Nonlinear Absorptions

Let (Ω,Bμ) be a σ-finite measure space, A be an m-completely accretive operator in L1(Ω) (A is m-accretive with resolvant Jλ= (I+λA)−1 satisfying u≤û+k ⇒Jλ≤Jλû + k for k≥0), and j: Ω × R → [0,∞] be measurable in x ∈ Ω. convex, and l.s.c. in r ∈ R with j(x, 0) = 0. We consider the operator A + B where B is the operator in L1(Ω) defined by w ∈ Bu iff j(r) ≥j(u(x)) + (r − u(x)) w(x) for any r ∈ R, a.e. x ∈ Ω.We define natural m-completely accretive extensions of A + B in L1(Ω) and study their dependence with respect to j and different cases where A + B is itself m-completely accretive; we consider also the evolution problem du/dt + Au + Bu ∋ ƒ, u(0)= u0.