Long-time asymptotics for coagulation equations with injection that do not have stationary solutions

In this paper we study a class of coagulation equations including a source term that injects in the system clusters of size of order one. The coagulation kernel is homogeneous, of homogeneity gamma<1, such that K(x,y) is approximately x(gamma+lambda)y(-lambda), when x is larger than y. We restrict the analysis to the case gamma+2 lambda >= 1. In this range of exponents, the transport of mass toward infinity is driven by collisions between particles of different sizes. This is in contrast with the case considered in Ferreira et al. (Annales de l'Institut Henri Poincar & eacute; C, Analyse Non Lin & eacute;aire,2023), where gamma+2 lambda<1. In that case, the transport of mass toward infinity is due to the collision between particles of comparable sizes. In the case gamma+2 lambda >= 1,the interaction between particles of different sizes leads to an additional transport term in the coagulation equation that approximates the solution of the original coagulation equation with injection for large times. We prove the existence of a class of self-similar solutions for suitable choices of gamma and lambda for this class of coagulation equations with transport. We prove that for the complementary case such self-similar solutions do not exist.

Automated summary

In this paper, we study a class of coagulation equations with a source term that injects clusters of size of order one into the system. The homogeneous coagulation kernel is characterized by a homogeneity exponent gamma<1, and is approximately given by x(gamma+lambda)y(-lambda) when x is larger than y. We focus on the case where gamma+2 lambda >= 1, where the transport of mass towards infinity is driven by collisions between particles of different sizes. In contrast, previous studies considered the case where gamma+2 lambda<1, where the transport of mass towards infinity is due to collisions between particles of comparable sizes. We prove the existence of a class of self-similar solutions for this class of coagulation equations with transport, under suitable choices of gamma and lambda. We also show that such self-similar solutions do not exist for the complementary case.