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tr ss rrstt sts trr trrt qts t

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SLIDE 1

❙♣❡❝tr❛❧ ❛♥❛❧②s✐s ❛♥❞ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ s♦❧✉t✐♦♥s ♦❢ ❱♦❧t❡rr❛ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ ❢r❛❝t✐♦♥❛❧ ❡①♣♦♥❡♥t✐❛❧ ❦❡r♥❡❧s

◆✳ ❆✳ ❘❛✉t✐❛♥ ❛♥❞ ❱✳ ❱✳ ❱❧❛s♦✈ ▲♦♠♦♥♦s♦✈ ▼♦s❝♦✇ ❙t❛t❡ ❯♥✐✈❡rs✐t② ❖❚❑❘✲✷✵✶✾✱ ❉❡❝❡♠❜❡r ✶✾✲✷✷✱ ✷✵✶✾

✶ ✴ ✹✽

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SLIDE 2

■♥tr♦❞✉❝t✐♦♥

❲❡ st✉❞② ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ ✉♥❜♦✉♥❞❡❞ ♦♣❡r❛t♦r ❝♦❡✣❝✐❡♥ts ✐♥ ❍✐❧❜❡rt s♣❛❝❡✳ ❚❤❡ ♠❛✐♥ ♣❛rt ♦❢ t❤❡s❡ ❡q✉❛t✐♦♥s ✐s ❛♥ ❛❜str❛❝t ❤②♣❡r❜♦❧✐❝ ❡q✉❛t✐♦♥s✱ ❞✐st✉r❜❡❞ ❜② t❤❡ t❡r♠s ❝♦♥t❛✐♥✐♥❣ ❛❜str❛❝t ✐♥t❡❣r❛❧ ❱♦❧t❡rr❛ ♦♣❡r❛t♦rs✳ ❚❤❡ ❡q✉❛t✐♦♥s ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡ ❛r❡ t❤❡ ❛❜str❛❝t ❢♦r♠ ♦❢ t❤❡ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ♦❢ ●✉rt✐♥✲P✐♣❦✐♥ ❞❡s❝r✐❜✐♥❣ t❤❡ ♣r♦❝❡ss ♦❢ ❤❡❛t ♣r♦♣❛❣❛t✐♦♥ ✐♥ ♠❡❞✐❛ ✇✐t❤ ♠❡♠♦r②✱ ♣r♦❝❡ss ♦❢ ✇❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ t❤❡ ✈✐s❝♦✲ ❡❧❛st✐❝ ♠❡❞✐❛✱ ❛♥❞ ❛❧s♦ ❛r✐s✐♥❣ ✐♥ t❤❡ ♣r♦❜❧❡♠s ♦❢ ♣♦r♦✉s ♠❡❞✐❛ ✭❉❛r❝✐ ❧❛✇✮✳ ✶✮ ❲❡ ♦❜t❛✐♥ ❝♦rr❡❝t s♦❧✈❛❜✐❧✐t② ♦❢ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠s ❢♦r t❤❡ ❞❡s❝r✐❜❡❞ ❡q✉❛t✐♦♥s ✐♥ t❤❡ ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡s ♦♥ t❤❡ ♣♦s✐t✐✈❡ s❡♠✐❛①✐s✳ ✷✮ ❲❡ st✉❞② t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ s♦❧✉t✐♦♥s ♦❢ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♦♥ t❤❡ ❜❛s✐s ♦❢ s♣❡❝tr❛❧ ❛♥❛❧②s✐s ♦❢ t❤❡✐r s②♠❜♦❧s✳ ✸✮ ❚♦ t❤✐s ❡♥❞✱ ✇❡ ♦❜t❛✐♥ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ str♦♥❣ s♦❧✉t✐♦♥s ♦❢ t❤❡s❡ ❡q✉❛t✐♦♥s ✐♥ t❤❡ ❢♦r♠ ♦❢ ❛ s✉♠ ♦❢ t❡r♠s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ r❡❛❧ ❛♥❞ ♥♦♥r❡❛❧ ♣❛rts ♦❢ t❤❡ s♣❡❝tr✉♠ ♦❢ t❤❡ ♦♣❡r❛t♦r ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ t❤❡ s②♠❜♦❧s ♦❢ t❤❡s❡ ❡q✉❛t✐♦♥s✳ ❚❤❡s❡ r❡♣r❡s❡♥t❛t✐♦♥s ❛r❡ ♥❡✇ ❢♦r t❤❡ ❝♦♥s✐❞❡r❡❞ ❝❧❛ss ♦❢ ✐♥t❡❣r♦✲ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳

