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We investigate the application of a posteriori error estimate to a fractional optimal control problem with pointwise control constraints. Specifically, we address a problem in which the state equation is formulated as an integral form of the fractional Laplacian equation, with the control variable embedded within the state equation as a coefficient. We propose two distinct finite element discretization approaches for an optimal control problem. The first approach employs a fully discrete scheme where the control variable is discretized using piecewise constant functions. The second approach, a semi-discrete scheme, does not discretize the control variable. Using the first-order optimality condition, the second-order optimality condition, and a solution regularity analysis for the optimal control problem, we devise a posteriori error estimates. Based on the established error estimates framework, an adaptive refinement strategy is developed to help achieve the optimal convergence rate. Numerical experiments are given to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a new coupled PDE-based configuration for image denoising. We elaborate a new class of coupled PDE that involves a $ p(x) $ p (x) -Laplace and a controlled fractional-type operator, which takes into account the texture and smooth components during the recovering process. We give some essential theoretical results and we establish the well-posedness of the suggested coupled equation based on Galerkin approximation. Finally, we perform a fully discretization part of the system and illustrate some various numerical realizations to ensure the efficiency of our coupled PDE by conducting some comparison experiments against state of art PDE denoising models. [ABSTRACT FROM AUTHOR]

In this paper we consider two classes of nonlinear partial differential equations with the fractional Laplacian, namely $$\begin{align*} \left(-\Delta\right)^{\frac{\alpha}{2}}\left(u^{m} \right) &=u\left|u\right|^{q-1}+w\left(x\right), \quad x\in \mathbb{R}^{N},\\ &\qquad 1\leq m (− Δ) α 2 (u m) = u | u | q − 1 + w (x) , x ∈ R N , 1 ≤ m < q , 0 < α ≤ 2 , N > α and $$\begin{align*} &\frac{\partial^{k}u}{\partial t^{k}}+\left(-\Delta\right)^{\frac{\alpha}{2}}u=u^{q}, \quad \left(x,t\right)\in \mathbb{R}^{N}\times\left(0,+\infty\right),\\ &\qquad k\geq 1, \ 0 ∂ k u ∂ t k + (− Δ) α 2 u = u q , (x , t) ∈ R N × (0 , + ∞) , k ≥ 1 , 0 < α ≤ 2 , N > α. Solutions defined for all $ x\in \mathbb {R}^{N} $ x ∈ R N of the first equation are referred to as entire solutions, while solutions defined for all $ \left (x,t\right)\in \mathbb {R}^{N}\times \left [0,+\infty \right) $ (x , t) ∈ R N × [ 0 , + ∞) of the second equation are referred to as global solutions. Several existence and nonexistence theorems are established over different ranges of q, and thus the respective relations between the existence, nonexistence of solutions for these equations and the index q in the nonlinear terms are obtained. It is illustrated that our results are sharp in cases of m = 1 and k = 1 respectively. In addition, we prove the positivity, symmetry and odevity of solutions we constructed for the first equation with m = 1 associated with the inhomogeneous term w. [ABSTRACT FROM AUTHOR]

This paper deals with a class of parabolic reaction-diffusion equations driven by the fractional Laplacian as the diffusion operator over a bounded domain with zero Dirichlet external condition. Using a comparison principle and monotone iteration method, we establish existence and uniqueness results. We apply the existence result to the logistic growth problems with constant yield harvesting by constructing an ordered pair of positive sub- and supersolution of the corresponding elliptic problem. [ABSTRACT FROM AUTHOR]

In this paper, we present and analyze fully discrete finite difference schemes designed for solving the initial value problem associated with the fractional Korteweg-de Vries (KdV) equation involving the fractional Laplacian. We design the scheme by introducing the discrete fractional Laplacian operator which is consistent with the continuous operator, and possesses certain properties which are instrumental for the convergence analysis. Assuming the initial data u 0 ∈ H 1 + α (R) , where α ∈ [ 1 , 2) , our study establishes the convergence of the approximate solutions obtained by the fully discrete finite difference schemes to a classical solution of the fractional KdV equation. Theoretical results are validated through several numerical illustrations for various values of fractional exponent α . Furthermore, we demonstrate that the Crank–Nicolson finite difference scheme preserves the inherent conserved quantities along with the improved convergence rates. [ABSTRACT FROM AUTHOR]

