Uniform exponential dichotomy and continuity of attractors for singularly perturbed damped wave equations
SM Bruschi, AN Carvalho, JW Cholewa… - Journal of Dynamics and …, 2006 - Springer
SM Bruschi, AN Carvalho, JW Cholewa, T Dlotko
Journal of Dynamics and Differential Equations, 2006•SpringerFor η\geqslant 0, we consider a family of damped wave equations u_ tt+ η Λ^ 1 2 u_t+ a u_t+
Λ u= f (u),\quad t> 0,\quad x ∈ Ω ⊂ R^ N, where− Λ denotes the Laplacian with zero
Dirichlet boundary condition in L 2 (Ω). For a dissipative nonlinearity f satisfying a suitable
growth restrictions these equations define on the phase space H^ 1_0 (Ω) * L^ 2 (Ω)
semigroups {T_ η (t): t\geqslant 0\} which have global attractors A η, η\geqslant 0. We show
that the family {A_ η\} _ η\geqslant 0, behaves upper and lower semicontinuously as the …
Λ u= f (u),\quad t> 0,\quad x ∈ Ω ⊂ R^ N, where− Λ denotes the Laplacian with zero
Dirichlet boundary condition in L 2 (Ω). For a dissipative nonlinearity f satisfying a suitable
growth restrictions these equations define on the phase space H^ 1_0 (Ω) * L^ 2 (Ω)
semigroups {T_ η (t): t\geqslant 0\} which have global attractors A η, η\geqslant 0. We show
that the family {A_ η\} _ η\geqslant 0, behaves upper and lower semicontinuously as the …
Abstract
For , we consider a family of damped wave equations $$u_{tt}+\eta \Lambda^{\frac{1}{2}} u_t + a u_t +\Lambda u = f(u),\quad t > 0, \quad x \in \Omega \subset \mathbb{R}^N$$, where − Λ denotes the Laplacian with zero Dirichlet boundary condition in L 2(Ω). For a dissipative nonlinearity f satisfying a suitable growth restrictions these equations define on the phase space semigroups which have global attractors A η, . We show that the family , behaves upper and lower semicontinuously as the parameter η tends to 0+.
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