Rate of convergence of global attractors of some perturbed reaction-diffusion problems
2013•projecteuclid.org
In this paper we treat the problem of the rate of convergence of attractors of dynamical
systems for some autonomous semilinear parabolic problems. We consider a prototype
problem, where the diffusion a_0(⋅) of a reaction-diffusion equation in a bounded domain Ω
is perturbed to a_ε(⋅). We show that the equilibria and the local unstable manifolds of the
perturbed problem are at a distance given by the order of ‖a_ε-a_0‖_∞. Moreover, the
perturbed nonlinear semigroups are at a distance ‖a_ε-a_0‖_∞^θ with θ<1 but arbitrarily …
systems for some autonomous semilinear parabolic problems. We consider a prototype
problem, where the diffusion a_0(⋅) of a reaction-diffusion equation in a bounded domain Ω
is perturbed to a_ε(⋅). We show that the equilibria and the local unstable manifolds of the
perturbed problem are at a distance given by the order of ‖a_ε-a_0‖_∞. Moreover, the
perturbed nonlinear semigroups are at a distance ‖a_ε-a_0‖_∞^θ with θ<1 but arbitrarily …
Abstract
In this paper we treat the problem of the rate of convergence of attractors of dynamical systems for some autonomous semilinear parabolic problems. We consider a prototype problem, where the diffusion of a reaction-diffusion equation in a bounded domain is perturbed to . We show that the equilibria and the local unstable manifolds of the perturbed problem are at a distance given by the order of . Moreover, the perturbed nonlinear semigroups are at a distance with but arbitrarily close to . Nevertheless, we can only prove that the distance of attractors is of order for some , which depends on some other parameters of the problem and may be significantly smaller than . We also show how this technique can be applied to other more complicated problems.
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