CC BY 4.0 UnportedMielke, AlexanderSchindler, Stefanie2026-03-262026-03-262023https://oa.tib.eu/renate/handle/123456789/33620https://doi.org/10.34657/32688We consider reaction-diffusion systems and other related dissipative systems on unbounded domains which would have a Liapunov function (and gradient structure) when posed on a finite domain. In this situation, the system may reach local equilibrium on a rather fast time scale but the infinite amount of mass or energy leads to persistent mass or energy flow for all times. In suitably rescaled variables the system converges to a steady state that corresponds to asymptotically self-similar behavior in the original system.enghttps://creativecommons.org/licenses/by/4.0/510Coupled parabolic systemsreaction-diffusion systemsscaling lawsself-similaritynon-equilibrium steady statesReport