Nonresonance and global existence of prestressed nonlinear elastic waves. (English) Zbl 0957.35126

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.


35Q72 Other PDE from mechanics (MSC2000)
74J30 Nonlinear waves in solid mechanics
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35L70 Second-order nonlinear hyperbolic equations
74H20 Existence of solutions of dynamical problems in solid mechanics
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