Abstract Evolution Equations, Periodic Problems and by D Daners

By D Daners

A part of the "Pitman learn Notes in arithmetic" sequence, this article covers: linear evolution equations of parabolic sort; semilinear evolution equations of parabolic style; evolution equations and positivity; semilinear periodic evolution equations; and purposes.

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Example text

Requirement (3), however, is not met since, so far, we have no information whatsoever on the norm of an operator T : E → F – between Banach couples E and F – when viewed as an operator T ∈ L F(E), F(F ) . To remedy this situation we need the following definition. 5 Definition An interpolation method F: B2 → B1 is called of exponent θ ∈ (0, 1) if there exists a positive constant c(F) such that (F 2) T L(F(E),F(F )) 1−θ L(E0 ,F0 ) ≤ c(F) T T θ L(E1 ,F1 ) holds for all E, F ∈ B2 and T ∈ L(E, F ). If we can choose c(F) = 1 then F is called exact of exponent θ.

We fix a number T>0 and consider a family A(t) 0≤t≤T of closed linear operators in X0 having the following properties: (A1) D(A(t)) = X1 for all t ∈ [0, T ]. (A2) For all t ∈ [0, T ] we have [Re µ ≥ 0] ⊂ ̺(−A(t)). Furthermore, there exists a constant M ≥ 1, independent of t ∈ [0, T ], such that M −1 λ + A(t) ≤ 1 + |λ| for all (λ, t) ∈ [Re µ ≥ 0] × [0, T ]. (A3) There exists a constant ρ ∈ (0, 1), such that A(·) ∈ C ρ [0, T ], L(X1 , X0 ) . 1 that for each t ∈ [0, T ] the operator −A(t) is the infinitesimal generator of a strongly continuous analytic semigroup on X0 .

N , there is given a semigroup (e−tAi )t≥0 . Set N Xi X := i=1 T (t)(x1 , . . , xN ) := (e−tA1 x1 , . . , e−tAN xN ) N D(A) := D(Ai ), i=1 30 for (x1 , . . , xN ) ∈ X, and A(x1 , . . , xN ) := (A1 x1 , . . , AN xN ) for (x1 , . . , xN ) ∈ D(A). Then: T (t) t≥0 is a C0 -semigroup on X with infinitesimal generator −A. If each of the semigroups (e−tAi )t≥0 , i = 1, . . , N , is analytic, then so is (T (t))t≥0 . Notes and references: A few books on semigroups are Cl´ement et al. [31], Butzer and Berens [26], Davies [43], Fattorini [55], Goldstein [63], Hille and Philips [69], Pazy [100].

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