论文标题
有界域上奇异非局部扩散的Cauchy-Dirichlet问题
The Cauchy-Dirichlet Problem for Singular Nonlocal Diffusions on Bounded Domains
论文作者
论文摘要
我们研究了$ \ \ partial_t u = - \ partial_t u = - \ mathcal {l} u^m $的非线性和非局部扩散方程的同质cauchy-dirichlet问题（CDP），该方程是在有界的euclidean uclidean romain $ω\ subset \ subset \ subset \ subseet \ mathbb {r r}^r and $ n $和$ n $ n $和$ n $ n $和$ n $中。线性扩散运算符$ \ MATHCAL {l} $是次马多维亚运算符，允许非局部类型，而非线性为单数类型，即$ u^m = | U | U |^{m-1} u $，$ 0 <m <m <m <1 $。原型方程是分数快速扩散方程（FFDE），当$ \ Mathcal {l} $是$ω$上的三个可能的dirichlet分数laplacians之一时。 我们的主要结果应为解决方案提供完整的基本理论：到目前为止，对于非负和签名解决方案，迄今为止已知的最大数据类别的存在和唯一性；尖锐的平滑估计：除了经典的$ l^p-l^\ infty $平滑效果外，我们还提供了新的加权估计，这在经过良好研究的本地情况下也代表了新颖性，即用于FDE $ u_t =Δu^m $的解决方案。我们比较了证明平滑效果的两种策略：Moser迭代与绿色功能方法。 由于奇异的非线性和非局部扩散算子的存在，解决方案如何满足横向边界条件的问题是微妙的。我们以定量的上边界估计来回答，以显示如何获取边界数据。 解决方案存在并受到界限后，我们表明它们在有限的时间内熄灭，并且我们提供了灭绝时间的上和较低估计，以及在不同规范中明确的尖锐灭绝率。 本文的方法是建设性的，从某种意义上说，估计中涉及的所有相关常数都是可计算的。
We study the homogeneous Cauchy-Dirichlet Problem (CDP) for a nonlinear and nonlocal diffusion equation of singular type of the form $\partial_t u =-\mathcal{L} u^m$ posed on a bounded Euclidean domain $Ω\subset\mathbb{R}^N$ with smooth boundary and $N\ge 1$. The linear diffusion operator $\mathcal{L}$ is a sub-Markovian operator, allowed to be of nonlocal type, while the nonlinearity is of singular type, namely $u^m=|u|^{m-1}u$ with $0<m<1$. The prototype equation is the Fractional Fast Diffusion Equation (FFDE), when $\mathcal{L}$ is one of the three possible Dirichlet Fractional Laplacians on $Ω$. Our main results shall provide a complete basic theory for solutions to (CDP): existence and uniqueness in the biggest class of data known so far, both for nonnegative and signed solutions; sharp smoothing estimates: besides the classical $L^p-L^\infty$ smoothing effects, we provide new weighted estimates, which represent a novelty also in well studied local case, i.e. for solutions to the FDE $u_t=Δu^m$. We compare two strategies to prove smoothing effects: Moser iteration VS Green function method. Due to the singular nonlinearity and to presence of nonlocal diffusion operators, the question of how solutions satisfy the lateral boundary conditions is delicate. We answer with quantitative upper boundary estimates that show how boundary data are taken. Once solutions exists and are bounded we show that they extinguish in finite time and we provide upper and lower estimates for the extinction time, together with explicit sharp extinction rates in different norms. The methods of this paper are constructive, in the sense that all the relevant constants involved in the estimates are computable.
