Related papers: Hyperbolic and Bi-hyperbolic solutions in the plan…

In this paper, we study the existence of non-planar periodic solutions for the following spatial restricted 3-body and 4-body problems: for $N=2 or 3$, given any masses $m_{1},...,m_{N}$, the mass points of $m_{1},...,m_{N}$ move on the $N$…

We solve the N-body problems in which the total potential energy is any function of the mass-weighted root-mean-square radius of the system of N point masses. The fundamental breathing mode of such systems vibrates non-linearly for ever. If…

In this paper we study higher order weakly hyperbolic equations with time dependent non-regular coefficients. The non-regularity here means less than H\"older, namely bounded coefficients. As for second order equations in \cite{GR:14} we…

We consider a restricted $(N+1)$-body problem, with $N \geq 3$ and homogeneous potentials of degree $-\a<0$, $\a \in [1,2)$. We prove the existence of infinitely many collision-free periodic solutions with negative and small Jacobi constant…

This paper is concerned with supersolutions to parabolic equations of the form \begin{equation} \partial_t U (x,t)-D(x)\Delta U(x,t)=0, \quad (x,t)\in \mathbb{R}^N \times (0,\infty), \end{equation} where $D\in C(\mathbb{R}^N)$ is positive.…

For the $N$-centre problem in the three dimensional space, $$ \ddot x = -\sum_{i=1}^{N} \frac{m_i \,(x-c_i)}{\vert x - c_i \vert^{\alpha+2}}, \qquad x \in \mathbb{R}^3 \setminus \{c_1,\ldots,c_N\}, $$ where $N \geq 2$, $m_i > 0$ and $\alpha…

In [Arch. Ration. Mech. Anal. 213 (2014), 981-991] it has been proved that in the Newtonian $N$-body problem, given a minimal central configuration $a$ and an arbitrary configuration $x$, there exists a completely parabolic orbit starting…

Well-posedness of the initial (boundary) value problem is an essential property, both of meaningful physical models and of numerical applications. To prove well-posedness of wave-type equations their level of hyperbolicity is an essential…

We find a one-parameter family of variables which recast the 3+1 Einstein equations into first-order symmetric-hyperbolic form for any fixed choice of gauge. Hyperbolicity considerations lead us to a redefinition of the lapse in terms of an…

We consider Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative, and a dissipative term whose coefficient may tend to 0 as t -> + infinity (weak dissipation). In this note we present some recent…

A new solvable many-body problem is identified. It is characterized by nonlinear Newtonian equations of motion ("acceleration equal force") featuring one-body and two-body velocity-dependent forces "of goldfish type" which determine the…

We give solutions to several decision problems in word hyperbolic groups

We study a 2+2 body problem introduced in a previous paper as the circular double Sitnikov problem. Since the secondary bodies are moving on the same perpendicular line where evolve the primaries, almost every solution is a collision orbit.…

We derive a first order formalism for solving the scattering of point sources in (2+1) gravity with negative cosmological constant. We show that their physical motion can be mapped, with a polydromic coordinate transformation, to a trivial…

For $n$-body problem with arbitrary positive masses, we prove there are regularizable collinear periodic solutions for any ordering of the masses, going from a simultaneous binary collision to another in half of a period with half of the…

We prove the existence of periodic solutions of the N=(n+1)-body problem starting with n bodies whose reduced motion is close to a non-degenerate central configuration and replacing one of them by the center of mass of a pair of bodies…

We revisit the dynamics of the post-Newtonian (PN) two-body problem for two inspiraling compact bodies. Starting from a matter-only reduced Hamiltonian, we present an adapted framework based on the Lie series approach, enabling the…

This paper deals with the global compactness and multiplicity of positive solutions to problems of the type $$ -\Delta_{\mathbb B^N} u -\lambda u=a(x) |u|^{2^*-2}u+f(x) \quad\text{in } \mathbb B^N, \quad u\in H^1(\mathbb B^N),$$ where…

We investigate the existence, non-existence, multiplicity of positive periodic solutions, both harmonic (i.e., $T$-periodic) and subharmonic (i.e., $kT$-periodic for some integer $k \geq 2$) to the equation \begin{equation*} \Biggl{(}…

In this paper we prove that if $u$ is a solution to second order hyperbolic equation $\partial^2_tu+a(x)\partial_tu-(div_x\left(A(x)\nabla_x u\right)+b(x)\cdot\nabla_x u+c(x)u)=0$ and $u$ is flat on a segment $\{x_0\}\times (-T,T)$ then $u$…