Oscillation or nonoscillation property for semilinear wave equations

We considerP±u=∂t2u−Δu±F(u)=0,where F(u)=au+f(u) with a⩾0 and f(0)=0. For the Cauchy problem of P+ if a>0, every nontrivial solution oscillates for any initial data. On the other hand for the Cauchy problem of P− if a⩾0, the solution does not change its sign for some initial data, namely it has nonoscillation property.