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The hybridizable discontinuous Galerkin (HDG) method has been applied to a nonlinear parabolic integro-differential equation. The nonlinear functions are considered to be Lipschitz continuous to analyze uniform in time a priori bounds. An extended type Ritz–Volterra projection is introduced and used along with the HDG projection as an intermediate projection to achieve optimal order convergence of O ⁢ (h k + 1) when polynomials of degree k ≥ 0 are used to approximate both the solution and the flux variables. By relaxing the assumptions in the nonlinear variable, super-convergence is achieved by element-by-element post-processing. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper involves a diffusive epidemic model whose domain has one free boundary with the Stefan boundary condition, and one fixed boundary subject to the usual homogeneous Dirichlet or Neumann condition. By using the standard upper and lower solutions method and the regularity theory, we first study some related steady-state problems which help us obtain the exact longtime behaviors of solution component (u, v). Then we prove there exists the unique classical solution whose longtime behaviors are governed by a spreading-vanishing dichotomy. Lastly, the criteria determining when spreading or vanishing happens are given with respect to the basic reproduction number R 0 , the initial habitat [ 0 , h 0 ] , the expanding rates μ 1 and μ 2 as well as the initial function (u 0 , v 0) . The criteria reveal the effect of the cooperative behaviors of agents and humans on spreading and vanishing. [ABSTRACT FROM AUTHOR]

We study the fractional p-Laplace equation (- Δ p) s u = 0 for 0 < s < 1 and in the subquadratic case 1 < p < 2 . We provide Hölder estimates with an explicit Hölder exponent. The inhomogeneous equation is also treated and there the exponent obtained is almost sharp for a certain range of parameters. Our results complement the previous results for the superquadratic case when p ≥ 2 . The arguments are based on a careful Moser-type iteration and a perturbation argument. [ABSTRACT FROM AUTHOR]

This article establishes a sufficient criteria for the existence of an optimal pair and a mild-solution of a non-autonomous fractional Sobolev-type integro-differential equation (FSIDE) with a generalized nonlocal condition. The fixed-point technique, tools from fractional calculus and the semigroup-theory played a valuable role in deriving the results. It is imperative to mention the distinguish attributes of this work that the developed results don't assume (i) that the generated semigroup is compact; (ii) that the linear operators - S (t) are bounded; and (iii) that the operator D is strongly continuous. We also provide an example to demonstrate the established results. [ABSTRACT FROM AUTHOR]

This study deals with the existence of nodal solutions for the following gauged nonlinear Schrödinger equation with zero mass: −Δu+hu2(∣x∣)∣x∣2+∫∣x∣+∞hu(s)su2(s)dsu=∣u∣p−2u,x∈R2,-\Delta u+\left(\frac{{h}_{u}^{2}\left(| x| )}{{| x| }^{2}}+\underset{| x| }{\overset{+\infty }{\int }}\frac{{h}_{u}\left(s)}{s}{u}^{2}\left(s){\rm{d}}s\right)u={| u| }^{p-2}u,\hspace{1.0em}x\in {{\mathbb{R}}}^{2}, where p>6p\gt 6 and hu(s)=12∫0sru2(r)dr{h}_{u}\left(s)=\frac{1}{2}{\int }_{0}^{s}r{u}^{2}\left(r){\rm{d}}r. By variational methods, we prove that for any integer k≥0k\ge 0, the above equation has a nodal solution wk{w}_{k} which changes sign exactly kk times. Moreover, we also prove that wk{w}_{k} belongs to L2(R2){L}^{2}\left({{\mathbb{R}}}^{2}) provided p>10p\gt 10.

