Journal of Soviet Mathematics

, Volume 56, Issue 2, pp 2348–2389

Norm estimates in besov and Lizorkin-Triebel spaces for the solutions of second-order linear hyperbolic equations

  • L. V. Kapitanskii
Article

DOI: 10.1007/BF01671936

Cite this article as:
Kapitanskii, L.V. J Math Sci (1991) 56: 2348. doi:10.1007/BF01671936

Abstract

In the paper one considers the nonhomogeneous hyperbolic equation
$$\partial _t^2 u + iB\left( t \right)\partial _t u + A\left( t \right)u = h$$
on\(\left[ {0, T} \right] \times \mathfrak{M}\), where\(\mathfrak{M}\)=Rn or\(\mathfrak{M}\) is a smooth closed manifold, A(t) and B(t) are pseudodifferential operators on\(\mathfrak{M}\), depending on t ε [0, T], of orders 2 and 1, respectively. For the solutions of equation (1) for small t one establishes estimates of the form
With arbitrary r εR and integer ℓ≥0, where for G:,. and E:,. one can take the Besov space B:,.\(\left( \mathfrak{M} \right)\) or the Lizorkin-Triebet space F:,.\(\left( \mathfrak{M} \right)\), depending on the values n, v, p, q1 q2, and the “Brenner number” m, which are determined from the principal symbols of the operators A(0) and B(0); also the actual form of the scalar function σv,p, n(t) depends on n, v, p, q1 q2, and m: it may be power-like ∣t∣v−n+2n/p, or logarithmic ¦log¦t¦∥, or a constant. In addition, one obtains estimates of the form
characterizing the integrability properties over timespace and the smoothing (for t>0) of the solutions of equation (1).

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • L. V. Kapitanskii

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