The Stability Of Solutions For A Nonlocal Equaiton With Laplacian Operator

In many realistic problems, such as stock prices, stochastic analysis, the anomalousdiffusion of particles in fluids etc, people usually use nonlocal equations to describe themodels of the problems’ dynamics behavior. We consider the nonlocal operator, becauseof its obvious theory and application background. From the physics, the nonlocal operatorpresents a anomalous diffusion; from the math, it is produced by Levy motion. So, wecould naturally think it as the extend of classical Laplacian operator. But, from the recentpapers, the nonlocal operator has its unique properties, and the properties are not the simpleextend from classical Laplacian operator. So it is necessary to study the basic problems ofequation with nonlocal operator.The paper is about to the stability of nonlocal equation with Laplacian operator, and thenonlocal operator is defined by singular integrals operator. The paper is based on Mendilaetc proved the nonlocal equation’s existence. Firstly, we introduce some concepts of nonlo-cal operator, it refers to the definition of integro-differential operator. So, when we considerthe stability of nonlocal operator, we should consider the operator is or not have extremumprinciple. And the nonlocal operator equation has Laplacian operator, and Laplacian op-erator work on the solution’s influence. We prove the extremum principle using proof bycontradiction. As we prove, we prove the strict extremum principle. And we obtain thenonlocal operator equation’s stability expression by considering extremum principle. Theresult is based on the existence and uniqueness theory of similar equation by others.The nonlocal equation we considering is the combination of nonlocal operator andLaplacian operator, so the properties are not the simple extend from classical Laplacianoperator. But there exist ε and Laplacian operator, so in the progress we proved, ε andLaplacian operator work on the solution’s influence and guarantee the exact result.Andwhen ε=0, the nonlocal operator will have the major influence, that time we will furtherstudy.