Optimal Decay Estimates for Time-Fractional and Other NonLocal Subdiffusion Equations via Energy Methods
Optimal Decay Estimates for Time-Fractional and Other NonLocal Subdiffusion Equations via Energy Methods
复制标题
DOI：
10.1137/130941900
复制
发表时间：
2013-10
期刊：
影响因子：
--
通讯作者：
Vicente Vergara;Rico Zacher
中科院分区：
文献类型：
--
作者：
Vicente Vergara;Rico Zacher
我们证明了尖锐的估计衰减的时间的解决方案，而一般类的非局部的时间subdiffusion方程的有界区域受到齐次Dirichlet边界条件。重要的特殊情况是时间分数阶和超慢扩散方程，在过去的几年里，这已经看到了很多的兴趣，主要是由于它们在异常扩散建模的应用。我们研究的情况下，方程的发散形式与有界可测系数。我们的证明依赖于能量估计，并利用一个新的和强大的不等式的形式$\partial_t（k\ast\cdot）$的积分微分算子。所得结果可推广到某些拟线性方程。我们说明了这一点，通过看时间分数$p $-拉普拉斯和多孔介质方程。在这里，事实证明，衰减行为是显着不同的，在经典的抛物线的情况下。
We prove sharp estimates for the decay in time of solutions to a rather general class of nonlocal in time subdiffusion equations on a bounded domain subject to a homogeneous Dirichlet boundary condition. Important special cases are the time-fractional and ultraslow diffusion equation, which have seen much interest during the last years, mostly due to their applications in the modeling of anomalous diffusion. We study the case where the equation is in divergence form with bounded measurable coefficients. Our proofs rely on energy estimates and make use of a new and powerful inequality for integro-differential operators of the form $\partial_t (k\ast \cdot)$. The results can be generalized to certain quasilinear equations. We illustrate this by looking at the time-fractional $p$-Laplace and porous medium equation. Here, it turns out that the decay behavior is markedly different from that in the classical parabolic case.