Using Mehler kernel, we give an example of nontrivial solution of the homogeneous Cauchy problem of the Hermite heat equation, which is, for each $t$, bounded in the space variables.
"An example of nonuniqueness of the Cauchy problem." Proc. Japan Acad. Ser. A Math. Sci. 81 (3) 37 - 39, March 2005. https://doi.org/10.3792/pjaa.81.37