✷ ✴ ✹✽

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SLIDE 3

▲❡t ✉s H ❜❡ ❛ s❡♣❛r❛❜❧❡ ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ A ❜❡ ❛ s❡❧❢✲❛❞❥♦✐♥t ♣♦s✐t✐✈❡ ♦♣❡r❛t♦r A∗ = A κ0 ✭κ0 > 0✮ ❛❝t✐♥❣ ✐♥ t❤❡ s♣❛❝❡ H ❛♥❞ ❤❛✈✐♥❣ ❛ ❝♦♠♣❛❝t ✐♥✈❡rs❡ ♦♣❡r❛t♦r✳ ▲❡t ✉s B ❜❡ ❛ s②♠♠❡tr✐❝ ♦♣❡r❛t♦r (Bx, y) = (x, By)✱ ❛❝t✐♥❣ ✐♥ t❤❡ s♣❛❝❡ H ❤❛✈✐♥❣ t❤❡ ❞♦♠❛✐♥ Dom (B) ✭Dom (A) ⊆ Dom (B)✮✳ ▼♦r❡♦✈❡r B ❜❡ ❛ ♥♦♥♥❡❣❛t✐✈❡ ♦♣❡r❛t♦r t❤❛t ✐s (Bx, x) 0 ❢♦r ❛♥② x, y ∈ Dom (B) ❛♥❞ s❛t✐s❢②✐♥❣ t♦ ✐♥❡q✉❛❧✐t② Bx κ Ax✱ 0 < κ < 1 ❢♦r ❛♥② x ∈ Dom (A) ❛♥❞ I ❜❡ t❤❡ ✐❞❡♥t✐t② ♦♣❡r❛t♦r ❛❝t✐♥❣ ✐♥ t❤❡ s♣❛❝❡ H✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠ ❢♦r ❛ s❡❝♦♥❞✲♦r❞❡r ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ♦♥ t❤❡ s❡♠✐❛①✐s R+ = (0, ∞)✿ d2u(t) dt2 +Au(t)+Bu(t)− t K(t − s)Au(s)ds− t Q(t − s)Bu(s)ds = = f(t), t ∈ R+, ✭✶✮ u(+0) = ϕ0, u(1)(+0) = ϕ1. ✭✷✮

✸ ✴ ✹✽

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SLIDE 4

❆ss✉♠❡ t❤❛t t❤❡ s❝❛❧❛r ❢✉♥❝t✐♦♥s K(t) ❛♥❞ Q(t) t❤❛t ❛r❡ t❤❡ ❦❡r♥❡❧s ♦❢ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦rs ❛❞♠✐ts t❤❡ ❢♦❧❧♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥s✿ K(t) = ∞ e−tτdµ(τ), Q(t) = ∞ e−tτdη(τ), ✭✸✮ ✇❤❡r❡ dµ ❛♥❞ dη ❛r❡ t❤❡ ♣♦s✐t✐✈❡ ♠❡❛s✉r❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❛♥ ✐♥❝r❡❛s✐♥❣ r✐❣❤t✲❝♦♥t✐♥✉♦✉s ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s µ ❛♥❞ η r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ✐♥t❡❣r❛❧ ✐s ✉♥❞❡rst♦♦❞ ✐♥ t❤❡ ❙t✐❡❧t❥❡s s❡♥s❡✳

