Properties of stationary solutions of a generalized Tjon-Wu equation (Q2642140)

The authors consider the following Tjon-Wu equation which is a version of the Boltzmann equation: \[ \begin{cases} \frac{\partial u(t,x)}{\partial t} + u(t,x) = \int_x^\infty \frac{dy}{y}\int_0^y u(t,y-z)u(t,z)\,dz, & t\geq 0, \;x\geq 0, \\ u(0,x)=f(x), & x\geq 0. \end{cases}\tag{1} \] Previous results on problem (1), when considered in the space of measures, have shown that an asymptotically stable stationary solution is a probability measure on \(\mathbb{R}^+ =[0,+\infty).\) The authors further explore the properties of this measure and prove that its support is one point or \(\mathbb{R}^+\) and if it is \(\mathbb{R}^+\) then it contains at most one atom. Also they show that in some cases the measure is absolutely continuous. Furthermore, the authors indicate that their results imply global asymptotic stability of solutions, and point out an open problem and a corresponding conjecture.