eBook ISBN:  9781470420307 
Product Code:  MEMO/234/1103.E 
List Price:  $68.00 
MAA Member Price:  $61.20 
AMS Member Price:  $40.80 
eBook ISBN:  9781470420307 
Product Code:  MEMO/234/1103.E 
List Price:  $68.00 
MAA Member Price:  $61.20 
AMS Member Price:  $40.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 234; 2014; 80 ppMSC: Primary 35; 37
The Hamiltonian \(\int_X(\lvert{\partial_t u}\rvert^2 + \lvert{\nabla u}\rvert^2 + \mathbf{m}^2\lvert{u}\rvert^2)\,dx\), defined on functions on \(\mathbb{R}\times X\), where \(X\) is a compact manifold, has critical points which are solutions of the linear KleinGordon equation.
The author considers perturbations of this Hamiltonian, given by polynomial expressions depending on first order derivatives of \(u\). The associated PDE is then a quasilinear KleinGordon equation. The author shows that, when \(X\) is the sphere, and when the mass parameter \(\mathbf{m}\) is outside an exceptional subset of zero measure, smooth Cauchy data of small size \(\epsilon\) give rise to almost global solutions, i.e. solutions defined on a time interval of length \(c_N\epsilon^{N}\) for any \(N\). Previous results were limited either to the semilinear case (when the perturbation of the Hamiltonian depends only on \(u\)) or to the one dimensional problem.
The proof is based on a quasilinear version of the Birkhoff normal forms method, relying on convenient generalizations of paradifferential calculus.

Table of Contents

Chapters

1. Introduction

2. Statement of the main theorem

3. Symbolic calculus

4. Quasilinear Birkhoff normal forms method

5. Proof of the main theorem

A. Appendix


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The Hamiltonian \(\int_X(\lvert{\partial_t u}\rvert^2 + \lvert{\nabla u}\rvert^2 + \mathbf{m}^2\lvert{u}\rvert^2)\,dx\), defined on functions on \(\mathbb{R}\times X\), where \(X\) is a compact manifold, has critical points which are solutions of the linear KleinGordon equation.
The author considers perturbations of this Hamiltonian, given by polynomial expressions depending on first order derivatives of \(u\). The associated PDE is then a quasilinear KleinGordon equation. The author shows that, when \(X\) is the sphere, and when the mass parameter \(\mathbf{m}\) is outside an exceptional subset of zero measure, smooth Cauchy data of small size \(\epsilon\) give rise to almost global solutions, i.e. solutions defined on a time interval of length \(c_N\epsilon^{N}\) for any \(N\). Previous results were limited either to the semilinear case (when the perturbation of the Hamiltonian depends only on \(u\)) or to the one dimensional problem.
The proof is based on a quasilinear version of the Birkhoff normal forms method, relying on convenient generalizations of paradifferential calculus.

Chapters

1. Introduction

2. Statement of the main theorem

3. Symbolic calculus

4. Quasilinear Birkhoff normal forms method

5. Proof of the main theorem

A. Appendix