Searcharxiv⌕ Search

arXiv subjects

Jesús Ildefonso Díaz

Publications and source records attributed to Jesús Ildefonso Díaz.

At least 19 recordsLinked to original sources
arXiv2026

Damped nonlinear Ginzburg-Landau equation with saturation. Part II. Strong Stabilization

We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schr{ö}dinger equation and dissipative parabolic dynamics through a complex timederivative prefactor, capturing the interplay between dispersion and dissipation. As a continuation of our previous study on the existence and uniqueness of solutions, we prove here some strong stabilization properties. In particular, we show the finite time extinction of solutions induced by the nonlinear saturation mechanism, which, sometimes, can be understood as a bang-bang control. The analysis relies on refined energy methods. Our results provide a rigorous justification of nonlinear dissipation as an effective stabilization mechanism for this class of complex equations where the maximum principle fails.

math.AP↗
arXiv2026

Damped nonlinear Ginzburg-Landau equation with saturation. Part I. Existence of solutions on general domains

We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schr{ö}dinger equation and dissipative parabolic dynamics through a complex time-derivative prefactor, capturing the interplay between dispersion and dissipation. Under suitable structural conditions on the complex coefficients, we establish the existence and uniqueness of global solutions. The analysis relies on the delicate proofs that the maximal monotone operator theory can be adapted to this framework, even for unbounded domains.

math.AP↗
arXiv2026

Strong stabilization of damped nonlinear Schr{ö}dinger equation with saturation on unbounded domains

We consider the damped nonlinear Schr\''{o}dinger equation with saturation: i.e., the complex evolution equation contains in its left hand side, besides the potential term $V(x)u,$ a nonlinear term of the form $\mathrm{i}μu(t,x)/|u(t,x)|$ for a given parameter $μ>0$ (arising in optical applications on non-Kerr-like fibers). In the right hand side we assume a given forcing term $f(t,x).$ The important new difficulty, in contrast to previous results in the literature, comes from the fact that the spatial domain is assumed to be unbounded. We start by proving the existence and uniqueness of weak and strong solutions according the regularity of the data of the problem. The existence of solutions with a lower regularity is also obtained by working with a sequence of spaces verifying the Radon-Nikodým property. Concerning the asymptotic behavior for large times we prove a strong stabilization result. For instance, in the one dimensional case we prove that there is extinction in finite time of the solutions under the mere assumption that the $L^\infty$-norm of the forcing term $f(t,x)$ becomes less than $μ$ after a finite time. This presents some analogies with the so called feedback \textit{bang-bang controls} $v$ (here $v=-\mathrm{i}μu/|u|+f).$

math.AP↗
arXiv2025

Electron beams: partially flat solutions of a nonlinear elliptic equation with a singular absorption term

In the so-called Child-Langmuir law, established since 1911, an electron beam is formed linking two electrodes, which are assumed to be two parallel plates of area $A$, separated to a finite distance $D.$ When $% D\ll \sqrt{A},$ \textquotedblleft edge effects\textquotedblright\ are negligible and the modelling is reduced to a nonlinear boundary problem for a singular ordinary differential equation\ in which a constant coefficient (the generated electric current $j$) must be found in order to get simultaneously Dirichlet and Neumann homogeneous boundary conditions in one of the extremes. If $D>\sqrt{A},$ then the problem becomes much more difficult since the \textquotedblleft edge effects\textquotedblright\ arise in the plane $(x,y)$ and the electric current (now $j(x)$ due to the presence of a very large perpendicular magnetic field) must be determined in order to get solutions $u(x,y)$ of a singular semilinear equation which are partially flat ($u=\frac{\partial u}{\partial n}=0$ on a part of the boundary). In this paper, we offer a rigorous mathematical treatment of some former studies (Joel Lebowitz and Alexander Rokhenko (2003) and Alexander Rokhenko (2006)), where several open questions were left open: for instance, the need for a singularity of $j(x)$ near the cathode edge to get such partially flat solutions.

math.AP↗
arXiv2025

Stochastic diffusive energy balance climate model with a multiplicative noise modeling the Solar variability

We prove the existence, uniqueness, and comparison of solutions for a nonlinear stochastic parabolic partial differential equation that includes the Solar variability in terms of a multiplicative Wiener cylindrical noise in the term of the absorbed radiative energy in a simplified diffusive one-dimensional Energy Balance Model. We introduce a hybrid co-albedo nonlinear term, which has the advantages of both the Sellers model, as it is a continuous function, and the Budyko model, as it has an infinite derivative at $u=-10^C$ (the temperature at which ice is white), allowing the location of the polar ice caps to be easily detected. We show that, despite the lack of differentiability of this function, the method of successive approximations can be satisfactorily applied.

math.AP↗
arXiv2025

On the compactness of the support of solitary waves of the complex saturated nonlinear Schr{ö}dinger equation and related problems

