A maximum principle for periodic solutions of the telegraph equation (Q1269090)

This paper deals with a maximum principle for the doubly \(2\pi\)-periodic solutions of the telegraph operator defined by \[ {\mathcal L}_\lambda u= u_{tt}- u_{xx}+ cu_t-\lambda u, \] with \(c>0\). It is proved that \({\mathcal L}_\lambda\) satisfies the maximum principle if and only if \(\lambda\in [-\nu,0)\) for some \[ \nu= \nu(c)\in\Biggl({c^2\over 4},{c^2\over 4}+ {1\over 4}\Biggr]. \] Applications of this interesting principle are given to the linear telegraph operator with variable coefficients, and to a method of upper and lower solutions for the periodic solutions of nonlinear telegraph equations, which include the sine-Gordon equation.