Free vibrations of \(n\)-dimensional balls. (English) Zbl 0656.35086
We consider the existence of non-trivial solutions of the problem:
\[
u_{tt}-\Delta u+g(u)=0,\quad u_{tt}-\Delta u-g(u)=0,\quad (x,t)\in B\times R
\]
under the boundary and periodicity conditions:
\[
u|_{\partial B^ n}=0,\quad u(x,t+T)=u(x,t),
\]
where \(B^ n\) is an n-dimensional ball centered at the origin with diameter \(D,T=(b/a)D\); a,b are coprime integers.
MSC:
| 35L70 | Second-order nonlinear hyperbolic equations |
| 35B10 | Periodic solutions to PDEs |
| 35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
References:
| [1] | Coron, J. M., Periodic solutions of a nonlinear wave equation without assumption of monotonicity,Mathematische Annalen,262 (1983), 273–285. · Zbl 0489.35061 · doi:10.1007/BF01455317 |
| [2] | Chang, K. C. and Sanchez, L., Nontrivial periodic solutions of a nonlinear beam equation.Math. Meth. in the Appl. Sci.,4 (1982), 194–205. · Zbl 0501.35004 · doi:10.1002/mma.1670040113 |
| [3] | Chang, K. C. and Hong Chong-wei, Periodic solutions for the semilinear spherical wave equation. to appear. |
| [4] | Triebel, H., Interpolation theory, function spaces, differential operators, North-Holland Publishing Co., New York, 1978. · Zbl 0387.46033 |
| [5] | Mclachlan, N. W., Bessel functions for engineers. Oxford, 1955. · JFM 61.1177.05 |
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