We study the vectorial stationary Schr{ö}dinger equation -$Δ$u + a U + b u = F, with a saturated nonlinearity U = u/|u| and with some complex coefficients (a, b) $\in$ C 2 . Besides the existence and uniqueness of solutions for the Dirichlet and Neumann problems, we prove the compactness of the support of the solution, under suitable conditions on (a, b) and even when the source in the right hand side F (x) is not vanishing for large values of |x|. The proof of the compactness of the support uses a local energy method, given the impossibility of applying the maximum principle. We also consider the associate Schr{ö}dinger-Poisson system when coupling with a simple magnetic field. Among other consequences, our results give a rigorous proof of the existence of ''solitons with compact support'' claimed, without any proof, by several previous authors.

math.AP↗
arXiv2024

Finite time extinction for a critically damped Schr{ö}dinger equation with a sublinear nonlinearity

This paper completes some previous studies by several authors on the finite time extinction for nonlinear Schr{ö}dinger equation when the nonlinear damping term corresponds to the limit cases of some ``saturating non-Kerr law'' $F(|u|^2)u=\frac{a}{\varepsilon+(|u|^2)^α}u,$ with $a\in\mathbb{C},$ $\varepsilon\geqslant0,$ $2α=(1-m)$ and $m\in[0,1).$ Here we consider the sublinear case $0 0 \text{ and } 2\sqrt{m}\mathrm{Im}(z)=(1-m)\mathrm{Re}(z)\big\}.$ Among other things, we know that this damping coefficient is critical, for instance, in order to obtain the monotonicity of the associated operator (see the paper by Liskevich and Perel'muter [16] and the more recent study by Cialdea and Maz'ya [14]). The finite time extinction of solutions is proved by a suitable energy method after obtaining appropiate a priori estimates. Most of the results apply to non-necessarily bounded spatial domains.

math.AP↗
arXiv2024

Existence of weak solutions to some stationary Schr{ö}dinger equations with singular nonlinearity

We prove some existence (and sometimes also uniqueness) of weak solutions to some stationary equations associated to the complex Schr\''{o}dinger operator under the presence of a singular nonlinear term. Among other new facts, with respect some previous results in the literature for such type of nonlinear potential terms, we include the case in which the spatial domain is possibly unbounded (something which is connected with some previous localization results by the authors), the presence of possible non-local terms at the equation, the case of boundary conditions different to the Dirichlet ones and, finally, the proof of the existence of solutions when the right-hand side term of the equation is beyond the usual $L^2$-space.

math.AP↗
arXiv2024

Finite time extinction for a class of damped Schr{ö}dinger equations with a singular saturated nonlinearity

We present some sharper finite extinction time results for solutions of a class of damped nonlinear Schr{ö}dinger equations when the nonlinear damping term corresponds to the limit cases of some ``saturating non-Kerr law'' $F(|u|^2)u=\frac{a}{\varepsilon+(|u|^2)^α}u,$ with $a\in\mathbb{C},$ $\varepsilon\geqslant0,$ $2α=(1-m)$ and $m\in[0,1).$ To carry out the improvement of previous results in the literature we present in this paper a careful revision of the existence and regularity of weak solutions under very general assumptions on the data. We prove that the problem can be solved in the very general framework of the maximal monotone operators theory, even under a lack of regularity of the damping term. This allows us to consider, among other things, the singular case $m=0.$ We replace the above approximation of the damping term by a different one which keeps the monotonicity for any $\varepsilon\geqslant0$. We prove that, when $m=0,$ the finite extinction time of the solution arises for merely bounded right hand side data $f(t,x).$ This is specially useful in the applications in which the Schr{ö}dinger equation is coupled with some other functions satisfying some additional equations.

math.AP↗
arXiv2024

Beyond the classical strong maximum principle: forcing changing sign near the boundary and flat solutions

We show that the classical strong maximum principle, concerning positive supersolutions of linear elliptic equations vanishing on the boundary of the domain $Ω$ can be extended, under suitable conditions, to the case in which the forcing term $f(x)$ is changing sign. In addition, in the case of solutions, the normal derivative on the boundary may also vanish on the boundary (definition of flat solutions). This leads to examples in which the unique continuation property fails. As a first application, we show the existence of positive solutions for a sublinear semilinear elliptic problem of indefinite sign. A second application, concerning the positivity of solutions of the linear heat equation, for some large values of time, with forcing and/or initial datum changing sign is also given.

math.AP↗
arXiv2023

Failure of the Hopf-Oleinik lemma for a linear elliptic problem with singular convection of non-negative divergence

