We study the asymptotic behavior of the Klein-Gordon equation with a nonlinear dissipative term $|\partial_{t}w(t)|^{p- 1}\partial_{t}w(t) (p\gt1)$ in $x\in \textbf{R}^{n} (n\geqq 1)$ and $t\geqq 0$. We prove that the energy of solutions does not converge to $0$ as $ t \rightarrow \infty$ for $p\gt1+2/n$ if Cauchy data are suffciently small. We also prove that solutions of the above equation converge to suitable solutions of the linear Klein-Gordon equation in the energy space as $t \rightarrow \infty$ for $p\gt1+4/n$ if $1\leqq n\leqq 6$ and $1+4/n \lt p \lt n/(n-6)$ if $n\geqq 7$.

In this paper, we will show that in each of the following compact symmetric spaces $SU(3),$ $SU(3)/SO(3),$ $SU(6)/Sp(3)$ and $E_{6}/F_{4}$, there are exactly 3 solutions to the minimal surface equations with a given prescribed boundary. The proof uses only elementary techniques in the calculus of variations.

In this paper we consider some generalizations of first countable spaces, called $w_{\kappa}$-spaces. When $\kappa=1,$ $\omega_{1},$ $\infty$, the spaces are respectively Fréchet spaces, $w$-spaces in the sense of $G$. Gruen-hage [5] and first countable spaces. We show that the $w_{\kappa}$-spaces are the images of metric spaces under certain kind of continuous maps, called $w_{\kappa}$-maps. For any cardinals $\kappa_{1} \lt \kappa_{2}$, we construct by forcing a model in which there is a countable space with character $\omega_{1}$ which is a $w_{\kappa_{1}}$-space but not $w_{\kappa_{2}}$-space.

The classical homological characterization theorem of (Lebesgue) covering dimension for compact Hausdorff spaces of finite covering dimension is generalized to proximity spaces in the sense of Efremovič.

A uniform proof is given for the following five assertions. Let $R$ be an integral domain such each overring of $R$ is a pseudovaluation domain (resp., divided domain; resp., going-down domain; resp., locally pseudovaluation domain; resp., locally divided domain). Then $R/P$ has the same property, for each prime ideal $P$ of $R$. The assertion for pseudovaluation domains was proved recently by Okabe-Yoshida by other methods.

It is proved that natural conditions on the weights $p_{n}$ all weighted Hölder and Cesaro means are equivalent to a generalized Abel method for bounded sequences. Furthermore it is shown that the induced $\infty$-limitation methods are equivalent to a uniform limitation method. At last some applications to the theory of uniform distribution of sequences in compact spaces are given.