Existence and nonexistence of solutions (both stationary and evolution) for a parabolic-elliptic system describing a cloud of self-interacting particles surrounding a fixed mass are studied. In particular, sharp conditions for local-in-time and global-in-time existence, as well as finite-time blow up are given.

The problem of the search for periodic solutions to certain second-order scalar subquadratic equations is reduced, by means of a variational argument, to the study of a real function of one variable, the "reduction function" of the problem. Existence and multiplicity of solutions for the original problem and for its perturbations are linked to the properties of the reduction function. Equivalent conditions for the perturbability of the problem as well as genericity results and descriptions of the range of the differential operator are obtained. Applications cover equations with oscillating or bounded nonlinearities or strongly resonant problems.

In this article we treat the Cauchy problem for the dispersive system of long waves, $$ \begin{cases} \partial_t u + \partial_{x}v + u\partial_{x}u = 0\\ \partial_t v - \partial_{x}^3u + \partial_{x}u + \partial_{x}(uv) = 0, \end{cases} $$ in weighted Sobolev spaces. It is shown that this problem is locally well-posed in $H^s(\Bbb R)\times H^{s-1}(\Bbb R) \cap L^2_{\gamma}(\Bbb R)\times H^{-1}_{\gamma}(\Bbb R)$, for $s>3/2$ and $0\leqq \gamma \leqq s$. The proof involves parabolic regularization and techniques of Bona-Smith. It is also determined, using the orbital stability of the special solitary-wave solutions of this system, that we can extend globally the local solution for data sufficiently close to the solitary wave in the norm $ H^1(\Bbb R)\times L^{2}(\Bbb R)$.

A dissipative nonlinear nonautonomous equation of hyperbolic type with translation-compact forcing term is studied, with particular reference to the problem of traveling waves in a strip and related attractor.

We consider a one-dimensional, monocharged plasma as described by the Vlasov--Poisson system and investigate the behaviour of the solutions for large times. Using a rescaling method we are able to determine an explicit solution of the system which corresponds to a globally attractive steady state for the rescaled system. We investigate in which sense and at which rate the solutions of the rescaled system converge to this global attractor and interpret the results for the original system.

In this paper, we give new optimality conditions in terms of Young measures for nonconvex minimization problems of type Inf${\int _\Omega} f \bigl(x, u(x), Du(x) \bigr)\,dx$. The analysis of these conditions allows us to find sufficient assumptions for the existence of a minimum.