Maximum principles for bounded solutions of the telegraph equation in space dimensions two and three and applications (Q705959)

The authors prove first an existence-uniqueness theorem and a maximum principle in space dimension three for the weak solutions \(u\in L^\infty(\mathbb{R}\times \mathbb{T}^3)\) of the telegraph equation \[ u_{tt}-\Delta_x u+ Cu_t+\lambda u= f(t,x)\quad\text{in }\mathbb{R}\times \mathbb{R}^3, \] where \(C> 0\), \(\lambda\in(0, C^2/4)\) and \(f\in L^\infty(\mathbb{R}\times \mathbb{T}^3)\). The result is then extended to a solution and a forcing belonging to a suitable space of bounded measures. Based on these results the authors apply the method of upper and lower solutions for the semilinear equation \[ u_{tt}-\Delta_x u+ Cu_t= F(t,x,u). \] A counterexample for the maximum principle in dimension four is also given.