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

Title: Continuous dependence of solutions for ill-posed evolution problems

Authors: Matthew Fury     (Bryn Mawr College, PA, USA)
         Rhonda J. Hughes (Bryn Mawr College, PA, USA)
 
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.