Nonlinear Cauchy problem with weakly oscillating initial data (Q5939011)

The authors study the Cauchy problem for nonlinear partial differential equations of higher-order with weakly oscillating data. The model under consideration is in general of super Kovalevskian type, that is, NEWLINE\[NEWLINED^m_t u= f(t, D^B u),\quad D^j_t u(0,x)= \varphi_j(\varepsilon x),\quad j= 0,\dots, m-1,NEWLINE\]NEWLINE where \(f\) is holomorphic in its arguments and the data belong to Gevrey classes of order \(d\). Under special assumptions the authors proveNEWLINENEWLINENEWLINE-- global existence of the solution with respect to \(t\) in a given interval of definition \([0,T]\) if the oscillations are small enough, the solutions are Gevrey with respect to \(x\);NEWLINENEWLINENEWLINE-- a stability result of solutions for vanishing data;NEWLINENEWLINENEWLINE-- an estimate of the life-span.NEWLINENEWLINENEWLINEThe proofs are basing on formal series, scale type estimates in Gevrey spaces, contraction principle by using the super-Kowalevskian structure. An interesting open question is that for global existence results for \(t\in [0,\infty)\) for such special Cauchy problems with characteristic roots of constant multiplicity \(m\).