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Construction of fundamental solutions for Huygens’ equations as invariant solutions. (English. Russian original) Zbl 0789.35093
Sov. Math., Dokl. 43, No. 2, 496-499 (1991); translation from Dokl. Akad. Nauk SSSR 317, No. 4, 786-789 (1991).
The author generalizes the group-theoretic method of constructing fundamental solutions to wave equations of an arbitrary dimension and to the general class of equations of the following hyperbolic equation \[ (\partial/\partial t)^ 2 u-(\partial/\partial x_ 1)^ 2 u- \sum_{i,j=2}^ n a^{ij}(x_ 1-t) (\partial^ 2/ \partial x_ i \partial x_ j)u=0, \;t>0, \;u(0,x)=\delta(x), \;x\in\mathbb{R}^ n,\tag{*} \] where \([a^{ij}(t)]\) is a positive definite matrix with \(C^ \infty(\mathbb{R}^ 1)\)-elements and \(\delta(x)\) is the Dirac delta function concentrated at 0. Then he proves that the solution of the problem (*) has the form \[ \begin{alignedat}{2} u(t,x) &= (2\pi)^{-(n-1)/2} \{| x_ 1- t|^{n-1} |\text{det } [A_{ij}]|^{-1}\}^{1/2} \delta^{((n-3)/2)}(\Gamma) &\;&(n \text{ odd})\\ &= (2\pi)^{-n/2} \{| x_ 1-t|^{n-1} |\text{det } [a_{ij}|^{- 1}\}^{1/2} \{\theta(\Gamma) \Gamma^{-1/2}\}^{((n-2))/2} &\;&(n \text{ even}),\end{alignedat} \] where \([A_{ij}]\) is the matrix of the primitives of the functions \(a_{ij}\) normalized so that \(A^{ij}(0)\), \(\Gamma= t^ 2- x_ 1^ 2+ (t-x_ 1) \sum_{i,j=2}^ n A_{ij} (x_ 1-t) x_ i x_ j\) is the square of the geodesic distance from the origin in \(\mathbb{R}^{n+1}\) of coordinates to the point \((t,x)\), and \(\delta(\Gamma)\) and \(\theta(\Gamma)\) are respectively the Dirac delta function and the unit Heaviside function defined on the characteristic conoid of equation (*) with the apex at origin, given by \(\Gamma(0,x)=0\).

35L15 Initial value problems for second-order hyperbolic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
35C05 Solutions to PDEs in closed form