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

Formulas for generalized principal Lyapunov exponent for parabolic PDEs

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.
Citation: Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete and 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.

[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.

[3]

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

[4]

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

[5]

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

[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.

[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.

[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.

show all references

References:
[1]

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

[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.

[3]

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

[4]

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

[5]

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

[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.

[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.

[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.

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