The author considers a general nonlinear hyperbolic equation in the region \(Q:=D\times [0,T]\) where D is a bounded domain in \({\mathbb{R}}^ n\) with smooth boundary \(\Gamma\). It is assumed that D is partitioned by a hypersurface \(\Gamma_ 1\) into regions \(D_ 1\) and \(D_ 2\) and the notation \(\gamma =\Gamma_ 1\times [0,T]\), \(S=\Gamma \times [0,T]\) and \(D_ 2=D_ k[0,T]\), \(k=1,2\) is introduced. The solutions u(x,t) of the considered equation are required to satisfy (i) a homogeneous initial value for u and \(u_ t\), (ii) a homogeneous Dirichlet condition on S, (iii) a transmission condition for u and its normal derivative on \(\gamma\). Conditions are established under which a unique weak solution to such a problem exists.