arXiv subjects

Wenxiong Chen

Publications and source records attributed to Wenxiong Chen.

At least 19 recordsLinked to original sources

arXiv2026

TopFeaRe: Locating Critical State of Adversarial Resilience for Graphs Regarding Topology-Feature Entanglement

Graph adversarial attacks are usually produced from the two perspectives of topology/structure and node feature, both of them represent the paramount characteristics learned by today's deep learning models. Although some defense countermeasures are proposed at present, they fails to disclose the intrinsic reasons why these two aspects necessitate and how they are adequately fused to co-learn the graph representation. Towards this question, we in this paper propose an adversarial defense approach through locating the graph's critical state of adversarial resilience, resorting to the equilibrium-point theory in the discipline of complex dynamic system (CDS). In brief, our work has three novelties: i) Adversarial-Attack Modeling, i.e. map a graph regime into CDS, and use the oscillation of dynamic system to model the behavior of adversarial perturbation; ii) 2D Topology-Feature-Entangled Function Design for Perturbed Graph, i.e. project graph topology and node feature as two characteristic spaces, and define two-dimensional entangled perturbation functions to represent the dynamic variance under adversarial attacks; and iii) Location of Critical State of Adversarial Resilience, i.e. utilize the equilibrium-point theory to locate the graph's critical state of attack resilience resorting to the perturbation-reflected 2D function. Finally, multi-facet experiments on five commonly-used realistic datasets validate the effectiveness of our proposed approach, and the results show our approach can significantly outperform the state-of-the-art baselines under four representative graph adversarial attacks.

cs.CR↗

arXiv2026

A convergence result for the master operator

In this paper, we establish a convergence result for the fully fractional heat operator $\ma{s}$, also known as the master operator, stated as follows: \[\mbox{If\ }u_i\to u\ \mbox{in}\ C^{2,1}_{x,t,loc}(\R^n\times\R),\ \mbox{then}\ \ma{s} u_i\to \ma{s}u-b\ \mbox{a.e. in}\ \R^n\times\R,\] for some nonnegative constant $b$. This result addresses a fundamental question in the blow-up and rescaling analysis, which are essential for establishing a priori estimates for solutions of master equations. Additionally, we present examples demonstrating that in certain cases, the constant $b$ can indeed be positive. This highlights a key distinction between nonlocal and local operators: for a local heat operator, such as $\partial_t - \lap$, it is well-known that $b \equiv 0$.

math.AP↗

arXiv2026

Monotonicity for the fractional semi-linear problem in a half space

In this paper, we study semilinear fractional equations $$(-Δ)^s u(x) = f(u(x))$$ in a half-space and prove that all positive solutions are strictly increasing in the $x_n$-direction. Previous results typically require the solution $u$ to be globally bounded in $\mathbb{R}^n$. We substantially weaken this condition by assuming only that $u$ be bounded in each slab. Moreover, our analysis relies solely on the local Lipschitz continuity of the nonlinearity $f$, which is weaker than the conditions imposed in earlier works. As a crucial ingredient, we obtained a boundary H\"{older} regularity estimate that requires only the boundedness of $u$ near the boundary. This represents a significant improvement over existing results, which often assumed global boundedness of $u$ throughout $\mathbb{R}^n$. The proof introduces a new idea that may be of independent interest. To derive the monotonicity, we employ the method of moving planes. We first obtain a narrow region principle in unbounded domains, which ensures that the moving plane procedure can be initiated from $x_n = 0$. We then establish two averaging effects for the solutions to fractional equations. These key ingredients guarantee that the planes can be moved continuously all the way to $x_n = \infty$. Previously, narrow region principle can only be applied to a single narrow region. It is for the first time that we establish a multiple narrow region principle that can be applied simultaneously to finitely many narrow regions. Compared with the traditional approaches, methods based on the {\em averaging effect} require substantially weaker regularity assumptions and can even accommodate unbounded solutions. We believe that these new ideas and techniques develop here will serve as powerful tools in studying qualitative properties of solutions to fractional equations.

