Uniqueness in the Cauchy problem for a class of partial differential operators degenerate on the initial surface (Q1067086)

Let \(\Omega\) be a neighborhood of the origin in \(R^ n_ x\times R^ 1_ t\). The author considers a linear partial differential operator of order m of the form \[ t^ kP(x,t,D_ x,D_ t)=(tD_ t)^ m+\Sigma a_{\alpha,j}(x,t)(t^{\nu}D_ x)^{\alpha}(tD_ t)^ j, \] where the sum is taken over all \(\alpha\) and j for which \(| \alpha | +j\leq m\) and \(j\leq m-1\). Here, k is a non-negative integer (0\(\leq k\leq m)\), \(\nu\) is a positive rational number, \(D_ t=(1/i)\partial /\partial t\), \(D_ x^{\alpha}=(1/i)^{| \alpha |}\partial^{| \alpha |}/\partial x_ 1^{\alpha_ 1}...\partial x_ n\) \(^{\alpha_ n}\) \((\alpha =(\alpha_ 1,...,\alpha_ n)\), \(| \alpha | =\alpha_ 1+...+\alpha_ n)\), and \(a_{\alpha,j}(x,t)\in C^{\infty}({\bar \Omega})\). The Cauchy problem \(Pu=0\) in \(\Omega\), \(\partial^ j_ tu=0\), \(j=0,1,...,\infty\), on \(t=0\), is studied. Let \(\tau =\lambda_ j(x,t,\xi)\) be the roots of \(\tau^ m+\sum_{| \alpha | +j=m}a_{\alpha,j}(x,t)\xi^{\alpha}\tau^ j=0\). The following assumptions are made: (1) real roots \(\lambda_ j\) are simple, and non-real roots \(\lambda_ j\) are at most double; (2) non-real roots \(\lambda_ j\) satisfy \(| Im \lambda_ j(x,t,\xi)| \geq \epsilon >0\); (3) distinct roots \(\lambda_ j,\lambda_ i\) satisfy \(| \lambda_ i(x,t,\xi)-\lambda_ j(x,t,\xi)| \geq \epsilon >0\). Under these assumptions it is proved that there exists a neighborhood \(\omega\) of the origin such that if \(u\in C^{\infty}(\Omega)\) is a solution of the above Cauchy problem, then \(u=0\) in \(\omega\). In the proof, \(P_ m+P_{m-1}\) is factored into a product of (at most) second order operators for which Carleman estimates are obtained. The paper is an extension of the results of \textit{G. B. Roberts} [J. Differ. Equations 38, 374-392 (1980; Zbl 0431.35003)] \textit{H. Uryu} [Tokyo J. Math. 5, 117-136 (1982; Zbl 0528.35002)], and \textit{S. Nakane} [Proc. Jap. Acad., Ser. A 58, 147-149 (1982; Zbl 0519.35012)].