In this paper we consider the semilinear damped wave problem of the form$\{\begin{array}{cc}{(\alpha (t)u_t)}_t-\beta (t)\mathrm{\Delta }u+\gamma (t)u_t+\delta (t)u=\beta (t)f(u),\hfill & x\in \mathrm{\Omega },t>\tau ,\hfill \\ u(x,t)=0,\hfill & x\in \partial \mathrm{\Omega },t⩾\tau ,\hfill \\ u(x,\tau )=u_{\tau }(x),u_t(x,\tau )=v_{\tau }(x),\hfill & x\in \mathrm{\Omega },\hfill \end{array}$ where Ω is a bounded smooth domain in ${\mathbb{R}}^N$, $N⩾3$, $\tau \in \mathbb{R}$, f is a real valued function of a real variable with some suitable conditions of growth, regularity and dissipativity, and $\alpha ,\beta ,\gamma$ and δ are continuous real valued functions of a real variable with some suitable conditions of growth, regularity and signs. Using rescaling of time we prove existence, regularity, gradient-like structure, upper and lower semicontinuity of the pullback attractors for the evolution processes associated with this boundary initial value problem in a suitable phase space.

在本文中，我们考虑以下形式的半线性阻尼波问题$\{\begin{array}{cc}{(\alpha (吨){你}_{吨})}_{吨}-\beta (吨)\mathrm{\Delta }你+\gamma (吨){你}_{吨}+\delta (吨)你=\beta (吨)F(你),\hfill & X\in \mathrm{\Omega },吨>\tau ,\hfill \\ 你(X,吨)=0,\hfill & X\in \partial \mathrm{\Omega },吨⩾\tau ,\hfill \\ 你(X,\tau )={你}_{\tau }(X),{你}_{吨}(X,\tau )=v_{\tau }(X),\hfill & X\in \mathrm{\Omega },\hfill \end{array}$ 其中 Ω 是有界光滑域 ${\mathbb{电阻}}^N$, $N⩾3$, $\tau \in \mathbb{电阻}$, f是具有一些合适的增长、规律性和耗散性条件的实变量的实值函数，并且$\alpha ,\beta ,\gamma$和δ是具有一些合适的增长、规律和符号条件的实变量的连续实值函数。使用时间的重新缩放，我们证明了在合适的相空间中与此边界初始值问题相关的演化过程的回拉吸引子的存在性、规律性、类梯度结构、上下半连续性。