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 $S^m_{0,0}$, $m\in\mathcal{R}$.