Uniqueness and non-uniqueness in the Cauchy problem for a class of operators of degenerate type. II
[For Part I see ibid., Ser. A 58, 147-149 (1982; Zbl 0519.35012) and J. Differ. Equations 51, 78-96 (1984; Zbl 0488.35002).] We consider uniqueness and non-uniqueness of \(C^{\infty}\)-solutions of the non-characteristic Cauchy problem for a class of partial differential operators whose characteristics degenerate on the initial surfaces. We explain the main result by the following examples: \[ P_{\pm}=\partial^ 2_ t\pm \exp (-2t^{-k})\partial^ 2_ x+at^{-m}\exp (-t^{-k})\partial_ x,\quad (a\in {\mathbb{C}}). \] For \(P_-\), which is a weakly hyperbolic operator, the Cauchy problem is \(C^{\infty}\)-well posed iff the following holds: \(m\leq k+1\) when Re \(a\neq 0\), or \(m\leq 2k+1\) when Re a\(=0.\) We show that uniqueness holds for \(P_{\pm}+b(t,x)\) for any \(C^ 0- function\) b if \(m\leq k+1\). Furthermore, we show that, if \(m>k+1\), there exist \(C^{\infty}\)-function \(u_{\pm}\) and \(C^ 0\)-function \(f_{\pm}\) near the origin satisfying \((P_{\pm}+f_{\pm})u_{\pm}=0\), \(0\in \sup p u_{\pm}\subset \{t\geq 0\}\). That is, uniqueness breaks. Thus, we may conclude uniqueness depends on the \(C^ k\)-regularity of coefficients of operators.
scientific article; zbMATH DE number 3923262