We will prove the existence and uniqueness of solutions of the Dirichlet problem $u_t=\Delta$ log $u$, $u>0$, in $\Omega\times (0,\infty)$, $u=g$ on $\partial \Omega\times (0,\infty)$, $u(x,0)=u_0(x)\ge 0$ on $\Omega$ where $\Omega\subset R^n$ is a smooth bounded domain. We will also prove the local and global existence and uniqueness of maximal solutions of the above equation in $R^n\times (0,\infty )$ for $n\ge 3$ under very general condition on $u_0$ and we will prove finite time extinction of solution for $u_0\in L_{loc}^{\infty}(R^n)$ satisfying $0\le u_0(x)\le C/|x|^2$ for all $|x|\ge R_0$ for some constants $C>0$, $R_0>0$.

Results on blow-up and global solvability of the modified Kuramoto-Sivashinsky equation proposed in [1] are presented.

In this paper, several sufficient conditions are established for the global stability of the positive steady state of a scalar functional differential equation $x'=-Lx_t+f(x_t), \ x\geq 0 \; (1)$. The basic idea of the paper is to reduce an infinite dimensional system generated by $(1)$ in some "friendly" spaces to the study of associated one-dimensional maps. In this way, we improve earlier results concerning not only the scalar Lasota-Wazewska and Mackey-Glass equations with infinite distributed delay but also the multidimensional Goodwin oscillator with infinite delay.