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Lectures on Nonlinear Evolution Equations : Initial Value Problems / by Reinhard Racke
版 | 2nd ed. 2015. |
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出版者 | (Cham : Springer International Publishing : Imprint: Birkhäuser) |
出版年 | 2015 |
大きさ | VIII, 306 p. 13 illus : online resource |
著者標目 | *Racke, Reinhard author SpringerLink (Online service) |
件 名 | LCSH:Differential equations FREE:Differential Equations |
一般注記 | Introduction -- 1. Global solutions to wave equations - existence theorems -- 2. L^p - L^q-decay estimates for the linear we equation -- 3. Linear symmetric hyperbolic systems -- 3.1 Energy estimates -- 3.2 A global existence theorem -- 3.3 Remarks on other methods -- 4. Some inequalities -- 5. Local existence for quasilinear symmetric hyperbolic -- 6. High energy estimates -- 7. Weighted a priori estimates -- 8. Global solutions to wave equations - proofs -- 8.1 Proof of Theorem 1.1 -- 8.2 Proof ot Theorem 1.2 -- 9. Other methods -- 10. Development of singularities -- 11. More evolutions equations -- 11.1 Equations of elasiticity -- 11.1.1 Initially isotropic media in R^3 -- 11.1.2 Initially cubic media in R^3 -- 11.2 Heat equations -- 11.3 Equations of thermoelasticity -- 11.4 Schrödinger equations -- 11.5 Klein-Gordon equations -- 11.6 Maxwell equations -- 11.7 Plate equations -- 12. Further aspects and questions -- 13. Evolution equations in waveguides -- 13.1 Nonlinear wave equations -- 13.1.1 Linear part -- 13.1.2 Nonlinear part -- 13.2. Schrödinger equations -- 13.3. Equations of elasticity and Maxwell equations -- 13.4 General waveguides -- Appendix -- A. Interpolation -- B. The Theorem of Cauchy-Kowalevsky -- C. A local existence theorem for hyperbolic-parabolic systems References Notation Index 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 HTTP:URL=https://doi.org/10.1007/978-3-319-21873-1 |
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電子ブック | 配架場所 | 資料種別 | 巻 次 | 請求記号 | 状 態 | 予約 | コメント | ISBN | 刷 年 | 利用注記 | 指定図書 | 登録番号 |
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電子ブック | オンライン | 電子ブック |
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Springer eBooks | 9783319218731 |
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電子リソース |
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EB00208226 |
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