A decay estimate for a class of hyperbolic pseudo-differential equations (Q1848123)

Summary: We consider the equation \(u_t- i\Lambda u= 0\), where \(\Lambda= \lambda(D_x)\) is a first-order pseudodifferential 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.