The coagulation-fragmentation partial differential equations.

摘要

The discrete coagulation-fragmentation system with diffusion is an infinite system of nonlinear partial differential equations, which is equivalent to a infinite system of nonlinear integral equations via a Green's function representation. The local existence and uniqueness of the truncated finite system of equations with initial and boundary conditions are first proved by the Contraction Mapping Theorem, then the local existence for the limit infinite system is proved by taking the limit on "N" which is the maximum cluster size. The asymptotic behavior of solutions is studied by applying "relaxed invariance principle" argument which was first exposited by M. Slemrod. One finds a metric with respect to which the Lyapunov function is continuous and non-decreasing and the positive orbit of a solution is relative compact as in "invariance principle". The advantage of this principle is that it ignores one of the necessary condition in the "invariance principle" which is the solution has appropriate continuous dependence on the initial data with respect to the same metric.