On the nonexistence of periodic radial solutions for semilinear wave equations in unbounded domain (Q1979122)

The authors deal with time-periodic radial solutions to a radial wave equation in \(\mathbb{R}^N\), \(N\geq 1\), with a nonlinearity which depends on time with the same periodicity as the solution in question. With the help of an integral identity which is satisfied by any solution of the equation \[ u_{tt} - u_{rr} - (N-1)u_r/r + g(t,r,u)=0,\quad r\in (0,R),\;R<\infty , \] they define some classes of function spaces in which no nontrivial time periodic solution to the above equation with~\(r\in (0,\infty)\) exists. The classes are defined as weighted spaces with weights which are the various powers of the variable~\(r\). These powers are in correspondence with the form of the function~\(g\) representing the nonlinearity. The detailed description is too technical to be reproduced here.