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arXiv 1903.09552math.AP
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一类高阶抛物型方程的同伦正则化

Homotopy regularization for a high-order parabolic equation

Pablo Álvarez-Caudevilla, Alejandro Ortega

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摘要：

本文研究一类拟线性退化高阶抛物型方程\begin{equation*} \left\{ \begin{tabular}{lcl} $u_t=(-1)^{m-1}\nabla\cdot(f^n(|u|)\nablaΔ^{m-1}u)$ & &in $\mathbb{R}^N\times\mathbb{R}_+$, $u(x,0)=u_0(x)$& & in $\mathbb{R}^N$, \end{tabular} \right. \end{equation*}的Cauchy问题的可解性，其中$m\in\mathbb{N},\ m>1$和$n>0$是固定指数.此外，$f$是一个连续单调递增的有界函数，其初始值$u_0(x)$是有界光滑的，且具有$f(0)=0$.因此，通过一个同伦参数的基础上的分析$\varepsilon$ -正则化的退化项$f^n(|u|)$，我们能够提取信息的解决方案继承的多调和方程时，$n=0$。

英文摘要：

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} $u_t=(-1)^{m-1}\nabla\cdot(f^n(|u|)\nablaΔ^{m-1}u)$ & &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 $n=0$.

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