Solvability of the heat equation on a half-space with a dynamical boundary condition and unbounded initial data

We study the linear heat equation on a halfspace with a linear dynamical boundary condition. We are interested in an appropriate choice of the function space of initial functions such that the problem possesses a solution. It was known before that bounded initial data guarantee solvability. Here we extend that result by showing that data from a weighted Lebesgue space will also do so.