Phase space analysis of semilinear parabolic equations

TL;DR

This paper develops a wave packet analysis for degenerate parabolic equations with variable coefficients, establishing local well-posedness in low-regularity modulation spaces and analyzing phase space flow effects.

Contribution

It introduces a novel wave packet approach to study degenerate parabolic equations, proving well-posedness in modulation spaces and examining phase space flow behavior.

Findings

Proves local well-posedness in modulation spaces for certain parabolic equations.

Shows phase space flow decreases the global wave front set.

Analyzes the action on spaces of analytic functions with analytic coefficients.

Abstract

We present a wave packet analysis of a class of possibly degenerate parabolic equations with variable coefficients. As a consequence, we prove local wellposedness of the corresponding Cauchy problem in spaces of low regularity, namely the modulation spaces, assuming a nonlinearity of analytic type. As another application, we deduce that the corresponding phase space flow decreases the global wave front set. We also consider the action on spaces of analytic functions, provided the coefficients are analytic themselves.