The solution manifold M of the equation ⎯φ + gφ3 = 0 in Minkowski space is studied from the standpoint of the establishment of differential-geometric structures therein. It is shown that there is an almost Kähler structure globally defined on M that is Poincaré invariant. In the vanishing curvature case g = 0 the structure obtained coincides with the complex Hilbert structure in the solution manifold of the real wave equation. The proofs are based on the transfer of the equation to an ambient universal...

While the physical properties of the gravitational field in the radiation regime are reasonably well understood, several mathematical questions remain unanswered. The question here is that of existence and properties of gravitational fields with asymptotic behavior compatible with existence of gravitational radiation. A framework to study those questions has been proposed by R. Penrose (R. Penrose, “Zero rest-mass fields including gravitation”, Proc. Roy. Soc. London A284 (1965), 159-203), and developed...

We shall give the local in time existence of the solutions in Gevrey classes to the Cauchy problem for Kirhhoff equations of $p$-laplacian type and investigate the propagation of analyticity of solutions for real analytic deta. When $p=2$, his equation as the global real analytic solution for the real analytic initial data.

In this paper we consider a nonlinear hyperbolic boundary value problem. We show that this problem admits weak solutions by using a lifting result for pseudomonotone operators and a surjectivity result concerning coercive and monotone operators.

Hysteresis operators are illustrated, and a weak formulation is studied for an initial- and boundary-value problem associated to the equation $\left({\partial }^2/\partial t^2\right)\left[u+\mathcal{F}\left(u\right)\right]+Au=f$; here $\mathcal{F}$ is a (possibly discontinuous) hysteresis operator, $A$ is a second order elliptic operator, $f$ is a known function. Problems of this sort arise in plasticity, ferromagnetism, ferroelectricity, and so on. In particular an existence result is outlined.

Existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced, nonlinear wave equations with periodic spatial boundary conditions is established. We consider both the cases the forcing frequency is (Case A) a rational number and (Case B) an irrational number.

In this paper, we consider the following initial-boundary value problem $$\left\}\begin{array}{c}{u}_{tt}(x,t)=\epsilon Lu(x,t)+f\left(u(x,t)\right)\text{in}\Omega \times (0,T),\hfill \\ u(x,t)=0\text{on}\partial \Omega \times (0,T),\hfill \\ u(x,0)=0\text{in}\Omega ,\hfill \\ u_t(x,0)=0\text{in}\Omega ,\hfill \end{array}\right.$$ where $\Omega$ is a bounded domain in ${ℝ}^N$ with smooth boundary $\partial \Omega$, $L$ is an elliptic operator, $\epsilon$ is a positive parameter, $f\left(s\right)$ is a positive, increasing, convex function for $s\in (-\infty ,b)$, ${lim}_{s\to b}f\left(s\right)=\infty$ and ${\int }_0^b\frac{ds}{f\left(s\right)}<\infty$ with $b=const>0$. Under some assumptions, we show that the solution of the above problem quenches in a finite time and its quenching time goes to that of the solution of the following differential equation $$\left\}\begin{array}{cc}{\alpha }^{\text{'}\text{'}}\left(t\right)=f\left(\alpha \left(t\right)\right),\hfill & t>0,\hfill \\ \alpha \left(0\right)=0,{\alpha }^{\text{'}}\left(0\right)=0,\hfill \end{array}\right.$$ as $\epsilon$ goes to zero. We also show that the above result remains...

We show that the set of nonnegative equilibrium-like states, namely, like $(y_d,0)$ of the semilinear vibrating string that can be reached from any non-zero initial state $(y_0,y_1)\in H_0^1(0,1)\times L^2(0,1)$, by varying its axial load and the gain of damping, is dense in the “nonnegative” part of the subspace $L^2(0,1)\times \left\{0\right\}$ of $L^2(0,1)\times H^{-1}(0,1)$. Our main results deal with nonlinear terms which admit at most the linear growth at infinity in $y$ and satisfy certain restriction on their total impact on (0,∞) with respect to the time-variable.

Dans cet article, on étudie la régularité d’une solution réelle, appartenant à $H^s$ pour $s$ assez grand, d’une équation aux dérivées partielles strictement hyperbolique et fortement non linéaire d’ordre deux. On suppose que les données de Cauchy sur une hypersurface spatiale lisse sont régulières en dehors d’un point, et ont une singularité conormale en ce point; on démontre alors que la réunion $\Gamma$ des bicaractéristiques nulles issues de ce point est, en dehors de ce point, une hypersurface lisse et...