Internal Modes and Radiation Damping for Quadratic Klein-Gordon in 3d

We consider Klein-Gordon equations with an external potential <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a quadratic nonlinearity in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3 plus 1"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">3+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> space dimensions. We assume that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is regular and decaying and that the (massive) Schrödinger operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H equals negative normal upper Delta plus upper V plus m squared"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:mo> − </mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">H = -\Delta + V + m^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a positive eigenvalue <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda squared greater-than m squared"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi> λ </mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>&gt;</mml:mo> <mml:msup> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda ^2 &gt; m^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with associated eigenfunction <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi> ϕ </mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This is a so-called internal mode and gives rise to time-periodic and spatially localized solutions of the linear flow. We address the classical question of whether such solutions persist under the full nonlinear flow, and describe the behavior of all solutions in a suitable neighborhood of zero. Provided a natural Fermi-Golden rule holds, our main result shows that a solution to the nonlinear Klein-Gordon equation can be decomposed into a discrete component <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a left-parenthesis t right-parenthesis phi"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi> ϕ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">a(t)\phi</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="a left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">a(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> decays over time, and a continuous component <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v"> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding="application/x-tex">v</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which has some weak dispersive properties. We obtain precise asymptotic information on these components such as the sharp rates of decay <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue a left-parenthesis t right-parenthesis EndAbsoluteValue almost-equals t Superscript negative 1 slash 2"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo> ≈ </mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">|a(t)| \approx t^{-1/2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar v left-parenthesis t right-parenthesis double-vertical-bar Subscript upper L Sub Subscript x Sub Superscript normal infinity Baseline almost-equals t Superscript negative 1"> <mml:semantics> <