The existence of time-dependent attractor for wave equation with fractional damping and lower regular forcing term

Abstract

<p style='text-indent:20px;'>We investigate the well-posedness and longtime dynamics of fractional damping wave equation whose coefficient <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> depends explicitly on time. First of all, when <inline-formula><tex-math id="M2">\begin{document}$ 1\leq p\leq p^{\ast\ast} = \frac{N+2}{N-2}\; (N\geq3) $\end{document}</tex-math></inline-formula>, we obtain existence of solution for the fractional damping wave equation with time-dependent decay coefficient in <inline-formula><tex-math id="M3">\begin{document}$ H_{0}^{1}(\Omega)\times L^{2}(\Omega) $\end{document}</tex-math></inline-formula>. Furthermore, when <inline-formula><tex-math id="M4">\begin{document}$ 1\leq p<p^{*} = \frac{N+4\alpha}{N-2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ u_{t} $\end{document}</tex-math></inline-formula> is proved to be of higher regularity in <inline-formula><tex-math id="M6">\begin{document}$ H^{1-\alpha}\; (t>\tau) $\end{document}</tex-math></inline-formula> and show that the solution is quasi-stable in weaker space <inline-formula><tex-math id="M7">\begin{document}$ H^{1-\alpha}\times H^{-\alpha} $\end{document}</tex-math></inline-formula>. Finally, we get the existence and regularity of time-dependent attractor.</p>

Asymptotic Regularity and Existence of Time-Dependent Attractors for Second-Order Undamped Evolution Equations with Memory

“…By using the ideas in [16,17], some breakthrough progress was made in the research of the existence of time-dependent attractors and the regularity of solutions for the evolution equation problems. The semilinear wave equations have been treated in many papers, see, for example, [16,[18][19][20][21]. Conti et al [16] proved the existence and regularity of a timedependent attractor, and they [18] obtained the asymptotic structure of a time-dependent attractor.…”