This paper addresses the electromagnetic inverse scattering problem of determining the location and shape of anisotropic objects from near-field Cauchy data. We investigate both cases involving the Helmholtz equation and Maxwell's equations for this inverse problem. Our study focuses on developing efficient imaging functionals that enable a fast and stable recovery of the anisotropic object. The implementation of the imaging functionals is simple and avoids the need to solve an ill-posed problem. The resolution analysis of the imaging functionals is conducted using the Green representation formula. Furthermore, we establish stability estimates for these imaging functionals when noise is present in the data. To illustrate the effectiveness of the methods, we present numerical examples showcasing their performance. [ABSTRACT FROM AUTHOR]

We study the regularity of weak solutions to evolution equations with distributed order fractional time derivative. We prove a weak Harnack inequality for nonnegative weak supersolutions and Hölder continuity of weak solutions to this problem. Our results substantially generalise analogous known results for the problem with single order fractional time derivative. [ABSTRACT FROM AUTHOR]

In this paper, we study a Schrödinger–Choquard equation involving the logarithmic Laplacian operator in R n : L ▵ u (x) + ω u (x) = C n , s (| x | 2 s - n ∗ u p) u r , x ∈ R n , where 0 < s < 1 , p > 1 , r > 0 , n ≥ 2 , ω > 0 . Using the direct method of moving planes, we prove that if u satisfies some suitable asymptotic properties, then u must be radially symmetric and monotone decreasing about some point in the whole space. The key ingredients of the proofs are the narrow region principle and decay at infinity theorem; the ideas can be applied to problems involving more general nonlocal operators. [ABSTRACT FROM AUTHOR]

The paper is concerned with a nonlocal problem for a partial Fredholm integro-differential equation (IDE) of hyperbolic type is investigated. This problem is reduced to a problem containing a family of boundary value problems for the ordinary Fredholm IDEs and some integral relationships. A novel concept of general solution to a family of the ordinary Fredholm IDEs is introduced, and its properties are discussed. A necessary and sufficient condition for the well-posedness of the nonlocal problem for the partial Fredholm integro-differential equation (IDE) of hyperbolic type is obtained, and an algorithm for finding its solution is offered. [ABSTRACT FROM AUTHOR]

In this paper, we consider a finite delay integro-differential equation with a nonlinear kernel in a general Banach space. The nonlinear part is assumed to be continuous with respect to a fractional power of the linear part in the second variable. We prove, using the semigroup theory, the local existence, continuous dependence of the initial data, the phenomena of blowing up, regularity, and compactness properties of the so-called mild solution. An application is provided to illustrate our results. [ABSTRACT FROM AUTHOR]

We provide bounds for the sequence of eigenvalues {λi(Ω)}i{\left\{{\lambda }_{i}\left(\Omega )\right\}}_{i} of the Dirichlet problem (I−Δ)lnu=λuinΩ,u=0inRN\Ω,{\left(I-\Delta )}^{\mathrm{ln}}u=\lambda u\hspace{1em}{\rm{in}}\hspace{0.33em}\Omega ,\hspace{1.0em}u=0\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N}\setminus \Omega , where (I−Δ)ln{\left(I-\Delta )}^{\mathrm{ln}} is the Klein-Gordon operator with Fourier transform symbol ln(1+∣ξ∣2)\mathrm{ln}\left(1+{| \xi | }^{2}). The purpose of this study is to obtain the upper and lower bounds for the sum of the first k-eigenvalues by extending the Li-Yau’s method and Kröger’s method, respectively.

Accelerated propagation is a new phenomenon associated with nonlocal diffusion problems. In this paper, we determine the exact rate of accelerated propagation in the Fisher-KPP equation with nonlocal diffusion and free boundaries, where the nonlocal diffusion operator is given by ∫ R J (x - y) u (t , y) d y - u (t , x) , and the kernel function J(x) behaves like a power function near infinity, namely lim | x | → ∞ J (x) | x | α = λ > 0 for some α ∈ (1 , 2 ] . This is the precise range of α where accelerated spreading can happen for such kernels. By constructing subtle upper and lower solutions, we prove that the location of the free boundaries x = h (t) and x = g (t) goes to infinity at exactly the following rates: lim t → ∞ h (t) t ln t = lim t → ∞ - g (t) t ln t = μ λ , when α = 2 , lim t → ∞ h (t) t 1 / (α - 1) = lim t → ∞ - g (t) t 1 / (α - 1) = 2 2 - α 2 - α μ λ , when α ∈ (1 , 2). Here μ > 0 is a given parameter in the free boundary condition. Accelerated propagation can also happen when lim | x | → ∞ J (x) | x | (ln | x |) β = λ > 0 for some β > 1 . For this case, we prove that - g (t) , h (t) = exp { [ ( 2 β μ λ β - 1 ) 1 / β + o (1) ] t 1 / β } as t → ∞. These results considerably sharpen the corresponding ones in [20], and the techniques developed here open the door for obtaining similar precise results for other problems. A crucial technical point is that such precise conclusions on the propagation are achievable by finding the correct improvements on the form of the lower solutions used in [20], even though the precise long-time profile of the density function u(t, x) is still lacking. [ABSTRACT FROM AUTHOR]

