We present an extension of the results given in the first part of this paper (2004) referring to the existence in the $3$D case of a free boundary between the saturated and unsaturated domains that may be evidenced during the water flow into a porous medium.

Sufficient conditions are given so that the initial value problem for the Shabat equation has a unique analytic solution, which, together with its first derivative, converges absolutely for $z\in ℂ:\left|z\right|<T$, $T>0$. Moreover, a bound of this solution is given. The sufficient conditions involve only the initial condition, the parameters of the equation, and $T$. Furthermore, from these conditions, one can obtain an upper bound for $T$. Our results are in consistence with some recently found results.

The time-dependent Ginzburg-Landau equations of superconductivity with a time-dependent magnetic field $\mathbf{H}$ are discussed. We prove existence and uniqueness of weak and strong solutions with $H^1$-initial data. The result is obtained under the “$\phi =-\omega \left(\nabla \cdot \mathbf{A}\right)$” gauge with $\omega >0$. These solutions generate a dynamical process and are uniformly bounded in time.

In this work, we study bifurcation in the von Kármán equations for a thin circular elastic plate which lies on the elastic base and is simply supported and subjected to a compressive force along the boundary. Applying analytical methods, we prove the existence of stable and unstable simple bifurcation points in the solution set of these equations.

We prove the exponential decay in the case $n>2$, as time goes to infinity, of regular solutions for the nonlinear beam equation with memory and weak damping $u_{tt}+{\Delta }^{2}u-$$M\left({\Vert \nabla u\Vert }_{L^2\left({\Omega }_t\right)}^2\right)\Delta u+{\displaystyle {\int }_0^{t}g}\left(t-s\right)\Delta u\left(s\right)ds+\alpha u_t=0\text{\hspace{0.17em}in\hspace{0.17em}}\widehat{Q}$ in a noncylindrical domain of ${ℝ}^{n+1}\left(n\ge 1\right)$ under suitable hypothesis on the scalar functions $M$ and $g$, and where $\alpha$ is a positive constant. We establish existence and uniqueness of regular solutions for any $n\ge 1$.