Hyperbolic equations with cofficients rapidly oscillating in time: a result of nonstability
The principle result of this article is a proof of the nonstability of solutions of the hyperbolic equation \[ (1)\quad u_{tt}- \sum^{N}_{i,j=1}\alpha_{ij}(k,t)u_{x_ ix_ j}=0,\quad u(x,0)=\phi (x),\quad u_ t(x,0)=\psi (x) \] as k approaches \(+\infty\). The initial data \(\phi\) and \(\psi\) are assumed analytic, since even in the \(C^{\infty}\) case examples can be given of problems having no solution in the space of distributions. It is proved that for all nonconstant \(L^ 1_{loc}\), periodic coefficients \(\alpha_{ij}(t)\), there exist analytic initial data \(\phi\) and \(\psi\) for which the sequence of solutions \(\{u_ j(\cdot,t)\}\) of (1) becomes unbounded in the space of distributions on \(R^ N.\) The proof uses a technique due to P. Ungar, which involves the study of the Hill's discriminant of the ordinary differential equation \(w''+\beta (t)w=0\), as a holomorphic function. An example is given to illustrate the main result, and it is conjectured that an analogous result holds more generally for a sequence of problems \[ u_{tt}-\Sigma a_{ij,k}(t)u_{x_ ix_ j}=0,\quad u(x,0)=\phi (x),\quad u_ t(x,0)=\psi (x). \]
scientific article; zbMATH DE number 3921042