Vibrating infinite string under general observation conditions and minimally smooth force

Existence of the classical solution u(x, t) ∈ C2(R2) to the Problem (1), (2) (shortly Problem A): Lu := utt(x, t)− auxx(x, t) = f (x, t), (x, t) ∈ R2, a > 0, (1) under the observation conditions (the observed states) given at t1, t2 ∈ R with variable coefficients A1, B1, A2, B2 such that A1(x)u|t=t1 + B1(x)ut|t=t1 = g1(x), x ∈ R, A2(x)u|t=t2 + B2(x)ut|t=t2 = g2(x), x ∈ R, (2) is proved. Here the coefficients Ai, Bi, i = 1, 2, and g1, g2 are given functions smooth enough, f ∈ C(R2), the directional derivative ∂ f /∂t exists and ∂ f /∂t ∈ C(R2).