Behavior of solutions to the initial-value problem for a class of integro-differential equations
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.
scientific article; zbMATH DE number 1788055
scientific article