In this work we develop methods for studying the Navier-Stokes equations in thin domains. We consider various boundary conditions and establish the global existence of strong solutions when the initial data belong to "large sets." Our work was inspired by the recent interesting results of G. Raugel and G. Sell [22, 23, 24] which, in the periodic case, give global existence for smooth solutions of the 3D Navier-Stokes equations in thin domains for large sets of initial conditions. We extend their results in several ways, we consider numerous boundary conditions and as it will appear hereafter, the passage from one boundary condition to another one is not necessarily straightforward. The proof of our improved results is based on precise estimates of the dependence of some classical constants on the thickness $\epsilon$ of the domain, e.g. Sobolev-type constants and the regularity constant for the corresponding Stokes problem.

The existence and nonexistence of positive time-periodic solutions of semilinear reaction diffusion systems of the type $$ \begin{cases} \partial_{t}u +\mathcal{A}_{1}u = au-bg_{1}(u)u - h_{1}(u,v)u\\ \partial_{t}v +\mathcal{A}_{2}v = dv-fg_{2}(v)v + h_{2}(u)v \end{cases} \quad \text{ in } \ \mathbb{r}^{N}\times (0,\infty ), $$ is discussed. Here the coefficients $a,b,d,f,$ and the coupling terms $g_{1},g_{2},h_{1},h_{2}$ depend on space and time and satisfy suitable conditions. The differential operators $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are elliptic with space- and time-dependent coefficients. In particular, the time dependence is always assumed to be periodic with a fixed common period. A topological method, based on the Leray-Schauder degree, is used to obtain the existence of positive time-periodic solutions. Finally, we apply our results to various population models.

We describe the behavior as $\varepsilon\to 0$ of minimizers for a Ginzburg-Landau functional $$ E_{\varepsilon}(u;\Omega )=\int_{\Omega}\bigl [ \frac {|\nabla u|^n}n +\frac 1{4\varepsilon^n}(1-|u|^2)^2\bigr ]\,dx $$ in the space $H^{1,n}_g(\Omega ;\Bbb R^n)$, where $\Omega\subset\Bbb R^n$ and the boundary data $g:\partial \Omega\to S^{n-1}$ has a nonzero topological degree. Some recent results of Bethuel, Brezis and H\'elein, and of Struwe on the two-dimensional problem, are extended to higher-dimensional cases. New proofs for their results are also presented in this paper.

Under homogeneous boundary conditions, the "energy" remains constant in time (conservative problem). We then introduce suitable dissipative feedback controls on the boundary (forces, shears, moments) and show that

(i) the resulting closed loop feedback problem generates a s.c. semigroup of contractions on a natural function space;

the corresponding "energy" (norm of the semigroup solutions) decays exponentially in the uniform topology.

As a consequence of the above uniform stabilization result, we obtain---via a result of [15]---a corresponding exact controllability result by explicit boundary controls, which improves upon the recent result of [4, 5]. Energy (multipliers) methods are used, along with semigroup methods. In the process of absorbing lower-order terms to obtain the final energy inequality, we establish a unique continuation result for the system of strongly coupled equations describing the dynamics of the shell, which is of interest in its own right. To this end, Carleman estimates are used after a suitable change of variable.

It is shown that, in general, the first four Melnikov functions have to be taken into account in order to obtain definitive results concerning the limit cycles in quadratic perturbations of Hamiltonian systems in the plane with degenerate cubic Hamiltonians. An application is done in completing the proof that no more than two limit cycles can bifurcate out of homoclinic loops of quadratic Hamiltonian systems.

This paper is devoted to global static bifurcation theory for a class of degenerateboundary value problems for nonlinear second-order elliptic differential operators which includes as particular cases the Dirichlet and Neumann problems. In the previous paper [13] we treated the asymptotic linear case, for example, such nonlinear terms as $u^p$, $p > 1$, near $u = 0$ but $u + 1/u$ near $u = +\infty$. The purpose of this paper is to study the asymptotic nonlinear case, for example, such nonlinear terms as $u^p$ also near $u = +\infty$. First we prove a general existence and uniqueness theorem of positive solutions for our nonlinear boundary value problems, by using the super-subsolution method, and then we study in great detail the asymptotic nonlinear case.