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In the paper "Bellman function for extremal problems in $\mathrm{BMO}$", theauthors built the Bellman function for integral functionals on the$\mathrm{BMO}$ space. The present paper provides a development of the subject.We abandon the majority of unwanted restrictions on the function that generatesthe functional. It is the new evolutional approach that allows us to treat theproblem in its natural setting. What is more, these new considerations lightendynamical aspects of the Bellman function, in particular, evolution of itspicture.

Bellman function for extremal problems in $\mathrm{BMO}$ II: evolution