In this paper we study existence, uniqueness, and integrability of solutions to the Dirichlet problem $-\mathrm{div}( M(x) \nabla u ) = -\mathrm{div} (E(x) u) + f$ in a bounded domain of $\mathbb R^N$ with $N \ge 3$. We are particularly interested in singular $E$ with $\mathrm{div} E \ge 0$. We start by recalling known existence results when $|E| \in L^N$ that do not rely on the sign of $\mathrm{div} E $. Then, under the assumption that $\mathrm{div} E \ge 0$ distributionally, we extend the existence theory to $|E| \in L^2$. For the uniqueness, we prove a comparison principle in this setting. Lastly, we discuss the particular cases of $E$ singular at one point as $Ax /|x|^2$, or towards the boundary as $\mathrm{div} E \sim \mathrm{dist}(x, \partial Ω)^{-2-α}$. In these cases the singularity of $E$ leads to $u$ vanishing to a certain order. In particular, this shows that the Hopf-Oleinik lemma, i.e. $\partial u / \partial n < 0$, fails in the presence of such singular drift terms $E$.

math.AP↗
arXiv2021

Stochastic energy balance climate models with Legendre weighted diffusion and a cylindrical Wiener process forcing

We consider a class of one-dimensional nonlinear stochastic parabolic problems associated with Sellers and Budyko diffusive energy balance climate models with a Legendre weighted diffusion and an additive cylindrical Wiener processes forcing. Our results use in an important way that, under suitable assumptions on the Wiener processes, a suitable change of variables leads the problem to a pathwise random PDE, hence an essentially "deterministic" formulation depending on a random parameter. Two applications are also given: the stability of solutions when the Wiener process converges to zero and the asymptotic behaviour of solutions for large time.

math.PR↗
arXiv2021

Finite time extinction for the strongly damped nonlinear Schr{ö}dinger equation in bounded domains

We prove the \textit{finite time extinction property} $(u(t)\equiv 0$ on $Ω$ for any $t\ge T_\star,$ for some $T_\star>0)$ for solutions of the nonlinear Schrödinger problem ${\rm i} u_t+Δu+a|u|^{-(1-m)}u=f(t,x),$ on a bounded domain $Ω$ of $\mathbb{R}^N,$ $N\le 3,$ $a\in\mathbb{C}$ with $\Im(a)>0$ (the damping case) and under the crucial assumptions $0<m<1$ and the dominating condition $2\sqrt m\,\Im(a)\ge(1-m)|\Re(a)|.$ We use an energy method as well as several a priori estimates to prove the main conclusion. The presence of the non-Lipschitz nonlinear term in the equation introduces a lack of regularity of the solution requiring a study of the existence and uniqueness of solutions satisfying the equation in some different senses according to the regularity assumed on the data.

math.AP↗
arXiv2020

Steiner symmetrization for anisotropic quasilinear equations via partial discretization

In this paper we obtain comparison results for the quasilinear equation $-Δ_{p,x} u - u_{yy} = f$ with homogeneous Dirichlet boundary conditions by Steiner rearrangement in variable $x$, thus solving a long open problem. In fact, we study a broader class of anisotropic problems. Our approach is based on a finite-differences discretization in $y$, and the proof of a comparison principle for the discrete version of the auxiliary problem $A U - U_{yy} \le \int_0^s f^*$, where $AU = (nω^{1/n}s^{1/n'} )^p (- U_{ss})^{p-1}$. We show that this operator is T-accretive in $L^\infty$. We extend our results for $-Δ_{p,x}$ to general operators of the form $-\mathrm{div} (a(|\nabla_x u|) \nabla_x u)$ where $a$ is non-decreasing and behaves like $| \cdot |^{p-2}$ at infinity.

math.AP↗
arXiv2020

Pointwise gradient estimates in multi-dimensional slow diffusion equations with a singular quenching term

We consider a slow diffusion equation with a singular quenching term, extending the mathematical treatment made in 1992 by B. Kawohl and R. Kersner for the one-dimensional case. Besides the existence of a very weak solution, we prove some pointwise gradient estimates, mainly when the absorption dominates over the diffusion. In particular, a new kind of universal gradient estimate is proved. Several qualitative properties (such as the finite time quenching phenomena and the finite speed of propagation) and the study of the Cauchy problem are also considered.

math.AP↗
arXiv2019

A nonlocal memory strange term arising in the critical scale homogenisation of a diffusion equation with a dynamic boundary condition

Our main interest in this paper is the study of homogenised limit of a parabolic equation with a nonlinear dynamic boundary condition of the micro-scale model set on a domain with periodically place particles. We focus on the case of particles (or holes) of critical diameter with respect to the period of the structure. Our main result proves the weak convergence of the sequence of solutions of the original problem to the solution of a reaction-diffusion parabolic problem containing a `strange term'. The novelty of our result is that this term is a nonlocal memory solving an ODE. We prove that the resulting system satisfies a comparison principle.

math.AP↗
arXiv2018

Some remarks on the coincidence set for the Signorini problem

We study some properties of the coincidence set for the boundary Signorini problem, improving some results from previous works by the second author and collaborators. Among other new results, we show here that the convexity assumption on the domain made previously in the literature on the location of the coincidence set can be avoided under suitable alternative conditions on the data.

math.AP↗