## Journal of the Mathematical Society of Japan

### Optimal decay rate of the energy for wave equations with critical potential

#### Abstract

We study the long time behavior of solutions of the wave equation with a variable damping term $V(x)u_t$ in the case of critical decay $V(x)\geq V_0(1+|x|^2)^{-1/2}$ (see condition (A) below). The solutions manifest a new threshold effect with respect to the size of the coefficient $V_0$: for $1 < V_0 < N$ the energy decay rate is exactly $t^{-V_0}$, while for $V_0\geq N$ the energy decay rate coincides with the decay rate of the corresponding parabolic problem.

#### Article information

Source
J. Math. Soc. Japan, Volume 65, Number 1 (2013), 183-236.

Dates
First available in Project Euclid: 24 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1359036453

Digital Object Identifier
doi:10.2969/jmsj/06510183

Mathematical Reviews number (MathSciNet)
MR3034403

Zentralblatt MATH identifier
1267.35034

#### Citation

IKEHATA, Ryo; TODOROVA, Grozdena; YORDANOV, Borislav. Optimal decay rate of the energy for wave equations with critical potential. J. Math. Soc. Japan 65 (2013), no. 1, 183--236. doi:10.2969/jmsj/06510183. https://projecteuclid.org/euclid.jmsj/1359036453

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