Cauchy problem for effectively hyperbolic operators with triple characteristics

Abstract

We study the Cauchy problem for effectively hyperbolic operators $P$ with principal symbol $p(t,x,\tau ,\xi )$ having triple characteristics on $t=0$. Under a condition (E) we show that such operators are strongly hyperbolic, that is the Cauchy problem is well posed for $p(t,x,D_t,D_x)+Q(t,x,D_t,D_x)$ with arbitrary lower order term $Q$. The proof is based on energy estimates with weight $t^{-N}$ for a first order pseudo-differential system, where $N$ depends on lower order terms. For our analysis we construct a non-negative definite symmetrizer $S(t)$ and we prove a version of Fefferman-Phong type inequality for $\mathrm{Re}(S(t)U,U{)}_{L^2({\mathbb{R}}^n)}$ with a lower bound $-C{t}^{-1}\Vert ⟨D{⟩}^{-1}U{\Vert }_{L^2({\mathbb{R}}^n)}$.