Rothe's method for a telegraph equation with integral conditions (Q1015821)

The paper deals with the telegraph equation \[ \tau\frac{\partial^2 v}{\partial t^2}+a\frac{\partial v}{\partial t}-b\frac{\partial^2 v}{\partial x^2}=f(x,t,v),\quad (x,t)\in (0,1)\times [0,T], \] subject to the initial conditions \[ v(x,0)=v_0(x),\quad \frac{\partial v}{\partial t}(x,0)=v_1(x), \] the Neumann condition \[ \frac{\partial v}{\partial x}(0,t)=G(t) \] and the integral condition \[ \int_0^1v(x,t)\,dx=E(t), \] where \(f,\,v_1,\,v_0,\,G,\,E\) are given functions which verify some assumptions, \(T,\,\tau,\,b\) are positive constants and \(a\geq 1\). By using the Rothe time discretization method, the authors prove the existence and uniqueness of the weak solutions of the above problem.