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Linear and Quasi-linear Evolution Equations in Hilbert Spaces (Paperback)
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Linear and Quasi-linear Evolution Equations in Hilbert Spaces (Paperback)
Series: Graduate Studies in Mathematics
Expected to ship within 12 - 17 working days
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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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