Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Almost Periodic Viscosity Solutions of Nonlinear Parabolic Equations | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 873526 15 pages doi 2009 873526 Research Article Almost Periodic Viscosity Solutions of Nonlinear Parabolic Equations Shilin Zhang and Daxiong Piao School of Mathematical Sciences Ocean University of China Qingdao 266071 China Correspondence should be addressed to Daxiong Piao dxpiao@ Received 26 March 2009 Accepted 9 June 2009 Recommended by Zhitao Zhang We generalize the comparison result 2007 on Hamilton-Jacobi equations to nonlinear parabolic equations then by using Perron s method to study the existence and uniqueness of time almost periodic viscosity solutions of nonlinear parabolic equations under usual hypotheses. Copyright 2009 S. Zhang and D. Piao. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction In this paper we will study the time almost periodic viscosity solutions of nonlinear parabolic equations of the form dtu H x u Du D2u f f x t e Q X R v u x t 0 x t e dQ X R where Q e RN is a bounded open subset and dQ is its boundary. Here H RN X R X RN X S N R and S N denotes the set of symmetric N X N matrices equipped with its usual order . for X Y e S N we say that X Y if and only if pfXp pfYp Vp e RN Du and D2u denote the gradient and Hessian matrix respectively of the function u the argument x. f is almost periodic in t. Most notations and notions of this paper relevant to viscosity solutions are borrowed from the celebrated paper of Crandall et al. 1 . Bostan and Namah 2 have studied the time periodic and almost periodic viscosity solutions of first-order Hamilton-Jacobi equations. Nunziante considered the existence and uniqueness of viscosity solutions of parabolic equations with discontinuous time dependence in 3 4 but the time almost periodic viscosity solutions of .