Analytic smoothing effect of the Cauchy problem for a class of ultra-parabolic equations

TL;DR

This paper proves that solutions to a class of strongly degenerate ultraparabolic equations with analytic coefficients become analytic in space for any positive time, demonstrating an analytic smoothing effect similar to uniformly parabolic equations.

Contribution

It establishes the analytic smoothing effect for a class of strongly degenerate ultraparabolic equations with analytic coefficients, extending known results to degenerate cases.

Findings

Solutions become analytic in all spatial variables for positive time

Unique solutions exist for initial data in Sobolev spaces

Smoothing effect parallels that of uniformly parabolic equations

Abstract

In this paper, we study a class of strongly degenerate ultraparabolic equations with analytic coefficients. We demonstrate that the Cauchy problem exhibits an analytic smoothing effect. This means that, with an initial datum belonging to the Sobolev space $H^s$ (of real index s), the associated Cauchy problem admits a unique solution that is analytic in all spatial variables for any strictly positive time. This smoothing effect property is similar to that of the Cauchy problem for uniformly parabolic equations with analytic coefficients.