论文标题
具有一般非本地时间和空间衍生物的扩散方程
Diffusion equations with general nonlocal time and space derivatives
论文作者
论文摘要
在本研究中，首先，基于连续时间随机步行（CTRW）理论，得出了一般扩散方程。时间衍生物被视为Kochubei引入的一般Caputo型衍生物和空间衍生物是通过从纸张中提出的一般分数拉普拉斯（Servadei and valdinoci，2012年）中删除条件（1.5）和（1.6）来定义的一般拉普拉斯式。其次，通过将一般laplacian的域扩展到一般的sobolev空间，证明了一般扩散方程的库奇问题的解决方案的存在。还获得了解决方案阳性和界限的结果。最后，通过使用书中引入的一般分数laplacian的Friedrichs扩展（Bisci，Radulescu和Servadei，2016年），建立了有界域上一般扩散方程的解决方案的存在结果。
In the present study, firstly, based on the continuous time random walk (CTRW) theory, general diffusion equations are derived. The time derivative is taken as the general Caputo-type derivative introduced by Kochubei and the spatial derivative is the general Laplacian defined by removing the conditions (1.5) and (1.6) from the definition of the general fractional Laplacian proposed in the paper (Servadei and Valdinoci, 2012). Secondly, the existence of solutions of the Cauchy problem for the general diffusion equation is proved by extending the domain of the general Laplacian to a general Sobolev space. The results for positivity and boundedness of the solutions are also obtained. In the last, the existence result for solutions of the initial boundary value problem (IBVP) for the general diffusion equation on a bounded domain is established by using the Friedrichs extension of the general fractional Laplacian introduced in the book (Bisci, Radulescu and Servadei, 2016).
