EUDML

We study oscillatory solutions of semilinear first order symmetric hyperbolic system $L u = f (t, x, u, \overline{u})$ , with real analytic $f$ . The main advance in this paper is that it treats multidimensional problems with profiles that are almost periodic in $T, X$ with only the natural hypothesis of coherence. In the special case where $L$ has constant coefficients and the phases are linear, the solutions have asymptotic description $u^{ϵ} = U (t, x, t / ϵ, x / ϵ) + o (1)$ where the profile $U (t, x, T, X)$ is almost periodic in $(T, X)$ . ...

Nonlinear Hyperbolic Smoothing at a Focal Point

Jean-Luc Joly; Guy Métivier; Jeffrey Rauch

Séminaire Équations aux dérivées partielles

The nonlinear dissipative wave equation $u_{t t} - Δ u + {| u_{t} |}^{h - 1} u_{t} = 0$ in dimension $d > 1$ has strong solutions with the following structure. In $0 \leq t < 1$ the solutions have a focusing wave of singularity on the incoming light cone $| x | = 1 - t$ . In ${t \geq 1}$ that is after the focusing time, they are smoother than they were in ${0 \leq t < 1}$ . The examples are radial and piecewise smooth in ${0 \leq t < 1}$

Optique géométrique non linéaire et équations de Maxwell-Bloch

Jean-Luc Joly; Guy Métivier; Jeffrey Rauch

Séminaire Équations aux dérivées partielles

Sharp Domains of Determinacy and Hamilton-Jacobi Equations

Jean-Luc Joly; Guy Métivier; Jeffrey Rauch

Séminaire Équations aux dérivées partielles

If $L (t, x, \partial_{t}, \partial_{x})$ is a linear hyperbolic system of partial differential operators for which local uniqueness in the Cauchy problem at spacelike hypersurfaces is known, we find nearly optimal domains of determinacy of open sets $Ω_{0} \subset {t = 0}$ . The frozen constant coefficient operators $L (\underset{̲}{t}, \underset{̲}{x}, \partial_{t}, \partial_{x})$ determine local convex propagation cones, $Γ^{+} (\underset{̲}{t}, \underset{̲}{x})$ . Influence curves are curves whose tangent always lies in these cones. We prove that the set of points $Ω$ which cannot be reached by influence curves beginning in the exterior of $Ω_{0}$ is a domain of...