✹ ✴ ✹✽

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SLIDE 5

❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ s❛t✐s✜❡❞✿ ∞ dµ(τ) τ < 1, ∞ dη(τ) τ < 1, ✭✹✮ ❍❡r❡ t❤❡ s✉♣♣♦rts µ ❛♥❞ η ❜❡❧♦♥❣ t♦ t❤❡ ✐♥t❡r✈❛❧ (d0, +∞)✱ d0 > 0✳ ❚❤❡ ❝♦♥❞✐t✐♦♥s ✭✹✮ ♠❡❛♥s t❤❛t K(t), Q(t) ∈ L1(R+)✱ KL1 < 1✱ QL1 < 1✳ ■❢ ❝♦♥❞✐t✐♦♥s ✭✹✮ ❛r❡ s✉♣♣❧❡♠❡♥t❡❞ ✇✐t❤ t❤❡ ❝♦♥❞✐t✐♦♥s K(0) = ∞ dµ(τ) ≡ ❱❛r µ|∞

0 < +∞,

Q(0) = ∞ dη(τ) ≡ ❱❛r η|∞

0 < +∞.

✭✺✮ t❤❡♥ t❤❡ ❦❡r♥❡❧s K(t) ❛♥❞ Q(t) ❜❡❧♦♥❣ t♦ t❤❡ s♣❛❝❡ W 1

1 (R+)✳

❋✉rt❤❡r ✇❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞ inf

||x||=1, x∈Dom(A)

((A + B)x, x) > 1. ✭✻✮

✺ ✴ ✹✽

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SLIDE 6

❚❤❡ ❡q✉❛t✐♦♥ ✭✶✮ ❝❛♥ ❜❡ r❡❣❛r❞❡❞ ❛s ❛♥ ❛❜str❛❝t ❢♦r♠ ♦❢ ❞②♥❛♠✐❝❛❧ ✈✐s❝♦❡❧❛st✐❝ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✇❤❡r❡ ♦♣❡r❛t♦rs A ❛♥❞ B ❛r❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐✛❡r❡♥t✐❛❧ ❡①♣r❡ss✐♦♥s A = −ρ−1µ (∆u + (1/3)❣r❛❞(❞✐✈u)) , B = −ρ−1λ(1/3)❣r❛❞(❞✐✈u), ❤❡r❡ u = u(x, t) ∈ R3 ✐s ❞✐s♣❧❛❝❡♠❡♥t ✈❡❝t♦r ♦❢ ✈✐s❝♦❡❧❛st✐❝ ❤❡r❡❞✐t❛r② ✐s♦tr♦♣✐❝ ♠❡❞✐❛ t❤❛t ✜❧❧ t❤❡ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ Ω ⊂ R3 ✇✐t❤ s♠♦♦t❤ ❜♦✉♥❞❛r②✱ ∂Ω✱ ρ ✐s ❛ ❝♦♥st❛♥t ❞❡♥s✐t②✱ ρ > 0✱ ▲❛♠❡ ♣❛r❛♠❡t❡rs λ, µ ❛r❡ t❤❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts✱ K(t)✱ Q(t) ❛r❡ t❤❡ r❡❧❛①❛t✐♦♥ ❢✉♥❝t✐♦♥s ❝❤❛r❛❝t❡r✐③✐♥❣ ❤❡r❡❞✐t❛r② ♣r♦♣❡rt✐❡s ♦❢ ♠❡❞✐❛✳ ❖♥ t❤❡ ❞♦♠❛✐♥ ❜♦✉♥❞❛r② ∂Ω t❤❡ ❉✐r✐❝❤❧❡t ❝♦♥❞✐t✐♦♥ u|∂Ω = 0. ✭✼✮ ✐s s❛t✐s✜❡❞✳ ❚❤❡ ❍✐❧❜❡rt s♣❛❝❡ H ❝❛♥ ❜❡ r❡❛❧✐③❡❞ ❛s t❤❡ s♣❛❝❡ ♦❢ t❤r❡❡ ❞✐♠❡♥s✐♦♥❛❧ ✈❡❝t♦r✲❢✉♥❝t✐♦♥s L2(Ω)✳ ❚❤❡ ❞♦♠❛✐♥ Dom(A) ❜❡❧♦♥❣s t♦ t❤❡ ❙♦❜♦❧❡✈ s♣❛❝❡ W 2

2 (Ω) ♦❢ ✈❡❝t♦r ❢✉♥❝t✐♦♥s s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✼✮✳ ❙❡❡

