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Annals of Mathematics, II. Series Vol. 151, No. 2, pp. 849874 (2000) 

Nonresonance and global existence of prestressed nonlinear elastic wavesThomas C. SiderisReview from Zentralblatt MATH: The existence of global classical solutions to the Cauchy problem in nonlinear elastodynamics for unbounded homogeneous isotropic hyperelastic medium is treated. The considered deformation is $\varphi (x,t)= \lambda x+u(t,x)$ $(\lambda>0)$ in which $u(t,x)$ represents a small displacement from a homogeneous dilatation. The longtime behaviour of solutions of quasilinear wave equations in $3D$ is determined by the structure of the quadratic portion on nonlinearity of the equations of motion. The nonlinear terms must obey a type of nonresonance or null condition. A new version of the null condition is introduced and the presented decay estimates make clear that the leading contribution of the resonant interactions along the characteristic cones is potentially dangerous. This permits the application of approximate local decomposition. Reviewed by I.Ecsedi Keywords: hyperbolicity; nonlinear elastic waves; existence of global classical solutions; Cauchy problem; nonlinear elastodynamics; longtime behaviour Classification (MSC2000): 35Q72 74J30 35A05 Full text of the article:
Electronic fulltext finalized on: 27 Apr 2001. This page was last modified: 22 Jan 2002.
© 2001 Johns Hopkins University Press
