Uniqueness of solutions to the initial value problem for an integro-differential equation (Q5933769)

The author establishes uniqueness of a solution to the initial-value problem for the integro-differential equation NEWLINE\[NEWLINE\frac{d}{dt}\int^1_0J(x)\mu_t(dx)=\frac{1}{2} \int^1_0 \int^1_0 \frac{J'(y)-J'(x)}{y-x} \cdot \frac{\mu_t(dx)\mu_t(dy)}{\mid y-x\mid^\gamma},\quad t >0NEWLINE\]NEWLINE 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 integral \((-1,1)\). Stationary solutions are given explicitly and the convergence to them of general time-dependent solutions is proved.