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Semilinear perturbations of Hille-Yosida operators

Horst R. ThiemeHauke Vosseler — 2003

Banach Center Publications

The semilinear Cauchy problem (1) u’(t) = Au(t) + G(u(t)), u ( 0 ) = x D ( A ) ¯ , with a Hille-Yosida operator A and a nonlinear operator G: D(A) → X is considered under the assumption that ||G(x) - G(y)|| ≤ ||B(x -y )|| ∀x,y ∈ D(A) with some linear B: D(A) → X, B ( λ - A ) - 1 x = λ 0 e - λ t V ( s ) x d s , where V is of suitable small strong variation on some interval [0,ε). We will prove the existence of a semiflow on [ 0 , ) × D ( A ) ¯ that provides Friedrichs solutions in L₁ for (1). If X is a Banach lattice, we replace the condition above by |G(x) - G(y)| ≤ Bv whenever...

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