Pseudo-differential equations applied to nonlinear systems of crystal optics (Q2704048)

In the first part of the paper the authors study decay estimates in weighted Sobolev spaces for solutions of pseudodifferential equations of the form NEWLINE\[NEWLINEu _{t}+ i \lambda (D) u=0.NEWLINE\]NEWLINE They apply their result to prove global existence for solutions of the quasilinear Maxwell system \(\partial_t(\varepsilon _{0} E+ \Phi (E))= \text{curl } H\), \(\partial _{t} H= - \text{curl} E\), \( \varepsilon _{0} = \text{diag} ( a ^{2}, b ^{2}, b ^{2})\), \(\Phi (E)= O(|E |^{4})\), for small initial data \(E _{0} \), \( H _{0} \) which are assumed to satisfy div \((\varepsilon _{0} E _{0} +\Phi ( E _{0}))=0\), \(\text{div } H _{0} =0\). (The system describes propagation of light in uniaxial crystals.).NEWLINENEWLINEFor the entire collection see [Zbl 0958.00030].