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

By Irina V. Melnikova, Alexei Filinkov

Appropriate to numerous mathematical types in physics, engineering, and finance, this quantity reports Cauchy difficulties that aren't well-posed within the classical experience. It brings jointly and examines 3 significant techniques to treating such difficulties: semigroup tools, summary distribution equipment, and regularization tools. even though largely constructed over the past decade, the authors offer a different, self-contained account of those equipment and exhibit the profound connections among them. available to starting graduate scholars, this quantity brings jointly many alternative rules to function a reference on sleek equipment for summary linear evolution equations.

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N! Since τ λn+1 e−λt 0 tn τn xdt = −λn e−λτ x + n! n! n = ... = − k=0 τ λn e−λt 0 tn−1 xdt (n − 1)! (λτ )k −λτ e x + x, k! then R(λ, τ )(λI − A)x = (λI − A)R(λ, τ )x = (I − G(λ))x, where n−1 G(λ)x = λn e−λτ V (τ )x + k=0 ©2001 CRC Press LLC ©2001 CRC Press LLC (λτ )k −λτ e x, k! x ∈ D(A), and G(λ) ≤ C(1 + |λ|)n e−τ Re λ , C = C(τ, n). We also have that G(λ) commutes with R(λ, τ ) on X and with A on D(A). Using this estimate for G(λ) , we can find a region Λ ⊂ C such that G(λ) < 1 for any λ ∈ Λ. After taking logarithms of the inequality C(1 + |λ|)n e−τ Re λ < γ < 1, we obtain that the estimates G(λ) < γ, (I − G(λ))−1 < 1 1−γ hold in the region Λ= λ ∈ C Re λ > n 1 C log(1 + |λ|) + log τ τ γ .

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. Let us consider a smaller space H01 (Ω) × L2 (Ω), and let 0 I A 0 Ψ= , D(Ψ) = D(A) × H01 (Ω). with From the calculations in Case 2 we deduce that the operators U (t) are bounded on H01 (Ω) × L2 (Ω), and they form a C0 -semigroup generated by Ψ. 21) in the variational sense.

Proof (I) =⇒ (II). Let x ∈ D(An+1 ), consider the function V (·)x. 2, it is (n + 1)-times continuously differentiable, V (n) (t)x ∈ D(A), and V (n) n n−1 k (t)x = V (t)A x + k=0 t k A x, k! d (n) V (t)x = AV (n) (t)x. dt Let u(t) := V (n) (t)x, then u(0) = x and u(t) ∈ D(A) for t ≥ 0. Furthermore, u(t) ≤ Keωt x An , and u (t) = Au(t). We now show that u(·) is unique. Let v(·) be a solution of (CP), then for n (λ)v(·) is the solution of (CP) with the initial value Rn (λ)x ∈ λ ∈ ρ(A), RA n+1 ). 1, we have D(A d n n V (t − s)RA (λ)v(s) ds n n n (λ)v(s) + V n (t − s)ARA (λ)v(s) = 0, = −AV (t − s)RA for 0 ≤ s ≤ t.

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