Remarks On The Asymptotic Behavior Of Solutions To Damped Wave Equations In Hilbert Space

Remarks on the Asymptotic Behavior of Solutions to Damped Wave Equations in Hilbert Space

Lower bounds are derived for the norms of solutions to a class of initial-value problems associated with the damped wave equation sub tt + Au sub t + Bu=0 in Hilbert space. Under appropriate assumptions on the linear operator B it is shown that even in the special strongly damped case where A = Gamma I, Gamma> 0, solutions are bounded way from zero as t approaches plus infinity, even when Gamma approaches plus infinity. (Author).

Linear and Quasi-linear Evolution Equations in Hilbert Spaces

This book considers evolution equations of hyperbolic and parabolic type. These equations are studied from a common point of view, using elementary methods, such as that of energy estimates, which prove to be quite versatile. The authors emphasize the Cauchy problem and present a unified theory for the treatment of these equations. In particular, they provide local and global existence results, as well as strong well-posedness and asymptotic behavior results for the Cauchy problem for quasi-linear equations. Solutions of linear equations are constructed explicitly, using the Galerkin method; the linear theory is then applied to quasi-linear equations, by means of a linearization and fixed-point technique. The authors also compare hyperbolic and parabolic problems, both in terms of singular perturbations, on compact time intervals, and asymptotically, in terms of the diffusion phenomenon, with new results on decay estimates for strong solutions of homogeneous quasi-linear equations of each type. This textbook presents a valuable introduction to topics in the theory of evolution equations, suitable for advanced graduate students. The exposition is largely self-contained. The initial chapter reviews the essential material from functional analysis. New ideas are introduced along with their context. Proofs are detailed and carefully presented. The book concludes with a chapter on applications of the theory to Maxwell's equations and von Karman's equations.

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