34.8. SOME EVOLUTION INCLUSIONS 1227

which shows that a high enough power of the mapping v→ uv is a contraction map on Hand so there exists a unique fixed point u. Thus uu = u and so

u′+ z+u = f +u, uv (0) = u0 ∈ D(φ) , z(t) ∈ ∂φ (t) a.e.

and so

u′+ z = f in H , u(t) ∈ D(∂φ) a.e., z(t) ∈ ∂φ (u(t)) a.e., u′ ∈H , and u(0) = u0

Note that in the above, the initial condition only needs to be in D(φ) , not in the smallerD(∂φ) , although the solution is in D(∂φ) for a.e. t. Also note that f has no smoothness.It only is in H . This is really a nice result.