American Institute of Mathematical Sciences

Advanced
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]

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

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

[1]

Sylvia Novo, Carmen Núñez, Rafael Obaya, Ana M. Sanz. Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4291-4321. doi: 10.3934/dcds.2014.34.4291
[2]

Bogdan Sasu, A. L. Sasu. Input-output conditions for the asymptotic behavior of linear skew-product flows and applications. Communications on Pure & Applied Analysis, 2006, 5 (3) : 551-569. doi: 10.3934/cpaa.2006.5.551
[3]

Davide Guidetti. Partial reconstruction of the source term in a linear parabolic initial problem with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5107-5141. doi: 10.3934/dcds.2013.33.5107
[4]

P.E. Kloeden, Victor S. Kozyakin. The perturbation of attractors of skew-product flows with a shadowing driving system. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 883-893. doi: 10.3934/dcds.2001.7.883
[5]

Saša Kocić. Reducibility of skew-product systems with multidimensional Brjuno base flows. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 261-283. doi: 10.3934/dcds.2011.29.261
[6]

Tomás Caraballo, Alexandre N. Carvalho, Henrique B. da Costa, José A. Langa. Equi-attraction and continuity of attractors for skew-product semiflows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2949-2967. doi: 10.3934/dcdsb.2016081
[7]

Tuhin Ghosh, Karthik Iyer. Cloaking for a quasi-linear elliptic partial differential equation. Inverse Problems & Imaging, 2018, 12 (2) : 461-491. doi: 10.3934/ipi.2018020
[8]

J. Húska, Peter Poláčik. Exponential separation and principal Floquet bundles for linear parabolic equations on $R^N$. Discrete & Continuous Dynamical Systems, 2008, 20 (1) : 81-113. doi: 10.3934/dcds.2008.20.81
[9]

Noboru Okazawa, Kentarou Yoshii. Linear evolution equations with strongly measurable families and application to the Dirac equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 723-744. doi: 10.3934/dcdss.2011.4.723
[10]

Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control & Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001
[11]

Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231
[12]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053
[13]

Fang Chen, Ning Gao, Yao- Lin Jiang. On product-type generalized block AOR method for augmented linear systems. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 797-809. doi: 10.3934/naco.2012.2.797
[14]

Janusz Mierczyński, Wenxian Shen. Time averaging for nonautonomous/random linear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 661-699. doi: 10.3934/dcdsb.2008.9.661
[15]

Juan A. Calzada, Rafael Obaya, Ana M. Sanz. Continuous separation for monotone skew-product semiflows: From theoretical to numerical results. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 915-944. doi: 10.3934/dcdsb.2015.20.915
[16]

Nguyen Dinh Cong, Doan Thai Son. On integral separation of bounded linear random differential equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 995-1007. doi: 10.3934/dcdss.2016038
[17]

Doan Thai Son. On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3113-3126. doi: 10.3934/dcdsb.2017166
[18]

Pedro Duarte, Silvius Klein, Manuel Santos. A random cocycle with non Hölder Lyapunov exponent. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4841-4861. doi: 10.3934/dcds.2019197
[19]

Robert Baier, Lars Grüne, Sigurđur Freyr Hafstein. Linear programming based Lyapunov function computation for differential inclusions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 33-56. doi: 10.3934/dcdsb.2012.17.33
[20]

Ting Guo, Xianhua Tang, Qi Zhang, Zu Gao. Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1563-1579. doi: 10.3934/cpaa.2020078

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (56)
  • HTML views (1)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences