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The moment and almost surely exponential stability of stochastic heat equations. (English) Zbl 1147.93395
Summary: The \( p\)-th moment and almost surely exponential stability of the strong solution to a stochastic heat equation driven by an \( m\)-dimensional Brownian motion is investigated by a simple method. In particular, the sharp top Lyapunov exponents are explicitly calculated based on the representation of the strong solution.

MSC:
93D20 Asymptotic stability in control theory
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35K05 Heat equation
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[1] Christopher T. H. Baker and Evelyn Buckwar, Exponential stability in \?-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations, J. Comput. Appl. Math. 184 (2005), no. 2, 404 – 427. · Zbl 1081.65011 · doi:10.1016/j.cam.2005.01.018 · doi.org
[2] Tomás Caraballo, Kai Liu, and Xuerong Mao, On stabilization of partial differential equations by noise, Nagoya Math. J. 161 (2001), 155 – 170. · Zbl 0986.60058
[3] Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. · Zbl 0761.60052
[4] Dariusz G tarek and Beniamin Gołdys, On weak solutions of stochastic equations in Hilbert spaces, Stochastics Stochastics Rep. 46 (1994), no. 1-2, 41 – 51. · Zbl 0824.60052
[5] U. G. Haussmann, Asymptotic stability of the linear Itô equation in infinite dimensions, J. Math. Anal. Appl. 65 (1978), no. 1, 219 – 235. · Zbl 0385.93051 · doi:10.1016/0022-247X(78)90211-1 · doi.org
[6] Akira Ichikawa, Stability of semilinear stochastic evolution equations, J. Math. Anal. Appl. 90 (1982), no. 1, 12 – 44. · Zbl 0497.93055 · doi:10.1016/0022-247X(82)90041-5 · doi.org
[7] Anna A. Kwiecińska, Stabilization of partial differential equations by noise, Stochastic Process. Appl. 79 (1999), no. 2, 179 – 184. · Zbl 0962.60052 · doi:10.1016/S0304-4149(98)00080-5 · doi.org
[8] Kai Liu, Stability of infinite dimensional stochastic differential equations with applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 135, Chapman & Hall/CRC, Boca Raton, FL, 2006. · Zbl 1085.60003
[9] Jelena Randjelović and Svetlana Janković, On the \?th moment exponential stability criteria of neutral stochastic functional differential equations, J. Math. Anal. Appl. 326 (2007), no. 1, 266 – 280. · Zbl 1115.60065 · doi:10.1016/j.jmaa.2006.02.030 · doi.org
[10] Takeshi Taniguchi, Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics Stochastics Rep. 53 (1995), no. 1-2, 41 – 52. · Zbl 0854.60051
[11] Takeshi Taniguchi, Kai Liu, and Aubrey Truman, Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Differential Equations 181 (2002), no. 1, 72 – 91. · Zbl 1009.34074 · doi:10.1006/jdeq.2001.4073 · doi.org
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