Minimal periods for solutions of semilinear wave equations in exterior domains and for solutions of the equations of nonlinear elasticity
The author examines i) time-periodic solutions of the semilinear wave equations over \({\mathbb{R}}^ n\), in particular for \(n=3\); ii) the equations of `semilinear elasticity' with spatially periodic elastic coefficients; iii) one-dimensional nonlinear elasticity; iv) nonlinear elasticity. For i), it is shown that radially symmetric time-periodic solutions decaying spatially rapidly enough must be time-independent. A hypothesis on the growth of the nonlinearity at zero is essential to achieve this. In ii), assuming decaying in time of spatial measures of solutions, these are seen to be independent of space. Cases iii) and iv) are similar in spirit to ii). iii) is a special case of iv) about which more is said - an example is given involving the failure of strict hyperbolicity and consequent failure of the above results. A similar point concerns case i) and the sine-Gordon equation.
scientific article; zbMATH DE number 4107299