论文标题
对于分数单相滞后热方程的弱解决方案的存在和唯一性
Existence and uniqueness of a weak solution to fractional single-phase-lag heat equation
论文作者
论文摘要
在本文中，我们研究了分数单相滞后热方程的弱解决方案的存在和独特性。 This model contains the terms $\cal{D}_t^α(u_t)$ and $\cal{D}_t^αu $ (with $α\in(0,1)$), where $\cal{D}_t^α$ denotes the Caputo fractional derivative in time of constant order $α\in(0,1)$.我们考虑温度的均匀Dirichlet边界数据。我们严格地表明在数据上的低规律性假设下存在独特的弱解决方案。我们的主要策略是根据Rothe的方法在及时使用变分的配方和半散制。我们获得了关于离散解决方案的先验估计，并显示了Rothe函数与弱解的收敛性。使用变分的方法来表明这种弱解决方案对问题的独特性。我们还考虑了一维问题，并得出了解决方案的表示公式。我们通过扩展多项式Mittag-Leffler函数的属性来在此显式解决方案及其时间导数上建立界限。
In this article, we study the existence and uniqueness of a weak solution to the fractional single-phase lag heat equation. This model contains the terms $\cal{D}_t^α(u_t)$ and $\cal{D}_t^αu $ (with $α\in(0,1)$), where $\cal{D}_t^α$ denotes the Caputo fractional derivative in time of constant order $α\in(0,1)$. We consider homogeneous Dirichlet boundary data for the temperature. We rigorously show the existence of a unique weak solution under low regularity assumptions on the data. Our main strategy is to use the variational formulation and a semidiscretisation in time based on Rothe's method. We obtain a priori estimates on the discrete solutions and show convergence of the Rothe functions to a weak solution. The variational approach is employed to show the uniqueness of this weak solution to the problem. We also consider the one-dimensional problem and derive a representation formula for the solution. We establish bounds on this explicit solution and its time derivative by extending properties of the multinomial Mittag-Leffler function.
