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In this note we consider the nonlinear heat equation associated to thefractional Hermite operator $H^\beta =(-\Delta+|x|^2)^\beta$, $0<\beta\leq 1$.We show the local solvability of the related Cauchy problem in the framework ofmodulation spaces. The result is obtained by combining tools from microlocaland time-frequency analysis. As a byproduct, we compute the Gabor matrix ofpseudodifferential operators with symbols in the H\"ormander class $S^m_{0,0}$,$m\in\mathcal{R}$.

On the local well-posedness of the nonlinear heat equation associated to the fractional Hermite operator in modulation spaces