The Schrödinger equation with fractional Laplacian on hyperbolic spaces and homogeneous trees

Global existence and finite-time blow-up of solutions for parabolic equations involving the fractional Musielak Laplacian

An extension of the spectral fractional Laplacian to non-homogeneous boundary condition on rectangular domains, with application to well-posedness for plate equation with structural damping

Second-order error analysis for FEM of fractional Laplacian on graded meshes via FDM auxiliary

Normalized solutions for the NLS equation with mixed fractional Laplacians and combined nonlinearities

Neural field dynamics in the cerebral cortex exhibit complex spatiotemporal patterns inadequately captured by classical integer-order diffusion models that assume exponentially decaying spatial interactions. This study establishes a stochastic fractional FitzHugh–Nagumo framework incorporating power-law spatial correlations through fractional Laplacian operators, providing explicit parameterization of non-local cortical connectivity characteristics. The inverse problem of estimating fractional orders and model parameters from electroencephalographic data is addressed through multi-objective optimization with rigorous train–test validation. Systematic sensitivity analysis across the parameter space (αu,αv)∈[1.0,2.0]×[1.0,2.0] identifies optimal subdiffusive characteristics at αu=αv=1.5, corresponding to power-law spatial kernels C(x)∼|x|−1.5 consistent with anatomical connectivity measurements. The optimized model achieves out-of-sample performance R2=0.973 on held-out test data, approaching the measurement noise ceiling. While classical FitzHugh–Nagumo models achieve comparable test accuracy, the fractional framework provides enhanced interpretability through explicit spatial interaction parameterization. The fractional orders serve as quantitative biomarkers of cortical network organization, enabling data-driven characterization across brain states and neurological conditions. The methodology establishes computational foundations for clinical applications in epilepsy monitoring, neurodegenerative disease detection, and brain–computer interfaces.

Recent research has demonstrated the importance of spatial diffusion and environmental heterogeneity in influencing the transmission dynamics of infectious diseases. At the same time, human mobility patterns have been shown to exhibit scale-free, nonlocal dynamics characterized by an anomalous Lévy process diffusion, which is mathematically represented by nonlocal equations involving fractional Laplacian operators. To investigate the effects of environmental heterogeneity and long-range geographical disease transmission, we propose a time-periodic susceptible-infectious-susceptible (SIS) epidemic model that incorporates anomalous diffusion and spatial heterogeneity. The key issues of this paper include the existence and stability of both disease-free and endemic periodic equilibria, as well as the impact of diffusion rates and fractional powers on the spatial distribution of these periodic states. Our analytical findings indicate that spatio-temporal heterogeneity promotes disease persistence and that the fractional power can modulate the transmission threshold.

Abstract We study the minimizers of $$\lambda _k^s(A) + |A|$$ λ k s ( A ) + | A | where $$\lambda ^s_k(A)$$ λ k s ( A ) is the k -th Dirichlet eigenvalue of the fractional Laplacian on A . Unlike in the case of the Laplacian, free boundary of minimizers exhibits distinct global behaviors. Our main results include: the existence of minimizers, optimal Hölder regularity for the corresponding eigenfunctions, and in the case where $$\lambda _k$$ λ k is simple, non-degeneracy, density estimates, separation of the free boundary, and free boundary regularity. We propose a combinatorial toy problem related to the global configuration of such minimizers.

Qualitative analysis of rupture set for a semilinear elliptic equation involving the fractional Laplacian

This paper deals with the obstacle problem for the fractional infinity Laplacian with nonhomogeneous term $f(u)$, where $f:\mathbb{R}^+ \mapsto \mathbb{R}^+$: $$\begin{cases} L[u]=f(u) &\qquad in \,\,\,\,\, \{u>0\}\\ u \geq 0 &\qquad in \,\,\,\,\, \Omega\\ u=g &\qquad on\,\,\,\,\, \partial \Omega\end{cases},$$ with [[EQUATION]]\\ Under the assumptions that $f$ is a continuous and monotone function and that the boundary datum $g$ is in $C^{0,\beta}(\partial\Omega)$ for some $0<\beta<\alpha$, we prove existence of a solution $u$ to this problem. Moreover, this solution $u$ is $\beta-$H\"olderian on $\overline{\Omega}$. Our proof is based on an approximation of $f$ by an appropriate sequence of functions $f_\varepsilon$ where we prove using Perron's method the existence of solutions $u_\varepsilon$, for every $\varepsilon>0$. Then, we show some uniform H\"older estimates on $u_\varepsilon$ that guarantee that $u_\varepsilon \rightarrow u$ where this limit function $u$ turns out to be a solution to our obstacle problem.

