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Register a new userOn global well-posedness for Klein-Gordon equation with concentrated nonlinearity
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Elena Kopylova
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We prove global well-posedness for the 3D Klein-Gordon equation with a concentrated nonlinearity.
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We prove global well-posedness for 3D Dirac equation with a concentrated nonlinearity.
The global attraction is proved for the nonlinear 3D Klein-Gordon equation with a nonlinearity concentrated at one point. Our main result is the convergence of each finite energy solution to the manifold of all solitary waves as $ttopminfty$. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersion radiation. We justify this mechanism by the following strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time-spectrum in the spectral gap $[-m,m]$ and satisfies the original equation. Then the application of the Titchmarsh Convolution Theorem reduces the spectrum of each omega-limit trajectory to a single frequency $omegain[-m,m]$.
We establish probabilistic small data global well-posedness of the energy-critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge for scaling super-critical random initial data. The proof relies on an induction on frequency procedure and a modified linear-nonlinear decomposition furnished by a delicate probabilistic parametrix construction. This is the first global existence result for a geometric wave equation for random initial data at scaling super-critical regularity.
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