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

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 &...

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Bibliographic Details
Published in:Journal of the Mathematical Society of Japan 2013-01, Vol.65 (1), p.183-236
Main Authors: IKEHATA, Ryo, TODOROVA, Grozdena, YORDANOV, Borislav
Format: Article
Language:English
Subjects:
35B33
35B40
35L05
35L70
critical potential
damped wave equation
diffusive structure
energy decay
finite speed of propagation
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Summary: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.
ISSN:0025-5645
1881-2333
DOI:10.2969/jmsj/06510183