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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:

${u}_{tt}\left(t\right)-\alpha {\Delta }{u}_{t}\left(t\right)-\beta {\Delta }u\left(t\right)-{\int }_{0}^{\infty }\mu \left(\zeta \right){\Delta }{\eta }^{t}\left(\zeta \right)d\zeta +g\left(u\left(t\right)\right)=f\left(t\right)\phantom{\rule{2.em}{0ex}}\left(1\right)$

on ${\Omega }×\left(\tau ,\infty \right)$ with $u\left(x,t\right)=0$ on $\partial {\Omega }×ℝ$, where $u\left(0,t\right)={u}_{0}\left(x,t\right)$ for $t\le \tau$ and ${\eta }^{t}\left(x,s\right)=u\left(x,t\right)-u\left(x,t-s\right)$. A variety of conditions are imposed on the data $\mu \left(\zeta \right)$, $g\left(u\right)$, $f\left(t\right)$. After some preparations, (1) is transformed into a system:

${z}_{t}=Lz+N\left(z\right),\phantom{\rule{4pt}{0ex}}z=\left(u,v,\eta \right),\phantom{\rule{4pt}{0ex}}v={u}_{t},\phantom{\rule{4pt}{0ex}}z\left(x,t\right)=0\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{1.em}{0ex}}\partial {\Omega }×ℝ,\phantom{\rule{4pt}{0ex}}z\left(x,\tau \right)={z}_{0}\left(x\right)·\phantom{\rule{2.em}{0ex}}\left(2\right)$

System (2) is then cast into a functional frame. With $A=-{\Delta }$, the Dirichlet Laplacian, fractional powerspaces ${V}_{2\sigma }=\text{dom}\left({A}^{\sigma }\right)$, ${〈,〉}_{2\sigma }$ are introduced for $\sigma \in ℝ$ in the usual way. A weighted Hilbert space ${L}_{\mu }^{2}\left({R}_{+},{V}_{\sigma }\right)$ is then defined via ${〈\varphi ,\psi 〉}_{\sigma ,\mu }={\int }_{0}^{\infty }\mu \left(\zeta \right){〈\varphi \left(\zeta \right),\psi \left(\zeta \right)〉}_{\sigma }d\zeta$. The phase space for (2) then is ${𝒱}_{\sigma }={V}_{1+\sigma }×{V}_{\sigma }×{L}_{\mu }^{2}\left({ℝ}_{+},{V}_{\sigma }\right)$.

A function $h\in {L}_{\text{loc}}^{p}\left(ℝ,{V}_{0}\right)$ $\left({V}_{0}={L}^{2}\left({\Omega }\right)\right)$ is called translation invariant if the closure $H\left(h\right)$ of all translates $h\left(·+\tau \right)={h}_{\tau }$, $\tau \in ℝ$ is compact in ${L}_{\text{loc}}^{p}\left(ℝ,{V}_{0}\right)$. Finally, a function space ${T}^{p}$ is defined, consisting of the $f\in {L}_{\text{loc}}^{p}\left(ℝ,{V}_{0}\right)$ with norm $sup\left({\int }_{\tau }^{\tau +1}{\parallel f\left(y\right)\parallel }^{p}{dy\right)}^{1/p}<\infty$.

##### MSC:
 35B41 Attractors (PDE) 35L70 Nonlinear second-order hyperbolic equations 35L20 Second order hyperbolic equations, boundary value problems
##### Keywords:
thermoelasticity; weighted Hilbert space