Lectures on Nonlinear Evolution Equations
Initial Value Problems
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Beschreibung
This book mainly serves as an elementary, self-contained introduction to several important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The book employs the classical method of continuation of local solutions with the help of a priori estimates obtained for small data. The existence and uniqueness of small, smooth solutions that are defined for all values of the time parameter are investigated. Moreover, the asymptotic behavior of the solutions is described as time tends to infinity. The methods for nonlinear wave equations are discussed in detail. Other
examples include the equations of elasticity, heat equations, the equations of thermoelasticity, Schrödinger equations, Klein-Gordon equations, Maxwell equations and plate equations. To emphasize the importance of studying the conditions under which small data problems offer global solutions, some blow-up results are briefly described. Moreover, the prospects for corresponding initial-boundary value problems and for open questions are provided.
In this second edition, initial-boundary value problems in waveguides are additionally considered.
This book mainly serves as an elementary, self-contained introduction to several important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The book employs the classical method of continuation of local solutions with the help of a priori estimates obtained for small data. The existence and uniqueness of small, smooth solutions that are defined for all values of the time parameter are investigated. Moreover, the asymptotic behaviour of the solutions is described as time tends to infinity. The methods for nonlinear wave equations are discussed in detail. Other examples include the equations of elasticity, heat equations, the equations of thermoelasticity, Schrödinger equations, Klein-Gordon equations, Maxwell equations and plate equations. To emphasize the importance of studying the conditions under which small data problems offer global solutions, some blow-up results are briefly described. Moreover, the prospects for corresponding initial boundary value problems and for open questions are provided.
In this second edition, initial-boundary value problems in waveguides are additionally considered.
It serves since the first edition as a self-contained introduction into methods dealing with Cauchy problems for nonlinear evolutions equations presenting details for graduate students Giving a survey for researchers in particular with an advanced topic on waveguides in the second edition Only existing in separate original papers before Includes supplementary material: sn.pub/extras
Autor*in
Reinhard Racke
Themen in »Lectures on Nonlinear Evolution Equations«
Cauchy problem Klein-Gordon equation Schrödinger equation heat equation plate equation thermoelasticity wave equation waveguide partial differential equations