2024/08/13 by Bianca Paolini Lorenzi, Lorenzi, Bianca P., Antônio Luíz Pereira +1 · 1 citation
Engineering · Mathematics · Computer Science · #Stability and Controllability of Differential Equations #Advanced Differential Equations and Dynamical Systems #Nonlinear Dynamics and Pattern Formation
paper · pdf · doi:10.48550/arxiv.2408.07204
Consider the family of semilinear parabolic problems \ ut(x,t) = Δu(x,t) - au(x,t) + f(u(x,t)), x ∈ Ωε, t · gt; 0,
(∂ u)/(∂ N) (x,t) = g(u(x,t)), x ∈ ∂ Ωε, t · gt; 0, . where a > 0, Ω is the unit square, Ωε = hε(Ω), hε is a family of Cm - diffeomorphisms, m ≥ 1, which converge to the identity of Ω in Cα norm, if α<1 but do not converge in the C1 - norm and, f,g: ℝ → ℝ are real functions. We show that a weak version of this problem, transported to the fixed domain Ω by a ``pull-back'' procedure, is well posed for 0 <ε≤ ε0, ε0 > 0, in a suitable phase space, the associated semigroup has a global attractor Aε and the family \ Aε \_0 < ε ≤ ε0 converges as ε→ 0 to the attractor of the limiting problem: \ ut(x,t) = Δu(x,t) - au(x,t) + f(u(x,t)), x ∈ Ω, t · gt; 0,
(∂ u)/(∂ N) (x,t) = g(u(x,t))μ, x ∈ ∂ Ω, t · gt; 0, . where μ is essentially the limit of the Jacobian determinant of the diffeomorphism hε| ∂ Ω : ∂ Ω→ ∂ hε(Ω) (but does not depend on the particular family hε).