On solutions of parabolic and elliptic type differential equations on (-∞, ∞) in a Banach space - Gorbachuk, V. M. (Methods of Functional Analysis and Topology, 2008)
We show that every classical solution of a parabolic or elliptic type homogeneous differential equation on (-∞, ∞) in a Banach space may be extended to an entire vector-valued function. The description of all the solutions is given, and necessary and sufficient conditions for a solution to be continued to a finite order and finite type entire vector-valued function are presented.