Cauchy problem for SG-hyperbolic equations with constant multiplicities
The goal of the authors is to study Cauchy problems for so-called SG-hyperbolic linear operators with constant multiplicities. This means that the principal symbol has \(\mu\) distinct real roots \(\lambda_j=\lambda_j(t;x,\xi)\) of constant multplicity which belong to \(C^\infty(J,SG^{(1,1)}_l)\) and satisfy \[ |\lambda_j (t;x,\xi)-\lambda_k(t;x,\xi)|\geq C\langle \xi\rangle\langle x\rangle. \] By \(SG^{(1,1)}_l\) the authors denote the symbol space \[ \{a= a(x,\xi)\in C^\infty(\mathbb{R}^n\times \mathbb{R}^n) : |\partial^\alpha_\xi \partial^\beta_x a(x,\xi)|\leq C\langle \xi\rangle^{1-|\alpha|}\langle x\rangle^{1-|\beta|}\}. \] They explain the symbolic calculus, Sobolev spaces and the notion of a pseudodifferential operator; the notion of ellipticity; the notion of parametrix; the notion of a Fourier integral operator and the notion of hyperbolicity basing on the SG symbol classes. The SG-hyperbolic operators with constant multiplicities have to satisfy some Levi conditions. Finally a reduction procedure is proposed which transfers the starting Cauchy problem to a Cauchy problem for a system \[ \partial_tU - i K(t)U = F(t),\quad U(0) = U_0, \] with diagonal principal part, this means, that a perfect diagonalization modulo lower order terms exists. This reduction bases on a well-known result of Mizohata. These considerations generate a lot of interesting questions. What about considerations in Gelfand-Shilov classes? What about considerations for the weakly hyperbolic case with characteristic roots of variable multiplicity?
perfect diagonalization modulo lower order terms
scientific article; zbMATH DE number 1370119