We consider reaction–diffusion systems and other related dissipative systems on unbounded domains with the aim of showing that self-similarity, besides the well-known exact self-similar solutions, can also occur asymptotically in two different forms. For this, we study systems on the unbounded real line that have the property that their restriction to a finite domain has a Lyapunov function (and a gradient structure). In this situation, the system may reach local equilibrium on a rather fast time scale, but on unbounded domains with an infinite amount of mass or energy, it leads to a persistent mass or energy flow for all times; hence, in general, no true equilibrium is reached globally. In suitably rescaled variables, however, the solutions to the transformed system converge to so-called non-equilibrium steady states that correspond to asymptotically self-similar behavior in the original system.

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