math.AP↗

arXiv2026

Methods in studying qualitative properties of fractional equations

In this paper, we systematically review a series of effective methods for studying the qualitative properties of solutions to fractional equations. Beginning with the pioneering extension method and the method of moving planes in integral forms, we introduce a variety of direct methods, including the direct method of moving planes, the method of moving spheres, blow-up and rescaling techniques, the sliding method, regularity lifting, and approaches for interior and boundary regularity estimates. To elucidate the core ideas behind these methods, we employ simple examples that demonstrate how they can be applied to investigate qualitative properties of solutions. We also provide a comparative discussion of their respective strengths and limitations. It is our hope that this paper will serve as a useful handbook for researchers engaged in the study of fractional equations.

math.AP↗

arXiv2026

Master equations with indefinite nonlinearities

In this paper, we consider the following indefinite fully fractional heat equation involving the master operator \begin{equation} (\partial_t -Δ)^{s} u(x,t) = x_1u^p(x,t)\ \ \mbox{in}\ \R^n\times\R , \end{equation} where $s\in(0,1)$, and $-\infty < p < \infty$. Under mild conditions, we prove that there is no positive bounded solutions. To this end, we first show that the solutions are strictly increasing along $x_1$ direction by employing the direct method of moving planes. Then by constructing an unbounded sub-solution, we derive the nonexistence of bounded solutions. To circumvent the difficulties caused by the fully fractional master operator, we introduced some new ideas and novel approaches that, as we believe, will become useful tool in studying a variety of other fractional elliptic and parabolic problems.

math.AP↗

arXiv2026

Regularity of solutions for fully fractional parabolic equations

In this paper, we study the fully fractional heat equation involving the master operator: $$ (\partial_t -Δ)^{s} u(x,t) = f(x,t)\ \ \mbox{in}\ \mathbb{R}^n\times\mathbb{R} , $$ where $s\in(0,1)$ and $f(x,t) \geq 0$. First we derive Hölder and Schauder estimates for nonnegative solutions of this equation. Due to the {\em nonlocality} of the master operator, existing results (cf. \cite{ST}) rely on global bounds of the solutions $u$ to control their higher local norms. However, such results are inadequate for blow-up and rescaling analysis aimed at obtaining a priori estimates for solutions to {\em nonlocal } equations on unbounded domains, as the global norms of the rescaled functions may diverge. This limitation raises to a natural and challenging question: {\em Can local bounds of solutions replace global bounds to control their higher local norms?} Here, we provide an affirmative answer to this question for nonnegative solutions. To achieve this, we introduced several new ideas and novel techniques. One of the key innovations is to use a {\em directional perturbation average} to derive an important estimate for the fully fractional heat kernel, as stated in Lemma \ref{key0}. We believe this estimate, along with other new techniques introduced here, will serve as powerful tools in regularity estimates for a wide range of nonlocal equations. Building on this breakthrough, we employ the blow-up and rescaling arguments to establish a priori estimates for solutions to a broader class of nonlocal equations in unbounded domains, such as $$(\partial_t -Δ)^{s} u(x,t) = b(x,t) |\nabla_x u (x,t)|^q + f(x, u(x,t))\ \ \mbox{in}\ \ \mathbb{R}^n\times\mathbb{R}.$$ Under appropriate conditions, we prove that all nonnegative solutions, along with their spatial gradients, are uniformly bounded.

math.AP↗

arXiv2025

Adverseness vs. Equilibrium: Exploring Graph Adversarial Resilience through Dynamic Equilibrium

Adversarial attacks to graph analytics are gaining increased attention. To date, two lines of countermeasures have been proposed to resist various graph adversarial attacks from the perspectives of either graph per se or graph neural networks. Nevertheless, a fundamental question lies in whether there exists an intrinsic adversarial resilience state within a graph regime and how to find out such a critical state if exists. This paper contributes to tackle the above research questions from three unique perspectives: i) we regard the process of adversarial learning on graph as a complex multi-object dynamic system, and model the behavior of adversarial attack; ii) we propose a generalized theoretical framework to show the existence of critical adversarial resilience state; and iii) we develop a condensed one-dimensional function to capture the dynamic variation of graph regime under perturbations, and pinpoint the critical state through solving the equilibrium point of dynamic system. Multi-facet experiments are conducted to show our proposed approach can significantly outperform the state-of-the-art defense methods under five commonly-used real-world datasets and three representative attacks.