A one-dimensional generalized backward stochastic differential equation with jumps and two barriers is the main objective of this paper. When the generators are monotone and the barriers are right continuous with left limits and completely separated, we prove the existence and uniqueness of a solution. As in application, we provide a probabilistic interpretation of a solution of a double obstacle problem of second-order parabolic integral-partial differential equations with nonlinear Neumann boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we establish Schauder's estimates for the following non-local equations in R d : ∂ t u = L κ , σ (α) u + b · ∇ u + f , u (0) = 0 , where α ∈ (1 / 2 , 2) and b : R + × R d → R is an unbounded local β -order Hölder function in x uniformly in t, and L κ , σ (α) is a non-local α -stable-like operator with form: L κ , σ (α) u (t , x) : = ∫ R d (u (t , x + σ (t , x) z) - u (t , x) - σ (t , x) z (α) · ∇ u (t , x)) κ (t , x , z) ν (α) (d z) , where z (α) = z 1 α ∈ (1 , 2) + z 1 | z | ≤ 1 1 α = 1 , κ : R + × R 2 d → R + is bounded from above and below, σ : R + × R d → R d ⊗ R d is a γ -order Hölder continuous function in x uniformly in t, and ν α) is a singular non-degenerate α -stable Lévy measure. [ABSTRACT FROM AUTHOR]

The focus of this investigation centers on the formulation of a novel spectral method deployed to numerically solve partial integro-differential equations that possess memory features. The approach leverages the hexic shifted Chebyshev polynomials, and addresses the nonlinearity of the computational output using iterative methods. A comprehensive explanation of the methodology is presented, and a thorough convergence analysis is conducted. This technique has the capacity to be effortlessly modified and used to solve a plethora of linear and nonlinear issues, while minimizing computational time. Ultimately, the effectiveness of this novel strategy is demonstrated through the successful resolution of two exemplary problems. The findings of this study suggest that this spectral approach holds significant promise for solving partial integro-differential equations. [ABSTRACT FROM AUTHOR]

In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation (−Δ)su+μu=(Iα*F(u))F′(u) inRN, ${\left(-{\Delta}\right)}^{s}u+\mu u=\left({I}_{\alpha }{\ast}F\left(u\right)\right){F}^{\prime }\left(u\right)\quad \text{in} {\mathbb{R}}^{N},$ (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), Iα∼1|x|N−α ${I}_{\alpha }\sim \frac{1}{\vert x{\vert }^{N-\alpha }}$ is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u∈Hs(RN) $u\in {H}^{s}\left({\mathbb{R}}^{N}\right)$ , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫RNu2=m>0 ${\int }_{{\mathbb{R}}^{N}}{u}^{2}=m{ >}0$ prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, “Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities,” Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations II: existence of infinitely many solutions,” Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C 1-regularity.

