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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Time averaging for nonautonomous/random linear parabolic equations

Pages: 661 - 699, Volume 9, Issue 3/4, May, June 2008      doi:10.3934/dcdsb.2008.9.661

 
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Janusz Mierczyński - Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław, Poland (email)
Wenxian Shen - Department of Mathematics & Statistics, Auburn University, Auburn, AL 36849, United States (email)

Abstract: 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.

Keywords:  Nonautonomous linear partial differential equation of parabolic type, random linear partial differential equation of parabolic type, principal spectrum, principal Lyapunov exponent, averaging.
Mathematics Subject Classification:  Primary: 35K15, 35P15; Secondary: 35R60, 37H15.

Received: December 2006;      Revised: July 2007;      Available Online: February 2008.