On the Gevrey wellposedness of the Cauchy problem for weakly hyperbolic equations of 4th order (Q1599797)

The authors consider the Cauchy problem for the fourth-order partial differential equation \(D^4_t-\sum_{|\alpha|=2}b_\alpha(t)D^\alpha_xD^2_tu+ \sum_{|\alpha|=4}c_\alpha(x)D^\alpha_xu=0\), \(D^h_tu(0,x)=u_h(x)\), \(h=0,1,2,3\). The equation is assumed to be weakly hyperbolic with coefficients in the Hölder class \(C^r\), \(0<r\leq 1\). Under suitable assumptions, allowing multiplicity 4 of the roots, the authors obtain the global well-posedness for data in the Gevrey class \(G^s\), \(1\leq s<1+(r/4)\). Under different hypotheses, which imply that the multiplicity is at most 2, well-posedness is proved for \(1\leq s<1+(r/2)\).