Diffusion, fragmentation and merging: rate equations, distributions and critical points (Q852898)

Some properties of the coagulation-fragmentation equation with size diffusion \[ \partial_t N(x,t) = D\;\partial_x^2 N(x,t) + Q_{\text{coag}} + Q_{\text{frag}}\,, \qquad (x,t)\in (0,\infty)\times (0,\infty)\,, \] with the boundary condition \(\partial_x N(0,t)=0\), the coagulation term \[ Q_{\text{coag}} = {1\over 2}\;\int_0^x K(x-x',x')\;N(x-x',t)\;N(x',t)\;dx' - N(x,t)\;\int_0^\infty K(x,x')\;N(x',t)\;dx'\,, \] and the fragmentation term \[ Q_{\text{frag}} = - N(x,t)\;\int_0^x F(x-x',x')\;dx' + 2\;\int_0^\infty F(x,x')\;N(x+x',t)\;dx'\,, \] are described for particular choices of the diffusion coefficient \(D\geq 0\) and the coagulation and fragmentation kernels \(K\) and \(F\). When \(D>0\), \(F=f>0\) and \(K=\beta\geq 0\), an explicit stationary solution is found for \(\beta=0\) and evidence for the existence of at least one stationary solution is given for \(\beta>0\). When \(D=0\), \(F=f>0\) and \(K(x,x')=\beta\;x\;x'\), \(\beta>0\), the description of the large time behaviour by approximate scale invariant solutions is investigated by asymptotic expansions.