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 gl
obal 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.
In this paper we prove local well-posedness in Orlicz spaces for the biharmonic heat equation $partial_{t} u+ Delta^2 u=f(u),;t>0,;xinR^N,$ with $f(u)sim mbox{e}^{u^2}$ for large $u.$ Under smallness condition on the initial data and for exponential
nonlinearity $f$ such that $f(u)sim u^m$ as $uto 0,$ $m$ integer and $N(m-1)/4geq 2$, we show that the solution is global. Moreover, we obtain a decay estimates for large time for the nonlinear biharmonic heat equation as well as for the nonlinear heat equation. Our results extend to the nonlinear polyharmonic heat equation.
We continue the study of low regularity behavior of the viscous nonlinear wave equation (vNLW) on $mathbb R^2$, initiated by v{C}anic and the first author (2021). In this paper, we focus on the defocusing quintic nonlinearity and, by combining a para
bolic smoothing with a probabilistic energy estimate, we prove almost sure global well-posedness of vNLW for initial data in $mathcal H^s (mathbb R^2)$, $s >- frac 15$, under a suitable randomization.