Presentiamo nuovi risultati di esistenza e molteplicità di soluzioni periodiche di piccola ampiezza per equazioni alle derivate parziali Hamiltoniane. Otteniamo soluzioni periodiche di equazioni «completamente risonanti» aventi nonlinearità generali grazie ad una riduzione di tipo Lyapunov-Schmidt variazionale ed usando argomenti di min-max. Per equazioni «non risonanti» dimostriamo l'esistenza di soluzioni periodiche di tipo Birkhoff-Lewis, mediante un'opportuna forma normale di Birkhoff e realizzando...

In this paper we investigate the role of spatial effects in determining the dynamics of a subclass of signalling pathways characterised by their ability to demonstrate oscillatory behaviour. To this end, we formulate a simple spatial model of the p53 network that accounts for both a negative feedback and a transcriptional delay. We show that the formation of protein density patterns can depend on the shape of the cell, position of the nucleus, and the protein diffusion rates. The temporal...

Partial differential equations endowed with a Hamiltonian structure, like the Korteweg–de Vries equation and many other more or less classical models, are known to admit rich families of periodic travelling waves. The stability theory for these waves is still in its infancy though. The issue has been tackled by various means. Of course, it is always possible to address stability from the spectral point of view. However, the link with nonlinear stability  - in fact, orbital stability, since we are...

A periodic BVP for a semilinear elliptic-parabolic equation in an unbounded domain $\mathrm{\Omega }$ contained in a half-space of ${ℝ}^n$ is considered, with Dirichlet boundary conditions on the finite part of $\partial \mathrm{\Omega }$. A theorem of uniqueness of periodic solutions is proved by showing that a suitable function of the "energy" $E(x)$ is subharmonic in $\mathrm{\Omega }$ and satisfies a Phragmèn-Lindelöf growth condition at infinity.

Lorsque tous les champs caractéristiques d’un système hyperbolique riche sont linéairement dégénérés, les opérateurs résolvants sont bien définis et opèrent sur l’ensemble des solutions de certains systèmes d’équations différentielles ordinaires. Celles-ci peuvent être implicites ou explicites. Dans le cas implicite, on montre que toutes les solutions sont presque-périodiques; de plus elles seront toutes périodiques pourvu que l’une d’entre elles le soit. Dans le cas explicite, on définit un opérateur...

In this paper, the existence of an $\omega$-periodic weak solution of a parabolic equation (1.1) with the boundary conditions (1.2) and (1.3) is proved. The real functions $f(t,r),h\left(t\right),a\left(t\right)$ are assumed to be $\omega$-periodic in $t,f\in L_2(S,H),a,h$ such that $a^{\text{'}}\in L_{\infty }\left(R\right),h^{\text{'}}\in L_{\infty }\left(R\right)$ and they fulfil (3). The solution $u$ belongs to the space $L_2(S,V)\cap L_{\infty }(S,H)$, has the derivative $u^{\text{'}}\in L_2(S,H)$ and satisfies the equations (4.1) and (4.2). In the proof the Faedo-Galerkin method is employed.

The existence of the Hopf bifurcation for parabolic functional equations with delay of maximum order in spatial derivatives is proved. An application to an integrodifferential equation with a singular kernel is given.

We study the motion of a viscous incompressible fluid filling the whole three-dimensional space exterior to a rigid body, that is rotating with constant angular velocity ω, under the action of external force f. By using a frame attached to the body, the equations are reduced to (1.1) in a fixed exterior domain D. Given f = divF with $F\in BUC(ℝ;L_{3/2,\infty }\left(D\right))$, we consider this problem in D × ℝ and prove that there exists a unique solution $u\in BUC(ℝ;L_{3,\infty }\left(D\right))$ when F and |ω| are sufficiently small. If, in particular, the external force for...

This paper is devoted to some nonlinear propagation phenomena in periodic and more general domains, for reaction-diffusion equations with Kolmogorov–Petrovsky–Piskunov (KPP) type nonlinearities. The case of periodic domains with periodic underlying excitable media is a follow-up of the article [7]. It is proved that the minimal speed of pulsating fronts is given by a variational formula involving linear eigenvalue problems. Some consequences concerning the influence of the geometry of the domain,...