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Critical Exponent for the Semilinear Wave Equation with Time or Space Dependent Damping

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Progress in Partial Differential Equations

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 44))

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Abstract

Since the damped wave equation has the diffusion phenomenon, the critical exponent is expected to be the same as that for the corresponding diffusive equation with semilinear term. Therefore, we first remember the basic facts on the diffusion phenomenon. Then, from this point of view, we can conjecture the critical exponent for the damped wave equation and state several results. Finally, the small data global existence of solutions is shown in the supercritical exponent, while no global existence for some data is done in the critical and subcritical exponents. The latter part will be applied to the semilinear damped wave equation with quadratically decaying potential.

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Acknowledgements

The author acknowledges the anonymous referee for his/her careful reading of the manuscript and pointing out several errors. This work was supported in part by Grant-in-Aid for Scientific Research (C) 20540219 of Japan Society for the Promotion of Science.

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Authors and Affiliations

  1. Waseda University, 1-6-1 Nishiwaseda, Shinjuku, Tokyo, 169-8050, Japan

    Kenji Nishihara

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  1. Kenji Nishihara
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Correspondence to Kenji Nishihara .

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Editors and Affiliations

  1. Institut of Applied Analysis, TU Bergakademie Freiberg, Prüferstr. 9, Freiberg, 09596, Sachsen, Germany

    Michael Reissig

  2. Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2AZ, United Kingdom

    Michael Ruzhansky

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Nishihara, K. (2013). Critical Exponent for the Semilinear Wave Equation with Time or Space Dependent Damping. In: Reissig, M., Ruzhansky, M. (eds) Progress in Partial Differential Equations. Springer Proceedings in Mathematics & Statistics, vol 44. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00125-8_11

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