In this paper we study the problem of prescribing fractional Q-curvature of order 2 σ for a conformal metric on the standard sphere S n with σ ∈ (0 , n / 2) and n ≥ 3 . Compactness and existence results are obtained in terms of the flatness order β of the prescribed curvature function K. Making use of integral representations and perturbation result, we develop a unified approach to obtain these results when β ∈ [ n - 2 σ , n) for all σ ∈ (0 , n / 2) . This work generalizes the corresponding results of Jin-Li-Xiong (Math Ann 369:109–151, 2017) for β ∈ (n - 2 σ , n) . [ABSTRACT FROM AUTHOR]

Let Σ (f) be the singular points of a polynomial f ∈ K [ x , y ] in the plane K 2 , where K is R or C . Our goal is to study the singular point map S d , it sends polynomials f of degree d to their singular points Σ (f) . Very roughly speaking, a polynomial f is essentially determined when any other g sharing the singular points of f satisfies that f = λ g ; here both are polynomials of degree d, λ ∈ K ∗ . In order to describe the degree d essentially determined polynomials, a computation of the required number of isolated singular points δ (d) is provided. A dichotomy appears for the values of δ (d) ; depending on a certain parity, the space of essentially determined polynomials is an open or closed Zariski set. We compute the map S 3 , describing under what conditions a configuration of 4 points leads to a degree 3 essentially determined polynomial. Furthermore, we describe explicitly configurations supporting degree 3 non essential determined polynomials. The quotient space of essentially determined polynomials of degree 3 up to the action of the affine group Aff (K 2) determines a singular K -analytic surface. [ABSTRACT FROM AUTHOR]

A mutualist model with nonlocal diffusions and a free boundary is first considered. We prove that this problem has a unique solution defined for t ≥ 0 , and its dynamics are governed by a spreading-vanishing dichotomy. Some criteria for spreading and vanishing are also given. Of particular importance is that we find that the solution of this problem has quite rich longtime behaviors, which vary with the conditions satisfied by kernel functions and are much different from those of the counterpart with local diffusion and free boundary. At last, we extend these results to the model with nonlocal diffusions and double free boundaries. [ABSTRACT FROM AUTHOR]

We establish the existence in the sense of sequences of solutions for some integro-differential type equations containing the drift term and the square root of the one dimensional negative Laplacian, on the whole real line or on a finite interval with periodic boundary conditions in the corresponding H 2 spaces. The argument relies on the fixed point technique when the elliptic equations involve first order differential operators with and without Fredholm property. It is proven that, under the reasonable technical assumptions, the convergence in L 1 of the integral kernels implies the existence and convergence in H 2 of solutions. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a nonlinear elliptic problems that incorporate a Hardy potential and a nonlinear convection term. We establish the existence and regularity of solutions under various assumptions concerning the summability of the source term f.

We study an abstract linear viscoelastic equation in Hilbert space. It fits some convolution type of integro-differential equations which govern dynamics in viscoelastic or thermoviscoelastic materials. A new semigroup approach is proposed which is adapted from a semigroup approach designed originally to a second-order abstract linear integro-differential equation. As compared with the existing semigroup approach, the system contraction can be obtained readily. Sufficient conditions for the system exponential stability are also established. An illustrative example is given, and numerical simulations are provided. They are in good agreement with the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper proposes two efficient numerical schemes to deal with time-fractional integro-differential reaction-convection-diffusion equations of the Volterra type in a rectangular domain. The fractional derivative is taken in the Caputo sense of order α ∈ (0 , 1). In order to construct the computational semi-discrete schemes, we discretize the fractional operator using finite difference techniques, namely the L1 and the L1-2, on a uniform mesh in the time direction. A composite trapezoidal rule approximates the integral part. The semi-discrete problems become a set of two-point boundary value problems (BVPs) in the spatial domain. The cubic B-spline collocation method is employed to solve the corresponding BVPs. The optimal error estimation and convergence analysis are established under suitable regularity assumptions on the initial data and the true solution of the considered model problem. It is shown in the theoretical establishment that the L1 based scheme achieves (2 - α) order of accuracy, while the rate of convergence of the L1-2 based scheme is quadratic. Finally, a numerical experiment is performed in support of the theoretical findings. [ABSTRACT FROM AUTHOR]

