Branching of singularities for degenerate hyperbolic operators
Let \(P(t,x,D_ t,D_ x)=\sum A_{k,\alpha}(t,x)D^ k_ tD_ x^{\alpha},\) \(k+| \alpha | \leq m\), be an m-th order linear hyperbolic partial differential operator with \(C^{\infty}\) coefficients defined for -T\(\leq t\leq T\) and \(x\in R^ n\). The following conditions are assumed: 1) the principal symbol \(P_ m(t,x,\tau,\xi)=\prod (\tau - t^{\ell}\lambda_ j(t,x,\xi)),\) \(1\leq j\leq m\), of P is smoothly factorizable, with \(\ell\) a given positive integer, 2) for \(j\neq k\), \(| \lambda_ j(t,x,\xi)-\lambda_ k(t,x,\xi)| \geq c| \xi |,\) \(c>0\) constant, \(3)\quad A_{k,\alpha}(t,x)=t^{i\ell +i+k- m}B_{k,\alpha}(t,x)\) for each (k,\(\alpha)\) such that \(| \alpha | =i\) and \(i\ell +i+k-m\geq 0\), with \(B_{k,\alpha}\) smooth. The authors generalize and complete previous results about the branching of singularities for solutions to the Cauchy problem: \(Pu=0\) in \(]s,T]\times R^ n,\) \(D^ k_ tu|_{t=s}=g_ k\), \(0\leq k\leq m- 1\), \(g_ k\) given distributions with compact support in \(R^ n\). By constructing suitable parametrices for the Cauchy problem, they obtain sufficient conditions of branching of singularities.
scientific article; zbMATH DE number 3884629
scientific article