cs.LG↗

arXiv2025

Refined regularity for nonlocal elliptic equations and applications

In this paper, we establish refined regularity estimates for nonnegative solutions to the fractional Poisson equation $$ (-Δ)^s u(x) =f(x),\,\, x\in B_1(0). $$ Specifically, we have derived Hölder, Schauder, and Ln-Lipschitz regularity estimates for any nonnegative solution $u,$ provided that only the local $L^\infty$ norm of $u$ is bounded. These estimates stand in sharp contrast to the existing results where the global $L^\infty$ norm of $u$ is required. Our findings indicate that the local values of the solution $u$ and $f$ are sufficient to control the local values of higher order derivatives of $u$. Notably, this makes it possible to establish a priori estimates in unbounded domains by using blowing up and re-scaling argument. As applications, we derive singularity and decay estimates for solutions to some super-linear nonlocal problems in unbounded domains, and in particular, we obtain a priori estimates for a family of fractional Lane-Emden type equations in $\mathbb{R}^n.$ This is achieved by adopting a different method using auxiliary functions, which is applicable to both local and nonlocal problems.

math.AP↗

arXiv2023

Liouville theorem for fully fractional master equations and its applications

In this paper, we study the fully fractional master equation \begin{equation}\label{pdeq1} (\partial_t-Δ)^s u(x,t) =f(x,t,u(x,t)),\,\,(x, t)\in \mathbb{R}^n\times \mathbb{R}. \end{equation} First we prove a Liouville type theorem for the homogeneous equation \begin{equation}\label{pdeq0} (\partial_t-Δ)^s u(x,t) = 0,\,\,(x, t)\in \mathbb{R}^n\times \mathbb{R}, \end{equation} where $0<s<1$. When $u$ belongs to the slowly increasing function space $$\mathcal{L}^{2s,s}(\mathbb{R}^n\times\mathbb{R})=\left\{u(x,t) \in L^1_{\rm loc} (\mathbb{R}^n\times\mathbb{R}) \mid \int_{-\infty}^{+\infty} \int_{\mathbb{R}^n} \frac{|u(x,t)|}{1+|x|^{n+2+2s}+|t|^{\frac{n}{2}+1+s}}\operatorname{d}\!x\operatorname{d}\!t<\infty\right\} $$ and satisfies an additional asymptotic assumption $$\liminf_{|x|\rightarrow\infty}\frac{u(x,t)}{|x|^γ}\geq 0 \; ( \mbox{or} \; \leq 0) \,\,\mbox{for some} \;0\leqγ\leq 1, $$ in the case $\frac{1}{2}<s < 1$, we prove that all solutions of (\ref{pdeq0}) must be constant. This result includes the previous Liouville theorems on harmonic functions \cite{ABR} and on $s$-harmonic functions \cite{CDL} as special cases. Then we establish the equivalence between nonhomogeneous pseudo-differential equations (\ref{pdeq1}) and the corresponding integral equations. We believe that these integral equations will become very useful tools in further analysing qualitative properties of solutions, such as regularity, monotonicity, and symmetry. In the process of deriving the Liouville type theorem, through very delicate calculations, we obtain an optimal estimate on the decay rate of $(\partial_t-Δ)_{\rm right}^s φ(x,t)$. This sharp estimate will become a key ingredient and an important tool in investigating master equations.

