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Kiss singularities of Green’s functions of non-strictly hyperbolic equations. (English) Zbl 0993.35006

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 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.
The 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 \(G,\) \[ \frac 1{2\pi(a-1)}\Biggl[\sum_{j=1}^2(-1)^jY(t-\rho_j)\Biggl( \sqrt{a^{j-1}(t^2-\rho_j^2)}-x_1\arctan\biggl(\sqrt{a^{j-1}(t^2- \rho_j^2)} \Big/x_1\biggr)\Biggr)\Biggr], \]
\[ \rho_1=\sqrt{x_1^2+x_2^2},\qquad \rho_2=\sqrt{\frac{x_1^2}a+x_2^2} \] The remaining three examples exemplify very well the asymptotic expansions obtained.

MSC:

35A20 Analyticity in context of PDEs
35A08 Fundamental solutions to PDEs
35L25 Higher-order hyperbolic equations
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