Asymptotic Behavior And Applications Of Several Nonlocal Partial Differential Equations

This dissertation thesis for Ph.D.considers applications and analysis of several nonlo-cal partial differential equations.The outline is as follows.In Chapter 1,we review the original definition and the importance of of nonlocal par-tial differential equations.We discuss the difference between nonlocal PDEs and classical PDEs.Some simple examples are served to illustrate the point.Then we presents the main research contents.In Chapter 2,we consider the transition density upper bounds for symmetric jump process with small jumps perturbed by slowly varying function.We convert the problem into the nonlocal operators without scaling property.For the transition density corresponding to the nonlocal operator,we first prove a new Nash-type inequality for the associated Dirichlet form and then we get an on-diagonal upper bound estimate.Finally we give an off-diagonal upper bound estimate by the Davies' method.In Chapter 3,we consider the asymptotical behavior of the density function of one dimensional nonsymmetric layered stable processes via Levy-Khinchin exponent.First,we develop a new analytical method to instead of the usual probability method,then we give the long-time and the short-time asymptotical behavior of the solutions to the corresponding Cauchy problem of the nonlocal equation respectively,and these imply the asymptotical behavior of transition density.Finally we localize the parabolic partial integro-differential equation to the bounded domains and give the error estimates due to the localization.In Chapter 4,we consider an option pricing problem in a pure jump model where the process X(t)models the logarithm of the stock price.We consider the linear nonlocal partial differential equation,the nonlinear partial differential equation(the nonlocal obstacle equa-tion)generated by European option,American option pricing problems.By the Schauder fixed point theorem,we show the existence and uniqueness of the solutions in Holder spaces for the European and American option pricing problems respectively.By the estimates of fractional heat kernel we give the regularity of the value functions uE(t,x)and uA(t,x)of the European option and the American option.In Chapter 5,we consider the periodic initial value problem associated to the gener-alized Benjamin-Bona-Mahony equation with the generalized damping on one dimensional torus in L2-type Sobolev spaces.By the fixed point theorem,we prove that the periodic initial value problem is locally well-posed.We also prove that if the solution exists globally in time,it exhibits some asymptotic behavior.In Chapter 6,We consider the well posedness of generalized Benjamin-Bona-Mahony equations on one dimensional torus in LP-type Sobolev spaces.