We investigate models for nonlinear ultrasound propagation in soft biological tissue based on the one that serves as the core for the software package k‐Wave. The systems are solved for the acoustic particle velocity, mass density, and acoustic pressure and involve a fractional absorption operator. We first consider a system that incorporates additional viscosity in the equation for momentum conservation. By constructing a Galerkin approximation procedure, we prove the local existence of its solutions. In view of inverse problems arising from imaging tasks, the theory allows for the variable background mass density, speed of sound, and the nonlinearity parameter in the systems. Second, under stronger conditions on the data, we take the vanishing viscosity limit of the problem, thereby rigorously establishing the existence of solutions for the limiting system as well. [ABSTRACT FROM AUTHOR]

We study numerical methods and algorithms for time-dependent fractional-in-space diffusion problems. The considered anomalous diffusion is modelled by the fractional Laplacian (− Δ) α , 0 < α < 1 , following the integral definition. Fractional diffusion is non-local, and the finite element method (FEM) discretization in space leads to a dense stiffness matrix. It is well known that numerically solving such non-local boundary value problems is expensive. Difficulties increase significantly when the problem is time-dependent. The aim of the article is to develop computationally efficient methods and algorithms. There are two main features of our approach. Hierarchical semi-separable (HSS) compression is applied for an approximate solution of the arising linear systems. For time discretization, we use an adaptive forward–backward Euler scheme. The properties of the composite algorithm thus obtained are investigated. In particular, the block representation of HSS compression allowed us to upgrade the HSS solver to efficiently handle varying diagonally perturbed transition matrices corresponding to changing time steps. The contribution of the paper is threefold. The methods are completely constructive, which allows for a clearly structured description of the algorithms. A theoretical estimate of the computational complexity is presented. It shows the advantages of the adaptive time stepping in combination with the HSS solver. Theoretical results are supported by representative numerical experiments. Both sequential and parallel scalability and efficiency are analyzed. The presented results provide convincing proof of the concept of the proposed methods and algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we extend, for the first time, part of the Weierstrass extremal field theory in the Calculus of Variations to a nonlocal framework. Our model case is the energy functional for the fractional Laplacian (the Gagliardo–Sobolev seminorm), for which such a theory was still unknown. We build a null-Lagrangian and a calibration for nonlinear equations involving the fractional Laplacian in the presence of a field of extremals. Thus, our construction assumes the existence of a family of solutions to the Euler–Lagrange equation whose graphs produce a foliation. Then the minimality of each leaf in the foliation follows from the existence of the calibration. As an application, we show that monotone solutions to fractional semilinear equations are minimizers. In a forthcoming work, we generalize the theory to a wide class of nonlocal elliptic functionals and give an application to the viscosity theory. [ABSTRACT FROM AUTHOR]

The paper studies the fractional order version of the reinforced membrane problem introduced in [A. Henrot and H. Maillot, 2001]. Existence and uniqueness of the solutions of the corresponding non-local equations has been proven for the relaxed problem. In addition, for the radial symmetric case the existence of the optimal domain has been shown. [ABSTRACT FROM AUTHOR]

We study semilinear third‐order (in time) evolution equations with fractional Laplacian (−Δ)σ$$ {\left(-\Delta \right)}^{\sigma } $$ and power nonlinearity |u|p$$ {\left|u\right|}^p $$, which was proposed by Bezerra–Carvalho–Santos (J. Evol. Equ. 2022) recently. In this manuscript, we obtain a new critical exponent p=pcrit(n,σ):=1+6σmax{3n−4σ,0}$$ p={p}_{\mathrm{crit}}\left(n,\sigma \right):= 1+\frac{6\sigma }{\max \left\{3n-4\sigma, 0\right\}} $$ for n⩽103σ$$ n\leqslant \frac{10}{3}\sigma $$. Precisely, the global (in time) existence of small data Sobolev solutions is proved for the supercritical case p>pcrit(n,σ)$$ p>{p}_{\mathrm{crit}}\left(n,\sigma \right) $$, and energy solutions blow up in finite time even for small data if 1

We construct the global existence for a wave equation involving the fractional Laplacian with a logarithmic nonlinear source by using the Galerkin approximations. The corresponding results for equations with classical Laplacian are considered as particular cases of our assertions. [ABSTRACT FROM AUTHOR]