❆✳❆✳ ■❧②✉s❤✐♥✱ ❇✳❊✳ P♦❜❡❞r②❛ ❇❛s❡s ♦❢ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ t❤❡♦r② ♦❢ t❤❡r♠♦✈✐s❝♦❡❧❛st✐❝✐t②✳ ✲ ▼✳ ◆❛✉❦❛✱ ✶✾✼✵✱ ❘✳▼✳ ❈❤r✐st❡♥s❡♥ ❚❤❡♦r② ♦❢ ✈✐s❝♦❡❧❛st✐❝✐t②✳ ❆♥ ✐♥tr♦❞✉❝t✐♦♥✳ ✲ ❆❝❛❞❡♠✐❝ Pr❡ss ◆❡✇ ❨♦r❦ ❛♥❞ ▲♦♥❞♦♥✱ ✶✾✼✶

✻ ✴ ✹✽

slide-7
SLIDE 7

■♥ ❝❛s❡ ♦♣❡r❛t♦r B = 0✱ ♣♦s✐t✐✈❡ ❛♥❞ s❡❧❢✲❛❞❥♦✐♥t ♦♣❡r❛t♦r A ❝❛♥ ❜❡ r❡❛❧✐③❡❞ ❛s ♦♣❡r❛t♦r Ay = −y′′(x)✱ ✇❤❡r❡ x ∈ (0, π)✱ y(0) = y(π) = 0✱ ♦r t❤❡ ♦♣❡r❛t♦r Ay = −∆y ✇✐t❤ ❉✐r✐❝❤❧❡t ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ Q ⊂ Rn ✇✐t❤ s♠♦♦t❤ ❜♦✉♥❞❛r② ✭H = L2(Q)✮ ♦r ♠♦r❡ ❣❡♥❡r❛❧ ❡❧❧✐♣t✐❝ s❡❧❢✲❛❞❥♦✐♥t ♦♣❡r❛t♦rs ✐♥ t❤❡ s♣❛❝❡ L2(Q)✳ ❚❤❡ ❡q✉❛t✐♦♥ ✭✶✮ ❝❛♥ ❜❡ r❡❣❛r❞❡❞ ❛s ❛♥ ❛❜str❛❝t ❢♦r♠ ♦❢ t❤❡ ●✉rt✐♥✲P✐♣❦✐♥ ❡q✉❛t✐♦♥ t❤❛t ❞❡s❝r✐❜❡s ❤❡❛t tr❛♥s❢❡r ✐♥ ♠❛t❡r✐❛❧s ✇✐t❤ ♠❡♠♦r② ✇✐t❤ ✜♥✐t❡ s♣❡❡❞✳ ❙❡❡

  • ✉rt✐♥ ▼✳ ❊✳✱ P✐♣❦✐♥ ❆✳ ❈✳ ●❡♥❡r❛❧ t❤❡♦r② ♦❢ ❤❡❛t ❝♦♥❞✉❝t✐♦♥ ✇✐t❤ ✜♥✐t❡ ✇❛✈❡

s♣❡❡❞✳ ✴✴ ❆r❝❤✐✈❡ ❢♦r ❘❛t✐♦♥❛❧ ▼❡❝❤❛♥✐❝s ❛♥❞ ❆♥❛❧②s✐s✳ ✖ ✶✾✻✽✳ ✖ ✸✶✳ ✖ P✳ ✶✶✸✕✶✷✻✳

✼ ✴ ✹✽

slide-8
SLIDE 8

❆♣♣❧②✐♥❣ t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ t♦ t❤❡ ❡q✉❛t✐♦♥ ✭✶✮ ✇✐t❤ ③❡r♦ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣❡r❛t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ) = λ2I + A + B − ˆ K(λ)A − ˆ Q(λ)B, ✭✽✮ ✇❤✐❝❤ ❛r❡ t❤❡ s②♠❜♦❧ ✭❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ q✉❛s✐✲♣♦❧②♥♦♠✐❛❧✮ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭✶✮✳ ❍❡r❡ ˆ K(λ) ❛♥❞ ˆ Q(λ) ❛r❡ t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠s ♦❢ ❦❡r♥❡❧s K(t) ❛♥❞ Q(t) r❡s♣❡❝t✐✈❡❧②✱ ❤❛✈✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥s ˆ K(λ) = ∞ dµ(τ) λ + τ , ˆ Q(λ) = ∞ dν(τ) λ + τ , ✭✾✮

