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April  1995, 1(2): 237-252. doi: 10.3934/dcds.1995.1.237

Asymptotic behavior of solutions in nonlinear dynamic elasticity

Irena Lasiecka 1, and  W. Heyman 2,

1. 

Department of Mathematics, University of Virginia, Charlottesville, VA 22903, United States
2. 

Department of Applied Mathematics, University of Virginia, Charlottesville, VA 22903, United States

Received  November 1994 Published  February 1995

A Dynamic system of 2-D nonlinear elasticity with nonlinear interior dissipation is considered. It is assumed that the principal part of elastic operator is perturbed by the unstructured lower order linear terms. Asymptotic behavior of solutions when time $t \rightarrow 0$ is analyzed. It is shown that in the case of zero load applied to the plate, the arbitrarily large decay rates can be achieved provided that both the "damping" coefficient and the "traction" coefficient are suitably large. This result generalizes and extends, to the nonlinear and multidimensional context, the earlier results obtained only for the one-dimensional linear wave equation. In the case of a loaded plate the existence of compact global attractor attracting all solutions is established.
Keywords: asymptotic behavior, nonlinear elasticity, nonlinear interior dissipation, decay rates..
Mathematics Subject Classification: 74K20, 93B05, 35K55, 35L7.
Citation: Irena Lasiecka, W. Heyman. Asymptotic behavior of solutions in nonlinear dynamic elasticity. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 237-252. doi: 10.3934/dcds.1995.1.237
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