We consider the initial -- boundary value problem for
semilinear wave equations with nonlinear damping:
$$
u_{tt}-\Delta u+a|u|^{m-1}u_t=b|u|^{p-1}u \quad
                              \text{in }(0, \infty)\times\Omega,
$$
where $ \Omega $ is a domain in $ \R^n $ with a smooth boundary,
$ m>1 $ and $ p>1 $, while $ a $ and $ b $ are positive constants.
We impose the Dirichlet condition on the boundary.
In this paper, we prove global existence and uniqueness 
of a certain weak solution to this problem under the condition 
$ p\le m$.