Let $\mathscr{K'}_{M}$ be the space of distributions on ${\mathbf R}^{n}$ which grow no faster than $e^{M(kx)}$ for some $k>0$ and an index function $M(x)$ and let ${K'}_{M}$ be the Fourier transform of $\mathscr{K'}_{M}$. We establish the characterizations of the space $\mathscr{O}_{M}(\mathscr{K'}_{M} ;\mathscr{{'}_M)$ of multipliers in $\mathscr{K'}_{M}$ and prove various types of the continuity from or into $\mathscr{O}_{M}(\mathscr{K'}_{M} ; \mathscr{K'}_{M})$. Also we define the space $\mathscr{O}_M({K'}_{M} ; {K'}_{M})$ of multipliers in ${K'}_{M}$ and find the relation between $\mathscr{O}_{M}({K'}_{M} ; {K'}_{M})$ and the space $\mathscr{O'}_{C}(\mathscr{K'}_{{M};\mathscr{K'}_M)$ of convolution operators in $\mathscr{K'}_{M}$ by the Fourier transformation.

L. Wen proved the $C^{1}$ closing lemma for endomorphisms with finitely many singularities. The arguments and the tools of Wen are available for endomorphisms which have infinitely many singularities but at most finitely many ones in the nonwandering sets. By refining the argument of Wen we prove the $C^{1}$ closing lemma for endomorphisms with finitely many singularities in the nonwandering sets. By using this lemma we can slightly improve characterization of $C^{1}$ absolutely $\Omega$-stable endomorphisms. That is, for an endomorphism $f$ with finitely many singularities in the nonwandering set, $f$ is $C^{1}$ absolutely $\Omega$-stable if and only if $f$ has a neighborhood $\mathscr{U}$ such that every $g$ in $\mathscr{U}$ satisfies weak Axiom A.

We consider radially symmetric selfsimilar solutions $u(x,t)=t^{-a}U(|x|t^{-\beta})$ of the porous medium equation $u_{t}-\Delta(|u|^{m-1}u)=0$. If $m\in (\frac{(N-2)_{+}}{N} , 1 )$, we show that the resulting ODE allows global solutions with rapid decay for a sequence of parameters $k=\alpha/\beta$, denoted by $\{k_{i}^{g}(m, N)\}_{i\in N}\subset {\mathbf R}^{+}$. The corresponding solution $U_{i}$ has exactly $(i-1)$ simple zeroes in ${\mathbf R}^{+}$ This case was left open by previous papers, where the result for the degenerate case was given. Besides the existence result in the singular case $m<1$ for arbitrary space dimension N we prove continuity of the $k_{i}^{g}(., N)$ at functions of m in $(\frac{(N-2)_{+}}{N}, 1)$. In one space dimension there also exist antisymmetric solutions with rapid decay for certain values $\{k_{i}^{u}(m)\}_{i\in N}$ . We show that these values as well as the $k_{i}^{g}(., 1)$ are continuous functions of m in ${\mathbf R}^{+}$ and identify their limits marrow\infty with compactly supported solutions of a limit problem.