Uniqueness of solutions to the coagulation-fragmentation equation with singular kernel (Q2174798)

Uniqueness of solutions to the coagulation-fragmentation equation \begin{align*} \partial_t f(t,x) & = \frac{1}{2} \int_0^x K(x-y,y) f(t,x-y) f(t,y)\ dy - f(t,x)\int_0^\infty K(x,y) f(t,y)\ dy \\ & \quad + \int_x^\infty b(x,y) S(y) f(t,y)\ dy - S(x) f(t,x)\ , \\ f(0,x) & = f_0(x)\ , \end{align*} for \((t,x)\in (0,T)\times (0,\infty)\) is studied, when the coagulation and fragmentation coefficients satisfy \[ 0 \le K(x,y) = K(y,x) \le k \frac{1+x^\theta+y^\theta}{(xy)^\mu}, \qquad S(x) \le S_1 x^\beta \] with \(k>0\), \(\mu\in [0,1/2]\), \(\theta\in [\mu,\mu+1]\), \(S_1>0\), and \(\beta>0\). The daughter distribution function \(b\) satisfies \[ \int_0^y x b(x,y)\ dx = y, \qquad \int_0^y x^{-\gamma} b(x,y)\ dx \le n_0 y^{-\gamma} \] for some \(n_0>0\) and \(\gamma\in (0,1)\), as well as a pointwise technical condition. Uniqueness is shown in the class of solutions \(f\in C([0,T]\times (0,\infty))\) satisfying \[ \sup_{t\in [0,T]} \int_0^\infty (e^{\lambda x} + x^{-r_2}) f(t,x)dx < \infty \] for some \(\lambda>0\) and \(r_2\in (0,1)\). Though not indicated in the statement of the main result, the additional assumption \(r_2>\mu\) seems required.