We discover new cancellations upon H 2 (R n) -estimate of the Hall term, n ∈ { 2 , 3 } . Consequently, first, we derive a regularity criterion for the 3-dimensional Hall-magnetohydrodynamics system in terms of horizontal components of velocity and magnetic fields. Second, we are able to prove the global regularity of the 2 1 2 -dimensional electron magnetohydrodynamics system with magnetic diffusion (- Δ) 3 2 (b 1 , b 2 , 0) + (- Δ) α (0 , 0 , b 3) for α > 1 2 despite the fact that (- Δ) 3 2 is the critical diffusive strength. Lastly, we extend this result to the 2 1 2 -dimensional Hall-magnetohydrodynamics system with - Δ u replaced by (- Δ) α (u 1 , u 2 , 0) - Δ (0 , 0 , u 3) for α > 1 2 . The sum of the derivatives in diffusion that our result requires is 11 + ϵ for any ϵ > 0 , while the sum for the classical 2 1 2 -dimensional Hall-magnetohydrodynamics system is 12 considering - Δ u and - Δ b , of which its global regularity issue remains an outstanding open problem. [ABSTRACT FROM AUTHOR]

In this article, we study the following fractional Kirchhoff-type problems with critical and sublinear nonlinearities: a+b∬RN×RN∣u(x)−u(y)∣2∣x−y∣N+2sdxdy(−Δ)su=λuq−1+u2s*−1,u>0,inΩ,u=0,inRN\Ω,∫RNu2dx=c2,\left\{\begin{array}{l}\left(a+b\mathop{\iint }\limits_{{{\mathbb{R}}}^{N}\times {{\mathbb{R}}}^{N}}\frac{{| u\left(x)-u(y)| }^{2}}{{| x-y| }^{N+2s}}{\rm{d}}x{\rm{d}}y\right){\left(-\Delta )}^{s}u=\lambda {u}^{q-1}+{u}^{{2}_{s}^{* }-1},\hspace{1em}u\gt 0,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0\left,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right,\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}}x={c}^{2},\end{array}\right. where (−Δ)s{\left(-\Delta )}^{s} is the fractional Laplacian, Ω⊂RN\Omega \subset {{\mathbb{R}}}^{N} is a bounded domain with Lipschitz boundary, 00,a>0,b>0,c>0\lambda \gt 0,a\gt 0,b\gt 0,c\gt 0. First, we prove that the bounded Palais-Smale sequence has a profile decomposition in the fractional Laplacian setting. Then, by utilizing decomposition techniques and variational methods, we acquire that there are two positive normalized solutions for the aforementioned problems.

In this article, we consider the global and local well-posedness of the mild solutions to the Cauchy problem of fractional drift diffusion system with higher-order nonlinearity. The main difficulty comes from the higher-order nonlinearity. Instead of the convention that people always focus on the properties of the solution in critical spaces, here we are interested in non-critical spaces such as supercritical Sobolev spaces and subcritical Lebesgue spaces. For the initial data in these non-critical spaces, using the properties of fractional heat semigroup and the classical Hardy-Littlewood-Sobolev inequality, we obtain the existence and uniqueness of the mild solution, together with the decaying rate estimates in terms of time variable.

We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian (− Δ) α 2 . Numerical analysis and experiments are provided to study its performance. Our method has the same symbol | ξ | α as the fractional Laplacian (− Δ) α 2 at the discrete level, and thus it can be viewed as the exact discrete analogue of the fractional Laplacian. This unique feature distinguishes our method from other existing methods for the fractional Laplacian. Note that our method is different from the Fourier pseudospectral methods in the literature which are usually limited to periodic boundary conditions (see Remark 1.1). Numerical analysis shows that our method can achieve a spectral accuracy. The stability and convergence of our method in solving the fractional Poisson equations were analyzed. Our scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms can be developed for efficient matrix-vector multiplications. The computational complexity is O (2 N log ⁡ (2 N)) , and the memory storage is O (N) with N the total number of points. Extensive numerical experiments verify our analytical results and demonstrate the effectiveness of our method in solving various problems. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak strong uniqueness of the Dirichlet type problems of fractional Laplace (Poisson) equations. We construct the Green’s function and the Poisson kernel. We then provide a somewhat sharp condition for the solution to be unique. We also show that the solution under such condition exists and must be given by our Green’s function and Poisson kernel. In doing these, we establish several basic and useful properties of the Green’s function and Poisson kernel. Based on these, we obtain some further a priori estimates of the solutions. Surprisingly those estimates are quite different from the ones for the local type elliptic equations such as Laplace equations. These are basic properties to the fractional Laplace equations and can be useful in the study of related problems. [ABSTRACT FROM AUTHOR]