A priori estimates for anti-symmetric solutions to a fractional Laplacian equation in a bounded domain

This paper is devoted to investigating the following fractional Choquard problem [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] is a small parameter. Here [Formula: see text] with [Formula: see text] being a smooth bounded domain in [Formula: see text] with [Formula: see text], and [Formula: see text] denotes the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. We construct a family of solutions to the above problem exhibiting volcano-like blow-up behavior as [Formula: see text] goes to zero. This result extends the study of critical Choquard equations to fractional settings with shrinking holes, highlighting the interplay between nonlocality and domain topology.

A Semianalytic Diagonalization Finite Element Method for the Spectral Fractional Laplacian

With the help of a new Picone-type identity, we prove that positive solutions u\,{\in}\, H^{s}(\mathbb{R}^{N}) to the equation (-\Delta )^{s} u+ u=u^{p} in \mathbb{R}^{N} are nonradially nondegenerate, for all s\in (0,1) , N\geq 1 and p>1 strictly smaller than the critical Sobolev exponent. By this we mean that the linearized equation (-\Delta )^{s} w+ w-pu^{p-1}w = 0 does not admit nonradial solutions besides the directional derivatives of u . Letting B be the unit centered ball and \lambda_{1}(B) the first Dirichlet eigenvalue of the fractional Laplacian (-\Delta )^{s} , we also prove that positive solutions to (-\Delta )^{s} u+\lambda u=u^{p} in {B} , with u=0 on \mathbb{R}^{N}\setminus B , are nonradially nondegenerate for any \lambda> -\lambda_{1}(B) in the sense that the linearized equation does not admit nonradial solutions. From these results, we then deduce uniqueness and full nondegeneracy of positive solutions in some special cases. In particular, in the case N=1 , we prove that the equation (-\Delta )^{s} u+ u=u^{2} in \mathbb{R} or in B , with zero exterior data, admits a unique even solution which is fully nondegenerate in the optimal range s \in (\frac{1}{6},1) , thus extending the classical uniqueness result of Amick and Toland on the Benjamin–Ono equation. Moreover, in the case N=1 , \lambda=0 , we also prove the uniqueness and full nondegeneracy of positive solutions for the Dirichlet problem in B with arbitrary subcritical exponent p . Finally, we determine the unique positive ground state solution of (-\Delta )^{\frac{1}{2}}u+ u=u^{p} in \mathbb{R}^{N} , N \ge 1 with p=1+\frac{2}{N+1} and compute the sharp constant in the associated Gagliardo–Nirenberg inequality \|u\|_{L^{p+1}(\mathbb{R}^N)}\le \rule{0pt}{9pt}C \|(-\Delta )^{\frac{1}{4}} u\|_{L^2(\mathbb{R}^N)}^{\frac{N}{N+2} \|u\|_{L^2(\mathbb{R}^N)}^{\frac{2}{N+2}}} .

In this note, we present a logarithmic-type upper bound for weak subsolutions to a class of integro-differential problems, whose prototype is the Dirichlet problem for the fractional Laplacian. The bound is slightly smaller than the classical one in this field.

Modified L2-1$$_{\sigma}$$ schemes coupled with a spectral method for solving the time-space fractional diffusion equation containing the fractional Laplacian

Stability of Traveling Wave Solutions for Degenerate Fisher Type Equations with Fractional Laplacian

This paper is concerned with the nonlinear strongly damped wave equations involving the fractional Laplacian and regional fractional Laplacian with various boundary conditions. We first prove the existence and uniqueness of weak solutions using the compactness method and weak convergence techniques in Orlicz spaces. Then we study the existence and regularity of global attractors of associated semigroups. The main novelty of the obtained results here is to improve and extend the previous results in [6, 7, A.N. Carvalho and J.W. Cholewa] and [24, J. Shomberg].

We study an elliptic equation, with homogeneous Dirichlet bound-ary conditions, driven by a mixed type operator (the sum of the Lapla-cian and the fractional Laplacian), involving a parametric reaction and an undetermined source term. Applying a recent abstract critical point theorem of Ricceri, we prove existence of a solution for a convenient source and small enough parameters.

The spreading phenomenon of solutions for reaction–diffusion equations with fractional Laplacian