Lectures on Nonlinear Evolution Equations : Initial Value Problems / by Reinhard Racke.

Format:

Book

Author/Creator:

Racke, Reinhard., Author.

Language:

English

Subjects (All):

Differential equations, Partial.

Partial Differential Equations.

Local Subjects:

Partial Differential Equations.

Genre:

Evolution equations, Nonlinear.

Initial value problems.

Physical Description:

1 online resource (315 p.)

Edition:

2nd ed. 2015.

Place of Publication:

Cham : Springer International Publishing : Imprint: Birkhäuser, 2015.

Language Note:

English

Summary:

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.

Contents:

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.

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

3-319-21873-5