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Spectral theory for random and nonautonomous parabolic equations and applications. (English) Zbl 1387.35007
Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 139. Boca Raton, FL: Chapman & Hall/CRC Press (ISBN 978-1-58488-895-6/hbk). xiii, 317 p. (2008).
Publisher’s description: Providing a basic tool for studying nonlinear problems, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications focuses on the principal spectral theory for general time-dependent and random parabolic equations and systems. The text contains many new results and considers existing results from a fresh perspective.
Taking a clear, unified, and self-contained approach, the authors first develop the abstract general theory in the framework of weak solutions, before turning to cases of random and nonautonomous equations. They prove that time dependence and randomness do not reduce the principal spectrum and Lyapunov exponents of nonautonomous and random parabolic equations. The book also addresses classical Faber-Krahn inequalities for elliptic and time-periodic problems and extends the linear theory for scalar nonautonomous and random parabolic equations to cooperative systems. The final chapter presents applications to Kolmogorov systems of parabolic equations.
By thoroughly explaining the spectral theory for nonautonomous and random linear parabolic equations, this resource reveals the importance of the theory in examining nonlinear problems.

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35P05 General topics in linear spectral theory for PDEs
35R60 PDEs with randomness, stochastic partial differential equations
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
92B05 General biology and biomathematics