❉❡✜♥✐t✐♦♥

❚❤❡ s❡t ♦❢ ✈❛❧✉❡s λ ∈ C ✐s ❝❛❧❧❡❞ t❤❡ r❡s♦❧✈❡♥t s❡t R(L) ♦❢ ♦♣❡r❛t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ) ✐❢ t❤❡r❡ ❡①✐sts L−1(λ) ✐s ❜♦✉♥❞❡❞ ❢♦r ❛♥② λ ∈ R(L)✳ ❚❤❡ s❡t σ(L) = {λ ∈ C\R(L) | L(λ) ❡①✐sts} ✐s ❝❛❧❧❡❞ t❤❡ s♣❡❝tr❛ ♦❢ ♦♣❡r❛t♦r✲✈❛❧✉❡❞ ❢✉❝♥t✐♦♥ L(λ)✳

✽ ✴ ✹✽

slide-9
SLIDE 9

❉❡♥♦t❡ ❜② A0 := A+B✳ ■t ✐s ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ♦♣❡r❛t♦rs A ❛♥❞ B t❤❛t t❤❡ ♦♣❡r❛t♦r A0 ✐s ♣♦s✐t✐✈❡ ❛♥❞ s❡❧❢✲❛❞❥♦✐♥t✳ ▼♦r❡♦✈❡r A0 ✐s r❡✈❡rs✐❜❧❡✱ ♦♣❡r❛t♦rs AA−1

0 ✱ BA−1

❛r❡ ❜♦✉♥❞❡❞ ❛♥❞ ♦♣❡r❛t♦r A−1 ✐s ❝♦♠♣❛❝t ✭s❡❡ ♠♦♥♦❣r❛♣❤ ❚✳ ❑❛t♦ P❡rt✉r❜❛t✐♦♥ ❚❤❡♦r② ❢♦r ▲✐♥❡❛r ❖♣❡r❛t♦rs✴✴ ❙♣r✐♥❣❡r✲ ❱❡r❧❛❣ ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣ ◆❡✇ ❨♦r❦✱ ✶✾✽✵✮✳ ▲❡t ✉s ❞❡♥♦t❡ ❜② W n

2,γ (R+, A0) t❤❡ ❙♦❜♦❧❡✈ s♣❛❝❡ ♦❢ t❤❡ ✈❡❝t♦r✲✈❛❧✉❡❞

❢✉♥❝t✐♦♥s ♦♥ t❤❡ ♣♦s✐t✐✈❡ s❡♠✐❛①✐s R+ = (0, ∞) ✇✐t❤ t❤❡ ✈❛❧✉❡s ✐♥ t❤❡ s♣❛❝❡ H ❡q✉✐♣❡❞ ❜② t❤❡ ♥♦r♠ uW n

2,γ(R+,A0) ≡

∞ e−2γt

  • u(n)(t)
  • 2

H + A0u(t)2 H

  • dt

1/2 , γ ≥ 0. ❋♦r ♠♦r❡ ❞❡t❛✐❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ s♣❛❝❡ W n

2,γ (R+, A0) s❡❡ t❤❡ ♠♦♥♦❣r❛♣❤

❏✳ ▲✳ ▲✐♦♥s ❛♥❞ ❊✳ ▼❛❣❡♥❡s ◆♦♥❤♦♠♦❣❡♥❡♦✉s ❇♦✉♥❞❛r②✲❱❛❧✉❡ Pr♦❜❧❡♠s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ✴✴ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✲❍❡✐❞❡❧❜❡r❣✲◆❡✇ ❨♦r❦✳ ✶✾✼✷✱ ❝❤❛♣t❡r ✶✳ ❋♦r n = 0 ✇❡ ❤❛✈❡ W 0

