In this paper, we give a necessary condition for the Cauchy problem for parabolic equation in order to be uniquely solvable in the analytic class.
Hironobu HONDA. "On a necessary condition for analytic wellposedness of the Cauchy problem for parabolic equation." J. Math. Soc. Japan 56 (3) 907 - 922, July, 2004. https://doi.org/10.2969/jmsj/1191334091