Related papers: Phase space analysis of the Hermite semigroup and …

In this note we consider the nonlinear heat equation associated to the fractional Hermite operator $H^\beta =(-\Delta+|x|^2)^\beta$, $0<\beta\leq 1$. We show the local solvability of the related Cauchy problem in the framework of modulation…

We develop a phase-space framework for fractional generalised anharmonic oscillators and their heat semigroups on weighted modulation spaces. We consider operators of the form \[ \mathcal{H}_{k,l}=(-\Delta)^{l}+V(x), \] where $V$ is a…

We establish some fixed-time decay estimates in Lebesgue spaces for the fractional heat propagator $e^{-tH^{\beta}}$, $t, \beta>0$, associated with the harmonic oscillator $H=-\Delta + |x|^2$. We then prove some local and global…

The Wigner-Weyl transform and phase space formulation of a density matrix approach are applied to a non-Hermitian model which is quadratic in positions and momenta. We show that in the presence of a quantum environment or reservoir, mean…

This study investigates a semilinear wave equation characterized by nonlinear damping $g(u_t) $ and nonlinearity $f(u)$. First, the well-posedness of weak solutions across broader exponent ranges for $g$ and $f$ is established, by utilizing…

This work investigates the semilinear wave equation featuring the displacement dependent term $\sigma(u)\partial_t u $ and nonlinearity $f(u)$. By developing refined space-time a priori estimates under extended ranges of the nonlinearity…

Let $\mathcal{L}= \partial_t- \Delta_x+|x|^2$. Consider its Poisson semigroup $e^{-y\sqrt{\mathcal{L}}}$. For $\alpha >0$ define the Parabolic Hermite-Zygmund spaces $$ \Lambda^\alpha_{\mathcal{L}}=\left\{f: \:f\in…

In this article, we introduce inhomogeneous variable Triebel-Lizorkin spaces, $F_{p(\cdot),q(\cdot)}^{\alpha(\cdot),H}(\mathbb R^n)$, associated with the Hermite operator $H:=-\Delta+|x|^2$, where $\Delta$ is the Laplace operator on…

We characterize weighted modulation spaces (data space) for which the heat semigroup $e^{-tL}f$ converges pointwise to the initial data $f$ as time $t$ tends to zero. Here $L$ stands for the standard Laplacian $-\Delta $ or Hermite operator…

Let $\mathbb{H}^n$ be the $n$-dimensional real hyperbolic space, $\Delta$ its nonnegative Laplace--Beltrami operator whose bottom of the spectrum we denote by $\lambda_{0}$, and $\sigma \in (0,1)$. The aim of this paper is twofold. On the…

We establish Hermite expansion characterizations for several subspaces of the Fr\'{e}chet space of functions on the real line satisfying \begin{equation*} |f(x)| \lesssim e^{-(\frac{1}{2} - \lambda ) x^{2}} , \qquad | \widehat{f}(\xi )|…

This paper is concerned with a non-conserved phase field system of Caginalp type in which the main operators are fractional versions of two fixed linear operators $A$ and $B$. The operators $A$ and $B$ are supposed to be densely defined,…

When $L$ is the Hermite or the Ornstein-Uhlenbeck operator, we find minimal integrability and smoothness conditions on a function $f$ so that the fractional power $L^\sigma f(x_0)$ is well-defined at a given point $x_0$. We illustrate the…

We consider the defocusing periodic fractional nonlinear Schr\"odinger equation $$ i \partial_t u +\left(-\Delta\right)^{\alpha}u=-\lvert u \rvert ^2 u, $$ where $\frac{1}{2}< \alpha < 1$ and the operator $(-\Delta)^\alpha$ is the…

In this paper we consider the problem: $\partial_{t} u- \Delta u=f(u),\; u(0)=u_0\in \exp L^p(\R^N),$ where $p>1$ and $f : \R\to\R$ having an exponential growth at infinity with $f(0)=0.$ We prove local well-posedness in $\exp L^p_0(\R^N)$…

The purpose of this work is to analyze the wellposedness and the blow-up of solutions of the higher-order parabolic semilinear equation \[ u_t+(-\Delta)^{d}u=|x|^{\alpha}|u|^{p}+\zeta(t){\mathbf w}(x) \ \quad\mbox{for }…

In view of the newly discovered and physically acceptable $PT$ symmetric and non-Hermitian models, we reinvestigated the phase structure of the ($g\phi^{4}+h\phi^{6}$)$_{1+1}$ Hermitian model. The reinvestigation concerns the possibility of…

We consider a nonlinear fourth order in space partial differential equation arising in the context of the modeling of nonlinear acoustic wave propagation in thermally relaxing viscous fluids. We use the theory of operator semigroups in…

We study a class of time-dependent (TD) non-Hermitian Hamiltonians $H(t)$ that can be transformed into a time-independent pseudo-Hermitian Hamiltonian $\mathcal{H}_{0}^{PH}$ using a suitable TD unitary transformation $F(t)$. The latter can…

In this paper, we are concerned with the well-posed issues of the fractional dissipative system in the framework of the Fourier--Besov spaces with variable regularity and integrability indices. By fully using some basic properties of these…