In the present paper, we consider the following fractional double phase problem with nonlocal reaction:

\begin{align*}\qquad \quad \left\{\begin{array}{lll}(- \Delta)_p^s u + (- \Delta)_q ^s u + V(\varepsilon x) (|u|^{p - 2} u + |u|^{q - 2} u) = \left(\frac{1}{|x|^\mu} * F(u)\right) f(u) & \ \text{in}\ \mathbb{R}^N, \\u \in W^{s,p}(\mathbb{R}^N) \cap W^{s,q}(\mathbb{R}^N) ,\ u \gt 0 & \ \text{in}\ \mathbb{R}^N,\end{array}\right\}\end{align*}

where ɛ is apositive parameter, $0 \lt s \lt 1$, $2 \leqslant p \lt q \lt \min\{2p, N / s\}$, $0 \lt \mu \lt sp$, $(- \Delta)_t^s$ $(t \in \left\{p,q\right\})$ is the fractional t-Laplace operator, the reaction term $f : \mathbb{R} \mapsto \mathbb{R}$ is continuous, and the potential $V \in C (\mathbb{R}^N , \mathbb{R})$ satisfying a local condition. Using a variational approach and topological tools (the non-standard C1-Nehari manifold analysis and the abstract category theory), multiplicity of positive solutions and concentration properties for the above problem are established. Our results extend and complement some previous contributions related to double phase variational integrals.

We prove the existence of solutions to the Kuramoto–Sivashinsky equation with low regularity data in function spaces based on the Wiener algebra and in pseudomeasure spaces. In any spatial dimension, we allow the data to have its antiderivative in the Wiener algebra. In one spatial dimension, we also allow data that are in a pseudomeasure space of negative order. In two spatial dimensions, we also allow data that are in a pseudomeasure space one derivative more regular than in the one-dimensional case. In the course of carrying out the existence arguments, we show a parabolic gain of regularity of the solutions as compared to the data. Subsequently, we show that the solutions are in fact analytic at any positive time in the interval of existence.

Coffee berry diseases (CBD) pose significant threats to coffee production worldwide, affecting the livelihoods of millions of farmers and the global coffee market. Fractional calculus provides a powerful framework for describing non-local and memory-dependent phenomena, making it suitable for modelling the long-range interactions inherent in CBD spread. This study aims to formulate and analyse fractional order model for CBD transmission dynamics in the sense of Atangana–Baleanu–Caputo. Fixed point theorems were utilised to test the existence and uniqueness of the model’s solutions using fractional order. The basic reproduction number was calculated utilising the next-generation matrix. The model has locally asymptotically stable equilibrium positions (disease-free and endemic). Furthermore, the Lyapunov function was used to conduct a global stability analysis of the equilibrium locations. A numerical simulation of the CBD model was created using the fractional Adam–Bashforth–Moulton approach to validate the analytical findings. Our findings contribute to the development of more accurate predictive models and inform the design of targeted interventions to mitigate the impact of CBD on coffee production systems.

We introduce a free boundary model to study the effect of vesicle transport onto neurite growth. It consists of systems of drift-diffusion equations describing the evolution of the density of antero- and retrograde vesicles in each neurite coupled to reservoirs located at the soma and the growth cones of the neurites, respectively. The model allows for a change of neurite length as a function of the vesicle concentration in the growth cones. After establishing existence and uniqueness for the time-dependent problem, we briefly comment on possible types of stationary solutions. Finally, we provide numerical studies on biologically relevant scales using a finite volume scheme. We illustrate the capability of the model to reproduce cycles of extension and retraction.

We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension,

\begin{equation*} u_t = D u_{xx} + u(1-\phi *u), \end{equation*}

$\phi *u$

$\phi (y) \equiv H\left (\frac{1}{4}-y^2\right )$

$u=1$

$D$

$\Delta _1 \approx 0.00297$

$D \ll 1$

$O(1)$

$u=O(1)$

$u$

$D \to 0^+$

$D \geq \Delta _1$

$2 \sqrt{D}$

$0 \lt D \lt \Delta _1$

$u=0$

$D$

We consider the following problem: the drift of the wealth process of two companies, modelled by a two-dimensional Brownian motion, is controllable such that the total drift adds up to a constant. The aim is to maximize the probability that both companies survive. We assume that the volatility of one company is small with respect to the other, and use methods from singular perturbation theory to construct a formal approximation of the value function. Moreover, we validate this formal result by explicitly constructing a strategy that provides a target functional, approximating the value function uniformly on the whole state space.

