On -solutions of the equation

In this article, we consider the initial value problem for the equation in the title with u ( x , 0 ) = u 0 ( x ) ∈ W 2 1 ( R ) ∩ W ∞ 1 ( R ) and u t ( x , 0 ) = u 1 ( x ) ∈ L 2 ( 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 ) , u t ( ⋅ , t ) ) ∈ ( W 2 1 ( R ) ∩ W ∞ 1 ( R ) ) × ( L 2 ( 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.