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Continuity of attractors for a non-autonomous singularly perturbed hyperbolic equation.
- Speaker(s)
- Piotr Kalita
- Affiliation
- Uniwersytet Jagielloński
- Date
- Nov. 17, 2016, 12:30 p.m.
- Room
- room 4060
- Seminar
- Seminar of Mathematical Physics Equations Group
We study the non-autonomous dynamical system given by
hyperbolic equation on a bounded domain $\Omega\subset\mathbb{R}^3$
$$
\epsilon u_{tt}+u_t-\Delta u=f_\epsilon(t,u).
$$
This dynamical system has uniform, pullback, and cocycle attractors.
For $\epsilon=0$ the limit parabolic equation
$$
u_t-\Delta u=f_0(u)
$$
has a global attractor $A_0$ in $H_0^1(\Omega)$ which can be naturally
embedded into a compact set $\mathcal{A}_0$ in $H_0^1(\Omega)\times
L^2(\Omega)$. We prove that all three types of non-autonomous
attractors converge, both upper- and lower-semicontinuously to
$\mathcal{A}_0$ as $\epsilon\to 0$.
hyperbolic equation on a bounded domain $\Omega\subset\mathbb{R}^3$
$$
\epsilon u_{tt}+u_t-\Delta u=f_\epsilon(t,u).
$$
This dynamical system has uniform, pullback, and cocycle attractors.
For $\epsilon=0$ the limit parabolic equation
$$
u_t-\Delta u=f_0(u)
$$
has a global attractor $A_0$ in $H_0^1(\Omega)$ which can be naturally
embedded into a compact set $\mathcal{A}_0$ in $H_0^1(\Omega)\times
L^2(\Omega)$. We prove that all three types of non-autonomous
attractors converge, both upper- and lower-semicontinuously to
$\mathcal{A}_0$ as $\epsilon\to 0$.
