论文标题
$ p $ -laplacian的双线非线性抛物线方程的可溶性
Solvability of Doubly Nonlinear Parabolic Equation with $p$-Laplacian
论文作者
论文摘要
在本文中，我们考虑了一个双重非线性抛物线方程$ \ partial_tβ（u） - \ nabla \cdotα（x，x，\ nabla u）\ ni f $，均匀dirichlet dirichlet边界条件在有界的域中，其中$β：\ bbb {r} $ ismimim is a p}在β（0）$中满足$ 0 \的单调图和$ \ nabla \cdotα（x，\ nabla u）$代表着广义的$ p $ -laplacian。对于单个价值，强制性或某些生长条件施加在$β$上的情况下，已经研究了该方程初始边界价值问题的解决方案。但是，在删除此类假设的情况下，有一些结果很难构建一个抽象理论，该理论涵盖了$ 1 <p <2 $的情况。本文的主要目的是显示任何$ p \ in（1，\ infty）$的初始边界价值问题的解决性，而没有任何条件的$β$，除了$ 0 \ inβ（0）$。我们还使用熵解决方案的性质讨论了解决方案的唯一性。
In this paper, we consider a doubly nonlinear parabolic equation $ \partial _t β(u) - \nabla \cdot α(x , \nabla u) \ni f$ with the homogeneous Dirichlet boundary condition in a bounded domain, where $β: \mathbb{R} \to 2 ^{ \mathbb{R} }$ is a maximal monotone graph satisfying $0 \in β(0)$ and $ \nabla \cdot α(x , \nabla u )$ stands for a generalized $p$-Laplacian. Existence of solution to the initial boundary value problem of this equation has been investigated in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on $β$. However, there are a few results for the case where such assumptions are removed and it is difficult to construct an abstract theory which covers the case for $1 < p < 2$. Main purpose of this paper is to show the solvability of the initial boundary value problem for any $ p \in (1, \infty ) $ without any conditions for $β$ except $0 \in β(0)$. We also discuss the uniqueness of solution by using properties of entropy solution.
