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In this work we consider the semigroup$e^{-t\mathcal{A}_{k,\,\ell}^{\gamma}}$ for $\gamma>0$ associated to ananharmonic oscillator of the form $\mathcal{A}_{k,\,\ell}=(-\Delta)^{\ell}+|x|^{2k}$ where $k,\ell$ are integers$\geq 1$. By introducing a suitable H\"ormander metric on the phase-space weanalyse the semigroup $e^{-t\mathcal{A}_{k,\,\ell}^{\gamma}}$ within theframework of H\"ormander $S(M,g)$ classes and obtain mapping properties in thescale of modulation spaces $M^{p,q},\, 0<p,q\leq \infty,$ with respect to ananharmonic modulation weight. As an application, we apply the obtained boundsto establish the well-posedness for the nonlinear heat equation associated with$\mathcal{A}_{k,\,\ell}^{\gamma}$. It is worth noting that the resultspresented in this paper are novel, even in the case where $\gamma=1.$

Anharmonic semigroups and applications to global well-posedness of nonlinear heat equations