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 $$\begin{aligned} \left\{ \begin{array}{ll} u_t=(-1)^{m-1}\nabla \cdot (f^n(|u|)\nabla \Delta ^{m-1}u) &{} \hbox { in }{\mathbb {R}}^N\times {\mathbb {R}}_+, u(x,0)=u_0(x)&{} \hbox { in }{\mathbb {R}}^N, \end{array} \right. \end{aligned}$$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 a 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 $$n=0$$.