We study the bounded and the a.p. (almost-periodic) solutions of forced second order systems with monotone fields and a linear damping term. A special class of such systems is the class of the second order Lagrangian systems with convex Lagrangians. We provide results of existence and uniqueness, we study the dependence of the bounded and a.p. solutions on the bounded and a.p. forcing terms, and finally we treat the case where an additional small nonlinear damping term is present in the equation.

We consider the 3-D Navier-Stokes equations with Coriolis force of order $\frac{1}{\epsilon}$ and vanishing vertical viscosity of order $\epsilon$. For suitable initial data, we prove some long-time existence results. Moreover, we obtain convergence as $\epsilon$ goes to 0 to the 2-D Navier-Stokes equations. We deal with periodic boundary conditions and nonhomogeneous stress. In this case, we compute and justify the corrector.

We study weakly coupled superlinear systems with different boundary conditions. The boundary conditions that we consider are the Sturm-Liouville and the three-point boundary conditions. We first prove a continuation theorem based on coincidence degree and explain how to compute the degree for some simple scalar differential equations. Then we apply those results to prove the existence of infinitely many solutions to the weakly coupled system.

We study the asymptotic behaviour of the bounded solutions of a hyperbolic conservation law with source term $$ \partial_t u(x,t)+ \partial_x f(u(x,t)) =g(u(x,t)),\quad x\in\mathbb{R},t\geq 0, $$ where the flux $f$ is convex and the source term $g$ has simple zeros. We assume that the initial value coincides outside a compact set with an initial value of Riemann type. We prove that the solutions converge in general to a sequence of travelling waves delimited by two shock waves. Some of the travelling waves are smooth and connect two consecutive zeros of the source term, while the remaining are discontinuous and oscillate around an unstable zero of the source. However, we prove that for a generic class of initial data the asymptotic profile contains only travelling waves of the first type. We also analyze the rate of convergence of the solutions to the asymptotic profile.

In this paper we study the Dirichlet problem for a class of ultraparabolic equations. More precisely, we prove the existence of a generalized Perron-Wiener solution and we provide a geometric condition for the regularity of the boundary points which extends the classical Zaremba exterior cone criterion to our setting. The main steps for deriving our results are: i) the introduction in $\mathbf{R}^{N+1}$ of a homogeneous structure; ii) the proof of some interior estimates in a suitable space of Hölder-continuous functions; iii) the construction of a basis of open subsets of $\mathbf{R}^{N+1}$ for which the Dirichlet problem is univocally solvable.