2,γ (R+, A0) ≡ L2,γ (R+, H)✱ ❛♥❞ ❢♦r γ = 0 ✇❡ s❤❛❧❧

✇r✐t❡ W n

2,0 = W n 2 ✳

✾ ✴ ✹✽

slide-10
SLIDE 10

❈♦rr❡❝t s♦❧✈❛❜✐❧✐t②

❲❡ ❡st❛❜❧✐s❤ ✇❡❧❧✲❞❡✜♥❡❞ s♦❧✈❛❜✐❧✐t② ♦❢ ✐♥✐t✐❛❧ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠ ✭✶✮✱ ✭✷✮ ✐♥ ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡s ♦♥ t❤❡ ♣♦s✐t✐✈❡ s❡♠✐✲❛①✐s ❛♥❞ ❡①❛♠✐♥❡ t❤❡ s♣❡❝tr❛ ❧♦❝❛❧✐③❛t✐♦♥ ♦❢ ♦♣❡r❛t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s L(λ) r❡♣r❡s❡♥t✐♥❣ s②♠❜♦❧ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭✶✮✳

❉❡✜♥✐t✐♦♥

❱❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ u ✐s ❝❛❧❧❡❞ t❤❡ str♦♥❣ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✮✱ ✭✷✮✱ ✐❢ ✐t ❜❡❧♦♥❣s t♦ t❤❡ s♣❛❝❡ W 2

2,γ(R+, A0) ❢♦r s♦♠❡ γ 0✱ s❛t✐s✜❡s t❤❡

❡q✉❛t✐♦♥ ✭✶✮ ❛❧♠♦st ❡✈❡r②✇❤❡r❡ ♦♥ t❤❡ s❡♠✐❛①✐s R+✱ ❛♥❞ ❛❧s♦ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✭✷✮✳ ▲❡t ✉s ❝♦♥✈❡rt t❤❡ ❞♦♠❛✐♥ Dom(Aβ

0) ♦❢ t❤❡ ♦♣❡r❛t♦r Aβ 0✱ ✭β > 0✮ ✐♥t♦ t❤❡

❍✐❧❜❡rt s♣❛❝❡ Hβ✱ ❜② ✐♥tr♦❞✉❝✐♥❣ t❤❡ ♥♦r♠ · β = Aβ

0 · ♦♥ t❤❡ s♣❛❝❡

Dom(Aβ

0) ✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t t❤❡ ❣r❛♣❤ ♥♦r♠ ♦❢ t❤❡ ♦♣❡r❛t♦r Aβ 0✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠ ♣r❡s❡♥t t❤❡ r❡s✉❧t ♦♥ t❤❡ ❝♦rr❡❝t s♦❧✉❜✐❧✐t② ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✮✱ ✭✷✮✳

✶✵ ✴ ✹✽

slide-11
SLIDE 11

❍❡r❡ ✇❡ ✉s❡ t❤❡ ❛♣♣r♦❛❝❤ s✐♠✐❧❛r t♦ t❤❡ ❛♣♣r♦❛❝❤ ♦❢ ▼✳❙✳ ❆❣r❛♥♦✈✐❝❤ ❛♥❞ ▼✳■✳ ❱✐s❤✐❦ ✐♥ t❤❡✐r ❢❛♠♦✉s ❛rt✐❝❧❡ ❊❧❧✐♣t✐❝ ♣r♦❜❧❡♠s ✇✐t❤ ❛ ♣❛r❛♠❡t❡r ❛♥❞ ♣❛r❛❜♦❧✐❝ ♣r♦❜❧❡♠s ♦❢ ❣❡♥❡r❛❧ t②♣❡✳✴✴ ✭❘✉ss✐❛♥✮ ❯s♣❡❤✐ ▼❛t✳ ◆❛✉❦ ✶✾✱ ✶✾✻✹ ✳ ✸ ✭✶✶✼✮✱ ✺✸â❶✏✶✻✶✳

❚❤❡♦r❡♠ ✭✶✮

❙✉♣♣♦s❡ t❤❛t f(1)(t) ∈ L2,γ0 (R+, H) ❢♦r s♦♠❡ γ0 0 ❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥s ✭✹✮✱ ✭✺✮ ❛r❡ s❛t✐s✜❡❞✱ ♠♦r❡♦✈❡r ϕ0 ∈ H1✱ ϕ1 ∈ H1/2✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts s✉❝❤ γ1 ≥ γ0 t❤❛t t❤❡ ♣r♦❜❧❡♠ ✭✶✮✱ ✭✷✮ ❤❛s t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ✐♥ t❤❡ s♣❛❝❡ W 2

