On the nonlinear wave equation \(u_{tt}-B(t,\|u_{x}\|^{2})u_{xx} = f(x,t,u,u_{x},u_{t})\) associated with the mixed homogeneous conditions (Q1856844)

In this paper the author considers a class of nonlinear wave equations with the Kirchhoff-Carrier operator of the form \[ u_{tt}-B(t, \|u_x\|^2)u_{xx}=f(x,t,u,u_x,u_t) \] \[ u_x(0,t)-h_0u(0,t)=0, \quad u_x(1,t)+h_1u(1,t)=0 \] \[ u(x,0)=\widetilde{u}_0(x),\quad u_t(x,0)=\widetilde{u}_1(x). \] The above problem is associated with a linear recursive scheme for which the existence of a local and unique solution is proved by using a standard compactness argument.