May  2008, 9(3&4, May): 661-699. doi: 10.3934/dcdsb.2008.9.661

Time averaging for nonautonomous/random linear parabolic equations

1. 

Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław, Poland

2. 

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

Received  December 2006 Revised  July 2007 Published  February 2008

Linear nonautonomous/random parabolic partial differential equations are considered under the Dirichlet, Neumann or Robin boundary conditions, where both the zero order coefficients in the equation and the coefficients in the boundary conditions are allowed to depend on time. The theory of the principal spectrum/principal Lyapunov exponents is shown to apply to those equations. In the nonautonomous case, the main result states that the principal eigenvalue of any time-averaged equation is not larger than the supremum of the principal spectrum and that there is a time-averaged equation whose principal eigenvalue is not larger than the infimum of the principal spectrum. In the random case, the main result states that the principal eigenvalue of the time-averaged equation is not larger than the principal Lyapunov exponent.
Citation: Janusz Mierczyński, Wenxian Shen. Time averaging for nonautonomous/random linear parabolic equations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 661-699. doi: 10.3934/dcdsb.2008.9.661
[1]

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

[2]

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

[3]

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

[4]

Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Principal Floquet subspaces and exponential separations of type Ⅱ with applications to random delay differential equations. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6163-6193. doi: 10.3934/dcds.2018265

[5]

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

[6]

Nobuyuki Kato, Norio Kikuchi. Campanato-type boundary estimates for Rothe's scheme to parabolic partial differential systems with constant coefficients. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 737-760. doi: 10.3934/dcds.2007.19.737

[7]

Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure and Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211

[8]

A. Rodríguez-Bernal. Perturbation of the exponential type of linear nonautonomous parabolic equations and applications to nonlinear equations. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 1003-1032. doi: 10.3934/dcds.2009.25.1003

[9]

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

[10]

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

[11]

Thomas Lorenz. Partial differential inclusions of transport type with state constraints. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1309-1340. doi: 10.3934/dcdsb.2019018

[12]

V. Balaji, I. Biswas and D. S. Nagaraj. Principal bundles with parabolic structure. Electronic Research Announcements, 2001, 7: 37-44.

[13]

Avner Friedman, Harsh Vardhan Jain. A partial differential equation model of metastasized prostatic cancer. Mathematical Biosciences & Engineering, 2013, 10 (3) : 591-608. doi: 10.3934/mbe.2013.10.591

[14]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221

[15]

Tôn Việt Tạ. Existence results for linear evolution equations of parabolic type. Communications on Pure and Applied Analysis, 2018, 17 (3) : 751-785. doi: 10.3934/cpaa.2018039

[16]

Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879

[17]

Susanna V. Haziot. Study of an elliptic partial differential equation modelling the Antarctic Circumpolar Current. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4415-4427. doi: 10.3934/dcds.2019179

[18]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks and Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9

[19]

Roberto Camassa, Pao-Hsiung Chiu, Long Lee, W.-H. Sheu. A particle method and numerical study of a quasilinear partial differential equation. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1503-1515. doi: 10.3934/cpaa.2011.10.1503

[20]

Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]