arXiv subjects

Raúl E. Vidal

Publications and source records attributed to Raúl E. Vidal.

7 recordsLinked to original sources

arXiv2026

Dynamics of Fractional Wave Equations with Nonlocal Damping

In this work we study a fractional wave equation with Balakrishnan--Taylor type damping posed on a bounded domain $Ω\subset\mathbb{R}^n$. The model couples the fractional Laplacian with a nonlinear damping coefficient depending on the fractional energy, leading to a doubly nonlocal evolution equation that extends the classical wave equation with energy-dependent dissipation. By means of the theory of linear operators, we establish the global well-posedness of both mild and regular solutions. We then investigate the long-time dynamics of the associated semigroup and prove that it is gradient and asymptotically smooth. As a consequence, we establish the existence of a compact global attractor and show that it coincides with the unstable manifold of the set of stationary solutions. To the best of our knowledge, this provides the first characterization of the asymptotic dynamics for fractional wave equations with Balakrishnan--Taylor type damping.

math.DS↗

arXiv2026

Characterization of weights for the variable fractional maximal operator and weighted inequalities for variable fractional rough operators

We characterize the class of weights related to the boundedness of variable fractional maximal operator $M_{β(\cdot),r(\cdot)}$ on variable Lebesgue spaces. This extend previously known results, including those corresponding to the fractional operator $M_{β(\cdot),1}$. In addition, we introduce a class of kernels $K$ satisfying a new variable Hörmander-type condition $H_{β(\cdot),r(\cdot)}$. For the fractional operator $T_{β(\cdot)}$ given by a kernel in $H_{β(\cdot),r(\cdot)}$, we prove a Coifman-Fefferman inequality and weighted inequalities in variable Lebesgue space. Finally, we provide examples of kernels in this variable Hörmander class.

math.FA↗

arXiv2024

Existence of positive solutions for a semipositone $p(\cdot)$-Laplacian problem

In this paper we find a positive weak solution for a semipositone $p(\cdot )$- Laplacian problem. More precisely, we find a solution for the problem \[ \left\{ \begin{array}{cc} -Δ_{p(\cdot )}u=f(u)-λ& \text{in }Ω\\ u>0 & \text{in }Ω\\ u=0 & \text{on }\partial Ω\end{array}% \right. , \] where $Ω\subset \mathbb{R}^{N}$, $N\geq 2$ is a smooth bounded domain, $f$ is a contiuous function with subcritical growth, $λ>0$ and $Δ_{p(\cdot )}u=\text{div}(\left\vert \nabla u\right\vert ^{p(\cdot )-2}\nabla u)$. Also, we assume an Ambrosetti-Rabinowitz type of condition and using the Mountain Pass arguments, comparision principles and regularity principles we prove the existence of positive weak solution for $λ$ small enough.

math.AP↗

arXiv2022

Existence of positive solutions for a parameter fractional $p$-Laplacian problem with semipositone nonlinearity

In this paper we prove the existence of at least one positive solution for the nonlocal semipositone problem \[ \displaystyle \left\{\begin{array}{rcll} (-Δ)_p^s(u) &=& λf(u) \qquad & \text{in} \ \ Ω \\u &=& 0 & \text{in} \ \ \mathbb{R}^N -Ω, \end{array}\right. \] whenever $λ>0$ is a sufficiently small parameter. Here $Ω\subseteq \mathbb{R}^N$ a bounded domain with $C^{1,1}$ boundary, $2\leqslant p 0$ is chosen sufficiently small the associated Energy Functional to the problem has a mountain pass structure and, therefore, it has a critical point $u_λ$, which is a weak solution. After that we manage to prove that this solution is positive by using new regularity results up to the boundary and a Hopf's Lemma.

math.AP↗

arXiv2021

Sharp bounds for fractional operator with $L^{α,r'}$-Hörmander conditions

In this paper we prove the sharp boundedness for a fractional type operator given by a kernel that satisfy a $L^{α,r'}$-Hörmander conditions and a fractional size condition, where $0<α<n$ and $1< r'\leq \infty$. To prove this result we use a new appropriate sparse domination which we provide in this work. For the case $r'=\infty$ we recover the sharp boundedness for the fractional integral, $I_α$, proved in [Lacey, M. T., Moen, K., Pérez, C., Torres, R. H. (2010). Sharp weighted bounds for fractional integral operators. Journal of Functional Analysis, 259(5), 1073-1097.]

math.CA↗

arXiv2020

Necessary condition on the weight for maximal and integral operators with rough kernels

Let $0\leq α<n$, $m\in \mathbb{N}$ and let consider $T_{α,m}$ be a of integral operator, given by kernel of the form $$K(x,y)=k_1(x-A_1y)k_2(x-A_2y)\dots k_m(x-A_my),$$ where $A_i$ are invertible matrices and each $k_i$ satisfies a fractional size and generalized fractional Hörmander condition. In [Ibañez-Firnkorn, G. H., and Riveros, M. S. (2018). Certain fractional type operators with Hörmander conditions. To appear in Ann. Acad. Sci. Fenn. Math.] it was proved that $T_{α,m}$ is controlled in $L^p(w)$-norms, $w\in A_{\infty}$, by the sum of maximal operators $M_{A_i^{-1},α}$. In this paper we present the class of weights $\mathcal{A}_{A,p,q}$, where $A$ is an invertible matrix. This class are the good weights for the weak-type estimate of $M_{A^{-1},α}$. For certain kernels $k_i$ we can characterize the weights for the strong-type estimate of $T_{α,m}$. Also, we give a the strong-type estimate using testing conditions.

math.CA↗

arXiv2019

Explicit fundamental solution for the operator $L+α|T|$ on the Gelfand pair $(\mathbb{H}_{n},U(n))$

By means of the spherical functions associated to the Gelfand pair $(\mathbb{H}_{n},U(n))$ we define the operator $L+α|T|$, where $L$ denotes the Heisenberg sublaplacian and $T$ denotes de central element of the Heisenberg Lie algebra, we establish a notion of fundamental solution and explicitly compute in terms of the Gauss hypergeometric function. For $α<n$ we use the Integral Representation Theorem to obtain a more detailed expression. Finally, we remark that when $α=0$ we recover the fundamental solution for the Heisenberg sublaplacian given by Folland.

math.FA↗