Exponential Decay Towards Equilibrium for the Inhomogeneous Aizenman-Bak Model

The Aizenman-Bak model for reacting polymers is considered for spatially inhomogeneous situations in which they diffuse in space with a nondegenerate size-dependent coefficient. Both the break-up and the coalescence of polymers are taken into account with fragmentation and coagulation constant kernels. We demonstrate that the entropy-entropy dissipation method applies directly in this inhomogeneous setting giving not only the necessary basic a priori estimates to start the smoothness and size decay analysis in one dimension, but also the exponential convergence towards global equilibria for constant diffusion coefficient in any spatial dimension or for non-degenerate diffusion in dimension one. We finally conclude by showing that solutions in the one dimensional case are immediately smooth in time and space while in size distribution solutions are decaying faster than any polynomial. Up to our knowledge, this is the first result of explicit equilibration rates for spatially inhomogeneous coagulationfragmentation models.