We show a perturbation theorem of Myiadera type for linear abstract non-autonomous Cauchy problems. We exhibit an application to a hyperbolic system with dynamic boundary conditions.

In this paper, by using the properties of decay character and the Fourier splitting method, we establish the time decay rates of solutions for three-dimensional incompressible electron inertial Hall-MHD equations. In particular, the optimal decay rates of the mixed space-time derivatives of the solution are obtained.

In this article, we consider the existence of multi-bump solutions for a class of the fractional Schrödinger equation with external magnetic field and critical nonlinearity in $\mathbb{R}^N$: $$(-\Delta)_A^su + (\lambda V(x) + Z(x))u = \beta f(|u|^2)u + |u|^{2_s^\ast-2}u,$$ where $f$ is a continuous function satisfying Ambrosetti-Rabinowitz condition, and $V: \mathbb{R}^N \rightarrow\mathbb{R}$ has a potential well $\Omega := \mbox{int}V^{-1}(0)$ which possesses $k$ disjoint bounded components $\Omega := \cup_{j=1}^k\Omega_j$. By using variational methods, we prove that if the parameter $\lambda > 0$ is large enough, then the equation has at least $2^k-1$ multi-bump solutions.