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In this paper, we investigate the fractional power (−Δν)γ/2, 0 −1/2. Our method uses a canonical representation for generalized Laplacia...

Let H = − Δ + | x | 2 be the harmonic oscillator on R n . In this paper, we prove estimates on Besov spaces associated to the operator H and the end-point maximal regularity estimates for the fractional harmonic oscillator H α , 0 α ≤ 1 , on the Besov spaces associated to the harmonic oscillator H. These spaces are the appropriate function spaces for the study of estimates on Besov type spaces and the end-point maximal regularity estimates for the fractional power H α in the sense that similar estimates might fail with the classical Besov spaces.

We study the solvability of certain linear nonhomogeneous elliptic equations and establish that, under some technical assumptions, the L2-convergence of the right-hand sides yields the existence and convergence of solutions in an appropriate Sobolev space. The problems involve differential operators with or without Fredholm property, in particular, the one-dimensional negative Laplacian in a fractional power, on the whole real line or on a finite interval with periodic boundary conditions. We prove that the presence of the transport term in these equations provides regularization of the solutions.

The article deals with the studies of the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power ...

International audience; The article deals with the existence of solutions of an integro-differential equation in the case of anomalous diffusion with the negative Laplace operator in a fractional power in the presence of the transport term. The proof of existence of solutions is based on a fixed point technique. Solvability conditions for elliptic operators without the Fredholm property in unbounded domains are used. We discuss how the introduction of the transport term impacts the regularity of solutions.

We study the well-posedness of a semilinear fractional diffusion equation and formulate an associated inverse problem. We determine fractional power type nonlinearities from the exterior partial measurements of the Dirichlet-to-Neumann map. Our arguments are based on a first order linearization as well as the parabolic Runge approximation property., 12 pages

The article is devoted to the consideration of the diagnostics of the controlled discrete-nonlinear Anishchenko-Astakhov system by the fractional-power spectra of the signals generated by the system. The conditions for the closeness of the system of fractional power-law functions of time, which reduce the errors in the representation of signals of discrete-nonlinear systems by fractional power series, are determined. The peculiarities of the representation of the signals of the discrete-nonlinear Anishchenko-Astakhov system by pulse sequences with expansion in fractional powers of time are revealed, its adequacy in the analysis of system signals is confirmed. Two-dimensional and three-dimensional fractional-power spectra of the signals of the discrete-nonlinear Anishchenko-Astakhov system have been constructed. The ranges of variation of indicators of fractional powers and normalized pulse durations have been determined. The dependence of the structure of fractional power-law spectra of signals of a controlled discrete-nonlinear Anishchenko-Astakhov system, modified by modulation of time sampling parameters, on the modulation depth of control actions, which reduce the correlation intervals of the generated pseudo-random signals, is revealed.

We show the existence of global and local solutions depending on the initial data in the Morrey space to the Keller–Segel system of parabolic–parabolic type. We solve this system by the successive approximation based on the estimation of the heat semigroup in the Morrey space and on the fractional power of the Laplace operator.

Gradient and stability type estimates of heat kernel associated with fractional power of a uniformly elliptic operator are obtained. $L^p$-operator norm of semigroups associated with fractional power of two uniformly elliptic operators are also obtained., 7 pages

In this paper, we concentrate on the regularity of the trajectories of the mild solution for the fractional evolution equations on the Hilbert space. By giving some critical estimates on the α-times resolvent generated by the operator A, we prove that under the condition of , the trajectories of the mild solution are Holder continuous both on the Hilbert space and on the domain of the fractional power of the operator with some appropriate index.

We derive the Moser-Trudinger-Onofri inequalities on the 2-sphere and the 4-sphere as the limiting cases of the fractional power Sobolev inequalities on the same spaces, and justify our approach as the dimensional continuation argument initiated by Thomas P. Branson., Comment: 17 pages

In this paper, we investigate the partial regularity of suitable weak solutions to the multi-dimensional stationary Navier–Stokes equations with fractional power of the Laplacian (−Δ) α (n/6 ≤ α < 1 and α ≠ 1/2). It is shown that the n + 2 − 6α (3 ≤ n ≤ 5) dimensional Hausdorff measure of the set of the possible singular points of suitable weak solutions to the system is zero, which extends a recent result of Tang and Yu [19] to four and five dimension. Moreover, the pressure in e-regularity criteria is an improvement of corresponding results in [1, 13, 18, 20].

