Related papers: A note on the multivariate generalization of a bas…
Let K be a non-Archimedean local field with the normalized absolute value $|\cdot |$. It is shown that a ``plane wave'' $f(t+\omega_1 x_1+... +\omega_nx_n)$, where f is a Bruhat-Schwartz complex-valued test function on K, $(t,x_1,...,…
Inequalities, asymptotics and, for some specific cases, asymptotical expansions were obtained for generalized Mathieu's series. A connection between inequalities for Mathieu's series and positive definite and completely monotonic functions.
We present a new exponential inequality as a generalization of that of Sung \textit{et al.} \cite{sun2011} for $M$-acceptable random variables, and hence for extended negative ones. Our result is based on the simple real inequality $e^{x}…
We consider an evolution equation with the regularized fractional derivative of an order $\alpha \in (0,1)$ with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables.…
We prove a general inequality for more than two sequences mirroring that of the discrete two-sequence Cauchy-Schwarz.
The well known Duhamel's principle allows to reduce the Cauchy problem for linear inhomogeneous partial differential equations to the Cauchy problem for the corresponding homogeneous equation. In the paper one of the possible…
We show that the Cauchy problem for the defocusing generalized Boussinesq equation $u_{tt}-u_{xx}+u_{xxxx}-(|u|^{2k}u)_{xx}=0$, $k\geq1$, on the real line is globally well-posed in $H^{s}(\R)$ for $s>1-({1}/{3k})$. We use the "$I$-method"…
A system of nonlinear Cauchy problem $\partial_t u_i=f_i(t,x, U, \nabla_xU )$ $u_i(0,x)= u_{i,0}(x)$ is studied in function spaces with asymptotic expansion with respect to $t$. To be specific, it is discussed in Borel summable or…
In this paper, the well-posedness of Cauchy's problem of fractional Schr\"odinger equations with a power type nonlinearity on $n$-dimensional manifolds with nonnegative Ricci curvature is studied. Under suitable volume conditions, the local…
In this paper, we deal with the following Cauchy problem \begin{equation*} \left\{ \begin{array}{lll} iu_t = \Delta u + 2uh'(|u|^2)\Delta h(|u|^2) + V(x)u,\ x\in \mathbb{R}^N,\ t>0\\ u(x,0) = u_0(x), \quad x \in \mathbb{R}^N.…
A fully non-linear kinetic Boltzmann equation for anyons and large initial data is studied in a periodic 1d setting. Strong L1 solutions are obtained for the Cauchy problem. The main results concern global existence, uniqueness, and…
We highlight overlap as one of the simplest inequalities in linear space that yields a number of useful results. One obtains the Cauchy-Schwarz inequality as a special case. More importantly, a variant of it is seen to work desirably in…
In this paper we study the Cauchy problem for diffusion equations associated to a class of strongly hypoelliptic pseudo-differential operators on graded Lie groups. To do so, we develop a global complex functional calculus on graded Lie…
The Cauchy problem in $\mathbb{R}^d,$ $d\geq 1,$ for a non-local in time p-Laplacian equations is considered. The nonexistence of nontrivial global weak solutions by using the test function method is obtained.
The objective of this paper is to introduce and study a complicated nonlinear system, called coupled variational-hemivariational inequalities, which is described by a highly nonlinear coupled system of inequalities on Banach spaces. We…
A classical inequality, which is known for families of monotone functions, is generalized to a larger class of families of measurable functions. Moreover we characterize all the families of functions for which the equality holds. We apply…
The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order $\beta \in (0,1)$. The fundamental solution for the Cauchy problem is…
We consider the Cauchy problem with smooth data for compressible Euler equations in many dimensions and concentrate on two cases: solutions with finite mass and energy and solutions corresponding to a compact perturbation of a nontrivial…
Our interest itself of this paper is strongly inspired from an open problem in the paper [1] published by D'Abbicco. In this article, we would like to study the Cauchy problem for a weakly coupled system of semi-linear structurally damped…
This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local-in-time Cauchy problem, which is relatively well understood, is…