Long Time Dynamics of Quasi-linear Hamiltonian Klein–Gordon Equations on the Circle Article Swipe
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.1007/s10884-024-10365-8
· OA: W4394993240
We consider a class of Hamiltonian Klein–Gordon equations with a quasilinear, quadratic nonlinearity under periodic boundary conditions. For a large set of masses, we provide a precise description of the dynamics for an open set of small initial data of size $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> showing that the corresponding solutions remain close to oscillatory motions over a time scale $$\varepsilon ^{{-\frac{9}{4}+\delta }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ε</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>9</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mi>δ</mml:mi> </mml:mrow> </mml:msup> </mml:math> for any $$\delta >0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . The key ingredients of the proof are normal form methods, para-differential calculus and a modified energy approach.
