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:
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