12 research outputs found

    Averaging of equations of viscoelasticity with singularly oscillating external forces

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    Given ρ[0,1]\rho\in[0,1], we consider for ε(0,1]\varepsilon\in(0,1] the nonautonomous viscoelastic equation with a singularly oscillating external force ttuκ(0)Δu0κ(s)Δu(ts)ds+f(u)=g0(t)+ερg1(t/ε) \partial_{tt} u-\kappa(0)\Delta u - \int_0^\infty \kappa'(s)\Delta u(t-s) d s +f(u)=g_{0}(t)+\varepsilon ^{-\rho }g_{1}(t/\varepsilon ) together with the {\it averaged} equation ttuκ(0)Δu0κ(s)Δu(ts)ds+f(u)=g0(t). \partial_{tt} u-\kappa(0)\Delta u - \int_0^\infty \kappa'(s)\Delta u(t-s) d s +f(u)=g_{0}(t). Under suitable assumptions on the nonlinearity and on the external force, the related solution processes Sε(t,τ)S_\varepsilon(t,\tau) acting on the natural weak energy space H{\mathcal H} are shown to possess uniform attractors Aε{\mathcal A}^\varepsilon. Within the further assumption ρ<1\rho<1, the family Aε{\mathcal A}^\varepsilon turns out to be bounded in H{\mathcal H}, uniformly with respect to ε[0,1]\varepsilon\in[0,1]. The convergence of the attractors Aε{\mathcal A}^\varepsilon to the attractor A0{\mathcal A}^0 of the averaged equation as ε0\varepsilon\to 0 is also established

    Attractors for equations of mathematical physics

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    One of the major problems in the study of evolution equations of mathematical physics is the investigation of the behavior of the solutions to these equations when time is large or tends to infinity. The related important questions concern the stability of solutions or the character of the instability if a solution is unstable. In the last few decades, considerable progress in this area has been achieved in the study of autonomous evolution partial differential equations. For a number of basic evolution equations of mathematical physics, it was shown that the long time behavior of their solut

    Attractors of non-autonomous partial differential equations and their dimension

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    Trajectory and global attractors of dissipative hyperbolic equations with memory

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    SIGLEAvailable from INIST (FR), Document Supply Service, under shelf-number : 22466, issue : a.2003 n.179 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc