On weak solutions of the initial value problem for the equation u_(tt) = a (x, t) u_(xx) + f (t, x, u, ut, u_x)

摘要

For the equation of the kind indicated in the title, it is assumed roughly speaking that a(?,t)?C(R;W_2 ~1)∩L_∞(R;W_∞ 1)∩C~1(R;L_2) and a~t(?,t)?L_∞(R;L_∞) and that there exist 0<~(a10 such that ~(a1)≤a(x,t)~(≤a2) and ?a(x,t)≤~(a3) for any x,t?R. The function f is assumed to be continuously differentiable and satisfying f(t,x,0,r,s)≡0. The initial data are assumed to be in (W_2~1∩W_∞~1)×(L~2∩L_∞). The existence and uniqueness of a local weak (W_2~1∩ W_∞~1)-solution is proved. In addition, in the special case f(t,x,u,u_t,u_x)=-~(up-1)u, p≥1, the existence of a global weak solution is proved.