This paper studies the following coupled k-Hessian system with different order fractional Laplacian operators: S k (D 2 w (x)) - A (x) (- Δ) α / 2 w (x) = f (z (x)) , S k (D 2 z (x)) - B (x) (- Δ) β / 2 z (x) = g (w (x)). Firstly, we discuss decay at infinity principle and narrow region principle for the k-Hessian system involving fractional order Laplacian operators. Then, by exploiting the direct method of moving planes, the radial symmetry and monotonicity of the nonnegative solutions to the coupled k-Hessian system are proved in a unit ball and the whole space, respectively. We believe that the present work will lead to a deep understanding of the coupled k-Hessian system involving different order fractional Laplacian operators. [ABSTRACT FROM AUTHOR]

The paper presents a family of efficient and high-accurate conservative schemes, which is achieved by using a combination of the diagonally implicit Runge-Kutta method and the quadratic auxiliary variable approach for the nonlocal Klein-Gordon Schrödinger system with Riesz fractional derivative. The process involves first expressing the equation as a new system with modified energy, original energy, and mass by introducing a new variable. The symplectic Runge-Kutta method is then applied to approximate the equivalent system and generate a class of semi-discrete systems with high-order accuracy in the temporal direction. The semi-discrete system is then discretized in space using the fourth-order difference method to obtain a fully-discrete scheme that preserves the original energy and mass. A fast solver is provided to implement the proposed methods in practical computations effectively. The accuracy and conservation properties of the new schemes are demonstrated through numerical experiments. [ABSTRACT FROM AUTHOR]

In this article, we analyze the stability of the parallel surface problem for semilinear equations driven by the fractional Laplacian. We prove a quantitative stability result that goes beyond the one previously obtained in [15]. Moreover, we discuss in detail several techniques and challenges in obtaining the optimal exponent in this stability result. In particular, this includes an upper bound on the exponent via an explicit computation involving a family of ellipsoids. We also sharply investigate a technique that was proposed in [14] to obtain the optimal stability exponent in the quantitative estimate for the nonlocal Alexandrov's soap bubble theorem, obtaining accurate estimates to be compared with a new, explicit example. [ABSTRACT FROM AUTHOR]

In this paper, we study a viscoelastic Kirchhoff equation with fractional Laplacian. We first prove local existence and uniqueness of solutions. By using the theory of potential wells, we further obtain the global existence of solutions. Moreover, in the case where the Kirchhoff function takes a common form, we obtain asymptotic behavior and blow-up of solutions. [ABSTRACT FROM AUTHOR]

Motivated by the notion of isotropic α -stable Lévy processes confined, by reflections, to a bounded open Lipschitz set D ⊂ R d , we study some related analytical objects. Thus, we construct the corresponding transition semigroup, identify its generator and prove exponential speed of convergence of the semigroup to a unique stationary distribution for large time. [ABSTRACT FROM AUTHOR]

This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with L 1 initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data. [ABSTRACT FROM AUTHOR]

This paper provides a variational treatment of the effect of external charges on the free charges in an infinite free-standing graphene sheet within the Thomas–Fermi theory. We establish existence, uniqueness and regularity of the energy minimizers corresponding to the free charge densities that screen the effect of an external electrostatic potential at the neutrality point. For the potential due to one or several off-layer point charges, we also prove positivity and a precise universal asymptotic decay rate for the screening charge density, as well as an exact charge cancellation by the graphene sheet. We also treat a simpler case of the non-zero background charge density and establish similar results in that case. [ABSTRACT FROM AUTHOR]

Laplace learning is a popular machine learning algorithm for finding missing labels from a small number of labeled feature vectors using the geometry of a graph. More precisely, Laplace learning is based on minimizing a graph-Dirichlet energy, equivalently a discrete Sobolev W2,1 seminorm, constrained to taking the values of known labels on a given subset. The variational problem is asymptotically ill-posed as the number of unlabeled feature vectors goes to infinity for finite given labels due to a lack of regularity in minimizers of the continuum Dirichlet energy in any dimension higher than one. In particular, continuum minimizers are not continuous. One solution is to consider higher-order regularization, which is the analogue of minimizing Sobolev Ws,2 seminorms. In this paper we consider the asymptotics of minimizing a graph variant of the Sobolev Ws,2 seminorm with pointwise constraints. We show that, as expected, one needs s > d/2, where d is the dimension of the data manifold. We also show that there must be an upper bound on the connectivity of the graph; that is, highly connected graphs lead to degenerate behavior of the minimizer even when s > d/2. [ABSTRACT FROM AUTHOR]

