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

Abstract

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

21 pages, comments are welcome