We consider a semi-linear integro-differential equation in dimension one associated to the half Laplacian whose solution represents the atom dislocation in a crystal. The equation comprises the evolutive version of the classical Peierls–Nabarro model. We show that for a large number of dislocations, the solution, properly rescaled, converges to the solution of a fully nonlinear integro-differential equation which is a model for the macroscopic crystal plasticity with density of dislocations. This leads to the formal derivation of the 1-D Groma–Balogh equations (Groma–Balogh in Acta Mater 47(13):3647–3654, 1999), a popular model describing the evolution of the density of positive and negative oriented parallel straight dislocation lines. This paper completes the work of Patrizi and Sangsawang (Nonlinear Anal 202:112096, 2021). The main novelty here is that we allow dislocations to have different orientation and so we have to deal with collisions of them. [ABSTRACT FROM AUTHOR]

In this paper, the existence and uniqueness of a strong solution of the inverse problem of determining a coefficient of right-hand side of the integro-differential Kelvin–Voigt equation are investigated. The unknown coefficient that we search defends on space variables. Additional information on a solution of the inverse problem is given here as an integral overdetermination condition. The original inverse problem is reduced to study an equivalent inverse problem with homogeneous initial condition. Then the equivalences of the last inverse problem to an operator equation of second kind is proved. We establish the sufficient conditions for the unique solvability of the operator equation of second kind. [ABSTRACT FROM AUTHOR]

This paper is concerned with the approximate controllability of a class of semi-linear neutral integro-differential equation with nonlocal condition. The basic tools of this study are the theory of resolvent operators of linear neutral integro-differential equation and fractional powers. The fundamental solution of linear neutral integro-differential equations is also constructed to deal with the non-uniform boundedness of the extra linear term. Sufficient conditions of approximate controllability are then obtained through the resolvent condition. Finally, an example is provided to illustrate the applications of the obtained results. [ABSTRACT FROM AUTHOR]

This article investigates the analytical and numerical solutions of a class of non-autonomous time-fractional integro-partial differential initial-boundary-value problems (IBVPs) with fractional derivative of Caputo-type. The existence and uniqueness of the analytical solution of the IBVP are established by using the Sumudu decomposition method and the maximum-minimum principle, respectively. To obtain the numerical solution, first, we semi-discretize the IBVP by discretizing the time fractional derivative by using the L1-scheme and the integral term by using the trapezoidal rule on a graded mesh, and then we approximate the spatial derivatives by using the cubic spline method over a uniform mesh. The stability and convergence analysis of the numerical method are established. The performance of the proposed technique is validated through numerical experiments, and the results are compared with the method presented in Santra and Mohapatra (J. Comput. Appl Math.400, 113746, 13, 2022). [ABSTRACT FROM AUTHOR]

The generalized Hirota-Satsuma coupled KdV system with fractional-order derivative plays a significant role to simulate the interaction of nearby identical-weight particles in a crystal lattice structure, two long waves interaction with different dispersion relationships, the description of shallow-water wave propagation, ion-acoustic waves, plasma physics and some other fields. In this study, diverse analytical wave solutions in general and standard structures are established of the stated equation by introducing two viable techniques, namely, the generalized Kudryashov scheme and the two variables G'/G,1/G-expansion approach. The solutions are established in terms of elementary functions, specifically trigonometric functions, rational functions, exponential functions, and hyperbolic functions. For definite values of free parameters, the obtained analytical wave solutions transform into solitary wave solutions. The graphical patterns of the wave solutions with there-, two-, and contour plots are depicted magnificently to elucidate the internal structure of the phenomenon. The methods contribute as powerful mathematical tools and appear to be further effective, computerized, and user-friendly to investigate nonlinear fractional equations as well as comprehensive analytical solutions of nonlinear evolution equations in engineering, technology, and mathematical physics.

