The long-time behaviour of the solutions to semilinear stochastic partial differential equations on the whole space

Ralf Manthey

Mathematica Bohemica (2001)

  • Volume: 126, Issue: 1, page 15-39
  • ISSN: 0862-7959

Abstract

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The Cauchy problem for a stochastic partial differential equation with a spatial correlated Gaussian noise is considered. The “drift” is continuous, one-sided linearily bounded and of at most polynomial growth while the “diffusion” is globally Lipschitz continuous. In the paper statements on existence and uniqueness of solutions, their pathwise spatial growth and on their ultimate boundedness as well as on asymptotical exponential stability in mean square in a certain Hilbert space of weighted functions are proved.

How to cite

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Manthey, Ralf. "The long-time behaviour of the solutions to semilinear stochastic partial differential equations on the whole space." Mathematica Bohemica 126.1 (2001): 15-39. <http://eudml.org/doc/248845>.

@article{Manthey2001,
abstract = {The Cauchy problem for a stochastic partial differential equation with a spatial correlated Gaussian noise is considered. The “drift” is continuous, one-sided linearily bounded and of at most polynomial growth while the “diffusion” is globally Lipschitz continuous. In the paper statements on existence and uniqueness of solutions, their pathwise spatial growth and on their ultimate boundedness as well as on asymptotical exponential stability in mean square in a certain Hilbert space of weighted functions are proved.},
author = {Manthey, Ralf},
journal = {Mathematica Bohemica},
keywords = {nuclear and cylindrical noise; existence and uniqueness of the solution; spatial growth; ultimate boundedness; asymptotic mean square stability; Cauchy problem; Cauchy problem; nuclear and cylindrical noise; existence and uniqueness of the solution; spatial growth; ultimate boundedness; asymptotic mean square stability},
language = {eng},
number = {1},
pages = {15-39},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The long-time behaviour of the solutions to semilinear stochastic partial differential equations on the whole space},
url = {http://eudml.org/doc/248845},
volume = {126},
year = {2001},
}

TY - JOUR
AU - Manthey, Ralf
TI - The long-time behaviour of the solutions to semilinear stochastic partial differential equations on the whole space
JO - Mathematica Bohemica
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 126
IS - 1
SP - 15
EP - 39
AB - The Cauchy problem for a stochastic partial differential equation with a spatial correlated Gaussian noise is considered. The “drift” is continuous, one-sided linearily bounded and of at most polynomial growth while the “diffusion” is globally Lipschitz continuous. In the paper statements on existence and uniqueness of solutions, their pathwise spatial growth and on their ultimate boundedness as well as on asymptotical exponential stability in mean square in a certain Hilbert space of weighted functions are proved.
LA - eng
KW - nuclear and cylindrical noise; existence and uniqueness of the solution; spatial growth; ultimate boundedness; asymptotic mean square stability; Cauchy problem; Cauchy problem; nuclear and cylindrical noise; existence and uniqueness of the solution; spatial growth; ultimate boundedness; asymptotic mean square stability
UR - http://eudml.org/doc/248845
ER -

References

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  10. Stochastic diffusion on an unbounded domain, Pacific J. Math. 84 (1979), 143–153. (1979) Zbl0423.60056MR0559632
  11. On the existence of a fundamental total biorthogonal sequence in every separable Banach space and related constructions of uniformly bounded orthonormal systems in 𝕃 2 , Studia Mathematica 54 (1975), 149–159. (1975) MR0394137
  12. Two contrasting properties of solutions for one-dimensional stochastic partial differential equations, Can. J. Math. 46 (1994), 415–437. (1994) Zbl0801.60050MR1271224

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