Pengyan Ding | Search arXiv

arXiv2022

Long-time dynamical behavior for a piezoelectric system with magnetic effect and nonlinear dampings

This paper is concerned with the long-time dynamical behavior of a piezoelectric system with magnetic effect, which has nonlinear damping terms and external forces with a parameter. At first, we use the nonlinear semigroup theory to prove the well-posedness of solutions. Then, we investigate the properties of global attractors and the existence of exponential attractors. Finally, the upper semicontinuity of global attractors has been investigated.

math.AP↗

arXiv2019

Global attractors and their upper semicontinuity for a structural damped wave equation with supercritical nonlinearity on $\mathbb{R}^{N}$

The paper investigates the existence of global attractors and their upper semicontinuity for a structural damped wave equation on $\mathbb{R}^{N}: u_{tt}-Δu+(-Δ)^αu_{t}+u_{t}+u+g(u)=f(x)$, where $α\in (1/2, 1)$ is called a dissipative index. We propose a new method based on the harmonic analysis technique and the commutator estimate to exploit the dissipative effect of the structural damping $(-Δ)^αu_{t}$ and to overcome the essential difficulty: "both the unbounded domain $\mathbb{R}^N$ and the supercritical nonlinearity cause that the Sobolev embedding loses its compactness"; Meanwhile we show that there exists a supercritical index $p_α\equiv\frac{N+4α}{N-4α}$ depending on $α$ such that when the growth exponent $p$ of the nonlinearity $g(u)$ is up to the supercritical range: $1\leqslant p 0$; (ii) the related solution semigroup possesses a global attractor $\mathcal{A}_α$ in natural energy space for each $α\in (1/2, 1)$; (iii) the family of global attractors $\{\mathcal{A}_α\}_{α\in (1/2, 1) }$ is upper semicontinuous at each point $α_0\in (1/2, 1)$.

math.AP↗