Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres

Abstract

The Hamiltonian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper X Endscripts left-parenthesis StartAbsoluteValue partial-differential Subscript t Baseline u EndAbsoluteValue squared plus StartAbsoluteValue nabla u EndAbsoluteValue squared plus bold m squared StartAbsoluteValue u EndAbsoluteValue squared right-parenthesis d x"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo> ∫ </mml:mo> <mml:mi>X</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mo fence="false" stretchy="false"> | </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>u</mml:mi> </mml:mrow> <mml:msup> <mml:mo fence="false" stretchy="false"> | </mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mo fence="false" stretchy="false"> | </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∇ </mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:msup> <mml:mo fence="false" stretchy="false"> | </mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">m</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo fence="false" stretchy="false"> | </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>u</mml:mi> </mml:mrow> <mml:msup> <mml:mo fence="false" stretchy="false"> | </mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\int _X(\lvert {\partial _t u}\rvert ^2 + \lvert {\nabla u}\rvert ^2 + \mathbf {m}^2\lvert {u}\rvert ^2)\,dx</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , defined on functions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R times upper X"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo> × </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}\times X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a compact manifold, has critical points which are solutions of the linear Klein-Gordon equation. We consider perturbations of this Hamiltonian, given by polynomial expressions depending on first order derivatives of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u"> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding="application/x-tex">u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The associated PDE is then a <italic>quasi-linear</italic> Klein-Gordon equation. We show that, when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the sphere, and when the mass parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold m"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {m}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is outside an exceptional subset of zero measure, smooth Cauchy data of small size <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon"> <mml:semantics> <mml:mi> ϵ </mml:mi> <mml:annotation encoding="application/x-tex">\epsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> give rise to almost global solutions, i.e. solutions defined on a time interval of length <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c Subscript upper N Baseline epsilon Superscript negative upper N"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:msup> <mml:mi> ϵ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">c_N\epsilon ^{-N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semanti