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The paper gives a comprehensive study of infinite-energy solutions and their long-time behavior for semi-linear weakly damped wave equations in $mathbb{R}^3$ with quintic nonlinearities. This study includes global well-posedness of the so-called Shatah-Struwe solutions, their dissipativity, the existence of a locally compact global attractors (in the uniformly local phase spaces) and their extra regularity.
We prove that solution of defocusing semilinear wave equation in $mathbb{R}^{1+3}$ with pure power nonlinearity is uniformly bounded for all $frac{3}{2}<pleq 2$ with sufficiently smooth and localized data. The result relies on the $r$-weighted energy
The dissipative wave equation with a critical quintic nonlinearity in smooth bounded three dimensional domain is considered. Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a compact global a
In this paper, we use Dafermos-Rodnianskis new vector field method to study the asymptotic pointwise decay properties for solutions of energy subcritical defocusing semilinear wave equations in $mathbb{R}^{1+3}$. We prove that the solution decays as
We report on new results concerning the global well-posedness, dissipativity and attractors of the damped quintic wave equations in bounded domains of R^3.
We consider finite energy solutions for the damped and driven two-dimensional Navier--Stokes equations in the plane and show that the corresponding dynamical system possesses a global attractor. We obtain upper bounds for its fractal dimension when t