# Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations

## TL;DR

This paper classifies the long-term behavior of solutions to viscous Hamilton-Jacobi equations, showing they asymptotically resemble specific self-similar solutions, providing a comprehensive understanding of their asymptotic profiles.

## Contribution

It offers a detailed classification of the asymptotic profiles of solutions to viscous Hamilton-Jacobi equations, identifying the roles of singular and viscosity self-similar solutions.

## Key findings

- Solutions asymptotically approach singular self-similar solutions
- Solutions also tend towards viscosity self-similar solutions
- The classification clarifies the long-term behavior of these equations

## Abstract

The large time behavior of solutions to Cauchy problem for viscous Hamilton-Jacobi equation is classified. The large time asymptotics are given by very singular self-similar solutions on one hand and by self-similar viscosity solutions on the other hand