数学与统计学院知识库

本文考虑如下时滞非经典扩散方程
u_{t}-\triangle u_{t}-\triangle u+f(u)=g(t,u^{t})+h(t),(x,t)\in\mathbb{R}^{N}\times[s,+\infty),
u(x,s+\theta)=\phi(x,\theta), x\in\mathbb{R}^{N}, \theta\in[-\ell,0],
在局部一致空间中解的长时间行为. 首先, 我们给出该方程弱解的定义及全局适定性; 然后给出局部一致空间中拉回吸引子的定义及存在性定理; 为了得到过程的拉回渐近紧性和渐近正则性, 我们分解该方程的弱解, 并要求临界非线性项也满足适当的分解条件; 最后, 我们得到(C_{\dot{H}^{1}_{lu}(\mathbb{R}^{N})},C_{H^{1}_{\rho}(\mathbb{R}^{N})})-拉回吸引子, 进一步可证明它是C_{H^{2}_{lu}(\mathbb{R}^{N})}中的一个有界子集, 且按C_{H^{1}_{\rho}(\mathbb{R}^{N})}-范数拉回吸引C_{\dot{H}^{1}_{lu}(\mathbb{R}^{N})}中的有界集.

In this paper, we consider the long-time behavior of the solutions for the following nonclassical diffusion equation with delays in locally uniform spaces
u_{t}-\triangle u_{t}-\triangle u+f(u)=g(t,u^{t})+h(t), \text{in}\mathbb{R}^{N}\times[s,+\infty),
u(x,s+\theta)=\phi(x,\theta), x\in\mathbb{R}^{N}, \theta\in[-\ell,0].
We first give the definition and the global well-posedness of weak solutions. Then we establish the definition and existence theorem of the pullback attractors in locally uniform spaces. In order to obtain pullback asymptotic compactness and asymptotic regularity for process, we decompose the solution for the above equation with initial conditions and the critical nonlinearity is required to have proper decomposition conditions. Finally we show there is a (C_{\dot{H}^{1}_{lu}(\mathbb{R}^{N})},C_{H^{1}_{\rho}(\mathbb{R}^{N})})-pullback attractor, which is a bounded (in C_{H^{2}_{lu}(\mathbb{R}^{N})}) subset and pullback attracts bounded subsets of C_{\dot{H}^{1}_{lu}(\mathbb{R}^{N})} with respect to the C_{H^{1}_{\rho}(\mathbb{R}^{N})}-norm.