Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping

Jun-Ren Luo; Ti-Jun Xiao

doi:10.3934/eect.2020009

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2020, Volume 9, Issue 2: 359-373. Doi: 10.3934/eect.2020009

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Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping

Jun-Ren Luo and
Ti-Jun Xiao^,

Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Corresponding author: Ti-Jun Xiao
Corresponding author: Ti-Jun Xiao

Received: December 31, 2018

Revised: March 31, 2019

Early access: August 2019

Published: June 2020

The work was supported partly by the NSF of China (11771091, 11831011), the Fudan University (IDH 1411016), and the Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900)

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Abstract

The paper is concerned with the Cauchy problem for second order hyperbolic evolution equations with nonlinear source in a Hilbert space, under the effect of nonlinear time-dependent damping. With the help of the method of weighted energy integral, we obtain explicit decay rate estimates for the solutions of the equation in terms of the damping coefficient and two nonlinear exponents. Specialized to the case of linear, time-independent damping, we recover the corresponding decay rates originally obtained in [3] via a different way. Moreover, examples are given to show how to apply our abstract results to concrete problems concerning damped wave equations, integro-differential damped equations, as well as damped plate equations.

Keywords:

Nonlinear hyperbolic equations,
nonlinear time-dependent damping,
nonlinear source,
decay rates; second order evolution equations,
Hilbert space.

Mathematics Subject Classification: 35B40, 35L70, 35L90.

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References

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