In this paper, we prove that a class of parabolic equations involving a second order fully nonlinear uniformly elliptic operator has a Fujita type exponent. These exponents are related with an eigenvalue problem in all R N and play the same role in blow-up theorems as the classical p * = 1 + 2/N introduced by Fujita for the Laplacian. We also obtain some associated existence results.

This paper is the first attempt to build CGC/saturation model based on the next-to-leading order corrections to linear and non-linear evolution in QCD. We assume that the renormalization scale is the saturation momentum and found that the scattering amplitude has geometric scaling behaviour deep in the saturation domain with the explicit formula of this behaviour at large τ = r 2 Q 2 s . We built a model that include this behaviour, as well as the ingredients that has been known: (i) the behaviour of the scattering amplitude in the vicinity of the saturation momentum, using the NLO BFKL kernel; (ii) the pre-asymptotic behaviour of ln Q 2 s (Y ) , as function of Y and (iii) the impact parameter behaviour of the saturation momentum, which has exponential behaviour ∝ exp (− m b) at large b. We demonstrated that the model is able to describe the experimental data for the deep inelastic structure function. Despite this, our model has difficulties that are related to the small value of the QCD coupling at Q s (Y 0 ) and the large values of the saturation momentum, which indicate the theoretical inconsistency of our description.Keywords: CGC/saturation approach, impact parameter dependence of the scattering amplitude, solution to non-linear equation, deep inelastic structure function, diffraction at high energies arXiv:1607.00832v2 [hep-ph]

In this paper, we show (i) that the NLO corrections do not change the power-like decrease of the scattering amplitude at large impact parameter (b 2 > r 2 exp (2ᾱSη(1 + 4ᾱS)), where r denotes the size of scattering dipole and η = ln (1/xBj) for DIS), and, therefore, they do not resolve the inconsistency with unitarity; and (ii) they lead to an oscillating behaviour of the scattering amplitude at large b, in direct contradiction with the unitarity constraints.However, from the more practical point of view, the NLO estimates give a faster decrease of the scattering amplitude as a function of b, and could be very useful for description of the experimental data. It turns out, that in a limited range of b, the NLO corrections generates the fast decrease of the scattering amplitude with b, which can be parameterized as N ∝ exp (− µ b) with µ ∝ 1/r in accord with the numerical estimates in Ref. [1].

In this paper we develop the DGLAP evolution for the system of produced gluons in the process of diffractive production in DIS, directly from the evolution equation in Color Glass Condensate approach. We are able to describe the available experimental data with small value of the QCD coupling (ᾱS ≈ 0.1). We conclude that in diffractive production, we have a dilute system of emitted gluons and in the order to describe them, we need to develop the next-to-leading order approach in perturbative QCD.

A class of constitutive relations for elastic bodies has been proposed recently, where the linearized strain tensor is expressed as a nonlinear function of the stress tensor. Considering this new type of constitutive equation, the initial boundary value problem for such elastic bodies has been expressed only in terms of the stress tensor. In this communication, this new type of nonlinear wave equation is studied for the case of a one-dimensional straight bar. Conditions for the existence of the travelling wave solutions are given and some self-similar solutions are obtained.