Narrow Results

In this paper, the Cauchy problem for convolution-type nonlocal linear and nonlinear Schrödinger equations (NSEs) is studied. The equations include the general differential operators. The existence, uniqueness, $L^p$-regularity properties of linear problem and the existence, uniqueness, and blow-up at finite time of the nonlinear problem are obtained. By choosing differential operators including in equations, the regularity properties of a different type of nonlocal Schrödinger equation are studied.

In this paper, the Cauchy problem for linear and nonlinear wave equations is studied. The equation involves abstract operator $A$ in a Banach space $E$ and convolution terms. Here, assuming enough smoothness on the initial data and on coefficients, the existence, uniqueness and regularity properties of local and global solutions are established in terms of fractional powers of a given sectorial operator $A$. We obtain the regularity properties of a wide class of wave equations by choosing the space $E$ and the operator $A$ which appear in the field of physics.

We consider fully discrete schemes for the one-dimensional linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model are presented in these approximations. In particular, Strichartz estimates and the local smoothing of the numerical solutions are analyzed. Using a backward Euler approximation of the linear semigroup we introduce a convergent scheme for the nonlinear Schrödinger equation with nonlinearities which cannot be treated by energy methods.

Let $A$ be a $n\times n$ expansive integral matrix with $|det(A)|=2$. This paper investigates matrix Fourier multipliers for $A$-dilation Parseval multi-wavelet frames, which are $N_1\times N$ matrices with $L^{\infty }$ function entries, map $A$-dilation Parseval multi-wavelet frames of length $N$ to $A$-dilation Parseval multi-wavelet frames of length $N_1$, where $N_1\ge N$. We completely characterize all matrix Fourier multipliers for $A$-dilation Parseval multi-wavelet frames and construct several numerical examples. As Fourier wavelet frame multiplier, matrix Fourier multipliers can be used to derive new $A$-dilation Parseval multi-wavelet frames and can help us better understand the basic properties of frame theory.

This work is concerned with the proof of L_p-L_q decay estimates for solutions of the Cauchy problem for u_tt - λ²(t)b²(t) Δ u = 0. The coefficient consists of an increasing smooth function λ and an oscillating smooth and bounded function b which are uniformly separated from zero. The authors' main interest is devoted to the critical case where one has an interesting interplay between the growing and the oscillating part.

We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: The first based on geometric properties of Banach spaces and the second on Littlewood–Paley and Hörmander-type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the $K$-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on ${𝕋}^d$, ${ℝ}^d$, $[-1,1]$, nonlinear wavelet approximation, etc.