# Blowup solutions of Grushin's operator

## TL;DR

This paper investigates the blowup behavior of solutions to Grushin's operator by deriving the heat kernel and demonstrating finite-time blowup using probabilistic methods and auxiliary functions.

## Contribution

It provides a new probabilistic approach to analyze blowup solutions of Grushin's operator and explicitly derives the heat kernel expression.

## Key findings

- Solutions blow up in finite time
- Heat kernel expression derived explicitly
- Probabilistic methods applied to PDE blowup analysis

## Abstract

In this note, we consider the blowup phenomenon of Grushin's operator. By using the knowledge of probability, we first get expression of heat kernel of Grushin's operator. Then by using the properties of heat kernel and suitable auxiliary function, we get that the solutions will blow up in finite time.