2,γ (R+, A0) ❢♦r ❛r❜✐tr❛r② γ > γ1✳ ▼♦r❡♦✈❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡ ✐s ✈❛❧✐❞

uW 2

2,γ(R+,A0) ≤ d

  • f(1)(t)
  • L2,γ(R+,H) + A0ϕ0H +
  • A1/2

ϕ1

  • H
  • ✭✶✵✮

✇✐t❤ ❛ ❝♦♥st❛♥t d t❤❛t ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ✈❡❝t♦r✲❢✉♥❝t✐♦♥ f ❛♥❞ ✈❡❝t♦rs ϕ0 ❛♥❞ ϕ1✳ ❚❤❡ r❡s✉❧ts ❛❜♦✉t ❝♦rr❡❝t s♦❧✈❛❜✐❧✐t② ♦❢ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♦❢ t❤❡ s❡❝♦♥❞ ♦❞❡r ✇✐t❤ ♦♣❡r❛t♦r ❝♦❡✣❝✐❡♥ts ✇❡r❡ ♦❜t❛✐♥❡❞ ❜② ▲✳ P❛♥❞♦❧✜✱ ❘✳ ▼✐❧❧❡r✱ ◆✳❉✳ ❑♦♣❛❝❤❡✈s❦②✱ ❉✳ ❩❛❦♦r❛✱ ❊✳ ❙②♦♠❦✐♥❛✳

✶✶ ✴ ✹✽

slide-12
SLIDE 12

❉❡✜♥✐t✐♦♥

❱❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ u ✐s ❝❛❧❧❡❞ t❤❡ ✇❡❛❦ ✭❣❡♥❡r❛❧✐③❡❞✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✮✱ ✭✷✮ ✐❢ ✐t ❜❡❧♦♥❣s t♦ t❤❡ s♣❛❝❡ W 1

2,γ(R+, A1/2

) ❢♦r s♦♠❡ γ 0✱ s❛t✐s✜❡s t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ u(+0) = ϕ0 ❛♥❞ ❛❧s♦ s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t② −

  • u′(t), v′(t)
  • L2,γ +
  • A1/2

u(t), A1/2 v(t)

  • L2,γ + 2γ
  • u′(t), v(t)
  • L2,γ−

− t K(t − s)A−1/2 Au(s)ds, A1/2 v(t)

  • L2,γ

− − t Q(t − s)A−1/2 Bu(s)ds, A−1/2 v(t)

  • L2,γ

− − f(t), v(t)L2,γ − (ϕ1, v(0)) = 0 ✭✶✶✮ ❢♦r ❡✈❡r② v(t) ∈ W 1

2,γ(R+, A1/2

)✱ s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥ lim

t→+∞ v(t)e−γt = 0.

✶✷ ✴ ✹✽

slide-13
SLIDE 13

❚❤❡♦r❡♠ ✭✷✮

❙✉♣♣♦s❡ t❤❛t f(t) ∈ L2,γ0 (R+, H) ❢♦r s♦♠❡ γ0 0 ❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥s ✭✹✮✱ ✭✺✮ ❛r❡ s❛t✐s✜❡❞✱ ♠♦r❡♦✈❡r ϕ0 ∈ H1/2✱ ϕ1 ∈ H✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts s✉❝❤ γ1 ≥ γ0 t❤❛t t❤❡ ♣r♦❜❧❡♠ ✭✶✮✱ ✭✷✮ ❤❛s t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ✐♥ t❤❡ s♣❛❝❡ W 1

2,γ

  • R+, A1/2
  • ❢♦r ❛r❜✐tr❛r② γ > γ1✳ ▼♦r❡♦✈❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡ ✐s

✈❛❧✐❞ uW 1

2,γ

  • R+,A1/2

≤ d

  • f(t)L2,γ(R+,H) +
  • A1/2

ϕ0

  • H + ϕ1H
  • ✭✶✷✮

✇✐t❤ ❛ ❝♦♥st❛♥t d t❤❛t ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ✈❡❝t♦r✲❢✉♥❝t✐♦♥ f ❛♥❞ ✈❡❝t♦rs ϕ0 ❛♥❞ ϕ1✳

