We use variational methods to study the nonexistence of positive solutions for the following nonlinear parabolic partial differential equations: \[ \begin{cases} \frac{\partial u}{\partial t}=\Delta( u^m)+V(x)u^m & \text{in}\quad \Omega \times (0, T ) ,\\ u(x,0)=u_{0}(x)\geq 0 & \text{in} \quad\Omega, \\ u(x,t)=0 & \text{on}\quad \partial\Omega\times (0, T), \end{cases} \] and \[ \begin{cases} \frac{\partial u}{\partial t}=\Delta_p u+V(x)u^{p-1} & \text{in}\quad \Omega \times (0, T ) ,\\ u(x,0)=u_{0}(x)\geq 0 & \text{in} \quad\Omega ,\\ u(x,t)=0 & \text{on}\quad \partial\Omega\times (0, T ), \end{cases} \] where $0 < m < 1$, $1 < p < 2$, $V\in L_{loc}^1(\Omega)$ and $\Omega$ is a bounded domain with smooth boundary in $ \mathbb{R}^N$.

We present bilinear and trilinear as well as quadrilinear null-form estimates arising in connection with the wave-maps problem. These estimates fit into the program of S. Klainerman begun in papers such as [4] and [1]. While the latter used the framework of $X^{s,b}$ spaces, our estimates involve the Banach spaces introduced by D. Tataru in [13] and further developed by T. Tao in [12]. In this paper we attempt to give a somewhat systematic account of the basic properties of (a certain brand of) these spaces in $n=3$ space dimensions. In particular, we solve some cases of the "division and summation problem" for semilinear wave equations whose nonlinearity has null-form structure.

The existence of nontrivial periodic solutions of semilinear fourth-and sixth-order differential equations arising in phase transition models is discussed via variational methods.

We consider the classical solutions to the multi-dimensional free-boundary problem in the Hele-Shaw cell with general boundary conditions on a given boundary. Exact regularity estimates in the Hölder space are established using the explicit form of the solution to the model linear problem in the half-space and a method for evaluating the convolution integrals. This method was suggested by V.~Solonnikov and is based on the use of Golovkin's theorem. We prove that if the free boundary $\Gamma ( t ) $ is initially $C^{l}$-regular$~(l>2$ is noninteger), then it preserves the same regularity ($\Gamma ( t ) \in C^{l})$ till some instant $T_{\ast}$ depending on the $C^{2}$-norm of the free boundary $\Gamma ( t ) $ and on the topology of $\Gamma ( t ) $. At this instant $T_{\ast}$, either the $C^{2}$-norm of the free boundary $\Gamma ( t ) $ tends to infinity or $\Gamma ( t ) $ changes its topology.