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Abstract Cauchy Problems: Three Approaches by Irina V. Melnikova, Alexei Filinkov PDF

By Irina V. Melnikova, Alexei Filinkov

ISBN-10: 1584882506

ISBN-13: 9781584882503

Proper to various mathematical versions in physics, engineering, and finance, this quantity reviews Cauchy difficulties that aren't well-posed within the classical feel. It brings jointly and examines 3 significant techniques to treating such difficulties: semigroup equipment, summary distribution equipment, and regularization tools. even though broadly built during the last decade, the authors supply a different, self-contained account of those tools and display the profound connections among them. available to starting graduate scholars, this quantity brings jointly many various principles to function a reference on sleek equipment for summary linear evolution equations.

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An x . 2 Let n ∈ N. The Cauchy problem (CP) is said to be (n, ω)-well-posed on E if for any x ∈ E ⊆ D(An+1 ) ©2001 CRC Press LLC ©2001 CRC Press LLC (a) there exists a unique solution u(·) ∈ C [0, ∞], D(A) ∩ C 1 [0, ∞], X ; (b) ∃K > 0, ω ∈ R : u(t) ≤ Keωt x An . If E = D(An+1 ), then we say that the problem (CP) is (n, ω)-well-posed. 4 Let A be a densely defined linear operator on X with nonempty resolvent set. Then the following statements are equivalent: (I) A is the generator of an n-times integrated semigroup {V (t), t ≥ 0}; (II) the Cauchy problem (CP) is (n, ω)-well-posed.

13) is satisfied as described in Cases 1–3 according to the choice of the initial data. 21) where u(t) u (t) w(t) = Φ= 0 I A 0 ∈ L2 (Ω) × L2 (Ω), D(Φ) = D(A) × L2 (Ω). 21) can be formally written as w(t) = U (t) = ©2001 CRC Press LLC ©2001 CRC Press LLC u0 u1 := C(t) C (t) C(t)u0 + S(t)u1 C (t)u0 + S (t)u1 , S(t) S (t) u0 u1 t ≥ 0, u0 , u1 ∈ L2 (Ω), and noting that for any v ∈ L2 (Ω), S (t)v = C(t)v, we write U in the form C(t) S(t) C (t) C(t) U (t) = t ≥ 0. , From the definition of C it is clear that C is not necessarily differentiable in t on L2 (Ω), implying that the operators U (t) are in general unbounded on L2 (Ω) × L2 (Ω), and therefore they do not form a C0 -semigroup on this space.

3) To prove that D(A) = X, we consider the set b U := U (τ )udτ, x ∈ X, b > a > 0 . va,b = a We show that U ⊂ D(A): h−1 U (h) − I va,b = b h−1 [U (h + τ ) − U (τ )] xdτ a = b+h h−1 b U (t)xdt − a+h = b+h h−1 ©2001 CRC Press LLC ©2001 CRC Press LLC a U (τ )xdτ − b → U (τ )xdτ a U (τ )xdτ a+h U (b) − U (a) x as h → 0. Now we will show that U = X. Suppose that there exists f ∈ X ∗ such that f (U) = 0, f ≡ 0. Then b ∀b > a > 0, f (va,b ) = f (U (τ )x)dτ = 0. a Hence ∀x ∈ X, f (U (τ )x) = 0, τ > 0, and f (U (τ )x) →τ →0 f (x) = 0, that is f ≡ 0.

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Abstract Cauchy Problems: Three Approaches by Irina V. Melnikova, Alexei Filinkov


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