Geodesic
On analytic solutions of the heat equation with an operator coefficient
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 36, Tome 355 (2008), pp. 139-162

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Let $A$ be a bounded linear operator on a Banach space and $g$ a vector-valued function analytic on a neighborhood of the origin of $\mathbb R$. We obtain conditions for the existence of analytic solutions for the Cauchy problem $$ \begin{cases} \dfrac{\partial u}{\partial t}=A^2\dfrac{\partial^2u}{\partial x^2},\\u(0,x)=g(x). \end{cases} $$ Moreover, we consider a representation of the solution of this problem as a Poisson integral and investigate the Cauchy problem for the corresponding nonhomogeneous equation. Bibl. – 22 titles. 
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     author = {A. Vershynina and S. L. Gefter},
     title = {On analytic solutions of the heat equation with an operator coefficient},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {139--162},
     publisher = {mathdoc},
     volume = {355},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_355_a4/}
}
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A. Vershynina; S. L. Gefter. On analytic solutions of the heat equation with an operator coefficient. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 36, Tome 355 (2008), pp. 139-162. http://geodesic.mathdoc.fr/item/ZNSL_2008_355_a4/