We consider the Cauchy problem posed in the whole space for the following nonlocal heat equation: ut = J ∗u−u , where J is a symmetric continuous probability density. Depending on the tail of J , we give a rather complete picture of the problem in optimal classes of data by: (i) estimating the initial trace of (possibly unbounded) solutions; (ii) showing existence and uniqueness results in a su...

We apply nonlocal symmetry-like operators to systems of two first and two second-order ordinary differential equations to seek reduction to quadratures. The reduction of order of such systems is carried out with the help of analytic continuation of scalar equations in the complex plane. Examples are taken from the literature. Precisely it is shown how the reduction to quadratures of a system of...

In this paper we study numerical approximations of the nonlocal p−Laplacian type diffusion equation, ut(t, x) = ∫ Ω J(x− y)|u(t, y)− u(t, x)|p−2(u(t, y)− u(t, x)) dy. First, we find that a semidiscretization in space of this problem gives rise to an ODE system whose solutions converge uniformly to the continuous one as the mesh size goes to zero. Moreover, the semidiscrete approximation shares ...

In [16], Córdoba, Córdoba, and Fontelos proved that for some initial data, the following nonlocaldrift variant of the 1D Burgers equation does not have global classical solutions ∂tθ + u ∂xθ = 0, u = Hθ, where H is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to t...

We consider a nonlocal scalar conservation law proposed by Andrew C. Fowler to describe the dynamics of dunes, and we develop a numerical procedure based on splitting methods to approximate its solutions. We begin by proving the convergence of the well-known Lie formula, which is an approximation of the exact solution of order one in time. We next use the split-step Fourier method to approximat...

A nonlinear, nonlocal dielectric continuum model, called the nonlocal modified PoissonBoltzmann equation (NMPBE), has been proposed to reflect the spatial-frequency dependence of dielectric permittivity in the calculation of electrostatics of ionic-solvated biomolecules. However, its analysis is difficult due to its definition involving Dirac delta distributions for modeling point charges, expo...