math.AP↗

arXiv2023

Gibbons' conjecture for entire solutions of master equations

In this paper, we establish a generalized version of Gibbons' conjecture in the context of the master equation \begin{equation*} (\partial_t-Δ)^s u(x,t)=f(t,u(x,t)) \,\, \mbox{in}\,\, \mathbb{R}^n\times\mathbb{R}. \end{equation*} We show that, for each $t\in\mathbb{R}$, the bounded entire solution $u(x,t)$ must be monotone increasing in one direction, and furthermore it is one-dimensional symmetric under certain uniform convergence assumption on $u$ and an appropriate decreasing condition on $f$. These conditions are slightly weaker than their counter parts proposed in the original Gibbons' conjecture. To overcome the difficulties in proving the Gibbons' conjecture and the impediments caused by the strong correlation between space and time of fully fractional heat operator $(\partial_t-Δ)^s$, we introduce some new ideas and provide several new insights. More precisely, we first derive a weighted average inequality, which not only provides a straightforward proof for the maximum principle in bounded domains, but also plays a crucial role in further deducing the maximum principle in unbounded domains. Such average inequality and maximum principles are essential ingredients to carry out the sliding method, and then we apply this direct method to prove the Gibbons' conjecture in the setting of the master equation. It is important to note that the holistic approach developed in this paper is highly versatile, and will become useful tools in investigating various qualitative properties of solutions as well as in establishing the Gibbons' conjecture for a broad range of fractional elliptic and parabolic equations and systems.

math.AP↗

arXiv2023

Qualitative properties of solutions for dual fractional nonlinear parabolic equations

In this paper, we consider the dual fractional parabolic problem in the right half space. We prove that the positive solutions are strictly increasing in $x_1$ direction without assuming the solutions be bounded. So far as we know, this is the first paper to explore the monotonicity of possibly unbounded solutions for the nonlocal parabolic problem involving both the fractional time derivative $\partial_t^α$ and the fractional Laplacian $(-Δ)^s$. To overcome the difficulties caused by the dual nonlocality in space-time and by the remarkably weak assumptions on solutions, we introduced several new ideas and our approaches are quite different from those in the previous literature. We first establish an unbounded narrow region principle without imposing any decay and boundedness assumptions on the antisymmetric functions at infinity by estimating the nonlocal operator $\partial_t^α+(-Δ)^s$ along a sequence of suitable auxiliary functions at their minimum points, which is an essential ingredient to carry out the method of moving planes at the starting point. Then in order to remove the decay or bounded-ness assumption on the solutions, we develop a new novel approach lies in establishing the {\em averaging effects} for such nonlocal operator and apply these {\em averaging effects} twice to guarantee that the plane can be moved all the way to infinity to derive the monotonicity of solutions. We believe that the new ideas and techniques developed here will become very useful tools in studying the qualitative properties of solutions, in particular of those unbounded solutions, for a wide range of fractional elliptic and parabolic problems.

math.AP↗

arXiv2021

Nonexistence of solutions for indefinite fractional parabolic equations

We study fractional parabolic equations with indefinite nonlinearities $$ \frac{\partial u} {\partial t}(x,t) +(-Δ)^s u(x,t)= x_1 u^p(x, t),\,\, (x, t) \in \mathbb{R}^n \times \mathbb{R}, $$ where $0<s<1$ and $1<p<\infty$. We first prove that all positive bounded solutions are monotone increasing along the $x_1$ direction. Based on this we derive a contradiction and hence obtain non-existence of solutions. These monotonicity and nonexistence results are crucial tools in a priori estimates and complete blow-up for fractional parabolic equations in bounded domains. To this end, we introduce several new ideas and developed a systematic approach which may also be applied to investigate qualitative properties of solutions for many other fractional parabolic problems.

math.AP↗

arXiv2021

Liouville theorems for fractional parabolic equations

In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition $u \to 0$ at infinity with respect to the spacial variables to a polynomial growth on $u$ by constructing auxiliary functions.Then we derive monotonicity for the solutions in a half space $\mathbb{R}_+^n \times \mathbb{R}$ and obtain some new connections between the nonexistence of solutions in a half space $\mathbb{R}_+^n \times \mathbb{R}$ and in the whole space $\mathbb{R}^{n-1} \times \mathbb{R}$ and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the non-locality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of non-local parabolic problems.

math.AP↗

arXiv2020

Hopf's lemmas for parabolic fractional Laplacians and parabolic fractional $p$-Laplacians

In this paper, we first establish Hopf's lemmas for parabolic fractional equations and parabolic fractional $p$-equations. Then we derive an asymptotic Hopf's lemma for antisymmetric solutions to parabolic fractional equations. We believe that these Hopf's lemmas will become powerful tools in obtaining qualitative properties of solutions for nonlocal parabolic equations.

