Logarithmic asymptotics of rapidly decreasing solutions of Petrovskij hyperbolic equations (Q912275)

The Cauchy problem for the homogeneous, in Petrovskij's sense hyperbolic equation \((1)\quad L(\partial /\partial t,\partial /\partial x)u=0,\) \(t>0\), \(x\in R^ n\) is considered. The existence of the limit \(\lim_{\lambda \to \infty}(\ell n(u(t,x))/\lambda)=S(t,x)\) is proved. Here \(\lambda\) is a parameter arising in the initial conditions. Also, it is shown that the function S is continuous and in the points of smoothness it satisfies the Hamilton-Jacobi equation \(L(\partial S/\partial t,\partial S/\partial x)u=0\) which is associated to (1).