Abstract
The goal of this paper is to derive L p − L q estimates away from the conjugate line for structural damped wave models. The damping term interpolates between exterior damping and viscoelastic damping. The crucial point is to derive at first L 1 − L 1 estimates. Depending on the behavior of the characteristic roots of the operator, one has to take into consideration oscillations in one part of the extended phase space. The radial symmetric behavior of the roots allows to apply the theory of modified Bessel functions. Oscillations may produce unbounded time-dependent constants (either for small times close to 0 or for large times close to infinity) in the L 1 − L 1 estimates. Some interpolation techniques imply the desired L p − L q estimates away from the conjugate line.
2010 Mathematics Subject Classification: 35L99, 35B40, 35A23.
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Acknowledgements
This project started during a sabbatical of the first author at TU Bergakademie Freiberg from September 2010 to March 2011. The first author thanks the faculty of Mathematics and Computer Science for the hospitality.
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Narazaki, T., Reissig, M. (2013). L 1 Estimates for Oscillating Integrals Related to Structural Damped Wave Models. In: Cicognani, M., Colombini, F., Del Santo, D. (eds) Studies in Phase Space Analysis with Applications to PDEs. Progress in Nonlinear Differential Equations and Their Applications, vol 84. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-6348-1_11
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DOI: https://doi.org/10.1007/978-1-4614-6348-1_11
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