Abstract
We consider the Cauchy problem of the semilinear wave equation with a damping term
where \(p>1\) and the coefficient of the damping term has the form
with some \(a_0 > 0\), \(\alpha < 0\), \(\beta \in (-1, 1]\). In particular, we mainly consider the cases
which imply \(\alpha + \beta < 1\), namely, the damping is spatially increasing and effective. Our aim is to prove that the critical exponent is given by
This shows that the critical exponent is the same as that of the corresponding parabolic equation
The global existence part is proved by a weighted energy estimates with an exponential-type weight function and a special case of the Caffarelli–Kohn–Nirenberg inequality. The blow-up part is proved by a test-function method introduced by Ikeda and Sobajima [15]. We also give an upper estimate of the lifespan.
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This work was supported by JSPS KAKENHI Grant Number JP18K134450 and JP16K17625.
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Nishihara, K., Sobajima, M. & Wakasugi, Y. Critical exponent for the semilinear wave equations with a damping increasing in the far field. Nonlinear Differ. Equ. Appl. 25, 55 (2018). https://doi.org/10.1007/s00030-018-0546-2
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DOI: https://doi.org/10.1007/s00030-018-0546-2