Periodic solutions of some semilinear wave equations on balls and on spheres
Radially symmetric solutions of the problem \[ u_{tt}- \Delta u+ g(u)= f(t,x), \qquad (t,x)\in \mathbb{R}\times B_ a^ n, \] \[ u(t,x)=0, \quad (t,x)\in\mathbb{R} \times S_ a^{n-1}, \qquad u(t+T,x)= u(t,x), \quad (t,x)\in \mathbb{R}\times B_ a^ n \] are considered where \(S_ a^{n-1}= \{x\in\mathbb{R}^ n\): \(\| x\|=a\}\), \(B_ a^ n= \{x\in \mathbb{R}^ n\): \(\| x\|<a\}\). In the case of even \(n\) it is supposed that \(g(t,x,u)\) satisfies nonresonance conditions of non-uniform type with respect to \(t\) and \(x\). In the case of odd \(n\) using recent existence theorems of the authors, a result of Smiley is improved by providing sharp conditions on the monotonicity constants of \(g(t,x,u)\) which insure the existence and uniqueness of the solution.
semilinear wave equations on balls and on spheres
scientific article; zbMATH DE number 482110
scientific article