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Hyperbolic Cauchy problem and Leray’s residue formula. (English) Zbl 0964.35091

The author considers a strictly hyperbolic operator \(P(D_t, D_x)\) of order \(m\) with constant coefficients, and the corresponding Cauchy problem \[ \begin{cases} P(D_t,D_x) u(t, x)=0,\\ D^{m-1}_tu(0, x)= v(x),\quad D^{m-j}_t u(0,x)= 0,\quad 2\leq j\leq m.\end{cases} \] The initial datum \(v(x)\) is assumed to be of a special form, namely the singular support is located on the cotangent bundle of a smooth algebraic surface. A precise asymptotic expansion is then computed for the solution \(u(t,x)\) in a neighborhood of the singular support, with the aid of Gauss-Nanin systems satisfied by Leray’s residues.
Reviewer: L.Rodino (Torino)

MSC:

35L30 Initial value problems for higher-order hyperbolic equations
35S05 Pseudodifferential operators as generalizations of partial differential operators
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
78A05 Geometric optics
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