No abstract

The paper gives a comprehensive study of infinite-energy solutions and their long-time behavior for semi-linear weakly damped wave equations in R 3 \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.

In this paper, the non-autonomous dynamical behavior of weakly damped wave equation with a sup-cubic nonlinearity is considered in locally uniform spaces. We first prove the global well-posedness of the Shatah-Struwe solutions, then establish the existence of the H 1 lu (R 3) × L 2 lu (R 3), H 1 ρ (R 3) × L 2 ρ (R 3)-pullback attractor for the Shatah-Struwe solutions process of this equation. The results are based on the recent extension of Strichartz estimates for the bounded domains.

In this paper, the dynamical behavior of weakly damped wave equations with a sup-cubic nonlinearity is considered in locally uniform spaces. We first prove the global well-posedness of the Shatah-Struwe solutions, then we obtain the existence of the H 1 lu (R 3) × L 2 lu (R 3), H 1 ρ (R 3) × L 2 ρ (R 3)-global attractor for the Shatah-Struwe solutions semigroup of this equation. The results are crucially based on the recent extension of Strichartz estimates to the case of bounded domains.

<p style='text-indent:20px;'>This paper gives a detailed study of long-time dynamics generated by a wave equation with time-dependent speed of propagation <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon(t) $\end{document}</tex-math></inline-formula> and time-dependent memory kernel <inline-formula><tex-math id="M2">\begin{document}$ h_{t}(\cdot) $\end{document}</tex-math></inline-formula>. Within the recent theory of process on time-dependent spaces, we first establish the global well-posedness result in extended time-dependent memory space <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{H}_{t} $\end{document}</tex-math></inline-formula>, then we develop further some new time-space (energy) estimates to overcome the mixed difficulties from <inline-formula><tex-math id="M4">\begin{document}$ \varepsilon(t) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ h_{t}(\cdot) $\end{document}</tex-math></inline-formula>, and obtain the dissipativity and regularity of the dynamical process. Furthermore, we study the existence and regularity of the time-dependent global attractor. And, when <inline-formula><tex-math id="M6">\begin{document}$ \varepsilon(t)\rightarrow0 $\end{document}</tex-math></inline-formula> and the time-dependent rescaled kernel approaches a multiple of the Dirac measure as <inline-formula><tex-math id="M7">\begin{document}$ t\rightarrow \infty $\end{document}</tex-math></inline-formula>, we show that the asymptotic structure of time-dependent attractor converges to the attractor of a nonclassical diffusion equation.</p>