arXiv subjects

Alexander Quaas

Publications and source records attributed to Alexander Quaas.

At least 19 recordsLinked to original sources

arXiv2026

Ergodic pairs for fractional Hamilton-Jacobi equations on bounded domains: large solutions

In this article, we study the ergodic problem associated to viscous Hamilton-Jacobi equation where the diffusion is governed by the censored fractional Laplacian, a nonlocal elliptic operator restricted to a bounded domain $\Omega \subset \mathbb{R}^N$. We restrict ourselves to the case in which the nonlinear gradient term has a scaling less or equal than the fractional order of the diffusion. In similarity to its second-order counterpart, we provide existence of ergodic pairs involving solutions that blow-up on $\partial \Omega$. We use the celebrated vanishing discount method, where the analysis of the approximated solutions have its own interest, leading to qualitative properties for the ergodic problem such as precise blow-up rates for the solution and characterization of the ergodic constant. The main difficulties arise from the state-dependency of the operator, from which the arguments of the local case based on well-known invariance properties of the Laplacian are not longer at disposal.

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arXiv2025

The moving patch model with fractional diffusion

In this paper we study the following one-dimensional reaction-diffusion problem $$ u_t+(-\Delta)^s u=f(x-c t, u) \;\:\textrm{ in } \mathbb{R}\times (0,+\infty), $$ where $s>\frac{1}{2}$, $c \in \mathbb{R}$ is a prescribed velocity, and $f$ is of KPP type, which describes the evolution of a population in an advective environment subjected to nonlocal diffusion. We suppose the environment is such that it is only advantageous in a bounded ``patch", outside of which the species dies at an asymptotically constant rate. We first derive an optimal solvability criteria for the corresponding traveling waves problem $$\Delta^s u+c u^{\prime}+f(x, u)=0 \;\:\textrm{ in } \mathbb{R},$$ through the first eigenvalue of the associated linearized elliptic operator with drift. Then we use this criteria to establish the long time behavior of the solution to the parabolic problem, for any continuous bounded nonnegative initial data, leading the species either through their extinction or survival. Moreover, assuming that for $c=0$ the population survives, we show that there exist two positive critical speeds $c^{*}$ and $c^{**}$ such that for all $|c| c^{**}$.

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arXiv2025

Propagation in the Fisher-KPP equation with Mixed Operator

Our investigation focuses on the asymptotic spreading behavior of the Fisher-KPP equation with a mixed local-nonlocal operator in the diffusion (see the work by X. Cabr\'e and J.-M. Roquejoffre, 2013, ref.[8]) to the setting of mixed diffusion, which involves both the classical and the fractional Laplacian in order to analyze the long-time dynamics of the equation. A key step in our approach involves the construction and detailed study of the heat kernel associated with the mixed operator, which we use to develop a theory of mild solutions and establish a comparison principle in suitable weighted function spaces. This framework allows us to rigorously establish the non-existence of traveling waves and characterize the large-time spreading rate of solutions. We show that the influence of the fractional Laplacian dominates over the classical Laplacian, especially in the initial layer, where it dictates the exponential propagation rate and the thickness of the solution tails.

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arXiv2025

Mixed local and nonlocal laplacian without standard critical exponent for Lane-Emden equation

In this paper, we investigate a mixed elliptic equation involving both local and nonlocal Laplacian operators, with a power-type nonlinearity. Specifically, we consider a Lane-Emden type equation of the form \[-\Delta u + (-\Delta)^s u = u^p,\quad\mbox{ in }\mathbb{R}^n.\] where the operator combines the classical Laplacian and the fractional Laplacian. We establish the existence of solutions for exponents slightly below the critical local Sobolev exponent, that is, for $p < \frac{n+2}{n-2}$, with $p$ close to $\frac{n+2}{n-2}$. Our results show that, due to the interaction between the local and nonlocal operators, this mixed Lane-Emden-Fowler equation does not admit a critical exponent in the traditional sense. The existence proof is carried out using a Lyapunov-Schmidt type reduction method and, as far as we know, provide the first example of an elliptic operator for which the duality between critical exponents fails.

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arXiv2024

Existence of solutions to a quasilinear nonlocal PDE

In this paper, we introduce a new class of quasilinear operators, which represents a nonlocal version of the operator studied by Stuart and Zhou [1], inspired by models in nonlinear optics. We will study the existence of at least one or two solutions in the cone $X=\{u\in H^s_0(\Omega): u\geq 0\}$ using variational methods. For this purpose, we analyze two scenarios: the asymptotic sublinear and linear growth cases for the reaction term. Additionally, in the sublinear case, we establish a nonexistence result.

