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August  2016, 9(4): 1189-1199. doi: 10.3934/dcdss.2016048

Formulas for generalized principal Lyapunov exponent for parabolic PDEs

Janusz Mierczyński 1, and  Wenxian Shen 2,

1. 

Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław
2. 

Department of Mathematics & Statistics, Auburn University, Auburn, AL 36849

Received  August 2015 Revised  February 2016 Published  August 2016

An integral formula is given representing the generalized principal Lyapunov exponent for random linear parabolic PDEs. As an application, an upper estimate of the exponent is obtained.
Keywords: Dirichlet form., measurable linear skew-product semiflow, Random linear parabolic partial differential equation, generalized principal Lyapunov exponent.
Mathematics Subject Classification: Primary: 37H15; Secondary: 35K10, 35R6.
Citation: Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048
References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide, third edition, Springer, Berlin, 2006.  Google Scholar

[2]

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[3]

L. C. Evans, Partial Differential Equations, Grad. Stud. Math., 19, American Mathematical Society, Providence, R.I., 1998.  Google Scholar

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U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985. doi: 10.1515/9783110844641.  Google Scholar

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J. Mierczyński, Estimates for principal Lyapunov exponents: A survey, Nonautonomous Dynamical Systems, 1 (2014), 137-162.  Google Scholar

[6]

J. Mierczyński and W. Shen, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2008. doi: 10.1201/9781584888963.  Google Scholar

[7]

J. Mierczyński and W. Shen, Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory, Trans. Amer. Math. Soc., 365 (2013), 5329-5365. doi: 10.1090/S0002-9947-2013-05814-X.  Google Scholar

[8]

J. Mierczyński and W. Shen, Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. III. Parabolic equations and delay systems, J. Dynam. Differential Equations, 28 (2016), 1039-1079. doi: 10.1007/s10884-015-9436-z.  Google Scholar

show all references

References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide, third edition, Springer, Berlin, 2006.  Google Scholar

[2]

J. Diestel and J. J. Uhl, Jr., Vector Measures, with a foreword by B. J. Pettis, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977.  Google Scholar

[3]

L. C. Evans, Partial Differential Equations, Grad. Stud. Math., 19, American Mathematical Society, Providence, R.I., 1998.  Google Scholar

[4]

U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985. doi: 10.1515/9783110844641.  Google Scholar

[5]

J. Mierczyński, Estimates for principal Lyapunov exponents: A survey, Nonautonomous Dynamical Systems, 1 (2014), 137-162.  Google Scholar

[6]

J. Mierczyński and W. Shen, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2008. doi: 10.1201/9781584888963.  Google Scholar

[7]

J. Mierczyński and W. Shen, Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory, Trans. Amer. Math. Soc., 365 (2013), 5329-5365. doi: 10.1090/S0002-9947-2013-05814-X.  Google Scholar

[8]

J. Mierczyński and W. Shen, Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. III. Parabolic equations and delay systems, J. Dynam. Differential Equations, 28 (2016), 1039-1079. doi: 10.1007/s10884-015-9436-z.  Google Scholar

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