In the paper we give an upper bound for the life-span of the mild solution to the Cauchy problem for semilinear equations u + ut = |u| 1+α (α > 0, constant) with certain small initial data. This shows the sharpness of the lower bound obtained in [2] on the life-span of classical solutions to the Cauchy problem for fully nonlinear wave equations with linear dissipation with small initial data.

Let R be a ring and M a right R-module. Let σ[M] be the full subcategory of Mod-R subgenerated by M. An M-natural class 𝒦 is a subclass of σ[M] closed under submodules, direct sums, isomorphic copies, and M-injective hulls. We present some equivalent conditions each of which describes when σ has the property that direct sums of (M-)injective modules in σ are (M-)injective. Specializing to particular M, and/or special subclasses we obtain many new results and known results as corollaries.

In this paper we extend DDVV-type inequalities involving the Frobenius norm of commutators from real symmetric and skew-symmetric matrices to Hermitian and skew-Hermitian matrices.2010 Mathematics Subject Classification. 15A45, 15B57, 53C42.

In this note, we show that a ring R is a left perfect ring if and only if every generating set of each left R-module contains a minimal generating set. This result gives a positive answer to a question on left perfect rings raised by Nashier and Nichols.

In this paper we generalize the known DDVV-type inequalities for real (skew-)symmetric and complex (skew-)Hermitian matrices into arbitrary real, complex and quaternionic matrices. Inspired by the Erdős-Mordell inequality, we establish the DDVV-type inequalities for matrices in the subspaces spanned by a Clifford system or a Clifford algebra. We also generalize the Böttcher-Wenzel inequality to quaternionic matrices.