arXiv subjects

Severino H. da Silva

Publications and source records attributed to Severino H. da Silva.

6 recordsLinked to original sources

arXiv2025

Asymptotic behavior for a class of damped second-order gradient systems via Lyapunov method

In this work we study the asymptotic behavior of a class of damped second-order gradient systems $$ \ddot{u}(t) + a\dot{u}(t) + \nabla W(u(t)) = 0, $$ under assumptions ensuring local convexity of the potential near equilibrium and coercivity at infinity. By introducing a Lyapunov functional adapted to the geometry of the system, we establish uniform asymptotic stability of the equilibrium for all $a \in (0,a_0]$, together with exponential decay when the potential satisfies a quadratic control near its minimum. Furthermore, complementary arguments based on semigroup theory reveal the existence of a global attractor. We also present numerical simulations for some $W$ potentials that illustrate the behavior of trajectories near equilibrium, in both dissipative and conservative regimes.

math.CA↗

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↗

arXiv2017

A Discontinuous Differential Calculus in the Framework Colombeau's Full Algebra

Starting from the Colombeau's full generalized functions, the sharp topologies and the notion of generalized points, we introduce a new kind differential calculus (for functions between totally disconnected spaces). We study generalized pointvalues, Colombeau's differential algebra, holomorphic and analytic functions. We show that the Embedding Theorem and the Open Mapping Theorem hold in this framework. Moreover, we study some applications in differential equations.

math.CA↗

arXiv2017

Dissipative property for a class of non local evolution equations

In this work we consider the non local evolution problem \[ \begin{cases} \partial_t u(x,t)=-u(x,t)+g(βK(f\circ u)(x,t)+βh), ~x \inΩ, ~t\in[0,\infty[;\\ u(x,t)=0, ~x\in\mathbb{R}^N\setminusΩ, ~t\in[0,\infty[;\\ u(x,0)=u_0(x),~x\in\mathbb{R}^N, \end{cases} \] where $Ω$ is a smooth bounded domain in $\mathbb{R}^N, ~g,f: \mathbb{R}\to\mathbb{R}$ satisfying certain growing condition and $K$ is an integral operator with symmetric kernel, $ Kv(x)=\int_{\mathbb{R}^{N}}J(x,y)v(y)dy.$ We prove that Cauchy problem above is well posed, the solutions are smooth with respect to initial conditions, and we show the existence of a global attractor. Futhermore, we exhibit a Lyapunov's functional, concluding that the flow generated by this equation has a gradient property.

math.DS↗

arXiv2014

Asymptotic behavior for a class of non autonomous non local problems

In this paper we consider the non local non autonomous evolution problem \[ \begin{cases} \partial_t u =- u + g \left(β(t)(Ku) \right)\ \ \mbox{in}\ \ Ω,\\ u = 0\ \ \mbox{in}\ \ \mathbb{R}^N\backslashΩ, \end{cases} \] where $Ω$ is a smooth bounded domain in $\mathbb{R}^N$, $β$ denotes the functional parameter given by a continuous bounded function on $\mathbb{R}$, and $K$ is an integral operator with symmetric kernel. We prove existence and some regularity properties of the pullback attractor.

math.DS↗

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↗