International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 825951, 27 pages
http://dx.doi.org/10.1155/2011/825951
Research Article

On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator

Marat V. Markin

Department of Partial Differential Equations, Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchekivs'ka Street, 01601 Kiev, Ukraine

Received 11 December 2010; Accepted 15 February 2011

Academic Editor: Palle E. Jorgensen

Copyright © 2011 Marat V. Markin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For the evolution equation 𝑦 ( 𝑡 ) = 𝐴 𝑦 ( 𝑡 ) with a scalar type spectral operator 𝐴 in a Banach space, conditions on 𝐴 are found that are necessary and sufficient for all weak solutions of the equation on [ 0 , ∞ ) to be strongly infinite differentiable on [ 0 , ∞ ) or [ 0 , ∞ ) . Certain effects of smoothness improvement of the weak solutions are analyzed.