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

TL;DR

This paper proves local well-posedness for a nonlinear heat equation linked to the fractional Hermite operator within modulation spaces, utilizing microlocal and time-frequency analysis techniques.

Contribution

It establishes the local solvability of the nonlinear heat equation associated with the fractional Hermite operator in modulation spaces, combining microlocal and time-frequency analysis methods.

Findings

Proves local well-posedness in modulation spaces.

Computes the Gabor matrix of pseudodifferential operators.

Integrates microlocal analysis with fractional Hermite operators.

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