A class of locally solvable semilinear equations of weakly hyperbolic type
The goal of the present paper is to study the following quasilinear weakly hyperbolic Cauchy problem \[ u_{tt}- \sum^n_{i, j= 1} (a_{ij}(t, x) u_{x_i})_{x_j}+ \sum^n_{j= 1} b_j(t, x) u_{x_j}= f(u, u_t), \] \[ u(0, x)= u_0(x),\;u_t(0, x)= u_1(x). \] Here, weakly hyperbolic means that \(a_{ij}= \overline{a_{ji}}\), \(\Lambda^2 |\xi|^2\geq \sum^n_{j= 1} a_{ij} \xi_i \xi_j\geq 0\). In general one cannot expect well-posedness in \(C^\infty\) if the coefficients, data and right-hand side are only \(C^\infty\). For this reason one has to prescribe so-called Levi conditions. The authors use the Oleijnik condition \[ B \Biggl( \sum^n_{j= 1} b_j(t, x) \xi_j\Biggr)^2\leq A \sum^n_{i, j= 1} a_{ij}(t, x) \xi_i \xi_j+ \sum^n_{i, j= 1} \partial_t a_{ij}(t, x) \xi_i \xi_j \] with some positive constants \(A\) and \(B\). Then they can prove local \(C^\infty\)-well posedness under some additional standard assumptions. Here well posedness means local existence and finite speed of propagation, cone of dependence respectively. Two different proofs are presented. If \(f\) is nonlinear only in \(u\), then some a-priori estimates and an iteration process give the existence result. If \(f\) is nonlinear in \(u_t\), too, then we have in the a-priori estimates some uniformly loss of Sobolev regularity. Consequently, the application of the Nash-Moser technique leads to the desired results.
quasilinear weakly hyperbolic Cauchy problem
scientific article; zbMATH DE number 878454