A one-dimensional space-time fractional reaction-diffusion problem is considered. We present a complete theory for the solution of the time-space fractional reaction-diffusion model, including existence and uniqueness in the case of using the spectral representation of the fractional Laplacian operator. An optimal error estimate is presented for the Galerkin spectral approximation of the problem under consideration. [ABSTRACT FROM AUTHOR]

In this paper, we study a Cauchy problem of fractional diffusion equation concerning fractional Laplacian. We prove the existence and uniqueness of solution to the problem under investigation in Hölder space. Then we apply the Monte Carlo method to solving this Cauchy problem. For the problem with free force term, we derive an unbiased scheme which only produces the statistic error. For the problem with inhomogeneous force term, we establish the simple and high-order jump-adapted schemes by approximating the trajectory of symmetric stable process. The error estimates of these two jump-adapted schemes are given based on the Hölder regularity of the solution. Numerical experiments including even one hundred dimensional case confirm the theoretical analysis and show the numerical efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we study the blow-up of solutions for hyperbolic equations involving the fractional Laplacian operator with damping and source terms. We obtain the global existence results. Then, we observe the blow-up of solutions using the concavity method. Finally, we present some numerical simulation results. [ABSTRACT FROM AUTHOR]

In the present paper, using the theory of boundary values of analytic semigroups, we find necessary and sufficient conditions to guarantee that the operator i (- Δ) α / 2 generates a strongly continuous semigroup in L p (R n) . [ABSTRACT FROM AUTHOR]

We establish sharp Adams type inequalities on Sobolev spaces W α , n / α (X) of any fractional order α < n on Riemannian symmetric space X of noncompact type with dimension n and of arbitrary rank. We also establish sharp Hardy–Adams inequalities on the Sobolev spaces W n / 2 , 2 (X) . We use Fourier analysis on the symmetric spaces to obtain these results. [ABSTRACT FROM AUTHOR]

This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular boundary data which are typical of fractional diffusion operators in space, and the other one is the consideration of the fractional-in-time Caputo and Riemann–Liouville derivatives in a unified way. We first construct classical solutions of our problems using the spectral theory and discussing the corresponding fractional-in-time ordinary differential equations. We take advantage of the duality between these fractional-in-time derivatives to introduce the notion of weak-dual solution for weighted-integrable data. As the main result of the paper, we prove the well-posedness of the initial and boundary-value problems in this sense. [ABSTRACT FROM AUTHOR]

In this paper, we provide several techniques to explicitly calculate fractional powers of 2×2$2\times 2$ operator matrices Λ=Λ11Λ12Λ21Λ22,$$\begin{equation*}\hspace*{10pc} \Lambda = \def\eqcellsep{&}\begin{bmatrix} \Lambda _{11} & \Lambda _{12} \\ \Lambda _{21} & \Lambda _{22} \end{bmatrix}, \end{equation*}$$focusing on creating a theory that can be applied to distinct situations. To illustrate the abstract results developed, we consider its application in systems of coupled reaction–diffusion equations and in (strongly damped) wave equations. We also discuss how these techniques can be applied to higher order matrices and we specifically calculate the fractional powers of a 4×4$4\times 4$ operator matrix associated to a weakly coupled system of wave equation. In addition, we deal with the applicability of this analysis with respect to solvability, stabilization, regularity, smooth dynamics, and connection with evolutionary classic equation and its fractional counterpart. [ABSTRACT FROM AUTHOR]

The theories of fractional Laplacians and of fractional calculus with respect to functions are combined to produce, for the first time, the concept of a fractional Laplacian with respect to a bijective function. The theory is developed both in the 1‐dimensional setting and in the general n$$ n $$‐dimensional setting. Fourier transforms with respect to functions are also defined, and the relationships between Fourier transforms, fractional Laplacians, and Marchaud‐type derivatives are explored. Function spaces for these operators are carefully defined, including weighted Lp$$ {L}&#x0005E;p $$ spaces and a new type of Schwartz space. The theory developed is then applied to construct solutions to some partial differential equations involving both fractional time derivatives and fractional Laplacians with respect to functions, with illustrative examples. [ABSTRACT FROM AUTHOR]

