Uniqueness of solutions to the initial value problem for an integro-differential equation

Abstract

We establish uniqueness of a solution to the initial-value problem for the integro-differential equation $$ \frac{d}{dt} \int_0^1J(x)\mu_t(dx) = \frac1{2}\int_0^1\int_0^1 \frac{J'(y)-J'(x)}{y -x} \cdot\frac{\mu_t(dx)\mu_t(dy)}{|y-x|^{\gamma}}, \quad\;\; t>0 $$ where the equality is required to hold for every smooth testing function $J$ with $J'(0) =J'(1) = 0$, and the solution $\mu_t=\mu_t(dx)$ is a finite measure on the unit interval $[0,1]$ for each $t$ and ${\gamma}$ a constant from the open interval $(-1,1).$ Stationary solutions are given explicitly and the convergence to them of general time-dependent solutions is proved.