A decay estimate for a class of hyperbolic pseudo-differential equations

Lucente, Sandra ; Ziliotti, Guido

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 10 (1999) no. 3, pp. 173-190

Résumé

We consider the equation \( u_{t} − i \Lambda u = 0 \), where \( \Lambda = \lambda(D_{x}) \) is a first order pseudo-differential operator with real symbol \( \lambda (\xi) \). Under a suitable convexity assumption on \( \lambda \) we find the decay properties for \( u(t,x) \). These can be applied to the linear Maxwell system in anisotropic media and to the nonlinear Cauchy Problem \( u_{t} − i \Lambda u = f (u) \), \( u(0,x) = g(x) \). If \( f(u) \) is a smooth function which satisfies \( f(u) \simeq |u|^{p} \) near \( u = 0 \), and \( g \) is small in suitably Sobolev norm, we prove global existence theorems provided \( p \) is greater than a critical exponent.