Well posedness of the Cauchy problem for nonlinear weakly hyperbolic equations (Q2759774)

The authors study the well posedness in Gevrey classes for the quasilinear problem NEWLINE\[NEWLINE\sum_{|\alpha|\leq m}a_\alpha (t,x,D^{m'}u) D^\alpha_{t,x} u=f(t,x,D^{m'} u),NEWLINE\]NEWLINE NEWLINE\[NEWLINED^j_t u|_{t=0}=g_j,\;0\leq j<m,NEWLINE\]NEWLINE where \((t,x)\in [-T,T] \times\mathbb{R}^n\) and \(D^{m'}u= (D^\alpha_{t,x}u; |\alpha|\leq m')\), \(m'<m\). Suppose that \(p_m(t,x, \zeta,\xi)= \sum_{|\alpha |=m} a_\alpha(t,x,\zeta) \xi^\alpha= \prod^d_{j=1}(\xi_0-\lambda_j (t,x,\zeta, \xi))^{m_j}\) has constant multiplicity, that the lower-order terms \(p_j(t,x, \zeta, \xi)\sum_{|\alpha|= j}a_\alpha (t,x,\zeta) \xi^\alpha\) satisfy NEWLINE\[NEWLINEP_j (t,x, \zeta,D)(he^{i\rho \varphi+ \rho^{1/ \sigma}\psi})= O(\rho^{m-m_j(1-1/ \sigma)}),\;\rho\to\infty,\;j=1,\dots,d,NEWLINE\]NEWLINE for every real smooth functions \(\psi, h\), and for every phase function \(\varphi\) associated to \(\lambda_j\), that the coefficients \(a_\alpha(t,x,\zeta)\) are \(k_0\)-times differentiable in \(t\), in the Gevrey class of index \(\sigma\) in \(x\) (resp. \(C^\infty)\) and analytic in \(\zeta\), and that \(m'<m-r (1-\sigma)\) where \(r=\max \{m_j\}\). Then the above Cauchy problem can be solved locally in Gevrey class (resp. Sobolev space) of index \({1\over \sigma}\) if \(\sigma>0\) (resp. \(\sigma=0)\).NEWLINENEWLINEFor the entire collection see [Zbl 0969.00056].