The nonlinear Schrödinger equations with harmonic potential in modulation spaces

Abstract

The aim of this article is two fold. First, we produce a finite time blow-up solution of the nonlinear fractional heat equation. Secondly, we prove space-time estimates for heat propagator e −tH β associated to fractional harmonic oscillator H β = (−∆ + |x| 2 ) β in modulation spaces M p,p (R d ) (1 ≤ p < ∞). As a consequence, the solution for free fractional heat equation ∂ t u + H β u = 0 grow-up in time near the origin and and decay exponentially in time at infinity.

On the local well-posedness of the nonlinear heat equation associated to the fractional Hermite operator in modulation spaces

“…[10,21] and the many contributions by Wong, for instance [31,32], see also [6]. In particular, heat equations associated to fractional Hermite operators were recently studied in [3], for related results see also [5]. Hermite multipliers are considered in [4], see also the textbook [25].…”

Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity

“…The Hermite operator (also known as quantum harmonic oscillator) H " ´∆ `|x| 2 plays a vital role in quantum mechanics and analysis (see e.g. [4,7] the references therein). The spectral decomposition of H on R d is given by…”