On the Heat equation involving a Grushin operator in Marcinkiewcz spaces

TL;DR

This paper investigates the heat equation with a Grushin operator in Marcinkiewicz spaces, establishing conditions for existence, uniqueness, and special solutions such as positive, symmetric, and self-similar ones.

Contribution

It provides new sufficient conditions for the existence and uniqueness of solutions to the heat equation involving a Grushin operator in Marcinkiewicz spaces.

Findings

Established conditions for existence and uniqueness of solutions.

Identified conditions for positive, symmetric, and self-similar solutions.

Extended analysis of heat equations with degenerate operators in Marcinkiewicz spaces.

Abstract

In this work, we give sufficient conditions for the existence and uniqueness of the heat equation involving the operator $$ \Delta_{\mathcal{G}}=\dfrac{1}{2}\left(\Delta_{x}+|x|^2\Delta_{y}\right) $$ in Marcinkiewicz spaces. Furthermore, we provide sufficient conditions for the existence of positive, symmetric, and self-similar solutions.