Electron. J. Differential Equations, Vol. 2016 (2016), No. 247, pp. 1-27.

Existence, uniqueness and exponential decay of solutions to Kirchhoff equation in $\mathbb{R}^n$

Flavio Roberto Dias Silva, Joao Manoel Soriano Pitot, Andre Vicente

Abstract:
We discuss the global well-posedness and uniform exponential stability for the Kirchhoff equation in $\mathbb{R}^n$
$$ u_{tt}-M\Big(\int_{\mathbb{R}^n}|\nabla u|^2dx\Big)\Delta u +\lambda u_t=0 \quad \text{in } \mathbb{R}^n\times (0,\infty). $$
The global solvability is proved when the initial data are taken small enough and the exponential decay of the energy is obtained in the strong topology $H^2(\mathbb{R}^n)\times H^1(\mathbb{R}^n)$, which is a different feature of the present article when compared with the prior literature. We also dedicate a section to discuss a model with the frictional damping term $\lambda u_t$, is replaced by a viscoelastic damping term $\int_0^tg(t-s)\Delta u(s)ds$.

Submitted June 17, 2016. Published September 12, 2016.
Math Subject Classifications: 35B35, 35B40, 35B45, 35B70.
Key Words: Kirchhoff equation; existence and uniqueness of solution; uniform stability; exponential decay; frictional damping; viscoelastic damping.

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Flávio Roberto Dias Silva
Universidade Estadual do Oeste do Paraná - CCET
Rua Universitária, 2069, Jd. Universitário
CEP: 85819-110, Cascavel, PR, Brazil
email: frdsilva@yahoo.com.br
João Manoel Soriano Pitot
Universidade Estadual Paulista
Rua Cristóvão Colombo, 2265, Jd. Nazareth
CEP: 15054-000, São José do Rio Preto, SP, Brazil
email: john.pitot@gmail.com
André Vicente
Universidade Estadual do Oeste do Paraná - CCET
Rua Universitária, 2069, Jd. Universitário
CEP: 85819-110, Cascavel, PR, Brazil
email: andre.vicente@unioeste.br

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