Self-similar solutions to a coagulation equation with multiplicative kernel

Abstract

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 ) − τ ψ τ ( y / s ( t ) ) for ( t , y ) ∈ ( 0 , T ) × ( 0 , ∞ ) , where T is some arbitrary positive real number, s ( t ) = ( ( 3 − τ ) ( T − t ) ) − 1 / ( 3 − τ ) and the parameter τ ranges in a given interval [ τ c , 3 ) . In addition, the second moment of these self-similar solutions blows up at time T . As for the profile ψ τ , it belongs to L 1 ( 0 , ∞ ; y 2 d y ) for each τ ∈ [ τ c , 3 ) but its behaviour for small and large y varies with the parameter τ .