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Uniform attractors for a strongly damped wave equation with linear memory. (English) Zbl 0936.35037
The authors consider a strongly damped wave equation with linear memory term: $$u_{tt} (t)-\alpha \Delta u_t(t)- \beta\Delta u(t)-\int^\infty_0 \mu(\zeta) \Delta\eta^t (\zeta) d \zeta+g \bigl(u(t) \bigr)=f(t)\tag 1$$ on $\Omega\times (\tau,\infty)$ with $u(x,t)=0$ on $\partial\Omega \times\bbfR$, where $u(0,t)= u_0(x,t)$ for $t\le\tau$ and $\eta^t(x,s) =u(x,t)- u(x,t-s)$. A variety of conditions are imposed on the data $\mu(\zeta)$, $g(u)$, $f(t)$. After some preparations, (1) is transformed into a system: $$z_t=Lz+N(z),\ z=(u,v,\eta),\ v=u_t,\ z(x,t)= 0\quad\text{on} \quad\partial \Omega\times\bbfR,\ z(x,\tau)= z_0 (x). \tag 2$$ System (2) is then cast into a functional frame. With $A=-\Delta$, the Dirichlet Laplacian, fractional powerspaces $V_{2\sigma}= \text{dom} (A^\sigma)$, $\langle , \rangle_{2\sigma}$ are introduced for $\sigma\in\bbfR$ in the usual way. A weighted Hilbert space $L^2_\mu(R_+, V_\sigma)$ is then defined via $\langle\varphi, \psi\rangle_{\sigma, \mu}=\int^\infty_0 \mu (\zeta) \langle \varphi(\zeta), \psi(\zeta) \rangle_\sigma d\zeta$. The phase space for (2) then is ${\cal V}_\sigma= V_{1+\sigma} \times V_\sigma \times L^2_\mu (\bbfR_+,V_\sigma)$. A function $h\in L^p_{\text{loc}}(\bbfR,V_0)$ $(V_0= L^2 (\Omega))$ is called translation invariant if the closure $H(h)$ of all translates $h(\cdot+ \tau)=h_\tau$, $\tau\in \bbfR$ is compact in $L^p_{\text{loc}} (\bbfR,V_0)$. Finally, a function space $T^p$ is defined, consisting of the $f\in L^p_{\text{loc}}(\bbfR,V_0)$ with norm $\sup(\int^{\tau+ 1}_\tau\|f(y)\|^p dy)^{1/p} <\infty$.

35B41Attractors (PDE)
35L70Nonlinear second-order hyperbolic equations
35L20Second order hyperbolic equations, boundary value problems