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arXiv2024

Nonexistence of positive supersolutions for semilinear fractional elliptic equations in exterior domains

This article is concerned with the nonexistence of positive solutions for nonlinear fractional elliptic inequalities in exterior domains of $\mathbb R^n$, $n\geq1$. We would like to highlight the fact that our results are new with very weak assumption on the nonlinear term appearing in the inequalities. The results are also novel in this generality even if the whole space is considered.

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arXiv2024

Nonlocal Liouville theorems with gradient nonlinearity

In this article we consider a large family of nonlinear nonlocal equations involving gradient nonlinearity and provide a unified approach, based on the Ishii-Lions type technique, to establish Liouville properties of the solutions. We also answer an open problem raised by [24]. Some applications to regularity issues are also studied.

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arXiv2024

The limit as $s\nearrow 1$ of the fractional convex envelope

We study the behavior of the fractional convexity when the fractional parameter goes to 1. For any notion of convexity, the convex envelope of a datum prescribed on the boundary of a domain is defined as the largest possible convex function inside the domain that is below the datum on the boundary. Here we prove that the fractional convex envelope inside a strictly convex domain of a continuous and bounded exterior datum converges when $s\nearrow 1$ to the classical convex envelope of the restriction to the boundary of the exterior datum.

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arXiv2023

The fundamental solution of the fractional p-laplacian

In this article, we find the fundamental solution of the fractional p-laplacian and use them to prove two different Liouville-type theorems. A non-existence classical Liouville-type theorem for p-superharmonic and a Louville type results for an Emden-Folder type equation with the fractional p-laplacian.

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arXiv2023

Fundamental solutions and critical Lane-Emden exponents for nonlinear integral operators in cones

In this article we study the fundamental solutions or "$\alpha$-harmonic functions" for some nonlinear positive homogeneous nonlocal elliptic problems in conical domains, such as \begin{eqnarray*}\label{ecbir1a1} {\mathcal F }(u)=0\ \ \hbox{in} \ \ \mathcal{C}_\omega,\quad u=0\ \ \hbox{in} \ \ \mathbb{R}^n\setminus \mathcal{C}_\omega ,\ \ \end{eqnarray*} where $\omega$ is a proper $C^2$ domain in $S^{N-1}$ for $ N\geq 2$, $\mathcal{C}_\omega:=\{x\,:\,x\neq 0, {|x|^{-1}}x\in \omega\}$ is the cone-like domain related to $\omega$, and ${\mathcal F }$ is an extremal fully nonlinear integral operator. We prove the existence of two fundamental solutions that are homogeneous and do not change signs in the cone; one is bounded at the origin and the other at infinity. As an application, we use the fundamental solutions obtained to prove Liouville type theorems in cones for supersolutions of the Lane-Emden-Fowler equation in the form \begin{eqnarray*}\label{eq 0.2} {\mathcal F }(u)+u^p = 0\ \ \hbox{in} \ \ \mathcal{C}_\omega, \quad u=0\ \ \hbox{in} \ \ \mathbb{R}^n\setminus \mathcal{C}_\omega.\ \ \end{eqnarray*} We also prove a generalized Hopf type lemma in domains with corners. Most of our results are new even when ${\mathcal F }$ is the fractional Laplacian operator.

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arXiv2023

Large harmonic functions for fully nonlinear fractional operators

We study existence, uniqueness and boundary blow-up profile for fractional harmonic functions on a bounded smooth domain $\Omega \subset \mathbb R^N$. We deal with harmonic functions associated to uniformly elliptic, fully nonlinear nonlocal operators, including the linear case $$ (-\Delta)^s u = 0 \quad \mbox{in} \ \Omega, $$ where $(-\Delta)^s$ denotes the fractional Laplacian of order $2s \in (0,2)$. We use the viscosity solution's theory and Perron's method to construct harmonic functions with zero exterior condition in $\bar \Omega^c$, and boundary blow-up profile $$ \lim_{x\to x_0, x \in \Omega}\mathrm{dist}(x, \partial \Omega)^{1-s}u(x)=h(x_0), \quad \mbox{for all} \quad x_0\in \partial \Omega, $$ for any given boundary data $h \in C(\partial \Omega)$. Our method allows us to provide blow-up rate for the solution and its gradient estimates. Results are new even in the linear case.

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arXiv2023

The evolution problem associated with the fractional first eigenvalue

In this paper we study the evolution problem associated with the first fractional eigenvalue. We prove that the Dirichlet problem with homogeneous boundary condition is well posed for this operator in the framework of viscosity solutions (the problem has existence and uniqueness of a solution and a comparison principle holds). In addition, we show that solutions decay to zero exponentially fast as $t\to \infty$ with a bound that is given by the first eigenvalue for this problem that we also study.

