We study the “hyperboloidal Cauchy problem” for linear and semi-linear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behavior at the boundary, or with polyhomogeneous initial data. Specifically, we consider nonlinear symmetric hyperbolic systems of a form which includes scalar fields with a $\lambda {\phi }^p$ nonlinearity, as well as wave maps, with initial data given on a hyperboloid; several of the results proved apply to general space-times admitting conformal...

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.

We report on new results concerning the global well-posedness, dissipativity and attractors for the quintic wave equations in bounded domains of ${ℝ}^3$ with damping terms of the form ${(-{\Delta }_x)}^{\theta }{\partial }_{t}u$, where $\theta =0$ or $\theta =1/2$. The main ingredient of the work is the hidden extra regularity of solutions that does not follow from energy estimates. Due to the extra regularity of solutions existence of a smooth attractor then follows from the smoothing property when $\theta =1/2$. For $\theta =0$ existence of smooth attractors is more complicated and follows...

In this paper we will give a brief survey of recent regularity results for Fourier integral operators with complex phases. This will include the case of real phase functions. Applications include hyperbolic partial differential equations as well as non-hyperbolic classes of equations. An application to an oblique derivative problem is also given.

The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P2DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations....

On considère un système semi-linéaire du premier ordre de taille $2\times 2$ dans un ouvert de ${ℝ}^n$, une hypersurface $S$ non caractéristique et une hypersurface $\Gamma$ de $S$. On suppose que, par $\Gamma$, passent deux hypersurfaces caractéristiques ${\Sigma }_1$, ${\Sigma }_2$ transverses et que les bicaractéristiqiues sur ${\Sigma }_1$, ${\Sigma }_2$ sont transverses à $\Gamma$. Soit $u$ une solution dans une demi-région $\Omega$ délimitée par $\sigma$. On suppose que $u$ est la restriction à $\Omega$ d’une distribution conormale par morceaux par rapport à ${\Sigma }_1$, ${\Sigma }_2$. Pour le problème de Cauchy, on montre...

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...