Nonresonance and global existence of nonlinear elastic wave equations with external forces
The nonlinear dynamical system \[ \partial _t^2u^i-c_2^2\Delta u^i-(c_1^2-c_2^2)\partial _i(\nabla \cdot u)=B_{lmn}^{ijk}\partial _l(\partial _mu^j\partial _nu^k)+h^i(x), \; i=1,2,3 \] is treated. The authors generalize the results of \textit{T. C. Sideris} [Ann. Math. (2) 151, No. 2, 849--874 (2000; Zbl 0957.35126)] (for \(h^i(x)\equiv 0\)) to the case of time-dependent external forces. First, they prove the existence and uniqueness of the stationary solution. Then they prove the global existence of the solution to the initial value problem by use of estimates, the invariance of the above system under translations, simultaneous rotations and changes of scaling.
scientific article; zbMATH DE number 6113074