In this work, we study early warning signs for stochastic partial differential equations (SPDEs), where the linearisation around a steady state is characterised by continuous spectrum. The studied warning sign takes the form of qualitative changes in the variance as a deterministic bifurcation threshold is approached via parameter variation. Specifically, we focus on the scaling law of the variance near the transition. Since we are dealing here, in contrast to previous studies, with the case of continuous spectrum and quantitative scaling laws, it is natural to start with linearisations of the drift operator that are multiplication operators defined by analytic functions. For a one-dimensional spatial domain, we obtain precise rates of divergence. In the case of the two- and three-dimensional domains, an upper bound to the rate of the early warning sign is proven. These results are cross-validated by numerical simulations. Our theory can be generically useful for several applications, where stochastic and spatial aspects are important in combination with continuous spectrum bifurcations.

We study the asymptotic behaviour of the least energy solutions to the following class of nonlocal Neumann problems:

$$ \begin{align*} \begin{cases} { d(-\Delta)^{s}u+ u= \vert u\vert^{p-1}u } & \text{in } \Omega, \\ {u>0} & \text{in } \Omega, \\ { \mathcal{N}_{s}u=0 } & \text{in } \mathbb{R}^{n}\setminus \overline{\Omega}, \end{cases} \end{align*} $$

where $\Omega \subset \mathbb {R}^{n}$ is a bounded domain of class $C^{1,1}$, $1<p<({n+s})/({n-s}),\,n>\max \{1, 2s \}, 0<s<1, d>0$ and $\mathcal {N}_{s}u$ is the nonlocal Neumann derivative. We show that for small $d,$ the least energy solutions $u_d$ of the above problem achieve an $L^{\infty }$-bound independent of $d.$ Using this together with suitable $L^{r}$-estimates on $u_d,$ we show that the least energy solution $u_d$ achieves a maximum on the boundary of $\Omega $ for d sufficiently small.

We consider a class of nonhomogeneous elliptic equations in the half-space with critical singular boundary potentials and nonlinear fractional derivative terms. The forcing terms are considered on the boundary and can be taken as singular measure. Employing a functional setting and approach based on localization-in-frequency and Littlewood–Paley decomposition, we obtain results on solvability, regularity, and symmetry of solutions.

We study a system of nonlocal aggregation cross-diffusion PDEs that describe the evolution of opinion densities on a network. The PDEs are coupled with a system of ODEs that describe the time evolution of the agents on the network. Firstly, we apply the Deterministic Particle Approximation (DPA) method to the aforementioned system in order to prove the existence of solutions under suitable assumptions on the interactions between agents. Later on, we present an explicit model for opinion formation on an evolving network. The opinions evolve based on both the distance between the agents on the network and the ’attitude areas’, which depend on the distance between the agents’ opinions. The position of the agents on the network evolves based on the distance between the agents’ opinions. The goal is to study radicalisation, polarisation and fragmentation of the population while changing its open-mindedness and the radius of interaction.

Infection mechanism plays a significant role in epidemic models. To investigate the influence of saturation effect, a nonlocal (convolution) dispersal susceptible-infected-susceptible epidemic model with saturated incidence is considered. We first study the impact of dispersal rates and total population size on the basic reproduction number. Yang, Li and Ruan (J. Differ. Equ. 267 (2019) 2011–2051) obtained the limit of basic reproduction number as the dispersal rate tends to zero or infinity under the condition that a corresponding weighted eigenvalue problem has a unique positive principal eigenvalue. We remove this additional condition by a different method, which enables us to reduce the problem on the limiting profile of the basic reproduction number into that of the spectral bound of the corresponding operator. Then we establish the existence and uniqueness of endemic steady states by a equivalent equation and finally investigate the asymptotic profiles of the endemic steady states for small and large diffusion rates to provide reference for disease prevention and control, in which the lack of regularity of the endemic steady state and Harnack inequality makes the limit function of the sequence of the endemic steady state hard to get. Finally, we find whether lowing the movements of susceptible individuals can eradicate the disease or not depends on not only the sign of the difference between the transmission rate and the recovery rate but also the total population size, which is different from that of the model with standard or bilinear incidence.

In this paper, we establish the sharp asymptotic decay of positive solutions of the Yamabe type equation $\mathcal {L}_s u=u^{\frac {Q+2s}{Q-2s}}$ in a homogeneous Lie group, where $\mathcal {L}_s$ represents a suitable pseudodifferential operator modelled on a class of nonlocal operators arising in conformal CR geometry.

