[Paper Review] Continuity of attractors for a nonlinear parabolic problem with terms concentrating in the boundary

Gleiciane S. Aragão, Antônio Luíz Pereira+1 more|arXiv (Cornell University)|Mar 31, 2012

Stability and Controllability of Differential Equations12 references11 citations

TL;DR

This paper investigates the asymptotic behavior of solutions to a nonlinear parabolic PDE with reaction and potential terms concentrating in a thin, oscillating boundary layer as the thickness parameter $\epsilon \to 0$. Using singular perturbation techniques and spectral analysis, it establishes upper and lower semicontinuity of the global attractors in $H^1(\Omega)$, proving that the attractor of the singular problem converges to that of a limit problem with nonlinear boundary conditions when all equilibria are hyperbolic.

ABSTRACT

We analyze the dynamics of the flow generated by a nonlinear parabolic problem when some reaction and potential terms are concentrated in a neighborhood of the boundary. We assume that this neighborhood shrinks to the boundary as a parameter εgoes to zero. Also, we suppose that the "inner boundary" of this neighborhood presents a highly oscillatory behavior. Our main goal here is to show the continuity of the family of attractors with respect to ε. Indeed, we prove upper semicontinuity under the usual properties of regularity and dissipativeness and, assuming hyperbolicity of the equilibria, we also show the lower semicontinuity of the attractors at ε=0.

Motivation & Objective

• To analyze the dynamics of a nonlinear parabolic problem where reaction and potential terms concentrate in a shrinking, oscillating neighborhood near the boundary.
• To investigate the asymptotic behavior of solutions as the concentration parameter $\epsilon \to 0$.
• To establish the continuity of the global attractors associated with the family of problems parameterized by $\epsilon$.
• To show that the limit problem is a parabolic PDE with nonlinear boundary conditions involving a flux term and potential on $\partial\Omega$.
• To prove lower semicontinuity of the attractors at $\epsilon = 0$ under the assumption of hyperbolic equilibria.

Proposed method

• Model the concentration of reaction and potential terms via a characteristic function $\mathcal{X}_{\omega_\epsilon}$ on a thin, oscillating $\epsilon$-strip $\omega_\epsilon$ near $\partial\Omega$.
• Use the method of periodic unfolding and weak convergence to identify the limit of the concentrated terms, showing $\frac{1}{\epsilon}\int_{\omega_\epsilon} V_\epsilon \varphi \to \int_{\partial\Omega} V_0 \varphi$ as $\epsilon \to 0$.
• Establish uniform bounds on the solutions and their derivatives in $H^1(\Omega)$, ensuring dissipativity and regularity of the semigroup.
• Prove upper semicontinuity of attractors using asymptotic compactness and convergence of the semigroups $T^\epsilon(t)$ to $T^0(t)$ in the strong operator topology.
• Apply spectral convergence results to show that the linearized operators $L_\epsilon(u^*)$ converge in norm to $L_0(u^*)$ as $\epsilon \to 0$, ensuring persistence of hyperbolicity.
• Use the continuity of unstable manifolds near equilibria (Proposition 5.2) and gradient system structure to deduce lower semicontinuity of the attractors (Theorem 5.3).

Experimental results

Research questions

• RQ1 Does the global attractor of the singular parabolic problem with boundary-concentrated terms converge to that of a limit problem with nonlinear boundary conditions as $\epsilon \to 0$?
• RQ2 Under what conditions is the family of attractors $\{\mathscr{A}_\epsilon\}$ upper semicontinuous at $\epsilon = 0$?
• RQ3 Can lower semicontinuity of the attractors be established when all equilibria of the limit problem are hyperbolic?
• RQ4 How does the oscillatory behavior of the concentration region $\omega_\epsilon$ affect the long-term dynamics of the system?
• RQ5 What is the relationship between the spectral properties of the linearized operators at equilibria and the continuity of the attractors?

Key findings

• The family of global attractors $\{\mathscr{A}_\epsilon\}_{\epsilon \in [0,\epsilon_0]}$ is upper semicontinuous at $\epsilon = 0$ in $H^1(\Omega)$, under standard regularity and dissipativity assumptions.
• Under the additional assumption that all equilibria of the limit problem ($\epsilon = 0$) are hyperbolic, the family of attractors is also lower semicontinuous at $\epsilon = 0$ in $H^1(\Omega)$.
• The limit problem is a parabolic PDE with a nonlinear boundary condition: $\frac{\partial u_0}{\partial N} + V_0 u_0 = \mu f(u_0)$ on $\partial\Omega$, where $\mu$ is the average of the oscillating function $g_\epsilon$ over its period.
• The convergence of the semigroups $T^\epsilon(t)$ to $T^0(t)$ is continuous in $H^1(\Omega)$, uniformly on bounded sets of $\mathbb{R}^+ \times H^1(\Omega)$, which supports the attractor continuity.
• The linearized operators $L_\epsilon(u^*)$ converge in operator norm to $L_0(u^*)$ as $\epsilon \to 0$, ensuring that hyperbolic equilibria persist under perturbation.
• The local unstable manifolds of the equilibria $u^*_{\epsilon,i}$ converge to those of $u^*_{0,i}$ in $H^1(\Omega)$ as $\epsilon \to 0$, which is a key ingredient in proving lower semicontinuity of the attractors.