math.AP↗

arXiv2020

Asymptotic method of moving planes for fractional parabolic equations

In this paper, we develop a systematical approach in applying an asymptotic method of moving planes to investigate qualitative properties of positive solutions for fractional parabolic equations. We first obtain a series of needed key ingredients such as narrow region principles, and various asymptotic maximum and strong maximum principles for antisymmetric functions in both bounded and unbounded domains. Then we illustrate how this new method can be employed to obtain asymptotic radial symmetry and monotonicity of positive solutions in a unit ball and on the whole space. Namely, we show that no matter what the initial data are, the solutions will eventually approach to radially symmetric functions. We firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal parabolic problems with more general operators and more general nonlinearities.

math.AP↗

arXiv2019

Monotonicity of solutions for fractional equations with De Giorgi type nonlinearities

In this paper, we develop a sliding method for the fractional Laplacian. We first obtain the key ingredients needed in the sliding method either in a bounded domain or in the whole space, such as narrow region principles and maximum principles in unbounded domains. Then using semi-linear equations involving the fractional Laplacian in both bounded domains and in the whole space, we illustrate how this new sliding method can be employed to obtain monotonicity of solutions. Some new ideas are introduced. Among which, one is to use Poisson integral representation of $s$-subharmonic functions in deriving the maximum principle, the other is to estimate the singular integrals defining the fractional Laplacians along a sequence of approximate maximum points by using a generalized average inequality. We believe that this new inequality will become a useful tool in analyzing fractional equations.

math.AP↗

arXiv2019

A maximum principle on unbounded domains and a Liouville theorem for fractional p-harmonic functions

In this paper, we establish the following Liouville theorem for fractional \emph{p}-harmonic functions. {\em Assume that $u$ is a bounded solution of $$(-\lap)^s_p u(x) = 0, \;\; x \in \mathbb{R}^n,$$ with $0<s<1$ and $p \geq 2$. Then $u$ must be constant.} A new idea is employed to prove this result, which is completely different from the previous ones in deriving Liouville theorems. For any given hyper-plane in $\mathbb{R}^n$, we show that $u$ is symmetric about the plane. To this end, we established a {\em maximum principle} for anti-symmetric functions on any half space. We believe that this {\em maximum principle}, as well as the ideas in the proof, will become useful tools in studying a variety of problems involving nonlinear non-local operators.

math.AP↗

arXiv2019

Maximum principles and monotonicity of solutions for fractional p-equations in unbounded domains

In this paper, we consider the following non-linear equations in unbounded domains $Ω$ with exterior Dirichlet condition: \begin{equation*}\begin{cases} (-Δ)_p^s u(x)=f(u(x)), & x\inΩ,\\ u(x)>0, &x\inΩ,\\ u(x)\leq0, &x\in \mathbb{R}^n\setminus Ω, \end{cases}\end{equation*} where $(-Δ)_p^s$ is the fractional p-Laplacian defined as \begin{equation} (-Δ)_p^s u(x)=C_{n,s,p}P.V.\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^{p-2}[u(x)-u(y)]}{|x-y|^{n+s p}}dy \label{0} \end{equation} with $0<s<1$ and $p\geq 2$. We first establish a maximum principle in unbounded domains involving the fractional p-Laplacian by estimating the singular integral in (\ref{0}) along a sequence of approximate maximum points. Then, we obtain the asymptotic behavior of solutions far away from the boundary. Finally, we develop a sliding method for the fractional p-Laplacians and apply it to derive the monotonicity and uniqueness of solutions. There have been similar results for the regular Laplacian \cite{BCN1} and for the fractional Laplacian \cite{DSV}, which are linear operators. Unfortunately, many approaches there no longer work for the fully non-linear fractional p-Laplacian here. To circumvent these difficulties, we introduce several new ideas, which enable us not only to deal with non-linear non-local equations, but also to remarkably weaken the conditions on $f(\cdot)$ and on the domain $Ω$. We believe that the new methods developed in our paper can be widely applied to many problems in unbounded domains involving non-linear non-local operators.

math.AP↗