On the behaviour of stochastic heat equations on bounded domains

Mohammud Foondun, Eulalia Nualart

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

Consider the following equation ∂tut(x) = 1 2 ∂xxut(x) + λσ(ut(x))W˙ (t, x) on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution grows exponentially fast if λ is large enough. But if λ is small, then the second moment eventually decays exponentially. If we replace the Dirichlet boundary condition by the Neumann one, then the second moment grows exponentially fast no matter what λ is. We also provide various extensions.

LanguageEnglish
Pages551-571
Number of pages21
JournalAlea
Volume12
Issue number2
Publication statusPublished - 2015
Externally publishedYes

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Stochastic Heat Equation
Bounded Domain
Moment
Dirichlet Boundary Conditions
Long-run
Decay
Interval

Keywords

  • stochastic partial differential equations
  • Dirichlet boundary condition

Cite this

Foondun, Mohammud ; Nualart, Eulalia. / On the behaviour of stochastic heat equations on bounded domains. In: Alea. 2015 ; Vol. 12, No. 2. pp. 551-571.
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Foondun, M & Nualart, E 2015, 'On the behaviour of stochastic heat equations on bounded domains' Alea, vol. 12, no. 2, pp. 551-571.

On the behaviour of stochastic heat equations on bounded domains. / Foondun, Mohammud; Nualart, Eulalia.

In: Alea, Vol. 12, No. 2, 2015, p. 551-571.

Research output: Contribution to journalArticle

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AU - Nualart, Eulalia

PY - 2015

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AB - Consider the following equation ∂tut(x) = 1 2 ∂xxut(x) + λσ(ut(x))W˙ (t, x) on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution grows exponentially fast if λ is large enough. But if λ is small, then the second moment eventually decays exponentially. If we replace the Dirichlet boundary condition by the Neumann one, then the second moment grows exponentially fast no matter what λ is. We also provide various extensions.

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KW - Dirichlet boundary condition

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