## Rothe’s method for a telegraph equation with integral conditions.(English)Zbl 1171.35306

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.

### MSC:

 35A35 Theoretical approximation in context of PDEs 35L70 Second-order nonlinear hyperbolic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35L20 Initial-boundary value problems for second-order hyperbolic equations
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### References:

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