Rate of convergence of global attractors of some perturbed reaction-diffusion problems. (English) Zbl 1331.35053

Arrieta, José M.; Bezerra, Flank D. M.; Carvalho, Alexandre N.

Topol. Methods Nonlinear Anal. 41, No. 2, 229-253 (2013).

The authors consider a prototype problem: \[ \begin{alignedat}{2} u_t^\varepsilon-\text{div}(a_\varepsilon(x)\nabla u^\varepsilon & =f(u^\varepsilon)\quad & \text{in }&(0,+\infty) \times \Omega,\\ u^\varepsilon(t,x) & =0 \quad &\text{in }& (0,+\infty) \times \partial\Omega,\\ u^\varepsilon(0,x)& =u_0^\varepsilon(x) \quad &\text{in }&\Omega.\end{alignedat} \] In this problem \(\Omega\) is a bounded, regular domain in \(\mathbb{R}^{N}\) with \(N \geq 2\), \(\varepsilon \in [0,1]\) is the perturbation parameter and \(f\) is a continuously differentiable, dissipative nonlinearity. As \(\varepsilon\) goes to zero, the diffusivity \(a_{\varepsilon}\) converges to \(a_0\) (the undisturbed diffusion coefficient) uniformly in \(\Omega\).
Quoting the authors in their introduction: “…we investigate the rate of convergence of the attractors of some gradient problems under (singular) perturbation. Our aim is to obtain the rate of convergence of attractors in terms of the rate of convergence of the semigroups and the later in terms of the parameters in the corresponding models”.

Reviewer: Jauber C. Oliveira (Florianopolis)

MSC:

35B41	Attractors
37L30	Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35K57	Reaction-diffusion equations
35B25	Singular perturbations in context of PDEs

Keywords:

attractors; rate of convergence; reaction-diffusion; singular perturbation