A remark on the existence of global BV solutions for a nonlinear hyperbolic wave equation

By means of a suitable change of variables we obtain, by application of a general result by Dafermos and Hsiao, cf. [2], an existence theorem in ${L^\infty } \cap {BV_{loc}}$ of a weak solution of the system corresponding to the quasilinear hyperbolic equation \[ {\phi _{tt}} - p’\left ( {\phi _x} \right ){\phi _{xx}} + {\phi _t} + F\left ( \phi \right ) = 0 \qquad in \qquad \mathbb {R} \times \left [ {0, + \infty } \left [ \right . \right .\], for small initial data in BV. This theorem is a partial extension of Dafermos’s result for the case with $F\left ( \phi \right ) \equiv 0$, proved in [1].