The Viscosity Solution For A Class Of Nonlinear Parabolic Equations In Unbounded Domains

In this thesis, we mainly study the viscosity solutions of a fully nonlinear parabolicpartial diferential equations and its related properties in unbounded area, including thedefinition and the classification of the viscosity solutions to the unbounded domain-s from the bounded domains, the ABP estimate, Harnack inequality, the maximumprinciple and the C~αregularity of the viscosity solutions in the unbounded cylindricalregional. The paper is organized in five chapters.In the first chapter, we briefly recall the historical background and the meaningsof the problem’s emergence, the most recent developments of the viscosity solutionsand the major work of the paper. The same times, we also briefly the basic knowledge,the basic definition and notation which is necessary for research.The second chapter, combined with the viscosity solution’s definition and contentof fully nonlinear parabolic equations in the bounded areas, using the method of com-parison demonstrates, we get the definition and classification of the viscosity solutionsin unbounded domains. In addition, another important task is to propose two struc-tural conditions of the parabolic operator F which laid the foundation for the laterstudy in this chapter.On the basis of structural conditions, the third chapter have a promotion aboutthe weak Harnack inequality and local extremum principle in the bounded domains,and get the interior weak Harnack inequality and interior local extremum principle ofparabolic equation in bounded domain. Then we extend them to boundary and getthe boundary weak Harnack inequality and boundary local extremum principle.We cite a measure-geometric condition in the forth chapter, and combine with theboundary weak Harnack inequality, we get the ABP estimate of the viscosity solutionsin unbounded cylinder area, then on the basis of this, we get the maximum principleof the viscosity solutions in unbounded cylinder area.We research the Harnack inequality of the viscosity solutions in unbounded cylin-der area in the fifth chapter, which is proved by the boundary weak Harnack inequalityand the boundary local extremum principle. In this chapter, we complement the proveof the C~αregularity theories of the viscosity solutions in bounded domains, then weconduct a preliminary inquiry on the regularity of the viscosity solutions in unboundedcylinder area, and obtain the C~αregularity theories of the parabolic equation.