Existence of the uniform attractors for a non-autonomous modified Swift-Hohenberg equation

摘要

The paper is concerned with the non-autonomous modified Swift-Hohenberg equation u t + △ 2 u + 2 △ u + a u + b | ∇ u | 2 + u 3 = g ( x , t ) $u_{t}+{riangle }^{2}u+2{riangle }u+au+b|abla u|^{2}+u^{3}=g(x,t)$ . It is shown that a uniform attractor exists in H 0 2 $H_{0}^{2}$ when the external force only satisfies the translation bounded condition instead of translation compactness. In order to overcome the difficulty caused by the critical nonlinearity terms u 3 $u^{3}$ and the parameter b belonging to the real set R $mathbb{R}$ , we take advantage of the Gagliardo-Nirenberg inequality several times.