Estimates for the solutions of the Cauchy problem for a hyperbolic equation of higher order in spaces \(H_ p\). (Russian) Zbl 0605.35049

Perzhan, A. V.

Mat. Issled. 88, 72-82 (1986).

Continuing former investigations of A. Miyachi on estimates for the wave equation in the spaces \(L_ p\) and \(H_ p\) [see J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 331-354 (1980; Zbl 0437.35042)] here sufficient and necessary conditions for the existence of a-priori \(H_ p\)-estimates (resp. for \(p>1\) \(L_ p\)-estimates) for the solutions of a Cauchy problem for a hyperbolic equation of fourth order and their derivatives by \(H_ p\)-norms (resp. for \(p>1\) \(L_ p\)-norms) are proved.
The proof is carried out using properties of Fourier \(H_ p\)- multipliers.

Reviewer: K.Barckow

MSC:

35L30	Initial value problems for higher-order hyperbolic equations
35B45	A priori estimates in context of PDEs
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

wave equation; a-priori \(H_ p\)-estimates; Cauchy problem; fourth order; Fourier \(H_ p\)-multipliers

Citations:

Zbl 0437.35042