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: ON BASIN OF ZERO-SOLUTIONS TO A SEMILINEAR PARABOLIC EQUATION WITH ORNSTEIN-UHLENBECK OPERATOR | ON BASIN OF ZERO-SOLUTIONS TO A SEMILINEAR PARABOLIC EQUATION WITH ORNSTEIN-UHLENBECK OPERATOR YASUHIRO FUJITA Received 27 April 2005 Accepted 10 July 2005 We consider the basin of the zero-solution to a semilinear parabolic equation on RN with the Ornstein-Uhlenbeck operator. Our aim is to show that the Ornstein-Uhlenbeck operator contributes to enlargement of the basin by using the logarithmic Sobolev inequality. Copyright 2006 Yasuhiro Fujita. 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 Let a B 0 be given constants. We consider the following semilinear parabolic problem Uị 7-Au - ax Du Bulogu in 0 to X RN 2 u 0 p in Rn where the initial data p satisfies y 0 in Rn y log p Lip Rn . When a 0 problem was considered by Samarskii et al. in 8 pages 93-99 . When a 0 the operator L defined by L 2 A - ax D is called the Ornstein-Uhlenbeck operator and has been studied by many authors 14 6 . In linear parabolic equations the Ornstein-Uhlenbeck operator contributes good properties to their solutions such as ergodicity and hypercontractivity. However to semi-linear parabolic equations a contribution of the Ornstein-Uhlenbeck operator is hardly known. Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 52498 Pages 1-10 DOI JIA 2006 52498 2 On the basin of zero-solutions Our motivation to study problem is that it provides an example of semilinear parabolic equations to which the Ornstein-Uhlenbeck operator contributes. Indeed in the Ornstein-Uhlenbeck operator L contributes to enlargement of the basin of the zero-solution. Our aim of this paper is to clarify this contribution by using the relation between the parameters a p. Our result states that if a is sufficiently larger than p 2 then the basin of the zero-solutions is .
