In this note we present the Pohozaev identity for the fractional Laplacian. As a consequence of this identity, we prove the nonexistence of nontrivial bounded solutions to semilinear problems with supercritical nonlinearities in starshaped domains. Résumé. Dans cette note, nous présentons l’identité de Pohozaev pour le Laplacien fractionnaire. Comme conséquence de cette identité, nous prouvons ...

We propose a discontinuous Galerkin method for fractional convection-diffusion equations with a superdiffusion operator of order α(1 < α < 2) defined through the fractional Laplacian. The fractional operator of order α is expressed as a composite of first order derivatives and a fractional integral of order 2 − α. The fractional convection-diffusion problem is expressed as a system of low order...

We show global and interior higher-order log-Hölder regularity estimates for solutions of Dirichlet integral equations where the operator has a nonintegrable kernel with singularity at origin that is weaker than any fractional Laplacian. As consequence, under mild assumptions on right hand side, we existence classical problems involving logarithmic Laplacian Schrödinger operator.

In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem (−∆)u = f(u) in Ω, u ≡ 0 in R\Ω. Here, s ∈ (0, 1), (−∆) is the fractional Laplacian in R, and Ω is a bounded C domain. To establish the identity we use, among other things, that if u is a bounded solution then u/δ|Ω is C up to the boundary ∂Ω, where δ(x) = dist(x, ∂Ω). In the fractional Pohozaev identity, the func...

In the paper, we present some applications and features related with the new notions of fractional derivatives with a time exponential kernel and with spatial Gauss kernel for gradient and Laplacian operators. Specifically, for these new models we have proved the coherence with the thermodynamic laws. Hence, we have revised the standard linear solid of Zener within continuum mechanics and the m...

Fractional-order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brownian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As magnetic resonance imaging is applied with increasing temporal and spatial resolution, the spin dynamics is being examined more closely; such examinations exte...

In this paper, we introduce a direct method of moving spheres for the spectral fractional Laplacian $ (-\Delta_D)^{\alpha/2} with 0&lt;\alpha&lt;2 on half Euclidean space. As one expected, key ingredient is narrow region maximum principle, which can be obtained via hide monotonicity kernel used in definition Laplacian. Using spheres, establish or symmetry results nonlinear equations

A set of wave equations with fractional loss operators in time and space are analyzed. The fractional Szabo equation, the power law wave equation and the causal fractional Laplacian wave equation are all found to be low-frequency approximations of the fractional Kelvin-Voigt wave equation and the more general fractional Zener wave equation. The latter two equations are based on fractional const...

The Kolmogorov scaling law of turbulences has been considered the most important theoretical breakthrough in the last century. It is an essential approach to analyze turbulence data present in meteorological, physical, chemical, biological and mechanical phenomena. One of its very fundamental assumptions is that turbulence is a stochastic Gaussian process in small scales. However, experiment da...

We develop a systematic study of the superpositions elliptic operators with different orders, mixing classical and fractional scenarios. For concreteness, we focus on sum Laplacian an...