Related papers: Globally solvable time-periodic evolution equation…
We study evolution equations associated to time-dependent dissipative non-selfadjoint quadratic operators. We prove that the solution operators to these non-autonomous evolution equations are given by Fourier integral operators whose…
The generalized time-dependent harmonic oscillator is studied. Though several approaches to the solution of this model have been available, yet a new approach is presented here, which is very suitable for the study of cyclic solutions and…
The logarithmic KdV (log-KdV) equation admits global solutions in an energy space and exhibits Gaussian solitary waves. Orbital stability of Gaussian solitary waves is known to be an open problem. We address properties of solutions to the…
We consider d-dimensional time dependent Schr\"odinger equations on the Hilbert space of square integrable functions. We assume magnetic and scalar potentials are almost critically singular with respect to spatial variables both locally and…
We study generalized solutions of an evolutionary equation related to a densely defined skew-symmetric operator in a real Hilbert space. We establish existence of a contractive semigroup, which provides generalized solutions, and find…
We study the representation theory of the solution space of the one-dimensional Schr\"{o}dinger equation with time-dependent potentials that posses $\mathfrak{sl}_2$-symmetry. We give explicit local intertwining maps to multiplier…
We consider evolution equations of the form \begin{equation*}\label{Abstract equation} \dot u(t)+ A(t)u(t)=0,\ \ t\in[0,T],\ \ u(0)=u_0, \end{equation*} where $A(t),\ t\in [0,T],$ are associated with a non-autonomous sesquilinear form…
We investigate the asymptotic behavior of the nonautonomous evolution problem generated by the Klein-Gordon equation in an expanding background, in one space dimension with periodic boundary conditions, with a nonlinear potential of…
We develop a theory of the Klein-Gordon equation on curved spacetimes. Our main tool is the method of (non-autonomous) evolution equations on Hilbert spaces. This approach allows us to treat low regularity of the metric, of the…
In this paper we consider the Zakharov system with periodic boundary conditions in dimension one. In the first part of the paper, it is shown that for fixed initial data in a Sobolev space, the difference of the nonlinear and the linear…
In this paper we consider Schr\"odinger equations with sublinear dispersion relation on the one-dimensional torus $\T := \R /(2 \pi \Z)$. More precisely, we deal with equations of the form $\partial_t u = \ii {\cal V}(\omega t)[u]$ where…
Given a class of nonautonomous elliptic operators $\A(t)$ with unbounded coefficients, defined in $\overline{I \times \Om}$ (where $I$ is a right-halfline or $I=\R$ and $\Om\subset \Rd$ is possibly unbounded), we prove existence and…
We study a class of elliptic operators $A$ with unbounded coefficients defined in $I\times\CR^d$ for some unbounded interval $I\subset\CR$. We prove that, for any $s\in I$, the Cauchy problem $u(s,\cdot)=f\in C_b(\CR^d)$ for the parabolic…
This article presents an investigation of global properties of a class of differential operators on $\mathbb{T}^1\times\mathbb{R}$. Our approach involves the utilization of a mixed Fourier transform, incorporating both partial Fourier…
A protocol for explicitly constructing the exact time-evolution operators generated by $2 \times 2$ time-dependent $PT$-symmetry Hamiltonians is reported. Its mathematical applicability is illustrated with the help of appropriate examples.…
We study the propagation in time of $1/2$-Gelfand-Shilov singularities, i.e. global analytic singularities, of tempered distributional solutions of the initial value problem \begin{align*} \begin{cases} u_t + q^w(x,D) u = 0 \\ u|_{t=0} =…
We prove a necessary and sufficient condition for the existence of a $T$-periodic solution for the time-periodic second order differential equation $\ddot{x}+f(t,x)+p(t,x,\dot x)=0$, where $f$ grows superlinearly in $x$ uniformly in time,…
We consider a time-fractional semilinear parabolic abstract Cauchy problem for a time-dependent sectorial operator $A(t)$ which satisfies the Acquistapace-Terreni conditions. We first prove local existence results for the mild solution of…
Given a smooth manifold $M$ (with or without boundary), in this paper we establish a global functional calculus (without the standard assumption that the operators are classical pseudo-differential operators) and the G\r{a}rding inequality…
We analytically derive novel explicit integral representations for the solution of nonhomogeneous initial-boundary-value problems for a large category of evolution partial differential equations of Sobolev-Galpern type with generic…