Fractional Calculus and Applied Analysis最新文献

Being based on coupling by change of measure and an approximation technique, the Harnack inequalities for a class of stochastic functional differential equations driven by fractional Ornstein-Uhlenbeck process with Hurst parameter (0<H<1/2) are established. By using a transformation formulas for fractional Brownian motion, the Harnack inequalities for stochastic functional differential equations driven by fractional Ornstein-Uhlenbeck process with Hurst parameter (1/2<H<1) are established.

In this investigation, we conduct a rigorous analysis of a class of non-homogeneous generalized double phase problems, characterized by the inclusion of the fractional (phi _{x ,y}^i(cdot ))-Laplacian operator (where (i=1,2)) and a Choquard-logarithmic nonlinearity, along with a real parameter. Our methodology involves establishing a set of precise conditions related to the Choquard nonlinearities and the continuous function (phi _{x ,y}^i), under which we are able to confirm the existence of multiple distinct solutions to the problem. The analysis is situated within the realm of fractional modular spaces. Key to our approach is the application of the mountain pass theorem, which allows us to circumvent the necessity of the Palais-Smale condition, beside this we lay in the strategic use of the Hardy-Littlewood-Sobolev inequality to underpin the theoretical framework of our study.

We consider fractional partial differential equations posed on the full space (mathbb {R}^d). Using the well-known Caffarelli-Silvestre extension to (mathbb {R}^d times mathbb {R}^+) as equivalent definition, we derive existence and uniqueness of weak solutions. We show that solutions to a truncated extension problem on (mathbb {R}^d times (0,mathcal {Y})) converge to the solution of the original problem as (mathcal {Y}rightarrow infty ). Moreover, we also provide an algebraic rate of decay and derive weighted analytic-type regularity estimates for solutions to the truncated problem. These results pave the way for a rigorous analysis of numerical methods for the full space problem.

We consider multi-term fractional differential equations with continuous variable coefficients and differential operators of Erdélyi–Kober type and multiple independent fractional orders. We solve such equations in a general framework, obtaining explicit solutions in the form of uniformly convergent series. By considering several particular cases, we verify the consistency of our results with others previously obtained in the literature.

This paper investigates the existence of infinitely many pairs of nontrivial solutions for a class of nonsmooth fractional Hamiltonian systems, where the energy functional associated with the system is not continuously differentiable and does not satisfy the Palais-Smale condition. By considering a potential function of the form (V(t,x)=-K(t,x)+W(t,x)), where K and W are continuously differentiable functions with specific growth conditions, we extend existing results to cover cases involving nonsmoothness and certain types of nonlocal interactions. The study is based on variational methods and critical point theory, and we establish several theorems that guarantee the existence of multiple solutions under appropriate hypotheses on the nonlinearities of the system. These results contribute to the understanding of nonsmooth fractional Hamiltonian systems, particularly when traditional compactness conditions fail.

This paper systematically treats the asymptotic behavior of many (linear/nonlinear) classes of higher-order fractional differential equations with multiple terms. To do this, we utilize the characteristics of Caputo fractional differentiable functions, the comparison principle, counterfactual reasoning, and the spectral analysis method (concerning the integral presentations of basic solutions). Some numerical examples are also provided to demonstrate the validity of the proposed results.

In this paper, we deal with a time dependent inverse source problem for a nonlinear p-Laplacian pseudoparabolic equation containing a fractional derivative in time of order (alpha in (0,1)). Moreover, the equation is perturbed by a power-law damping (reaction) term, which, depending on whether its sign is positive or negative, may account for the presence of a source or an absorption within the system. The equation is supplemented with a measurement in a form of an integral over space domain along with the initial and Dirichlet boundary conditions, to determine both the solution of the equation and the unknown source term. For the associated inverse source problem, under suitable assumptions on the data, we establish global and local in time existence and uniqueness of weak solutions for different values of exponents and coefficients.

This paper investigates the Cauchy problem of time-space fractional Keller-Segel-Navier-Stokes system in ({mathbb {R}}^d~(dge 2)), which describes both memory effect and Lévy process of the system. The local and global existence of mild solutions are obtained by the (L^p-L^q) estimates of Mittag-Leffler operators combined with Banach fixed point theorem and Banach implicit function theorem, respectively. Furthermore, some properties are established, such as mass conservation, decay estimates, stability and self-similarity of mild solutions.

In this note, we consider a pseudo-differential operator (T_a) defined as

It is well-known that (T_a) is not bounded on (L^2) in general when a belongs to the forbidden Hörmander class (S^{n(rho -1)/2}_{rho ,1},0le rho le 1). In this note, when (s>0,0le rho le 1,1le rle 2) and (ain S^{n(rho -1)/r}_{rho ,1}), we prove that (T_a) is bounded on the Triebel-Lizorkin space (F^s_{p,q}) if (r<p,q<infty ) or (r<ple infty ,q=infty ). As the most important special example, when (ain S^{n(rho -1)/2}_{rho ,1}) and (s>0), if (2<p,q<infty ) or (2<ple infty ,q=infty ), then (T_a) is bounded on (F^s_{p,q}). When (rho <1), this result is entirely new.

In this article, using that the fractional Laplacian can be factored into a product of the divergence operator, a Riesz potential operator and the gradient operator, we introduce an anomalous fractional diffusion operator, involving a matrix K(x), suitable when anomalous diffusion is being studied in a non homogeneous medium. For the case of K(x) a constant, symmetric positive definite matrix we show that the fractional Poisson equation is well posed, and determine the regularity of the solution in terms of the regularity of the right hand side function.