# Homotopy regularization for a high-order parabolic equation

## TL;DR

This paper investigates the solvability of a high-order degenerate parabolic equation using homotopy and regularization techniques, extending understanding from polyharmonic equations to more complex nonlinear cases.

## Contribution

It introduces a homotopy regularization method for analyzing solutions of a quasilinear degenerate high-order parabolic equation with nonlinear degeneracy.

## Key findings

- Established solvability results for the equation
- Developed an analytic regularization approach
- Extended insights from polyharmonic to nonlinear degenerate equations

## Abstract

In this work we study the solvability of the Cauchy Problem for a quasilinear degenerate high-order parabolic equation \begin{equation*} \left\{ \begin{tabular}{lcl} $u_t=(-1)^{m-1}\nabla\cdot(f^n(|u|)\nabla\Delta^{m-1}u)$ & &in $\mathbb{R}^N\times\mathbb{R}_+$, $u(x,0)=u_0(x)$& & in $\mathbb{R}^N$, \end{tabular} \right. \end{equation*} with $m\in\mathbb{N},\ m>1$ and $n>0$ a fixed exponent. Moreover, $f$ is a continuous monotone increasing positive bounded function with $f(0)=0$ and the initial data $u_0(x)$ is bounded smooth and compactly supported. Thus, through an homotopy argument based on an analytic $\varepsilon$-regularization of the degenerate term $f^n(|u|)$ we are able to extract information about the solutions inherited from the polyharmonic equation when $n=0$.