Effects of the noise level on nonlinear stochastic fractional heat equations

Kexue Li

doi:10.3934/dcdsb.2019065

Effects of the noise level on nonlinear stochastic fractional heat equations

Kexue Li

School of Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

Abstract

We consider the stochastic fractional heat equation ∂tu = 4^α/ ²u+ λσ(u)w on [0, L] with Dirichlet boundary conditions, where w denotes the space-time white noise. For any λ > 0, we prove that the pth moment of sup_x∈[0,L_] |u(t, x)| grows at most exponentially. If λ is small, we prove that the pth moment of sup_x∈[0,L_] |u(t, x)| is exponentially stable. At last, we obtain the noise excitation index of pth energy of u(t, x) is (Formula presented.).

Original language	English
Pages (from-to)	5437-5460
Number of pages	24
Journal	Discrete and Continuous Dynamical Systems - Series B
Volume	24
Issue number	10
DOIs	https://doi.org/10.3934/dcdsb.2019065
State	Published - Oct 2019

Keywords

Excitation index
Fractional heat kernel
Mittag-Leffler function
Stochastic fractional heat equations

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title = "Effects of the noise level on nonlinear stochastic fractional heat equations",

abstract = "We consider the stochastic fractional heat equation ∂tu = 4α/ 2u+ λσ(u)w on [0, L] with Dirichlet boundary conditions, where w denotes the space-time white noise. For any λ > 0, we prove that the pth moment of supx∈[0,L] |u(t, x)| grows at most exponentially. If λ is small, we prove that the pth moment of supx∈[0,L] |u(t, x)| is exponentially stable. At last, we obtain the noise excitation index of pth energy of u(t, x) is (Formula presented.).",

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language = "英语",

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AU - Li, Kexue

PY - 2019/10

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N2 - We consider the stochastic fractional heat equation ∂tu = 4α/ 2u+ λσ(u)w on [0, L] with Dirichlet boundary conditions, where w denotes the space-time white noise. For any λ > 0, we prove that the pth moment of supx∈[0,L] |u(t, x)| grows at most exponentially. If λ is small, we prove that the pth moment of supx∈[0,L] |u(t, x)| is exponentially stable. At last, we obtain the noise excitation index of pth energy of u(t, x) is (Formula presented.).

AB - We consider the stochastic fractional heat equation ∂tu = 4α/ 2u+ λσ(u)w on [0, L] with Dirichlet boundary conditions, where w denotes the space-time white noise. For any λ > 0, we prove that the pth moment of supx∈[0,L] |u(t, x)| grows at most exponentially. If λ is small, we prove that the pth moment of supx∈[0,L] |u(t, x)| is exponentially stable. At last, we obtain the noise excitation index of pth energy of u(t, x) is (Formula presented.).

KW - Excitation index

KW - Fractional heat kernel

KW - Mittag-Leffler function

KW - Stochastic fractional heat equations

UR - https://www.scopus.com/pages/publications/85072569947

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DO - 10.3934/dcdsb.2019065

M3 - 文章

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SN - 1531-3492

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SP - 5437

EP - 5460

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

IS - 10

ER -

Effects of the noise level on nonlinear stochastic fractional heat equations

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Keywords

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