Long-time dynamics for a class of Kirchhoff models with memory

Author(s)

Jorge Silva, Marcio Antonio ^[1] ; Ma, To Fu ^[2]

Affiliation

^[1] Univ Estadual Londrina, Dept Matemat, BR-86051990 Londrina, Parana - Brazil

^[2] Univ Sao Paulo, Inst Ciencias Matemat & Comp, BR-13560970 Sao Paulo - Brazil

• Document type: Journal article
• Source: Journal of Mathematical Physics; v. 54, n. 2 FEB 2013.
• Web of Science Citations: 19

Abstract

This paper is concerned with a class of Kirchhoff models with memory effects u(tt) + alpha Delta(2)u - div(vertical bar del u vertical bar(p-2)del u) - integral(infinity)(0) mu(s)Delta(2)u(t - s)ds - Delta u(t) + f(u) = h, defined in a bounded domain of RN. This non-autonomous equation corresponds to a viscoelastic version of Kirchhoff models arising in dynamics of elastoplastic flows and plate vibrations. Under assumptions that the exponent p and the growth of f(u) are up to the critical range, it turns out that the model corresponds to an autonomous dynamical system in a larger phase space, by adding a component which describes the relative displacement history. Then the existence of a global attractor is granted. Furthermore, in the subcritical case, this global attractor has finite Hausdorff and fractal dimensions. (C) 2013 American Institute of Physics. {[}http://dx.doi.org/10.1063/1.4792606] (AU)

• FAPESP's process: 08/00123-7 - Asymptotic stability of nonlocally defined evolution equations.
• Grantee: Marcio Antonio Jorge da Silva
• Support Opportunities: Scholarships in Brazil - Doctorate
• FAPESP's process: 10/12202-9 - Asymptotic stability of nonlinear hyperbolic equations
• Grantee: Ma To Fu
• Support Opportunities: Regular Research Grants