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

In this work we consider the semigroup $e^{-t{\mathcal{A}}_{k,\mathrm{\ell }}^{\gamma }}$ for $\gamma >0$ associated to an anharmonic oscillator of the form ${\mathcal{A}}_{k,\mathrm{\ell }}=(-\mathrm{\Delta }{)}^{\mathrm{\ell }}+\mathrm{\mid }x{\mathrm{\mid }}^{2k}$ where $k,\mathrm{\ell }$ are integers $\ge 1$. By introducing a suitable Hörmander metric on the phase-space we analyse the semigroup $e^{-t{\mathcal{A}}_{k,\mathrm{\ell }}^{\gamma }}$ within the framework of Hörmander $S(M,g)$ classes and obtain mapping properties in the scale of modulation spaces $M^{p,q},0<p,q\le \mathrm{\infty },$ with respect to an anharmonic modulation weight. As an application, we apply the obtained bounds to establish the well-posedness for the nonlinear heat equation associated with ${\mathcal{A}}_{k,\mathrm{\ell }}^{\gamma }$. It is worth noting that the results presented in this paper are novel, even in the case where $\gamma =1.$