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 ∈ L 2 ( 0 , T ; V ′ )andu 0 ∈ C , we investigate when the solution of the non‐autonomous evolutionary problemu ′ ( t ) + A ( t ) u ( t ) = f ( t ) , u ( 0 ) = u 0 , remains in C and show that this is the case ifP u ( t ) ∈ V and Re a ( t , P u ( t ) , u ( t ) − P u ( t ) ) ≥ Re ⟨ f ( t ) , u ( t ) − P u ( t ) ⟩ for a.e. t ∈ [ 0 , T ] . Moreover, we examine necessity of this condition and apply this result to a semilinear problem.