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Published in:
2013 | OriginalPaper | Chapter
14. Perturbation of diffusion and continuity of global attractors with rate of convergence
Authors : Alexandre N. Carvalho, José A. Langa, James C. Robinson
Published in: Attractors for infinite-dimensional non-autonomous dynamical systems
Publisher: Springer New York
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Abstract
In this chapter we consider a parabolic problem in which the diffusion coefficient depends on a parameter, The final goal in this direction would be to compare the asymptotic dynamics of systems with different ‘parameter values’ by comparing their attractors and the flow on them. We assume that \(m \leq {a}_{\epsilon }(x) \leq M\) and converges to a continuously differentiable function \({a}_{0} : \Omega \rightarrow \mathbb{R}\) as ε goes to zero. For this problem we prove that the attractors are continuous at ε = 0 and can be characterised as the union of the unstable sets of its global hyperbolic solutions.
$${u}_{t} - {({a}_{\epsilon }(x){u}_{x})}_{x} = f(u).$$