We consider a bounded linear operator A in a Hilbert space with a Hilbert-Schmidt Hermitian component (A − A*)/2i. A sharp norm estimate is established for functions of A nonregular on the convex hull of the spectrum. The logarithm, fractional powers and meromorphic functions of operators are examples of such functions. Our results are based on the existence of a sequence An (n = 1, 2, ...) of finite dimensional operators strongly converging to A, whose spectra belongs to the spectrum of A. Besides, it is shown that the resolvents and holomorphic functions of An strongly converge to the resolvent and corresponding function of A.

We discuss the existence and regularity of solutions to some fractional evolution equations in the q-norm. The linear part generates a noncompact semigroup, and the nonlinear part satisfies some conditions with respect to the fractional power norm of the linear part. In the end, we apply the obtained results to a control system.

In this work we study decay rates for the L 2 -norm of solutions to the plate equation with fractional damping and a pseudo fractional rotational inertia term. We also show that the decay rates depending on the fractional power of the damping term are optimal using an asymptotic expansion of the corresponding solution of the related equation in the Fourier space.

In this paper, we are concerned with the local regularity of suitable weak solutions to the 3D Navier–Stokes equations and its generalized case with fractional power of the Laplacian ( − Δ ) α ( 3 / 4 ≤ α 1 ). In the first part, we derive some ϵ -regularity criteria in terms of the deformation tensor D ( u ) for the classical Navier–Stokes equations. In the second part, for the fractional case, we obtain some regularity conditions for suitable weak solutions including the velocity u , the gradient of the velocity ∇ u , the rotation tensor Ω ( u ) and the deformation tensor D ( u ) . This generalizes the results obtained by Gustafson et al. (2007).

The aim of this paper is to prove Heisenberg-Pauli-Weyl inequality for a fractional power of the Dunkl transform on the real line for which there is an index law and a Plancherel theorem.

In this paper, we establish the results on existence and uniqueness of mild solution of impulsive neutral stochastic integrodifferential equations driven by a fractional Brownian motion. Further, by using an impulsive integral inequality, some novel sufficient conditions are derived to ensure the exponential stability of mild solution in the mean square moment. The results are obtained by utilizing the fractional power of operators and the semigroup theory. Finally, an example is presented to demonstrate the effectiveness of the proposed result.

We consider approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions. In this study, we use the theory of fractional powers and ?-norms, so our results can be applied to systems where nonlinear terms include derivatives of spatial variables. We formulate and prove sufficient conditions for approximate controllability. We also give a sample application of our results.

This paper deals with the approximate controllability of semilinear neutral functional differential systems with state-dependent delay. The fractional power theory and α-norm are used to discuss the problem so that the obtained results can apply to the systems involving derivatives of spatial variables. By methods of functional analysis and semigroup theory, sufficient conditions of approximate controllability are formulated and proved. Finally, an example is provided to illustrate the applications of the obtained results.

We study the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power in two dimensions. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for non Fredholm elliptic operators in unbounded domains are used.

This paper is aimed at constructing fractional power series (FPS) solutions of time-space fractional Boussinesq equations using residual power series method (RPSM). Firstly we generalize the idea of RPSM to solve any-order time-space fractional differential equations in high-dimensional space with initial value problems inRn. Using RPSM, we can obtain FPS solutions of fourth-, sixth-, and 2nth-order time-space fractional Boussinesq equations inRand fourth-order time-space fractional Boussinesq equations inR2andRn. Finally, by numerical experiments, it is shown that RPSM is a simple, effective, and powerful method for seeking approximate analytic solutions of fractional differential equations.

We consider the Cauchy problem for an abstract Euler–Poisson–Darboux equation in a Banach space and prove a necessary and a sufficient condition for the solvability of this problem. The conditions are stated in terms of an estimate for the norm of a fractional power of the resolvent and its derivatives. The properties of solutions are established, and examples are given.

This paper is concerned with the initial-boundary value problem of the n -dimensional Carrier equation with an internal damping. This damping is a fractional power of the velocity of the points of the material. The Faedo–Galerkin method, Tartar approach and compactness arguments provide the global existence of solutions of the above problem with restriction on the size of the initial data. The decay of solutions is obtained by the perturbation method.

