Solutions of a non‐local aggregation equation: Universal bounds, concavity changes, and efficient numerical solutions

Universal Bounds for Global Solutions of a Diffusion Equation with a Mixed Local-nonlocal Reaction Term

In this paper we prove the existence of a universal, i.e. independent of the initial data, bound for global positive solutions of a diffusion equation with a mixed local-nonlocal reaction term. Such results are already known in some cases of local or nonlocal reaction terms.

Unbounded solutions of the nonlocal heat equation

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...

Concavity and B-concavity of Solutions of Quasilinear Filtration Equations

The nonlocal bistable equation : Stationary solutions on a bounded interval ∗

We discuss instability and existence issues for the nonlocal bistable equation. This model arises as the Euler-Lagrange equation of a nonlocal, van der Waals type functional. Taking the viewpoint of the calculus of variations, we prove that for a class of nonlocalities this functional does not admit nonconstant C local minimizers. By taking variations along nonsmooth paths, we give examples of ...

Numerical Solutions of the Schrödinger Equation

for the energy eigenvalues En and the associated energy eigenfunctions (stationary states) Ψn. There are a number of important cases for which the stationary Schrödinger equation can be solved analytically, e.g., the harmonic oscillator (in any number of dimensions) and the hydrogen atom. However, in most cases of practical interest (in, e.g., atomic, molecular, and solid-state physics) exact o...