Behavior of solutions to the initial-value problem for a class of integro-differential equations
type: Item
instance of: scholarly article
title: Behavior of solutions to the initial-value problem for a class of integro-differential equations
zbMATH Open document ID: 1005.45006
DOI: 10.1016/S0362-546X(01)00801-X
author: Kohei Uchiyama
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-08-25T00:00:00Z
review text: The author studies the behavior of non-negative solutions to the initial value problem for the nonlinear integro-differential equation \[ \frac{\partial }{\partial t}\rho(x, t)= \frac{\partial }{\partial x}\left(\rho(x, t)\int_{-\infty}^\infty \frac{\rho(y, t)dy}{|y-x|^\gamma (y-x)}\right) \] for \(t>0\) and \(-\infty<x< \infty\). Here \(\gamma\) is a real constant such that \(-1 < \gamma < 1\). That equation expresses the conservation law of a flow. Actually, it is obtained as a continuum limit of a system of a large number of particles moving on \((-\infty, \infty)\) according to the classical equation of motion. Existence and uniqueness of regular solutions is proved. Moreover, solution estimates are derived. In particular, it is shown that the solution starting from the delta measure at the origin is given by a simple elementary function (similar to the Barenblatt solutions to the porous medium equation) and approximately represents large-time behavior of all solutions of unit total mass.
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 76S05
Mathematics Subject Classification ID: 45M05
Mathematics Subject Classification ID: 45M20
zbMATH DE Number: 1788055
zbMATH Keywords: Barenblatt solution
zbMATH Keywords: nonlinear integro-differential equation
zbMATH Keywords: finite speed of propagation
zbMATH Keywords: shifting comparison principle
zbMATH Keywords: large dynamical systems
zbMATH Keywords: potential of long range
zbMATH Keywords: non-negative solutions
zbMATH Keywords: initial value problem
zbMATH Keywords: equation of motion
zbMATH Keywords: porous medium equation
reviewed by: Michael I. Gil'
MaRDI profile type: Publication
cites work: Q3030503
cites work: A Brownian-Motion Model for the Eigenvalues of a Random Matrix
cites work: The Asymptotic Behavior of Gas in an n-Dimensional Porous Medium
cites work: Q3725277
cites work: Interacting Brownian particles and the Wigner law
cites work: Scaling limit for a mechanical system of interacting particles
cites work: Q4224972
cites work: Uniqueness of solutions to the initial value problem for an integro-differential equation
cites work: Scaling limit for a mechanical system of interacting particles. II
cites work: A Non-Linear Evolution Equation Driven By Logarithmic Potential
cites work: Asymptotic Behaviour and Propagation Properties of the One-Dimensional Flow of Gas in a Porous Medium
full work available at URL: s0362-546x(01)00801-x
OpenAlex ID: W2055519167
description: scientific article; zbMATH DE number 1788055