Lipschitz continuous solutions of some doubly nonlinear parabolic equations
Lipschitz continuous solutions of some doubly nonlinear parabolic equations
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一些双非线性抛物型方程的 Lipschitz 连续解
DOI：
10.3934/dcds.2002.8.647
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发表时间：
2002
影响因子：
1.1
通讯作者：
Y. Sugiyama
中科院分区：
文献类型：
--
作者：
M. Otani;Y. Sugiyama
本文研究了两类非线性抛物型方程，
这是由非牛顿流体的非线性过滤问题引起的。这些
作为特殊情况，方程包括多孔介质方程$u_t =$ div $(u^l\nabla u)$
和p-拉普拉斯方程$u_t =$ div $(|\nabla u|^{p-2}\nabla u)$。因为
对于由$u^l$和$|\nabla u|^{p-2}$两项引起的简并或奇点，我们不能
期望这些方程的整体（在时间上）经典解的存在，除了
对于特殊情况。因此，大多数作品都致力于对弱者的研究
解决方案。本文的主要目的是探讨多的存在性
更常规（不一定是全局）的解决方案。的局部解的存在性
在假设初始数据是非负函数的情况下，确保$W^{1,\infty}(\Omega)$
在$W_0^{1,\infty}(\Omega)$中，表示域边界$\partial \Omega$的平均曲率
$\Omega$为非正数。本文介绍了一种新的方法“$L^\infty$ -能量法”
为我们的参数提供了一个主要工具，在其他情况下也很有用。
This paper is concerned with two types of nonlinear parabolic equations,
which arise from the nonlinear filtration problems for non-Newtonian fluids. These
equations include as special cases the porous medium equations $u_t =$ div$(u^l\nabla u)$
and the evolution equation governed by p-Laplacian $u_t =$ div $(|\nabla u|^{p-2}\nabla u)$. Because
of the degeneracy or singularity caused by the terms $u^l$ and $|\nabla u|^{p-2}$, one can not
expect the existence of global (in time) classical solutions for these equations except
for special cases. Therefore most of works have been devoted to the study of weak
solutions. The main purpose of this paper is to investigate the existence of much
more regular (not necessarily global) solutions. The existence of local solutions in
$W^{1,\infty}(\Omega)$ is assured under the assumption that initial data are non-negative functions
in $W_0^{1,\infty}(\Omega)$, and that the mean curvature of the boundary $\partial \Omega$ of the domain
$\Omega$ is non-positive. We here introduce a new method "$L^\infty$-energy method", which
provides a main tool for our arguments and would be useful for other situations.