On (W21(R)∩W∞1(R))-solutions of the equation utt=(a(u)ux)x+f(x,t)

In this article, we consider the initial value problem for the equation in the title with u(x,0)=u0(x)∈W21(R)∩W∞1(R) and ut(x,0)=u1(x)∈L2(R)∩L∞(R) in the case when this equation is uniformly hyperbolic. We prove the existence and uniqueness of a local weak solution u(x,t) of this problem such that in particular (u(⋅,t),ut(⋅,t))∈(W21(R)∩W∞1(R))×(L2(R)∩L∞(R)) for any fixed t in the interval of existence. For smooth initial data, it is proved that the life-time of the smooth solution coincides with the life-time of our weak solution.