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

Abstract: In this note we consider the nonlinear heat equation associated to the fractional 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 of modulation spaces. The result is obtained by combining tools from microlocal and time-frequency analysis. As a byproduct, we compute the Gabor matrix of pseudodifferential operators with symbols in the H\"ormander class $S^m_{0,0}$, $m\in\mathcal{R}$.