On the Cauchy problem for second-order hyperbolic operators with the coefficients of their principal parts depending only on the time variable
The author studies the Cauchy problem for second-order hyperbolic operators whose coefficients of principal part depend only on the time variable, and prove \(C^\infty\) well-posedness of the Cauchy problem under the assumptions that the coeffients of principal part are real analytic functions and the coefficient of the lower-order terms are \(C^\infty\) functions of the whole variables. Put \(P=\tau^2- a(t,\xi)+ b(t,x,\xi)+ c(t,x)\), where NEWLINE\[NEWLINEa(t,\xi)= \sum^n_{j,k=1} a_{jk} \xi_j \xi_k,\quad b(t,\xi)= \sum^n_{j=1} b_j(t,x)\xi_jNEWLINE\]NEWLINE and \(a_{jk}(t)= a_{kj}(t)\) are real-valued and real analytic in \(t\) and \(b_j(t,x)\) and \(c(t, x)\) are in \(C^\infty([0, \infty))\times\mathbb R^n)\). Define \(V= \{\xi\in\mathbb R^n\); \(a(t,\xi)\equiv 0\) in \(t\in [0,\infty)\}\). We may assume that \(V= \{\xi\in\mathbb R^n\); \(\xi'= (\xi_1,\dots, \xi_{n'})= 0\}\), \(1\leq n'\leq n\). Moreover we may assume that \(a(t,\xi)\geq 0\) and \(b(t,\xi)\equiv b(t,\xi')\) does not depend on \((\xi_{n'+1},\dots, \xi_n)\). Let \(\Omega\) be a complex open neighborhood of \([0,\infty)\). For \(\xi'\in \mathbb R^{n'}\setminus \{0\}\) we define \(R(\xi')= \{({\mathfrak R}\lambda)_+;\lambda\in\Omega\) and \(a(\lambda,\xi')= 0\}\), where \(a_+=\max\{a,0\}\). We further assume so-called Levi condition such that for any \(T\geq 0\) and \(x\in\mathbb R^n\) there is \(C>0\) satisfying NEWLINE\[NEWLINE\min\Biggl\{\min_{\tau\in R(\xi')}|t- \tau|, 1\Biggr\}\cdot|b(t, x,\xi')|\leq C\sqrt{a(t,\xi')}NEWLINE\]NEWLINE for any \((t,\xi')\in [0,T]\times \{|\xi'|= 1\}\). Then the author proves that the Cauchy problem for \(P\) is \(C^\infty\) well-posed and, moreover, that the dependence domain for \(P\) is determined exactly.
scientific article; zbMATH DE number 6053042