We investigate the existence of mild solutions for a class of fractional partial neutral stochastic functional integro-differential inclusions with state-dependent delay and analytic α-resolvent operators in Hilbert spaces. Sufficient conditions for the existence are established by using the nonlinear alternative of Leray–Schauder type for multivalued maps due to O’Regan and the fractional power of operators. An example is provided to illustrate the obtained theory.

This paper is concerned with stochastic fractional power dissipative equations with multiplicative noise in n-dimension ( n ≥ 1 ) space. The well-posedness for the subcritical nonlinearities is proved in appropriate space–time space by the contraction mapping principle and Strichartz estimates. The main result can be applied to various types of SPDEs such as stochastic reaction–diffusion equations, stochastic fractional Burgers equation and stochastic fractional Navier–Stokes equation.

In this article, a new general form of fractional power series is introduced.Some results of classical power series are circulated and proved to this new form.A new general form of the generalized Taylor's formula is also obtained.Applications including fractional power series solutions for FDEs are analyzed.Results reveal that, the new expansion is effective for formulating the solutions. In this article, a new general form of fractional power series is introduced in the sense of the Caputo fractional derivative. Using this approach some results of the classical power series are circulated and proved to this fractional power series, whilst a new general form of the generalized Taylor's formula is also obtained. Some applications including fractional power series solutions for higher-order linear fractional differential equations subject to given nonhomogeneous initial conditions are provided and analyzed to guarantee and to confirm the performance of the proposed results. The results reveal that the new fractional expansion is very effective, straightforward, and powerful for formulating the exact solutions in the form of a rapidly convergent series with easily computable components.

Erratum to: On a Finite Range Decomposition of the Resolvent of a Fractional Power of the Laplacian

In this work, we study in the \(\alpha \)-norm, the existence, the continuity dependence, regularity and compactness of solutions for some partial functional integro-differential equations by using the operator resolvent theory. We suppose that the linear part has a resolvent operator in the sense of Grimmer and Pritchard (J Diff Equ 50:234–259, 1983). The nonlinear part is assumed to be continuous with respect to a fractional power of the linear part in the second variable. An application is provided to illustrate our results.

The stochastic resonance phenomenon in overdamped systems with fractional power nonlinearity is thoroughly investigated. The first kind of nonlinearity is a general fractional power function. The second kind of nonlinearity is a fractional power function with deflection. For the first case, the response is clearly divergent for some fractional exponent values. The curve of the spectral amplification factor versus the fractional exponent presents some discrete regions. For the second case, the response will not be divergent for any fractional exponent value. The spectral amplification factor decreases with the increase in the fractional exponent. For both cases, the nonlinearity is the necessary ingredient to induce stochastic resonance. However, it is not the sufficient cause to amplify the weak signal. On the one hand, the noise cannot induce stochastic resonance in the corresponding linear system. On the other hand, the spectral amplification factor of the nonlinear system is lower than that of the corresponding linear system. Through the analysis carried out in this paper, we are able to find that the system with fractional deflection nonlinearity is a better stochastic resonance system, especially when an appropriate exponent value is chosen. The results in this paper might have a certain reference value for signal processing problems in relation with the stochastic resonance method.

The paper deals with the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the Laplace operator in a fractional power. The proof of existence of solutions relies on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains are used.

We show the nonexistence of traveling wave solutions in the combustion model with fractional Laplacian $$\displaystyle (-\Delta )^s$$ when $$\displaystyle s\in (0,1/2]$$ . Our method can be used to give a direct and simple proof of the nonexistence of traveling fronts for the usual Fisher-KPP nonlinearity. Also we prove the existence and nonexistence of traveling wave solutions for different ranges of the fractional power $$s$$ for the generalized Fisher–KPP type model.

In this work, we discuss the existence of mild solutions in the fi-norm for some partial functional integrodifierential equa- tions with inflnite delay. We assume that the linear part generates an analytic semigroup on a Banach space X and the nonlinear part is a Lipschitz continuous function with respect to the fractional power norm of the linear part.

The bounded and smooth solitary wave solutions of 10 nonlinear evolution equations with a positive fractional power term of dependent variable are successfully obtained by homogeneous balance principle and with the aid of sub-ODEs that admits a solution of sech-power or tanh-power type. In the special cases that the fractional power equals to 1 and 2, the solitary wave solutions of more than 10 important model equations arisen from mathematical physics are easily rediscovered.

