On quasilinear hyperbolic equations with degenerate principal part (Q1293141)

The goal of the present paper is to study the quasilinear weakly hyperbolic Cauchy problem \[ u_{tt}= \sum_{i,j} (a_{ij}(t,x)u_{x_i})_{x_j}+ f(t,x,u,\nabla_x u), \] \[ u(0,x)= u_0(x),\quad u_t(0,x)= u_1(x). \] Here weakly hyperbolic means \[ 0\leq \sum^n_{i,j= 1} a_{ij}(t,x)\xi_j \xi_j\leq \Lambda|\xi|^2. \] In general one cannot expect well-posedness in \(C^\infty\) if the coefficients, the data and the function \(f\) are only \(C^\infty\) with respect to its arguments. One has to prescribe Levi conditions, the authors use the Oleijnik condition. To pove the local existence for \(C^\infty_0\)-data they use the Nash-Moser theorem basing on a priori estimates with an uniform loss of derivatives for the solutions of a linearized Cauchy problem. As a tool they prove a special case of Nersesjan's lemma (Lemma 2 of the present paper). The proof of uniqueness, cone of dependence is sketched by using a perturbation argument. It would be interesting to prove local existence for Sobolev data and to determine the loss of derivatives basing on constants appearing in the Levi condition.