Related papers: Dispersion estimates for 1D discrete equations

We prove dispersive estimates for solutions to the wave equation with a real-valued potential $V\in L^\infty({\bf R}^n)$, $n\ge 4$, satisfying $V(x)=O(|x|^{-(n+1)/2-\epsilon})$, $|x|>1$, $\epsilon>0$.

We investigate the decay estimates of global solutions for a class of one-dimensional inhomogeneous nonlinear Schr\"odinger equations. While most existing results focus on spatial dimensions $d\geq2$, the decay properties in one dimension…

We consider wave models with lower order terms and recollect some recent results on energy and dispersive estimates for their solution based on symbolic type estimates for coefficients and partly stabilisation conditions. The exposition is…

We study the dispersive behaviors of two-particles Schr\"odinger and wave equations in the Aharonov-Bohm field. In particular, we prove the Strichartz estimates for Schr\"odinger and wave equations in this setting. The key point is to…

We consider the initial-value problem of abstract wave equations with weak dissipation. We show that under conditions on the dissipation coefficient and its derivative the solutions to the abstract dissipative equation are closely related…

We prove dispersive decay, pointwise in time, for solutions to the mass-critical nonlinear Schr\"odinger equation in spatial dimensions $d=1,2,3$.

We study damped wave propagation problems phrased as abstract evolution equations in Hilbert spaces. Under some general assumptions, including a natural compatibility condition for initial values, we establish exponential decay estimates…

We investigate wavepacket solutions for time-dependent Schoedinger equation in the presence of an exponentially decaying potential. Assuming for travelling wave solutions the phase to be a linear combination of the space and time…

We establish a dispersive estimate (with a decay of 1/t), valid for all times, for the Schroedinger evolution on a non-compact 2-dimensional manifold with a trapped geodesic.

This paper proves $L^p$ decay estimates for Schr\"{o}dinger's and wave equations with scalar potentials on three-dimensional Riemannian manifolds. The main result regards small perturbations of a metric with constant negative sectional…

We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time…

We perform a complete Fourier analysis of the semi-discrete 1-d wave equation obtained through a P1 discontinuous Galerkin (DG) approximation of the continuous wave equation on an uniform grid. The resulting system exhibits the interaction…

We derive a $L^1_x (\mathbb R^d)-L^{\infty}_x ( \mathbb R^d)$ decay estimate of order $\mathcal O \left( t^{-d/2}\right)$ for the linear propagators $$\exp \left( {\pm it \sqrt{ |D|\left(1+ \beta |D|^2\right) \tanh |D | } }\right), \qquad…

Evolution PDEs for dispersive waves are considered in both linear and nonlinear integrable cases, and initial-boundary value problems associated with them are formulated in spectral space. A method of solution is presented, which is based…

The possibility of finite-time, dispersive blow up for nonlinear equations of Schroedinger type is revisited. This mathematical phenomena is one of the possible explanations for oceanic and optical rogue waves. In dimension one, the…

In this paper, we consider the Cauchy problem for the fractional Schr\"odinger equation $i D_t^\alpha u + (-\Delta)^{\frac{\beta}{2}} u =0$ with $0<\alpha<1$, $\beta>0$. We establish the dispersive estimates for the solutions. In…

We establish Strichartz estimates (both reversed and some direct ones), pointwise decay estimates, and weighted decay estimates for the linear wave equation in dimension two with an almost scaling-critical potential, in the case when there…

We consider an anisotropic model case for a strictly convex domain of dimension $d\geq 2$ with smoothboundary and we describe dispersion forthe semi-classical Schr{\"o}dinger equation with Dirichlet boundary condition. More specifically, we…

We investigate the dependence of the $L^1\to L^\infty$ dispersive estimates for one-dimensional radial Schr\"o\-din\-ger operators on boundary conditions at $0$. In contrast to the case of additive perturbations, we show that the change of…

This is the first of a series of papers in which we investigate the decay estimates for dispersive equations with Aharonov-Bohm solenoids in a uniform magnetic field. In this first starting paper, we prove the local-in-time dispersive…