Nonresonance and global existence of prestressed nonlinear elastic waves

Thomas C. Sideris

Review 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 long-time 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; long-time behaviour

Classification (MSC2000): 35Q72 74J30 35A05

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