In this article, we examine variational inequalities of the form 〈 A (u) , v − u 〉 + 〈 F (u) , v − u 〉 ≥ 0 , ∀ v ∈ K u ∈ K , , where  A  is a generalized fractional  Φ -Laplace operator, K is a closed convex set in a fractional Musielak–Orlicz–Sobolev space, and  F  is a multivalued integral operator. We consider a functional analytic framework for the above problem, including conditions on the multivalued lower order term  F  such that the problem can be properly formulated in a fractional Musielak–Orlicz–Sobolev space, and the involved mappings have certain useful monotonicity–continuity properties. Furthermore, we investigate the existence of solutions contingent upon certain coercivity conditions. [ABSTRACT FROM AUTHOR]

Motivated by the study of relativistic atoms, we consider the Hardy operator (− Δ) α / 2 − κ | x | − α acting on functions of the form u (| x |) | x | ℓ Y ℓ , m (x / | x |) in L 2 (R d) , when κ ≥ 0 and α ∈ (0 , 2 ] ∩ (0 , d + 2 ℓ). We give a ground state representation of the corresponding form on the half-line (Theorem 1.5). For the proof we use subordinated Bessel heat kernels. [ABSTRACT FROM AUTHOR]

The numerical solution of a 1D fractional Laplacian boundary value problem is studied. Although the fractional Laplacian is one of the most important and prominent nonlocal operators, its numerical analysis is challenging, partly because the problem’s solution has in general a weak singularity at the boundary of the domain. To solve the problem numerically, we use piecewise linear collocation on a mesh that is graded to handle the boundary singularity. A rigorous analysis yields a bound on the maximum nodal error which shows how the order of convergence of the method depends on the grading of the mesh; hence, one can determine the optimal mesh grading. Numerical results are presented that confirm the sharpness of the error analysis. [ABSTRACT FROM AUTHOR]

Consider the stochastic partial differential equation$ \begin{equation*} \frac{\partial }{\partial t}u_t({\boldsymbol {x}}) = -(-\Delta)^{\frac{\alpha}{2}}u_t({\boldsymbol {x}}) +b\left(u_t({\boldsymbol {x}})\right)+\sigma\left(u_t({\boldsymbol {x}})\right) \dot F(t, {\boldsymbol {x}}), \ \ \ t\ge0, {\boldsymbol {x}}\in \mathbb{R}^d, \end{equation*} $where $ -(-\Delta)^{\frac{\alpha}{2}} $ denotes the fractional Laplacian with the power $ \alpha/2\in (1/2, 1] $, and the driving noise $ \dot F $ is a centered Gaussian random field which is white in time and with a spatial homogeneous covariance given by the Riesz kernel. We study the detailed behavior of the approximation spatial gradient $ u_t({\boldsymbol {x}})-u_t({\boldsymbol {x}}-{{\varepsilon}} \boldsymbol e) $ at any fixed time $ t>0 $, as $ {{\varepsilon}}\downarrow 0 $, where $ \boldsymbol e $ is a unit vector in $ \mathbb{R}^d $. As applications, we deduce the law of iterated logarithm and the behavior of the $ q $-variations of the solution in space. [ABSTRACT FROM AUTHOR]

Let 0 < θ ≤ 2 , N ≥ 1 and T > 0 . We are concerned with the Cauchy problem for the fractional semilinear parabolic equation ∂ t u + (- Δ) θ / 2 u = f (u) in R N × (0 , T) , u (x , 0) = u 0 (x) ≥ 0 in R N. Here, f ∈ C [ 0 , ∞) denotes a rather general growing nonlinearity and u 0 may be unbounded. We study local in time solvability in the so-called critical and doubly critical cases. In particular, when f (u) = u 1 + θ / N log (u + e) a , we obtain a sharp integrability condition on u 0 which explicitly determines local in time existence/nonexistence of a nonnegative solution. [ABSTRACT FROM AUTHOR]

This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G/K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for L 1 initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-K-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-K-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G/K. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as L 1 asymptotic convergence without the assumption of bi-K-invariance. [ABSTRACT FROM AUTHOR]

