Levi conditions to the Gevrey well-posedness for hyperbolic operators of higher order (Q839309)

The author is interested in the weakly hyperbolic Cauchy problem \[ P(t, D_t, D_x)u = L(t,D_t,D_x)u,\;(t,x)\in[0, T]\times\mathbb{R}^n, \] \[ D^j_t u(0,x)= u_j(x),\;x\in\mathbb{R}^n,\;j = 0,1,\dots,m- 1, \] and in Gevrey well-posedness. In general one cannot expect Gevrey well-posedness, but only in Gevrey spaces of order \(s< {r\over r-1}\), where \(r\) is the highest multiplicity of the characteristic roots \(\tau_k= \tau_k(t,\xi)\) of \(P(t,\tau,\xi)= 0\). One needs Levi conditions to overcome this bound \({r\over r-1}\). The strategy of the author is the following: \(\bullet\) the single point of degeneracy is \(t= 0\), outside \(t= 0\) the above Cauchy problem is strictly hyperbolic, \(\bullet\) the characteristic roots increasing for \(t> 0\) with different speeds, \(\bullet\) \(\lambda_j(t)|\xi|\lesssim|\tau_j(t, \xi)- \tau_k(t,\xi)|\) for \(j< k\), that is, different characteristic roots may coincide in \(t= 0\) with different speeds. The author proposes Gevrey type Levi conditions which seem to be sharp (at least for well-known examples). To derive the desired a-priori estimates the author uses WKB analysis, the method of zones with definition of suitable micro-energies.