Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights

Vanessa Barros; Carlos Nonato; Carlos Raposo; Vanessa Barros; Carlos Nonato; Carlos Raposo

doi:10.3934/era.2020014

Electronic Research Archive

2020, Volume 28, Issue 1: 205-220. doi: 10.3934/era.2020014

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Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights

1.
Department of Mathematics, Federal University of Bahia, Salvador, 40170-115, Bahia, Brazil
2.
Department of Mathematics, Federal University of São João del-Rei, São João del-Rei, 36307-352, Minas Gerais, Brazil

Received: 01 January 2020 Revised: 01 February 2020
Primary: 35D05, 35E15; Secondary: 35Q35

We consider the wave equation with a weak internal damping with non-constant delay and nonlinear weights given by
$ \begin{eqnarray*} \label{NLS} u_{tt}(x,t) - u_{xx}(x,t)+\mu_1(t)u_t(x,t) +\mu_2(t)u_t(x,t-\tau(t)) = 0 \end{eqnarray*} $
in a bounded domain. Under proper conditions on nonlinear weights $ \mu_1(t), \mu_2(t) $ and non-constant delay $ \tau(t) $, we prove global existence and estimative the decay rate for the energy.
- Wave equation,
- non-constant delay and weights,
- exponential stability
Citation: Vanessa Barros, Carlos Nonato, Carlos Raposo. Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights[J]. Electronic Research Archive, 2020, 28(1): 205-220. doi: 10.3934/era.2020014

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Abstract

We consider the wave equation with a weak internal damping with non-constant delay and nonlinear weights given by

$ \begin{eqnarray*} \label{NLS} u_{tt}(x,t) - u_{xx}(x,t)+\mu_1(t)u_t(x,t) +\mu_2(t)u_t(x,t-\tau(t)) = 0 \end{eqnarray*} $

in a bounded domain. Under proper conditions on nonlinear weights $ \mu_1(t), \mu_2(t) $ and non-constant delay $ \tau(t) $, we prove global existence and estimative the decay rate for the energy.

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