# American Institute of Mathematical Sciences

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