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In Fialowski's classification for algebras of maximal class, there are threeLie algebras of maximal class with 1-dimensional homogeneous components:$\mathfrak{m}_0$, $L_1$ and $\mathfrak{m}_2$. In this paper, we studied theirbiderivations by considering the embedded mapping to derivation algebras. Thenwe determined commuting mappings on these algebras as an application ofbiderivations. Finally,  local and 2-local derivations for these three algebras were characterized asthe given gradings.

Biderivations, commuting mappings and (2-)local derivations of $\mathbb{N}$-graded Lie algebras of maximal class