The authors consider the following integro-partial differential equation which describes Smoluchowski’s coagulation equation: \[ \partial_t c(t,x) = \frac{1}{2} \int_0^x a(y,x-y)c(t,x-y)dy -c(t,x)\int_0^\infty a(x,y)c(t,y)dy, \] where \(c(t,x)\geq 0\) is the concentration of particles of masses \(x\in (0,\infty)\) at time \(t\geq 0\) and \((t,x)\in (0,\infty)^2.\) The coagulation kernel \( a(x,y)=a(y,x)\geq 0,\) which is also homogeneous with degree \(\lambda \in (-\infty,2]\diagdown{\{0\}},\) is the probability that two particles with masses \(x\) and \(y\) merge into a single particle with mass \(x+y.\)
Under suitable conditions, the authors prove the existence of a unique measure-valued solution to (1). Furthermore, for constant coagulation kernels, a contraction property as well as a comparison principle are proved.