Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation |
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Authors: | Jack K Hale Geneviéve Rauge |
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Institution: | (1) Center for Dynamical Systems and Nonlinear Studies, School of Mathematics, Ga. Tech., 30332 Atlanta, Georgia;(2) Laboratoire d'Analyse Numérique (U.A. au CNRS D760), Université de Paris-Sud, Bât 425, 91405 Orsay Cedex, France |
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Abstract: | For a smooth, bounded domain R, n 3, and a real, positive parameter, we consider the hyperbolic equationu
tt
+u
t
–u=–f(u) –g in with Dirichlet boundary conditions. Under certain conditions onf, this equation has a global attractorA
inH
0
1
() ×L
2(). For=0, the parabolic equation also has a global attractor which can be naturally embedded into a compact setA
0 inH
0
1
() ×L
2(). If all of the equilibrium points of the parabolic equation are hyperbolic, it is shown that the setsA
are lower semicontinuous at=0. Moreover, we give an estimate of the symmetric distance betweenA
0 andA
. |
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Keywords: | Attractors hyperbolic equation singular perturbations lower semicontinuity |
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