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arXiv2022

Periodic fractional Ambrosetti-Prodi for one-dimensional problem with drift

We establish Ambrosetti -Prodi type results for periodic solutions of one -dimensional nonlinear problems with drift term and drift -less whose principal operator is the fractional Laplacian of order $s\in(0,1)$. We establish conditions for the existence and nonexistence of solutions. The proofs of the existence results are based on the sub-supersolution method combined with topological degree type arguments. We also establish a priori bounds in order to get multiplicity results. We also prove that the solutions are $C^{1,\alpha}$ under some regularity assumptions in the nonlinearities, that is, the solutions of equations are classical. We finish the work obtaining existence results for problems with the fractional Laplacian with singular nonlinearity. In particular, we establish an Ambrosetti-Prodi type problem with singular nonlinearities.

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arXiv2022

On large solutions for fractional Hamilton-Jacobi equations

We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated to fully nonlinear elliptic equations of order $2s$, with $s\in (1/2,1)$, and a coercive gradient term with subcritical power $0<p<2s$. Due to the nonlocal nature of the diffusion, new blow-up phenomena arise within the range $0<p<2s$, involving a continuum family of solutions and/or solutions blowing-up to $-\infty$ on the boundary. This is in striking difference with the local case studied by Lasry-Lions for the case subquadratic case $1<p<2$.

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arXiv2021

Large-time behavior of unbounded solutions of the viscous Hamilton-Jacobi equation: quadratic and subquadratic cases

We determine the large-time behavior of unbounded solutions for the so-called viscous Hamilton Jacobi equation, $u_t - \Delta u + |Du|^m = f(x)$, in the quadratic and subquadratic cases (i.e., for $1<m\leq 2$), with a particular focus on allowing arbitrary growth at infinity for $f$ and the prescribed initial data. The lack of a comparison principle for the associated ergodic problem is overcome by proving that a generalized simplicity holds for sub- and supersolutions of the ergodic problem. Moreover, as the uniqueness of solutions of the parabolic problem remains open in the current setting, our result on large-time holds for any solution, even if multiple solutions exist.

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arXiv2021

Traveling fronts for the generalized Fisher-KPP equation with nonlocal diffusion

The aim of this paper is to study the generalized Fisher-KPP equation with nonlocal diffusion. In specific we prove the existence of a critical speed so that traveling front type solutions exist up to this critical speed and non-existence of traveling fronts below this critical value. Moreover, we obtain uniqueness, up to translation, and decay estimates of these traveling fronts.

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arXiv2021

A note on one-dimensional symmetry for Hamilton-Jacobi equations with extremal Pucci operators and application to Bernstein type estimate

We prove a Liouville-type theorem that is one-dimensional symmetry and classification results for non-negative $L^q$-viscosity solutions of the equation \begin{equation*} -\mathcal{M}_{\lambda, \Lambda}^{\pm}(D^2u)\pm |Du|^p=0, x\in \mathbb{R}_+^n, \end{equation*} with boundary condition $u(\tilde{x},0)=M\geq 0, \tilde{x}\in \mathbb{R}^{n-1}$, where $\mathcal{M}_{\lambda, \Lambda}^{\pm}$ are the Pucci's operators with parameters $\lambda, \Lambda \in \mathbb{R}_+$ $0<\lambda\leq \Lambda$ and $p>1$. The results are an extension of the results by Porreta and Ver\'on in arXiv:0805.2533 for the case $p\in (1,2]$ and by o Filippucci, Pucci and Souplet in arXiv:1906.05161 for the case $p>2$, both for the Laplacian case (i.e. $\lambda=\Lambda=1$). As an application in the case $p>2$, we prove a sharp Bernstein estimation for $L^q$-viscosity solutions of the fully nonlinear equation \begin{equation*} -\mathcal{M}_{\lambda, \Lambda}^{\pm}(D^2u)= |Du|^p+f(x), \quad x\in \Omega, \label{ecuacion1} \end{equation*} with boundary condition $u=0$ on $\partial \Omega$, where $\Omega \subset \mathbb{R}^n$.

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arXiv2020

Periodic solutions for one-dimensional nonlinear nonlocal problem with drift including singular nonlinearities

In this paper, we prove existence results of a one-dimensional periodic solution to equations with the fractional Laplacian of order $s\in(1/2,1)$, singular nonlinearity, and gradient term under various situations, including nonlocal contra-part of classical Lienard vector equations, as well other nonlocal versions of classical results know only in the context of second-order ODE. Our proofs are based on degree theory and Perron's method, so before that we need to establish a variety of priori estimates under different assumptions on the nonlinearities appearing in the equations. Besides, we obtain also multiplicity results in a regime where a priori bounds are lost and bifurcation from infinity occurs.

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