Research On Solutions For A Class Of Fractional Nonlinear Partial Differential Equations

Partial differential equation theory is widely applied in some branches of mathematics,physics,natural sciences and other fields.Many scholars at home and abroad have studied the properties of solutions for partial differential equation.Laplacian is one of the origins of partial differential equation,and it has important applications.With the research on arbitrary integer order Laplacian,fractional Laplacian has also been discussed in combination with some actual situations.Fractional order refers to the differential number is not integer.The fractional order operator plays an important role not only in mathematics,but also in other fields such as mechanics,physics,biomedical engineering and finance.There are many nonlinear problems in real life.Therefore,it's very necessary to study the fractional nonlinear partial differential equations.In this thesis,we mainly discuss the properties of solutions to Dirichlet problems associated to a class of fractional nonlinear partial differential equations.In the first part,we give the development of partial differential equation,the background knowledge and the research progress at home and abroad of fractional Laplacian.And we state the main research contents of this thesis.Then we give the definitions,lemmas and symbolic expressions related to this thesis.In the second part,we give a class of fractional nonlinear partial differential equations.We first give the comparison principle of classical solutions inRⁿ with the method of reduction to absurdity.Then,we discuss the Lipschitz continuity of classical solutions satisfying certain boundary conditions.In the third part,we give the simple maximum theorem,the maximum theorem for anti-symmetric functions,narrow region theorem and decay at infinity.And we introduce the application of the theorems and the method of moving planes in proving the symmetry of positive solutions.Then we give the symmetry for the positive solutions inB₁?0?andRⁿ.Finally we apply the method of moving planes to the upper half spaceR₊ⁿ to prove non-existence of positive solutions in this space.