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Équipartition de l’énegie pour les systèmes hyperboliques et formes compatibles. (Equipartition of energy for hyperbolic systems and compatible forms). (French) Zbl 0619.35068
Given a hyperbolic system I \(\partial_ t\psi -\sum^{n}_{i=1}A_ i\partial_ x\psi\) and a sesquilinear form f, \(I(t)=\int f(\psi (t,x),\psi (t,x))dx\) tends to zero when \(| t| \to \infty\) for any finite energy solution \(\psi\) if and only if f is compatible with the system in the sense of B. Hanouzet and J. L. Joly, i.e. f(Ker\(\sum^{n}_{i=1}\xi_ iA_ i)=0\) for a.e. \(\xi \in {\mathbb{R}}^ n\). If n is odd and the multiplicity of the system is constant, \(I(t)=0\) after a finite time for solutions having initial data with compact support. We also study the hermitian systems I \(\partial_ t\psi - \sum^{n}_{i=1}A_ i\partial_{x_ i}\psi +iB\psi\). We prove the equipartition of energy for the hyperbolic equations \(\partial^ 2_{tt}\psi -A^ 2(d)\psi +B^ 2\psi =0\), the wave equation, the elastic waves in anisotropic media, the magneto-elastic waves, the Klein- Gordon equation, Maxwell’s equations, the Dirac system and the Neutrino equation.

MSC:
35L40 First-order hyperbolic systems
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