Open Access

On basin of zero-solutions to a semilinear parabolic equation with Ornstein-Uhlenbeck operator

Journal of Inequalities and Applications20062006:52498

https://doi.org/10.1155/JIA/2006/52498

Received: 27 April 2005

Accepted: 10 July 2005

Published: 2 May 2006

Abstract

We consider the basin of the zero-solution to a semilinear parabolic equation on 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.

[12345678]

Authors’ Affiliations

(1)
Department of Mathematics, Toyama University

References

  1. Cerrai S: Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach, Lecture Notes in Mathematics. Volume 1762. Springer, Berlin; 2001:x+330.View ArticleMATHGoogle Scholar
  2. Da Prato G, Goldys B: Elliptic operators onwith unbounded coefficients. Journal of Differential Equations 2001,172(2):333–358. 10.1006/jdeq.2000.3866MathSciNetView ArticleMATHGoogle Scholar
  3. Da Prato G, Goldys B: Erratum: Elliptic operators onwith unbounded coefficients. Journal of Differential Equations 2002,184(2):620.MathSciNetView ArticleMATHGoogle Scholar
  4. Da Prato G, Lunardi A: On the Ornstein-Uhlenbeck operator in spaces of continuous functions. Journal of Functional Analysis 1995,131(1):94–114. 10.1006/jfan.1995.1084MATHMathSciNetView ArticleGoogle Scholar
  5. Friedman A: Partial Differential Equations of Parabolic Type. Prentice-Hall, New Jersey; 1964:xiv+347.MATHGoogle Scholar
  6. Fujita Y, Ishii H, Loreti P: Asymptotic solutions of viscous Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. to appear in Communications in PDE to appear in Communications in PDEGoogle Scholar
  7. Gross L: Logarithmic Sobolev inequalities. American Journal of Mathematics 1975,97(4):1061–1083. 10.2307/2373688MathSciNetView ArticleMATHGoogle Scholar
  8. Samarskii AA, Galaktionov VA, Kurdyumov SP, Mikhailov AP: Blow-Up in Quasilinear Parabolic Equations, De Gruyter Expositions in Mathematics. Volume 19. Walter de Gruyter, Berlin; 1995:xxii+535.View ArticleGoogle Scholar

Copyright

© Yasuhiro Fujita. 2006

This article is published under license to BioMed Central Ltd. 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.