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Strictly Hyperbolic Cauchy Problems with Coefficients Low-Regular in Time and Space

  • Daniel LorenzEmail author
Chapter
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Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

We consider the strictly hyperbolic Cauchy problem
$$\displaystyle \begin {cases} D_t^m u - \sum \limits _{j = 0}^{m-1} \sum \limits _{|\gamma |+j = m} a_{m-j,\,\gamma }(t,\,x) D_x^\gamma D_t^j u = 0,\\ D_t^{k-1}u(0,\,x) = g_k(x),\,k = 1,\,\ldots ,\,m, \end {cases} $$
for \((t,\,x) \in [0,\,T]\times \mathbb {R}^n\) with coefficients belonging to the Zygmund class \(C_\ast ^s\) in x and having a modulus of continuity below Lipschitz in t. Imposing additional conditions to control oscillations, we obtain a global (on [0, T]) L2 energy estimate without loss of derivatives for \(s \geq \max \{1+\varepsilon ,\,\frac {2m_0}{2-m_0}\}\), where m0 is linked to the modulus of continuity of the coefficients in time.

Keywords

Cauchy problem Modulus of continuity Zygmund space Low-regular Strictly hyperbolic Higher order 

Notes

Acknowledgements

The author wants to express his gratitude to Michael Reissig for many fruitful discussions and suggestions. Furthermore, he wants to thank Daniele Del Santo for his hospitality and the suggested improvements during the authors stay at Trieste University.

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Authors and Affiliations

  1. 1.TU Bergakademie Freiberg, Faculty of Mathematics and Computer ScienceInstitute of Applied AnalysisFreibergGermany

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