We provide bounds for the sequence of eigenvalues { λ i ⁢ (Ω) } i of the Dirichlet problem L Δ ⁢ u = λ ⁢ u ⁢ in ⁢ Ω , u = 0 ⁢ in ⁢ ℝ N ∖ Ω , where L Δ is the logarithmic Laplacian operator with Fourier transform symbol 2 ⁢ ln ⁡ | ζ | . The logarithmic Laplacian operator is not positively defined if the volume of the domain is large enough. In this article, we obtain the upper and lower bounds for the sum of the first k eigenvalues by extending the Li–Yau method and Kröger's method, respectively. Moreover, we show the limit of the quotient of the sum of the first k eigenvalues by k ⁢ ln ⁡ k is independent of the volume of the domain. Finally, we discuss the lower and upper bounds of the k-th principle eigenvalue, and the asymptotic behavior of the limit of eigenvalues. [ABSTRACT FROM AUTHOR]

This paper is concerned with the spreading speed of a food-limited population model with delay. First, the existence of the solution of Cauchy problem is proved. Then, the spreading speed of solutions with compactly supported initial data is investigated by using the general Harnack inequality. Finally, we present some numerical simulations and investigate the dynamical behavior of the solution. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solution of Volterra integral equations of the third kind with non-smooth solutions based on the recursive approach of the spectral Tau method. To this end, a new set of the fractional version of canonical basis polynomials (called FC-polynomials) is introduced. The approximate polynomial solution (called Tau-solution) is expressed in terms of FC-polynomials. The fractional structure of Tau-solution allows recovering the standard degree of accuracy of spectral methods even in the case of non-smooth solutions. The convergence analysis of the method is carried out. The obtained numerical results show the accuracy and efficiency of the method compared to other existing methods. [ABSTRACT FROM AUTHOR]

This manuscript proposes an approximate solution for weakly singular kernel partial integrodifferential equations with tempered ψ -Caputo fractional derivative of order α ∈ (0 , 1) . Our method employs a second-order time-difference approximation and uses the tempered fractional integral operator together with piecewise linear interpolation to compute the singularity of the kernel arising during the discretization process. In addition, the stability of the method is evaluated by using Von Neumann analysis. To show the reliability and applicability of the proposed approach, numerical examples have been solved. [ABSTRACT FROM AUTHOR]

This work is concerned with both higher integrability and differentiability for linear nonlocal equations with possibly very irregular coefficients of VMO-type or even coefficients that are merely small in BMO. In particular, such coefficients might be discontinuous. While for corresponding local elliptic equations with VMO coefficients such a gain of Sobolev regularity along the differentiability scale is unattainable, it was already observed in previous works that gaining differentiability in our nonlocal setting is possible under less restrictive assumptions than in the local setting. In this paper, we follow this direction and show that under assumptions on the right-hand side that allow for an arbitrarily small gain of integrability, weak solutions u ∈ W s , 2 in fact belong to W loc t , p for any s ≤ t < min { 2 s , 1 } , where p > 2 reflects the amount of integrability gained. In other words, our gain of differentiability does not depend on the amount of integrability we are able to gain. This extends numerous results in previous works, where either continuity of the coefficient was required or only an in general smaller gain of differentiability was proved. [ABSTRACT FROM AUTHOR]

We consider equations involving a combination of local and nonlocal degenerate p-Laplace operators. The main contribution of the paper is almost Lipschitz regularity for the homogeneous equation and Hölder continuity with an explicit Hölder exponent in the general case. For certain parameters, our results also imply Hölder continuity of the gradient. In addition, we establish existence, uniqueness and local boundedness. The approach is based on an iteration in the spirit of Moser combined with an approximation method. [ABSTRACT FROM AUTHOR]

This paper is devoted to establishing the compactness and existence results of the solutions to the prescribing fractional Q-curvature problem of order 2 σ on n-dimensional standard sphere when n - 2 σ = 2 , σ = 1 + m / 2 , m ∈ N + . The compactness results are novel and optimal. In addition, we prove a degree-counting formula of all solutions to achieve the existence. From our results, we can know where blow up occur. Furthermore, the sequence of solutions that blow up precisely at any finite distinct location can be constructed. It is worth noting that our results include the case of multiple harmonic. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study a nonlocal elliptic problem in R N (denoted as (P λ) below) involving the fractional Laplacian, a linear Hardy potential term and a critical nonlinear term. According to suitable assumptions on the set of extremal points of the functional coefficients, we prove that (P λ) has multiple positive solutions, and we determine their precise behavior near the extremal points. This work extends a previous article by the first author et al. on the local case. [ABSTRACT FROM AUTHOR]