This paper considers the approximate controllability for a class of neutral control systems governed by semilinear neutral delayed equations.Sufficient conditions for approximate controllability are established by constructing fundamental solutions and using resolvent condition, Banach contraction mapping principle and techniques on fractional power operators. The obtained results improve some existing analogous ones on this topic to some extent. An example is presented finally to illustrate the applications of the obtained results.

In this paper, some theorems of the classical power series are generalized for the fractional power series. Some of these theorems are constructed by using Caputo fractional derivatives. Under some constraints, we proved that the Caputo fractional derivative can be expressed in terms of the ordinary derivative. A new construction of the generalized Taylor’s power series is obtained. Some applications including approximation of fractional derivatives and integrals of functions and solutions of linear and nonlinear fractional differential equations are also given. In the nonlinear case, the new and simple technique is used to find out the recurrence relation that determines the coefficients of the fractional power series.

In this paper, the existence of mild solutions for a class of impulsive fractional partial neutral functional integro-differential inclusions with infinite delay and analytic α-resolvent operators in Banach spaces is investigated. Sufficient conditions for the existence are derived with the help of the fixed-point theorem for discontinuous multi-valued operators due to Dhage and the fractional power of operators combined with approximation techniques. An example is provided to illustrate the theory.

We consider the Schrodinger operator A = − Δ + V + − V − on L p ( R N , w d x ) where N ≥ 3 , and w is a weight in some Muckenhoupt class. We study the boundedness of Riesz transform type operators ∇ A − 1 2 and ∣ V ∣ 1 2 A − 1 2 on L p ( R N , w d x ) . Our result extends the one of Bui (2010) [14] to signed potentials and treat the case where p ≥ 2 . It also gives a weighted version of our earlier results Assaad (2011) [1] , Assaad and Ouhabaz (2012) [2] and of the result (Auscher and Ben Ali 2007) [4] to weighted Lebesgue spaces. We study also the boundedness from L p ( R N , w p d x ) to L q ( R N , w q d x ) of the fractional power A − α / 2 and the L p ( R N , w d x ) -boundedness of the H ∞ -functional calculus of A .

In this work, we study the existence, regularity and stability of solutions for some nonlinear class of partial neutral functional differential equations. We assume that the linear part generates a compact analytic semigroup on a Banach space X, the delayed part is assumed to be continuous with respect to the fractional power of the generator. For illustration, some application is provided for some model with diffusion and nonlinearity in the gradient.

In a recent paper we presented a general perturbation result for generators of \(C_0\)-semigroups, c.f. Theorem 2.1 below. The aim of the present work is to replace, in case the unperturbed semigroup is analytic, the various admissibility conditions appearing in this result by simpler inclusion assumptions on the domain and the range of the perturbation. This is done in Theorem 2.4 and allows to apply our results also in situations which are only in part governed by analytic semigroups. The power of our approach to treat in a unified and systematic way wide classes of PDE’s is illustrated by a generic example, a degenerate differential operator with generalized Wentzell boundary conditions, a reaction diffusion equation with unbounded delay and a perturbed Laplacian.

The article deals with the existence of solutions of a system of integro-differential equations in the case of anomalous diffusion with the Laplacian in a fractional power. The proof of existence of solutions is based on a fixed point technique. Solvability conditions for non Fredholm elliptic operators in unbounded domains are used.

The paper is devoted to local and global solvability in locally uniform spaces and to the existence of a global attractor for a parabolic problem containing fractional power of the minus Laplace operator. Several useful technical tools and estimates are also collected in this paper.

In this paper, we deal with the semilinear wave equation with strong damping. By choosing a suitable state space, we characterize the interpolation and extrapolation spaces of the operator matrix $\mathbf{A}_{\theta}$, analysis the criticality of the $\varepsilon$-regular nonlinearity with critical growth. Finally, we investigate the global existence of the $\varepsilon$-regular solutions which have bounded $X^{1/2}\times X$ norms on their existence intervals.

The aim of this research paper is to establish sufficient conditions for the nonexistence of global solutions for the following nonlinear fractional differential equation D?0|tu + (??)?/2|u|m?1u + a(x)??|u|q?1u = h(x, t)|u|p, (t,x) ? Q, u(0, x) = u0(x), x ? RN where (??)?/2, 0 < ? < 2 is the fractional power of ??, and D?0|t, (0 < ? < 1) denotes the time-derivative of arbitrary ? ? (0; 1) in the sense of Caputo. The results are shown by the use of test function theory and extended to systems of the same type.