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Flank D. M. Bezerra

Publications and source records attributed to Flank D. M. Bezerra.

15 recordsLinked to original sources
arXiv2025

Local well-posedness for a class of semilinear Moore-Gibson-Thompson equations with subcritical nonlinearities

In this paper, we study a class of higher-order semilinear evolution equations inspired by the Moore-Gibson-Thompson model introduced by Dell'Oro, Liverani and Pata (2023), involving strongly elliptic operators of order ($2m$) with homogeneous boundary conditions. The associated unbounded linear operator is a sectorial operator with zero belonging to the resolvent set, allowing the construction of fractional powers spaces, and the analysis of their spectral properties. We prove the local well-posedness of the corresponding semilinear Cauchy problem under subcritical nonlinearities. Our framework clarifies the role of extrapolation spaces and fractional domains in handling the lack of accretivity and bounded imaginary powers of the operator.

math.DS↗
arXiv2025

Non-autonomous problem for a $2m$-th order semilinear nonlocal parabolic equation

In this paper we consider a $2m$-th order non autonomous quasilinear parabolic equation. Under suitable conditions of growth and regularity for the nonlinear functions present in the model, we prove a result of existence and characterization of pullback attractors. Moreover, we consider an autonomous version from the $2m$-th order non autonomous quasilinear parabolic equation in question.

math.AP↗
arXiv2025

Pullback dynamics for a semilinear heat equation with homogeneous Neumann boundary conditions on time-varying domains

We are interested in studying a non-autonomous semilinear heat equation with homogeneous Neumann boundary conditions on time-varying domains. Using a differential geometry approach with coordinate transformations technique, we will show that the non-autonomous problem on a time-varying domain is equivalent, in some sense, to a non-autonomous problem on a fixed domain. Furthermore, we intend to show the local existence and uniqueness of solutions to this problem, as well as, to extend these solutions globally. Finally, we will show the existence of pullback attractors. To the best of our knowledge, results on attractors are new even for non-autonomous semilinear heat equations with homogeneous Neumann boundary conditions on time-varying domains subject to conditions with more restrictive assumptions

math.AP↗
arXiv2024

Upper semicontinuity for a class of nonlocal evolution equations with Neumann condition

In this paper we consider the following nonlocal autonomous evolution equation in a bounded domain $Ω$ in $\mathbb{R}^N$ \[ \partial_t u(x,t) =- h(x)u(x,t) + g \Big(\int_Ω J(x,y)u(y,t)dy \Big) +f(x,u(x,t)) \] where $h\in W^{1,\infty}(Ω)$, $g: \mathbb{R} \to \mathbb{R}$ and $f:\mathbb{R}^N\times\mathbb{R} \to \mathbb{R}$ are continuously differentiable function, and $J$ is a symmetric kernel; that is, $J(x,y)=J(y,x)$ for any $x,y\in\mathbb{R}^N$. Under additional suitable assumptions on $f$ and $g$, we study the asymptotic dynamics of the initial value problem associated to this equation in a suitable phase spaces. More precisely, we prove the existence, and upper semicontinuity of compact global attractors with respect to kernel $J$.

math.AP↗
arXiv2021

Fractional powers approach of operators for higher order abstract Cauchy problems

In this paper we explore the theory of fractional powers of non-negative (and not necessarily self-adjoint) operators and its amazing relationship with the Chebyshev polynomials of the second kind to obtain results of existence, regularity and behavior asymptotic of solutions for linear abstract evolution equations of $n$-th order in time, where $n\geqslant3$. We also prove generalizations of classical results on structural damping for linear systems of differential equations.

math.AP↗
arXiv2021

Well-posedness for some third-order evolution differential equations: A semigroup approach

In this paper, we discuss the well-posedness of the Cauchy problem associated with the third-order evolution equation in time $$ u_{ttt} +A u + ηA^{\frac13} u_{tt} +ηA^{\frac23} u_t=f(u) $$ where $η>0$, $X$ is a separable Hilbert space, $A:D(A)\subset X\to X$ is an unbounded sectorial operator with compact resolvent, and for some $λ_0>0$ we have $\mbox{Re}σ(A)>λ_0$ and $f:D(A^{\frac13})\subset X\to X$ is a nonlinear function with suitable conditions of growth and regularity.

math.AP↗
arXiv2020

Fractional oscillon equations; solvability and connection with classical oscillon equations

In this paper we are concerned with the asymptotic behavior of nonautonomous fractional approximations of oscillon equation $$ u_{tt}-μ(t)Δu+ω(t)u_t=f(u),\ x\inΩ,\ t\in\mathbb{R}, $$ subject to Dirichlet boundary condition on $\partial Ω$, where $Ω$ is a bounded smooth domain in $\mathbb{R}^N$, $N\geqslant 3$, the function $ω$ is a time-dependent damping, $μ$ is a time-dependent squared speed of propagation, and $f$ is a nonlinear functional. Under structural assumptions on $ω$ and $μ$ we establish the existence of time-dependent attractor for the fractional models in the sense of Carvalho, Langa, Robinson \cite{CLR}, and Di Plinio, Duane, Temam \cite{DDT1}.

math.AP↗
arXiv2018

Continuity of the set equilibria of non-autonomous damped wave equations with terms concentrating on the boundary

In this paper we are interested in the behavior of the solutions of non-autonomous damped wave equations when some reaction terms are concentrated in a neighborhood of the boundary and this neighborhood shrinks to boundary as a parameter \varepsilon goes to zero. We prove the conti- nuity of the set equilibria of these equations. Moreover, if an equilibrium solution of the limit problem is hyperbolic, then we show that the per- turbed equation has one and only one equilibrium solution nearby.

math.AP↗
arXiv2016

Pullback attractors for a class of non-autonomous thermoelastic plate systems

In this article we study the asymptotic behavior of solutions, in sense of global pullback attractors, of the evolution system $$ \begin{cases} u_{tt} +ηΔ^2 u+a(t)Δθ=f(t,u), & t>τ,\ x\inΩ,\\ θ_t-κΔθ-a(t)Δu_t=0, & t>τ,\ x\inΩ, \end{cases} $$ subject to boundary conditions $$ u=Δu=θ=0,\ t>τ,\ x\in\partialΩ, $$ where $Ω$ is a bounded domain in $\mathbb{R}^N$ with $N\geqslant 2$, which boundary $\partialΩ$ is assumed to be a $\mathcal{C}^4$-hypersurface, $η>0$ and $κ>0$ are constants, $a$ is an Hölder continuous function, $f$ is a dissipative nonlinearity locally Lipschitz in the second variable.

math.AP↗
arXiv2014

Pullback attractor for a non local non-autonomous evolution equation in an unbounded domain

In this work we consider the non local evolution equation with time-dependent terms which arises in models of phase separation in $\mathbb{R}^N$ \[ \partial_t u=- u + g \left(β(J*u) +βh(t,u)\right) \] under some restrictions on $h$, growth restrictions on the nonlinear term $g$ and $β>1$. We prove, under suitable assumptions, existence, regularity and upper-semicontinuity of pullback attractors with respect to functional parameter $h(t)$ in some weighted spaces.

math.DS↗