In this article, we study the fractional Choquard–Kirchhoff equation with subcritical or critical nonlinearities. We obtain some existence results of solutions using variational methods and the Nehari manifold under two different cases. [ABSTRACT FROM AUTHOR]

Differential buoyancy sources at an ocean surface may induce a density-driven flow that joins faster flow components to create a multi-scale, 3D flow. Potential temperature and salinity are active tracers that determine the ocean's potential density: their distribution strongly affects the density-driven component, while the overall flow affects their distribution. We present a robust framework that allows one to study the effects of a general prescribed 3D flow on a density-driven velocity component through temperature and salinity transport, by constructing a modular 3D model of intermediate complexity. The model contains an incompressible velocity that couples two advection–diffusion equations for the two tracers. Instead of solving the Navier–Stokes equations for the velocity, we consider a prescribed flow composed of several spatially predetermined modes. One of these modes models the density-driven flow: its spatial form describes a density-driven flow structure and its strength is determined dynamically by averaged density differences. The other modes are completely predetermined, consisting of any incompressible, possibly unsteady, 3D flow, e.g., as determined by kinematic models, observations, or simulations. The result is a hybrid kinematic–dynamic model, formulated as a nonlinear, weakly coupled system of two non-local PDEs. We prove its well-posedness in the sense of Hadamard and obtain a priori rigorous bounds regarding analytical solutions. When the relevant Rayleigh number is small enough, we show, both rigorously and numerically, that for all initial conditions, the corresponding solutions converge to a unique steady state. Motivated by the Atlantic Meridional Overturning Circulation, the model's relevance to oceanic systems is demonstrated by tuning the parameters to mimic the North Atlantic ocean. We show that in one limit the model may recover a simplified oceanic box model, including a bi-stable regime, and in another limit a kinematic model of oceanic chaotic advection, suggesting it can be utilized to study spatially dependent feedback processes in the ocean. [ABSTRACT FROM AUTHOR]

We study the existence, nonexistence and qualitative properties of the solutions to the problem (P) (- Δ) s u - θ u | x | 2 s = u p - u q in R N u > 0 in R N u ∈ H ˙ s (R N) ∩ L q + 1 (R N) , where s ∈ (0 , 1) , N > 2 s , q > p ≥ (N + 2 s) / (N - 2 s) , θ ∈ (0 , Λ N , s) and Λ N , s is the sharp constant in the fractional Hardy inequality. For qualitative properties of the solutions, we mean both the radial symmetry, that is obtained by using the moving plane method in a nonlocal setting on the whole R N , and a suitable upper bound behavior of the solutions. To this last end, we use a representation result that allows us to transform the original problem into a new nonlocal problem in a weighted fractional space. [ABSTRACT FROM AUTHOR]

In this work, we study the vibration control of a flexible mechanical system. The dynamic of the problem is modeled as a viscoelastic nonlinear Euler–Bernoulli beam. To suppress the undesirable transversal vibrations of the beam, we adopt a control at the right boundary of the beam. This control law is simple to implement. We prove uniform stability of the system using a viscoelastic material, the multiplier method and some ideas introduced in [20]. It is shown that a large range of rates of decay of the energy can be achieved through a determined class of kernels. Unlike most of the existing classes in the market, ours are not necessarily strictly decreasing. [ABSTRACT FROM AUTHOR]

We deal with the existence of solutions having L 2 -regularity for a class of non-autonomous evolution equations. Associated with the equation, a general non-local condition is studied. The technique we used combines a finite dimensional reduction together with the Leray–Schauder continuation principle. This approach permits to consider a wide class of nonlinear terms by allowing demicontinuity assumptions on the nonlinearity. [ABSTRACT FROM AUTHOR]