Eventual partition of conserved quantities in wave motion.

*(English)*Zbl 0591.35036Summary: Let u be a classical solution to the wave equation in an odd number n of space dimensions, with compact spatial support at each fixed time. R. J. Duffin [ibid. 32, 386-391 (1970; Zbl 0223.35055)] uses the Paley- Wiener theorem of Fourier analysis to show that, after a finite time, the (conserved) energy of u is partitioned into equal kinetic and potential parts. The wave equation actually has \((n+2)(n+3)/2\) independent conserved quantities, one for each of the standard generators of the conformal group of \((n+1)\)-dimensional Minkowski space. Of concern in this paper is the ”zeroth inversional quantity” \(I_ 0\), which is commonly used to improve decay estimates which are obtained using conservation of energy. We use Duffin’s method to partition \(I_ 0\) into seven terms, each of which, after a finite time, is explicitly given as a constant-coefficient quadratic function of the time, E. C. Zachmanoglou [ibid. 39, 296-297 (1972; Zbl 0233.35059)] has shown that under the above assumptions if \(n\geq 3\), the spatial \(L^ 2\) norm of u is eventually constant. A consequence of the analysis here is a bound on this constant in terms of the energy and the radius of the support of the Cauchy data of u at a fixed time.

##### MSC:

35L05 | Wave equation |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

35Q99 | Partial differential equations of mathematical physics and other areas of application |

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##### References:

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[2] | Duffin, R.J, Equipartition of energy in wave motion, J. math. anal. appl., 32, 386-391, (1970) · Zbl 0223.35055 |

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[5] | Zachmanoglou, E.C, Integral constants in wave motion, J. math. anal. appl., 39, 296-297, (1972) · Zbl 0202.37104 |

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