Kiss singularities of Green's functions of non-strictly hyperbolic equations (Q2748097)

Starting from Borovikov's integral representations of the Green's function ( = fundamental solution) of hyperbolic differential operators with constant coefficients the authors deduce asymptotic representations of it, first, in the neighborhood of a regular point of the characteristic cone. This case was settled already by \textit{V. A. Borovikov} [Am. Math. Soc. Transl. (2) 25, 11-76 (1963); translation from Tr. Mosk. Mat. O.-va 8, 199-257 (1959; Zbl 0090.31202)]. Second, an asymptotic approximation of the fundamental solution \(G(t,x)\) is derived in the neighborhood of a so-called kiss singularity defined as manifold where two sheets of the slowness surface meet tangentially. The results rely essentially on a suitable decomposition of the Fourier transform (with respect to the time variable \(t)\) of \(G(t,x)\) and a subsequent application of the stationary phase principle.NEWLINENEWLINENEWLINEThe general theory is illustrated with four examples. The first and simplest one determines asymptotically the fundamental solution of the operator \((\partial_t^2-\partial_1^2-\partial_2^2)(\partial_t^2-a\partial_1^2- \partial_2^2)\) (with support in \(t\geq 0).\) Let me remark that in this case the authors' results of asymptotic analysis (3 pages) can be checked by the explicitly known form of \(G,\) NEWLINE\[NEWLINE\frac 1{2\pi(a-1)}\Biggl[\sum_{j=1}^2(-1)^jY(t-\rho_j)\Biggl( \sqrt{a^{j-1}(t^2-\rho_j^2)}-x_1\arctan\biggl(\sqrt{a^{j-1}(t^2- \rho_j^2)} \Big/x_1\biggr)\Biggr)\Biggr],NEWLINE\]NEWLINE NEWLINE\[NEWLINE\rho_1=\sqrt{x_1^2+x_2^2},\qquad \rho_2=\sqrt{\frac{x_1^2}a+x_2^2}NEWLINE\]NEWLINE The remaining three examples exemplify very well the asymptotic expansions obtained.