Existence, uniqueness and exponential decay of solutions to Kirchhoff equation in R^n

• Main Authors: Flavio Roberto Dias Silva, Joao Manoel Soriano Pitot, Andre Vicente
• Format: Article
• Language: English
• Published: Texas State University 2016-09-01
• Series: Electronic Journal of Differential Equations
• Subjects: Kirchhoff equation; existence and uniqueness of solution; uniform stability; exponential decay; frictional damping; viscoelastic damping
• Online Access: http://ejde.math.txstate.edu/Volumes/2016/247/abstr.html

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 $\int_0^tg(t-s)\Delta u(s)ds$.