Homotopy regularization for a high-order parabolic equation

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 $\epsilon$-regularization of the degenerate term $f^n(|u|)$ we are able to extract information about the solutions inherited from the polyharmonic equation when $n=0$.