arXiv subjects

Pricila S. Barbosa

Publications and source records attributed to Pricila S. Barbosa.

3 recordsLinked to original sources

arXiv2024

Elliptic Equations in Weak Oscillatory Thin Domains: Beyond Periodicity with Boundary-Concentrated Reaction Terms

In this paper we analyze the limit behavior of a family of solutions of the Laplace operator with homogeneous Neumann boundary conditions, set in a two-dimensional thin domain which presents weak oscillations on both boundaries and with terms concentrated in a narrow oscillating neighborhood of the top boundary. The aim of this problem is to study the behavior of the solutions as the thin domain presents oscillatory behaviors beyond the classical periodic assumptions,including scenarios like quasiperiodic or almost-periodic oscillations. We then prove that the family of solutions converges to the solution of a 1 dimensional limit equation capturing the geometry and oscillatory behavior of boundary of the domain and the narrow strip where the concentration terms take place. In addition, we include a series of numerical experiments illustrating the theoretical results obtained in the quasiperiodic context.

math.AP↗

arXiv2019

Continuity of attractors for $\mathcal{C}^1$ perturbations of a smooth domain

We consider a family of semilinear parabolic problems with nonlinear boundary conditions \[ \left\{ \begin{aligned} u_t(x,t) &=Δu(x,t) -au(x,t) + f(u(x,t)),\ x \in Ω_ε\mbox{ and } t>0\,,\\ \displaystyle\frac{\partial u}{\partial N}(x,t) &=g(u(x,t)),\ x \in \partialΩ_ε\mbox{ and } t>0\,, \end{aligned} \right. \] where $Ω_0 \subset \mathbb{R}^n$ is a smooth (at least $\mathcal{C}^2$) domain , $Ω_ε = h_ε(Ω_0)$ and $h_ε$ is a family of diffeomorphisms converging to the identity in the $\mathcal{C}^1$-norm. Assuming suitable regularity and dissipative conditions for the nonlinearites, we show that the problem is well posed for $ε>0$ sufficiently small in a suitable scale of fractional spaces, the associated semigroup has a global attractor $\mathcal{A}_ε$ and the family $\{\mathcal{A}_ε\}$ is continuous at $ε= 0$.

math.DS↗

arXiv2016

Continuity of attractors for a family of $C^1$ perturbations of the square

We consider here the family of semilinear parabolic problems \begin{equation*} \begin{array}{rcl} \left\{ \begin{array}{rcl} u_t(x,t)&=&Δu(x,t) -au(x,t) + f(u(x,t)) ,\,\,\ x \in Ω_ε\,\,\,\mbox{and}\,\,\,\,\,\,t>0\,, \\ \displaystyle\frac{\partial u}{\partial N}(x,t)&=&g(u(x,t)), \,\, x \in \partialΩ_ε\,\,\,\mbox{and}\,\,\,\,\,\,t>0\,, \end{array} \right. \end{array} \end{equation*} where $ Ω $ is the unit square, $Ω_ε=h_ε(Ω)$ and $h_ε$ is a family of diffeomorphisms converging to the identity in the $C^1$-norm. We show that the problem is well posed for $ε>0$ sufficiently small in a suitable phase space, the associated semigroup has a global attractor $\mathcal{A}_ε$ and the family $\{\mathcal{A}_ε\}$ is continuous at $ε= 0$.

math.DS↗