On weak solutions of the initial value problem for the equation utt=a(x,t)uxx+f(t,x,u,ut,ux)utt=a(x,t)uxx+f(t,x,u,ut,ux)

For the equation of the kind indicated in the title, it is assumed roughly speaking that a(⋅,t)∈C(R;W21)∩L∞(R;W∞1)∩C1(R;L2) and at(⋅,t)∈L∞(R;L∞)at(⋅,t)∈L∞(R;L∞) and that there exist 00a3>0 such that a1≤a(x,t)≤a2a1≤a(x,t)≤a2 and |∇a(x,t)|≤a3|∇a(x,t)|≤a3 for any x,t∈Rx,t∈R. The function ff is assumed to be continuously differentiable and satisfying f(t,x,0,r,s)≡0f(t,x,0,r,s)≡0. The initial data are assumed to be in (W21∩W∞1)×(L2∩L∞). The existence and uniqueness of a local weak (W21∩W∞1)-solution is proved. In addition, in the special case f(t,x,u,ut,ux)=−|u|p−1uf(t,x,u,ut,ux)=−|u|p−1u, p≥1p≥1, the existence of a global weak solution is proved.