In this article, we consider the initial value problem for the equation in the title with u(x, 0) = u(0)(x) is an element of W-2(1)(R) boolean AND W-infinity(1) (R) and u(t)(x, 0) = u(1)(x) is an element of L-2(R) boolean AND L-infinity (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(center dot, t), ut(center dot, t)) is an element of (W-2(1)(R) boolean AND W-infinity(1)(R)) x (L-2(R) boolean AND L infinity (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. (C) 2014 Elsevier Inc. All rights reserved.