This paper investigates the separation property in binary phase-segregation processes modelled by Cahn-Hilliard type equations with constant mobility, singular entropy densities and different particle interactions. Under general assumptions on the entropy potential, we prove the strict separation property in both two and three-space dimensions. Namely, in 2D, we notably extend the minimal assumptions on the potential adopted so far in the literature, by only requiring a mild growth condition of its first derivative near the singular points $\pm 1$, without any pointwise additional assumption on its second derivative. For all cases, we provide a compact proof using De Giorgi’s iterations. In 3D, we also extend the validity of the asymptotic strict separation property to the case of fractional Cahn-Hilliard equation, as well as show the validity of the separation when the initial datum is close to an ‘energy minimizer’. Our framework offers insights into statistical factors like particle interactions, entropy choices and correlations governing separation, with broad applicability.

Flowering plants depend on some animals for pollination and contribute to nourish the animals in natural environments. We call these animals pollinators and build a plants-pollinators cooperative model with impulsive effect on a periodically evolving domain. Next, we define the ecological reproduction index for single plant model and plants-pollinators system, respectively, whose threshold dynamics, including the extinction, persistence and coexistence, is established by the method of upper and lower solutions. Theoretical analysis shows that a large domain evolution rate has a positive influence on the survival of pollinators whether or not the impulsive effect occurs, and the pulse eliminates the pollinators even when the evolution rate is high. Moreover, some selective numerical simulations are still performed to explain our theoretical results.

We show a result on propagation of the anisotropic Gabor wave front set for linear operators with a tempered distribution Schwartz kernel. The anisotropic Gabor wave front set is parametrized by a positive parameter relating the space and frequency variables. The anisotropic Gabor wave front set of the Schwartz kernel is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by any partial differential operator with constant coefficients, real symbol and order at least two.

We propose a new fractional Laplacian for bounded domains, expressed as a conservation law and thus particularly suited to finite-volume schemes. Our approach permits the direct prescription of no-flux boundary conditions. We first show the well-posedness theory for the fractional heat equation. We also develop a numerical scheme, which correctly captures the action of the fractional Laplacian and its anomalous diffusion effect. We benchmark numerical solutions for the Lévy–Fokker–Planck equation against known analytical solutions. We conclude by numerically exploring properties of these equations with respect to their stationary states and long-time asymptotics.

Strong unique continuation properties and a classification of the asymptotic profiles are established for the fractional powers of a Schrödinger operator with a Hardy-type potential, by means of an Almgren monotonicity formula combined with a blow-up analysis.

In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential,

\[ (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R}^n, \]

$n \geq 1$

$0< s<1$

$\omega >-\lambda _{1,s}$

$2< p< {2n}/{(n-2s)^+}$

$\lambda _{1,s}>0$

$(-\Delta )^s + |x|^2$

$(-\Delta )^s$

$\mathcal {F}((-\Delta )^{s}u)(\xi )=|\xi |^{2s} \mathcal {F}(u)(\xi )$

$\xi \in \mathbb {R}^n$

$\mathcal {F}$

This article is concerned with the problem of determining an unknown source of non-potential, external time-dependent perturbations of an incompressible fluid from large-scale observations on the flow field. A relaxation-based approach is proposed for accomplishing this, which makes use of a nonlinear property of the equations of motions to asymptotically enslave small scales to large scales. In particular, an algorithm is introduced that systematically produces approximations of the flow field on the unobserved scales in order to generate an approximation to the unknown force; the process is then repeated to generate an improved approximation of the unobserved scales, and so on. A mathematical proof of convergence of this algorithm is established in the context of the two-dimensional Navier–Stokes equations with periodic boundary conditions under the assumption that the force belongs to the observational subspace of phase space; at each stage in the algorithm, it is shown that the model error, represented as the difference between the approximating and true force, asymptotically decreases to zero in a geometric fashion provided that sufficiently many scales are observed and certain parameters of the algorithm are appropriately tuned.

In this paper, we consider the following non-linear system involving the fractional Laplacian0.1

\begin{equation} \left\{\begin{array}{@{}ll} (-\Delta)^{s} u (x)= f(u,\,v), \\ (-\Delta)^{s} v (x)= g(u,\,v), \end{array} \right. \end{equation}

$0< s<1$

$x_n$