Nonlocal gravity cosmology: An overview

We discuss some main aspects of theories of gravity containing nonlocal terms in view of cosmological applications. In particular, we consider various extensions of general relativity based on geometrical invariants as [Formula: see text], [Formula: see text] and [Formula: see text] gravity where [Formula: see text] is the Ricci curvature scalar, [Formula: see text] is the Gauss–Bonnet topological invariant, [Formula: see text] the torsion scalar and the operator [Formula: see text] gives rise to nonlocality. After selecting their functional form by using Noether symmetries, we find out exact solutions in a cosmological background. It is possible to reduce the dynamics of selected models and to find analytic solutions for the equations of motion. As a general feature of the approach, it is possible to address the accelerated expansion of the Hubble flow at various epochs, in particular the dark energy issues, by taking into account nonlocality corrections to the gravitational Lagrangian. On the other hand, it is possible to search for gravitational nonlocal effects also at astrophysical scales. In this perspective, we search for symmetries of [Formula: see text] gravity also in a spherically symmetric background and constrain the free parameters, Specifically, by taking into account the S2 star orbiting around the Galactic Center SgrA[Formula: see text], it is possible to study how nonlocality affects stellar orbits around such a massive self-gravitating object.

High perturbations of a new Kirchhoff problem involving the p-Laplace operator

In the present work we are concerned with the existence and multiplicity of solutions for the following new Kirchhoff problem involving the p-Laplace operator: $$ \textstyle\begin{cases} - (a-b\int _{\Omega } \vert \nabla u \vert ^{p}\,dx ) \Delta _{p}u = \lambda \vert u \vert ^{q-2}u + g(x, u), & x \in \Omega , \\ u = 0, & x \in \partial \Omega , \end{cases} $$ { − ( a − b ∫ Ω | ∇ u | p d x ) Δ p u = λ | u | q − 2 u + g ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where $a, b > 0$ a , b > 0 , $\Delta _{p} u := \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ Δ p u : = div ( | ∇ u | p − 2 ∇ u ) is the p-Laplace operator, $1 < p < N$ 1 < p < N , $p < q < p^{\ast }:=(Np)/(N-p)$ p < q < p ∗ : = ( N p ) / ( N − p ) , $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N ($N \geq 3$ N ≥ 3 ) is a bounded smooth domain. Under suitable conditions on g, we show the existence and multiplicity of solutions in the case of high perturbations (λ large enough). The novelty of our work is the appearance of new nonlocal terms which present interesting difficulties.

Asymmetrical cell division with exponential growth

An advanced pantograph-type partial differential equation, supplemented with initial and boundary conditions, arises in a model of asymmetric cell division. Methods for solving such problems are limited owing to functional (nonlocal) terms. The separation of variables entails an eigenvalue problem that involves a nonlocal ordinary differential equation. We discuss plausible eigenvalues that may yield nontrivial solutions to the problem for certain choices of growth and division rates of cells. We also consider the asymmetric division of cells with linear growth rate which corresponds to "exponential growth” and exponential rate of cell division, and show that the solution to the problem is a certain Dirichlet series. The distribution of the first moment of the biomass is shown to be unimodal. doi:10.1017/S1446181121000109

ASYMMETRICAL CELL DIVISION WITH EXPONENTIAL GROWTH

An advanced pantograph-type partial differential equation, supplemented with initial and boundary conditions, arises in a model of asymmetric cell division. Methods for solving such problems are limited owing to functional (nonlocal) terms. The separation of variables entails an eigenvalue problem that involves a nonlocal ordinary differential equation. We discuss plausible eigenvalues that may yield nontrivial solutions to the problem for certain choices of growth and division rates of cells. We also consider the asymmetric division of cells with linear growth rate which corresponds to “exponential growth” and exponential rate of cell division, and show that the solution to the problem is a certain Dirichlet series. The distribution of the first moment of the biomass is shown to be unimodal.

On existence and uniqueness of solutions to a pantograph type equation

We show existence and uniqueness of solutions to an initial boundary value problem that entails a pantograph type functional partial differential equation with two advanced nonlocal terms. The problem models cell growth and division into two daughter cells of different sizes. There is a paucity of information about the solution to the problem for an arbitrary initial cell distribution. doi:10.1017/S144618112100002X

ON EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A PANTOGRAPH TYPE EQUATION

We show existence and uniqueness of solutions to an initial boundary value problem that entails a pantograph type functional partial differential equation with two advanced nonlocal terms. The problem models cell growth and division into two daughter cells of different sizes. There is a paucity of information about the solution to the problem for an arbitrary initial cell distribution.

Fractional Schrödinger equation and anomalous relaxation: Nonlocal terms and delta potentials

We review and extend some results for the fractional Schrödinger equation by considering nonlocal terms or potential given in terms of delta functions. For each case, we have obtained the solution in terms of the Green function approach.