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

In this work we consider the semigroup e−tAk,ℓγ for γ > 0 associated to an anharmonic oscillator of the form Ak,ℓ=(−Δ)ℓ+|x|2k where k, ℓ are integers ≥1. By introducing a suitable Hörmander metric on the phase-space we analyze the semigroup e−tAk,ℓγ within the framework of Hörmander S(M, g) classes and obtain mapping properties in the scale of modulation spaces Mp,q,0<p,q≤∞, 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 Ak,ℓγ. It is worth noting that the results presented in this paper are novel, even in the case where γ = 1.