On the semilinear heat equation with the Grushin operator

TL;DR

This paper investigates the heat equation involving the Grushin operator, deriving its heat kernel, analyzing its properties in Lebesgue spaces, and applying these results to establish well-posedness and blowup behavior of nonlinear solutions.

Contribution

It provides explicit expressions for the heat kernel of the Grushin operator and applies these to study nonlinear PDE solutions, including existence, uniqueness, and blowup criteria.

Findings

Derived the heat kernel for the Grushin operator.

Proved decay estimates and approximation properties in Lebesgue spaces.

Established existence, uniqueness, and blowup criteria for nonlinear solutions.

Abstract

In this work, we study the heat equation with Grushin's operator. We present an expression for its heat kernel, prove its decay in $L^p$ spaces, and that it is an approximation of the identity. As a consequence, the heat semigroup associated with Grushin's operator using this heat kernel is estimated in Lebesgue spaces. Next, we use the results to prove the existence, uniqueness, continuous dependence, and blowup alternative of mild solutions of a nonlinear Cauchy problem associated with Grushin's operator. A global existence result is also presented.