Self-similar solutions to a coagulation equation with multiplicative kernel

Existence of self-similar solutions to the Oort–Hulst–Safronov coagulation equation with multiplicative coagulation kernel is established. These solutions are given by s(t)−τψτ(y/s(t)) for (t,y)∈(0,T)×(0,∞)(t,y)∈(0,T)×(0,∞), where TT is some arbitrary positive real number, s(t)=((3−τ)(T−t))−1/(3−τ)s(t)=((3−τ)(T−t))−1/(3−τ) and the parameter ττ ranges in a given interval [τc,3)[τc,3). In addition, the second moment of these self-similar solutions blows up at time TT. As for the profile ψτψτ, it belongs to L1(0,∞;y2dy) for each τ∈[τc,3)τ∈[τc,3) but its behaviour for small and large yy varies with the parameter ττ.