A Study On Viscosity Solution For Fully Nonlinear Uniformly Elliptic And Parabolic Equations

In this paper,we study the existence,boundary behavior and uniqueness of large viscosity solutions of fully nonlinear elliptic equations.Our results are based on the two following aspects.Firstly,from the last century to the present,the problem of boundary blowup for semilinear and quasilinear elliptic equations has developed well(see[2-6,8-12,23-24,26-27,59-62,64-67,106-109,155-157]),and Keller[57]and Osserman[132]found a necessary and sufficient condition to guarantee the existence of solutions.Secondly,S.Alarco and A.Quaas[133]applied the results to fully nonlinear elliptic equations in the viscosity solutions theory.Firstly,we study the existence,uniqueness and asymptotic behavior near the boundary of the viscosity solutions for the problem where ? is a C2 bounded domain in RN,F is a fully nonlinear elliptic operator depending on Du and ? is a nondecreasing continuous function.Then,we study the existence,uniqueness and asymptotic behavior near the boundary of the viscosity solutions for the problem where ? is a C2 bounded domain in RN,F is a fully nonlinear elliptic operator,a(x)? C(?)is a nonnegative function and ? is a nondecreasing continuous function.Further,we study the existence,uniqueness and asymptotic behavior near the boundary of the viscosity solutions for the problem with continuous weight where ? is a C2 bounded domain in RN,F is a fully nonlinear elliptic operator depending on Du,a(x)? C(?)is a nonnegative function and ? is a nondecreasing continuous function.Finally,we are concerned with fully nonlinear degeierate parabolic partial differential equations with a superlinear gradient nonlinearity where F:RN ×(0,T)×R×RN×SN?R,H:RN×(0,T)RN?R and f:RN ×(0,T)? R are given functions.The unknown u is a real-valued function defined in RN ×[0,T].Du and D2u denote,respectively,its gradient and Hessian matrix with respect to spacial variable x.? is a given initial condition.We prove a comparison result between viscosity subsolutions and supersolutions having superlinear growth.We extend our result to monotone systems of parabolic equations.