Non‐autonomous forms and invariance

We generalize the Beurling–Deny–Ouhabaz criterion for parabolic evolution equations governed by forms to the non‐autonomous, non‐homogeneous and semilinear case. Let V,H be Hilbert spaces such that V is continuously and densely embedded in H and let A(t):V→V′ be the operator associated with a bounded H‐elliptic form a(t,.,.):V×V→C for all t∈[0,T]. Suppose C⊂H is closed and convex and P:H→H the orthogonal projection onto C. Given f∈L2(0,T;V′) and u0∈C, we investigate when the solution of the non‐autonomous evolutionary problem u′(t)+A(t)u(t)=f(t),u(0)=u0,remains in C and show that this is the case if Pu(t)∈VandRea(t,Pu(t),u(t)−Pu(t))≥Re⟨f(t),u(t)−Pu(t)⟩for a.e. t∈[0,T]. Moreover, we examine necessity of this condition and apply this result to a semilinear problem.