Linear recursive schemes associated with the nonlinear wave equation involving Besselʹs operator

We prove the existence and uniqueness of a weak solution of a semilinear wave equation involving Besselʹs operator, the nonlinear term being in the form f(r, t, u, ▿u), while the boundary condition at r = 0 takes the weak form . proof, the Faedo-Galerkin method associated with a linear recursive scheme in appropriate Sobolev spaces with weight is used.