t)}{dt} = f(t,D)v(t) \quad 0\leq t<T \cr v(0) = \chi }$$
approximate known solutions of the original ill-posed problem, thereby establishing continuous dependence on modelling for the problems under consideration.

Published September 25, 2010.
Math Subject Classifications: 47A52, 42C40.
Key Words: Continuous dependence on modelling; time-dependent problems; Ill-posed problems.

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Matthew Fury
Department of Mathematics, Bryn Mawr College
Bryn Mawr, PA 19010, USA
email: mfury@brynmawr.edu
Rhonda J. Hughes
Department of Mathematics, Bryn Mawr College
Bryn Mawr, PA 19010, USA
email: rhughes@brynmawr.edu

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