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Non‐autonomous forms and invariance
Author(s) -
Dier Dominik
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201700090
Subject(s) - mathematics , bounded function , homogeneous , regular polygon , operator (biology) , pure mathematics , hilbert space , projection (relational algebra) , mathematical analysis , combinatorics , geometry , biochemistry , chemistry , repressor , algorithm , transcription factor , gene
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.