带有分数阶Laplace算子的偏微分方程解的存在性研究进展

RESEARCH PROGRESS ON THE EXISTENCE OF SOLUTIONS FOR SOME PARTIAL DIFFERENTIAL EQUATIONS WITH FRACTIONAL LAPLACE OPERATOR

带有分数阶Laplace算子的偏微分方程是一类典型的分数阶偏微分方程,它在科学及工程领域有着重要的应用.分数阶Laplace算子是一类非局部拟微分算子,是Lévy稳态过程的无穷小生成元,它与经典的Laplace算子有着本质的区别,从而导致一些经典性质的消失,这就给此类问题的研究带来困难.一般来说,求解带有分数阶Laplace算子的偏微分方程的显式解是十分困难的.因此,研究带有分数阶Laplace算子的偏微分方程解的存在性是一项具有现实意义但又具有挑战性的工作.本文主要简述几类带有分数阶Laplace算子的偏微分方程解的存在性的研究进展与动态,其中也包括了作者近年来在这一领域所做的部分工作.

The partial differential equation with fractional Laplace operator is a typical fractional partial differential equation,which has important applications in science and engineering.The fractional Laplace operator is a nonlocal operator and can be viewed as the infinitesimal generator of Lévy stable diffusion processes.However,due to the fact that the fractional Laplace operator is nonlocal,there are important differences between the classical and the fractional Laplace operator.In general,it is very difficult to obtain the apparent solution to the partial differential equations with fractional Laplace operator.This fact makes it necessary and significant to study the existence of solutions to the partial differential equations with fractional Laplace operator.This paper briefly describes the progress and dynamics of the existence of solutions for some partial differential equations with fractional Laplace operator.It also includes the authors′recent work in this field.