Time-dependent global attractor for extensible Berger equation

In this paper, we investigate the asymptotic behavior of the nonautonomous Berger equation

$\epsilon (t)u_{tt}+{\mathrm{\Delta }}^{2}u?(Q+\underset{\mathrm{\Omega }}{\int }|\mathrm{?}u{|}^{2}dx)\mathrm{\Delta }u+g(u_t)+\phi (u)=f,t>\tau ,$

on a bounded smooth domain $\mathrm{\Omega }?{\mathbb{R}}^N$ with hinged boundary condition, where $\epsilon (t)$ is a decreasing function vanishing at infinity. Under suitable assumptions, we establish an invariant time-dependent global attractor within the theory of process on time-dependent space.     

On Hr(curl,Ω) estimates for a Maxwell type system in convex domains

In bounded convex domains, the regularity of a vector field u with its $\mathrm{div}\mathbf{u}$, $\mathrm{curl}\mathbf{u}$ in $L^r$ space and the tangential component or the normal component of u over the boundary in $L^r$ space, is established for $1<r<\infty$. As an application, we derive an $H^r(\mathrm{curl},\mathrm{\Omega })$ estimate for solutions to a Maxwell type system with an inhomogeneous boundary condition in convex domains. In contrast to the well-posed region of r in the space $H^r(\mathrm{curl},\mathrm{\Omega })$ for the Maxwell type system in Lipschitz domains given by Kar and Sini (2016) [16], we extend the well-posed region to be optimal.