We investigate the Cauchy problem for the sixth order p-generalized Benney-Luke equation. The local well-posedness is established in the energy space $\dot{H}^{1}\left(\mathbb{R}^{n}\right) \cap \dot{H}^{3}\left(\mathbb{R}^{n}\right)$ for $1 \leq n \leq 10$, by means of the Sobolev multiplication law and the contrac-tion mapping principle. Moreover, we establish the energy identity of solutions and provide the sufficient conditions of the global existence of solutions by analyzing the properties of the energy functional.

It is well known that weak n-tilting modules are vital in tilting theory, which are generalizations of n-tilting and n-cotilting modules. The aim of this paper is to give a new characterization of weak n-tilting modules. In order to do that, we introduce the notion of weak n-star modules. We study more deeply the properties of them. Moreover, connections between (co) star and weak n-star modules are given.

In this paper, we consider the boundedness and compactness of the multi-linear singular integral operator on Morrey spaces, which is defined by $T_{A} f(x)=\text { p.v. } \int_{\mathbb{R}^{n}} \frac{\Omega(x-y)}{|x-y|^{n+1}} R(A ; x, y) f(y) d y,$ where $R(A ; x, y)=A(x)-A(y)-\nabla A(y) \cdot(x-y)$ with $D^{\beta} A \in B M O\left(\mathbb{R}^{n}\right)$ for all $|\beta|=1$ We prove that T_A is bounded and compact on Morrey spaces $L^{p, \lambda}\left(\mathbb{R}^{n}\right) \text { for all } 1<p<\infty$ with $\Omega \text { and } A$ satisfying some conditions. Moreover, the boundedness and compactness of the maximal multilinear singular integral operator T_A,∗ on Morrey spaces are also given in this paper.

As a generalization of global mappings, we study a class of non-global map-pings in this note. Let $\mathcal{A} \subseteq B(\mathcal{H})$ be a von Neumann algebra without abelian direct summands. We prove that if a map δ : A → A satisfies δ([[A,B]∗,C]) = [[δ(A),B]∗,C]+[[A,δ(B)]∗,C]+[[A,B]∗,δ(C)] for any A,B,C ∈ A with A∗ B∗C = 0, then δ is an additive ∗-derivation. As applications, our results are applied to factor von Neumann algebras, standard operator algebras, prime ∗-algebras and so on.

This paper is concerned with blow-up dynamics of solutions to coupled sys- tems of damped inhomogeneous wave equations with power nonlinearities related to weight function t^α|x|^β and variable boundary conditions on an exterior domain. The damping terms investigated in this work contain weak damping terms and convection terms. In terms of the Neumann-type boundary conditions and Dirichlet-type bound- ary conditions, the non-existence of global solutions to the problems is demonstrated by constructing appropriate test functions and applying contradiction arguments, re- spectively. Our main new contributions are that the effects of damping terms and non- linear terms on behaviors of solutions to the coupled inhomogeneous wave equations are analyzed. As far as the authors know, the results in Theorems 1.1-1.4 are new

The initial boundary value problem for a compressible Euler system out-side a ball in R³ is considered in this paper. Assuming the initial data have small and compact supported perturbations near a constant state, we show that the solution will blow up in a finite time, and the lifespan estimate can be estimated by the small pa-rameter of the initial perturbations. To this end, a “tricky” test function admitting good behavior is introduced.

We present a formula for the third derivative of the distribution function of a regular function on a domain of Rⁿ⁺¹, and a further discussion of the extra assumption that the function is harmonic. The present work builds on [3, 5].