Related papers: Refined Strichartz inequalities for the wave equat…
In this {\bf draft version} we prove inhomogeneous Strichartz estimates with spherical symmetry in the abstract setting via duality arguments. Then we derive some new explicit estimates in the context of the wave equation. This allows us to…
Fundamental rules and definitions of Fractional Differintegrals are outlined. Factorizing 1-D and 2-D Helmholtz equations four fractional eigenfunctions are determined. The functions exhibit incident and reflected plane waves as well as…
This paper is concerned with derivation of the global or local in time Strichartz estimates for radially symmetric solutions of the free wave equation from some Morawetz-type estimates via weighted Hardy-Littlewood-Sobolev (HLS)…
A nonlinear Schr\"odinger equation with variable coefficients for surface waves on a large-scale steady nonuniform current has been derived without the assumption of a relative smallness of the velocity of the current. This equation can…
We use exponential sums to study the fractal dimension of the graphs of solutions to linear dispersive PDE. Our techniques apply to Schr\"odinger, Airy, Boussinesq, the fractional Schr\"odinger, and the gravity and gravity-capillary water…
We prove inverse Strichartz theorems at $L^2$ regularity for a family of Schr\"{o}dinger evolutions in one space dimension. Prior results rely on spacetime Fourier analysis and are limited to the translation-invariant equation $i\partial_t…
We obtain partial improvement toward the pointwise convergence problem of Schr\"odinger solutions, in the general setting of fractal measure. In particular, we show that, for $n\geq 3$, $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ almost…
Our first aim of this article is to establish several new versions of refined Bohr inequalities for bounded analytic functions in the unit disk involving Schwarz functions. Secondly, %as applications of these results, we obtain several new…
We study possible Lie and non-classical reductions of multidimensional wave equations and the special classes of possible reduced equations - their symmetries and equivalence classes. Such investigation allows to find many new conditional…
The Dirichlet problem for the wave equation is a classical example of a problem which is not well-posed. Nevertheless, it has been used to model internal waves oscillating sinusoidally in time, in various situations, standing internal waves…
As a classical time-stepping method, it is well-known that the Strang splitting method reaches the first-order accuracy by losing two spatial derivatives. In this paper, we propose a modified splitting method for the 1D cubic nonlinear…
This paper can be considered as a sequel of [BS14] by Bernicot and Samoyeau, where the authors have proposed a general way of deriving Strichartz estimates for the Schr{\"o}dinger equation from a dispersive property of the wave propagator.…
In this paper we show a general Strichartz estimate for certain perturbed wave equation, and here we can drop the nontrapping hypothesis and handle trapping obstacles with some loss of derivatives for data in the local energy decay…
We prove an extension of the Stein-Weiss weighted estimates for fractional integrals, in the context of $L^{p}$ spaces with different integrability properties in the radial and the angular direction. In this way, the classical estimates can…
Improved estimates on the constants $L_{\gamma,d}$, for $1/2<\gamma<3/2$, $d\in N$ in the inequalities for the eigenvalue moments of Schr\"{o}dinger operators are established.
We provide a probabilistic characterization of criticality, subcriticality, and supercriticality for subordinated Schr\"{o}dinger operators. We also investigate the relationship between the subcriticality of these operators and the uniform…
Global existence for small data Cauchy problem of semilinear wave equations with scaling invariant damping in 3-D is established in this work, assuming that the data are radial and the constant in front of the damping belongs to $[1.5, 2)$.…
The aim of the paper is twofold. We establish refined Strichartz estimates for the Schr\"odinger equation on tori within the framework of partial regularity. As a result, we reveal that the solution of the free Schr\"odinger equation has…
We prove Strichartz estimates in similarity coordinates for the radial wave equation with a self similar potential in dimensions $d\geq 3$. As an application of these, we establish the asymptotic stability of the ODE blowup profile of the…
We study the well-posedness of the Cauchy problem with Dirichlet or Neumann boundary conditions associated to an H 1 -critical semilinear wave equation on a smooth bounded 2D domain {\Omega}. First, we prove an appropriate Strichartz type…