In this article, we consider the following Kirchhoff equation involving fractional Laplacian a + b [ u ] s 2 (- Δ) s u + u = | u | p - 2 u in Ω , u = 0 on R 3 \ Ω , where a , b > 0 are constants, 3 4 < s < 1 , [ u ] s is the so-called Gagliardo (semi)norm of u, 4 < p < 2 s ∗ = 6 3 - 2 s and Ω ⊂ R 3 is an exterior domain with smooth boundary ∂ Ω ≠ ∅. By establishing a global compactness lemma of the fractional Kirchhoff equation in exterior domains, we verify the compactness of Palais–Smale sequences corresponding to above problem at higher energy level interval. Then combining some crucial estimates and barycentric function, we determine the existence of positive bound state solutions provided that R 3 \ Ω is contained in a small ball. In addition, we point out that the main result can be extended to fractional Sobolev critical case with small parameter. [ABSTRACT FROM AUTHOR]

In this paper, let α be any real number between 0 and 2, we study the following semi-linear elliptic system involving the fractional Laplacian: (−Δ)α/2u(x)=f(u(x),v(x)),x∈Rn,(−Δ)α/2v(x)=g(u(x),v(x)),x∈Rn. $\begin{cases}{\left(-{\Delta}\right)}^{\alpha /2}u\left(x\right)=f\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n},\quad \hfill \\ {\left(-{\Delta}\right)}^{\alpha /2}v\left(x\right)=g\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n}.\quad \hfill \end{cases}$ Under nature structure conditions on f and g, we classify the positive solutions for the semi-linear elliptic system involving the fractional Laplacian by using the direct method of the moving spheres introducing by W. Chen, Y. Li, and R. Zhang (“A direct method of moving spheres on fractional order equations,” J. Funct. Anal., vol. 272, pp. 4131–4157, 2017). In the half space, we establish a Liouville type theorem without any assumption of integrability by combining the direct method of moving planes and moving spheres, which improves the result proved by W. Dai, Z. Liu, and G. Lu (“Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space,” Potential Anal., vol. 46, pp. 569–588, 2017).

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 summarize some of the recent developments in the area of fractional elliptic and parabolic equations with focus on how to apply the sliding method and the method of moving planes to obtain qualitative properties of solutions. We will compare the two methods and point out the pros and cons of each. We will demonstrate how to modify the ideas and techniques in studying fractional elliptic equations and then to employ them to investigate fractional parabolic problems. Besides deriving monotonicity of solutions, some other applications of the sliding method will be illustrated. These results have more or less appeared in a series of previous literatures, in which the ideas were usually submerged in detailed calculations. What we are trying to do here is to single out these ideas and illuminate the inner connections among them by using figures and intuitive languages, so that the readers can see the whole picture and quickly grasp the essence of these useful methods and will be able to apply them to solve a variety of other fractional elliptic and parabolic problems.

We investigate the Dirichelt problem involving the fractional Laplacian in the upper half-space R+n=x∈Rn∣x1>0 ${\mathbb{R}}_{+}^{n}=\left\{x\in {\mathbb{R}}^{n}\mid {x}_{1}{ >}0\right\}$ : (−Δ)su(x)=f(u(x)),x∈R+n, u(x)>0,x∈R+n, u(x)=0,x∉R+n. \begin{cases}\quad \hfill & {\left(-{\Delta}\right)}^{s}u\left(x\right)=f\left(u\left(x\right)\right),\qquad x\in {\mathbb{R}}_{+}^{n},\hfill \\ \quad \hfill & \qquad u\left(x\right){ >}0,\qquad x\in {\mathbb{R}}_{+}^{n},\hfill \\ \quad \hfill & \qquad u\left(x\right)=0,\qquad x\notin {\mathbb{R}}_{+}^{n}.\hfill \end{cases}. . We prove the positive solutions are monotonic increasing in the x 1-direction assuming u(x) grows no faster than |x|γ with γ ∈ (0, 2s) for |x| large. To start with, we develop a maximum principle on the narrow region. Then we apply a direct method of the moving planes for the fractional Laplacian to derive the monotonicity. As an application of the monotonicity result, we use it to prove nonexistence of bounded positive solutions in R+n ${\mathbb{R}}_{+}^{n}$ for f(u) = u p, p∈1,n−1+2sn−1−2s $p\in \left(1,\frac{n-1+2s}{n-1-2s}\right)$ .