Well-posedness of parabolic differential and difference equations

Ashyralyev A., Sozen Y.

COMPUTERS & MATHEMATICS WITH APPLICATIONS, vol.60, no.3, pp.792-802, 2010 (SCI-Expanded) identifier identifier


We consider the abstract parabolic differential equation u'(t) + Au(t) = f(t), -infinity < t < infinity in a Banach space E with -A the infinitesimal generator of an analytic, exponentially decreasing semigroup exp{-tA} (t >= 0). The main purpose of this paper is to establish the well-posedness of this equation in C-beta (R, E-alpha), (alpha, beta is an element of [0, 1]), and the well-posedness of the corresponding Rothe difference scheme in C-beta(R-iota, E-alpha), (alpha, beta is an element of [0, 1]). Moreover, we apply our theoretical results to obtain new coercivity inequalities for the solution of parabolic difference equations. Crown Copyright (C) 2010 Published by Elsevier Ltd. All rights reserved.