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

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.

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References

  1. 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.

    Google Scholar 

  2. 2.

    Da Prato G, Goldys B: Elliptic operators onwith unbounded coefficients. Journal of Differential Equations 2001,172(2):333–358. 10.1006/jdeq.2000.3866

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Da Prato G, Goldys B: Erratum: Elliptic operators onwith unbounded coefficients. Journal of Differential Equations 2002,184(2):620.

    MathSciNet  Article  MATH  Google Scholar 

  4. 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.1995.1084

    MATH  MathSciNet  Article  Google Scholar 

  5. 5.

    Friedman A: Partial Differential Equations of Parabolic Type. Prentice-Hall, New Jersey; 1964:xiv+347.

    Google Scholar 

  6. 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 PDE

  7. 7.

    Gross L: Logarithmic Sobolev inequalities. American Journal of Mathematics 1975,97(4):1061–1083. 10.2307/2373688

    MathSciNet  Article  MATH  Google Scholar 

  8. 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.

    Google Scholar 

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Correspondence to Yasuhiro Fujita.

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Fujita, Y. On basin of zero-solutions to a semilinear parabolic equation with Ornstein-Uhlenbeck operator. J Inequal Appl 2006, 52498 (2006). https://doi.org/10.1155/JIA/2006/52498

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Keywords

  • Parabolic Equation
  • Sobolev Inequality
  • Logarithmic Sobolev Inequality
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