We consider, for $\rho\in[0,1]$ and $\varepsilon>0$ small, the nonautonomous weakly damped wave equation with a singularly oscillating external force $$ \partial _{t}^{2}u-\Delta u+\gamma \partial_{t}u =-f(u)+g_{0}(t)+\varepsilon ^{-\rho }g_{1}(t/\varepsilon ), $$ together with the {\it averaged} equation $$ \partial _{t}^{2}u-\Delta u+\gamma \partial_{t}u =-f(u)+g_{0}(t). $$ Under suitable assumptions on the nonlinearity and the external force, we prove the uniform (w.r.t.\ $\varepsilon$) boundedness of the attractors $\mathcal{A}^\varepsilon$ in the weak energy space. If $\rho<1$, we establish the convergence of the attractor $\mathcal{A}^\varepsilon$ of the first equation to the attractor $\mathcal{A}^0$ of the second one, as $\varepsilon\to 0^+$. On the other hand, if $\rho=1$, this convergence may fail. When $\mathcal{A}^0$ is exponential, then the convergence rate of $\mathcal{A}^\varepsilon$ to $\mathcal{A}^0$ is controlled by $M\varepsilon^\eta$, for some $M\geq 0$ and some $\eta=\eta(\rho)\in(0,1)$.

Averaging of nonautonomous damped wave equations with singularly oscillating external forces

PATA, VITTORINO;
2008-01-01

Abstract

We consider, for $\rho\in[0,1]$ and $\varepsilon>0$ small, the nonautonomous weakly damped wave equation with a singularly oscillating external force $$ \partial _{t}^{2}u-\Delta u+\gamma \partial_{t}u =-f(u)+g_{0}(t)+\varepsilon ^{-\rho }g_{1}(t/\varepsilon ), $$ together with the {\it averaged} equation $$ \partial _{t}^{2}u-\Delta u+\gamma \partial_{t}u =-f(u)+g_{0}(t). $$ Under suitable assumptions on the nonlinearity and the external force, we prove the uniform (w.r.t.\ $\varepsilon$) boundedness of the attractors $\mathcal{A}^\varepsilon$ in the weak energy space. If $\rho<1$, we establish the convergence of the attractor $\mathcal{A}^\varepsilon$ of the first equation to the attractor $\mathcal{A}^0$ of the second one, as $\varepsilon\to 0^+$. On the other hand, if $\rho=1$, this convergence may fail. When $\mathcal{A}^0$ is exponential, then the convergence rate of $\mathcal{A}^\varepsilon$ to $\mathcal{A}^0$ is controlled by $M\varepsilon^\eta$, for some $M\geq 0$ and some $\eta=\eta(\rho)\in(0,1)$.
2008
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/518938
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