✶✸ ✴ ✹✽

slide-14
SLIDE 14

❙♣❡❝tr❛❧ ❆♥❛❧②s✐s

❲❡ ❢♦r♠✉❧❛t❡ t❤❡ r❡s✉❧ts ❛❜♦✉t t❤❡ s♣❡❝tr✉♠ ❧♦❝❛❧✐③❛t✐♦♥ ♦❢ ♦♣❡r❛t♦r✲ ❢✉♥❝t✐♦♥ L(λ) ✇❤❡♥ t❤❡ ♠❡❛s✉r❡s dµ(τ)✱ dη(τ) ❤❛✈❡ ❝♦♠♣❛❝t s✉♣♣♦rts✳

✶✹ ✴ ✹✽

slide-15
SLIDE 15

❚❤❡♦r❡♠ ✭✸✱ ❚❤❡ ♠❛✐♥ s♣❡❝tr❛❧ t❤❡♦r❡♠✮

❙✉♣♣♦s❡ t❤❛t ❝♦♥❞✐t✐♦♥s ✭✹✮✱ ✭✺✮ ❤♦❧❞s ❛♥❞ t❤❡ s✉♣♣♦rts ♦❢ ♠❡❛s✉r❡s dµ(τ)✱ dη(τ) ❜❡❧♦♥❣ t♦ t❤❡ s❡❣♠❡♥t [d1, d2]✱ 0 < d1 < d2 < +∞✳ ❚❤❡♥ ❢♦r ❛♥② ❛r❜✐tr❛r② s♠❛❧❧ ♥✉♠❜❡r θ0 > 0 t❤❡r❡ ❡①✐sts s✉❝❤ ♥✉♠❜❡r R0 > 0 t❤❛t s♣❡❝tr✉♠ ♦❢ ♦♣❡r❛t♦r✲❢✉♥❝t✐♦♥ L(λ) ❜❡❧♦♥❣s t♦ t❤❡ s❡t Ω := {λ ∈ C : Re λ < 0, |λ| < R0} ∪ {λ ∈ C : α1 Re λ α2} ✇❤❡r❡ α1 = α0 − θ0✱ R0 max(d2, −α0 + θ0)✱ α0 = −1 2 sup

||f||=1

((K(0)A + Q(0)B) f, f) ((A + B)f, f) , f ∈ D(A), α2 = −1 2 inf

||f||=1

((K(0)A + Q(0)B) f, f)

  • (A + B + d2

2I)f, f

  • ,

f ∈ D(A). ▼♦r❡♦✈❡r✱ t❤❡r❡ ❡①✐sts s✉❝❤ ♥✉♠❜❡r γ0 > 0 t❤❛t ❢♦r ❛♥② λ ∈ {λ ∈ C : Re λ < −R0} ∪ {λ ∈ C : Re λ > γ0} t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡ ♦♣❡r❛t♦r✲❢✉♥❝t✐♦♥ L−1(λ) ✐s ✈❛❧✐❞ L−1(λ)

const |λ|| Re λ|.

✶✺ ✴ ✹✽

slide-16
SLIDE 16

❘❡♠❛r❦

❚❤❡ q✉❛♥t✐t② α0 ✐♥ t❤❡ st❛t❡♠❡♥t ♦❢ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❛s α0 −1 2

  • A−1/2

(K(0)A + Q(0)B) A−1/2

  • .

❚❤❡♦r❡♠ ✭✹✮

▲❡t ✉s s✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ❤♦❧❞✳ ❚❤❡♥ t❤❡ ♥♦♥r❡❛❧ s♣❡❝tr✉♠ ♦❢ t❤❡ ♦♣❡r❛t♦r✲❢✉♥❝t✐♦♥ L(λ) ✐s s②♠♠❡tr✐❝ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ r❡❛❧ ❛①✐s ❛♥❞ ❝♦♥s✐st ♦❢ ❡✐❣❡♥✈❛❧✉❡s ♦❢ ✜♥✐t❡ ❛❧❣❡❜r❛✐❝ ♠✉❧t✐♣❧✐❝✐t②✱ ♠♦r❡♦✈❡r ❢♦r ❛♥② ε > 0 ✐♥ t❤❡ ❞♦♠❛✐♥ Ωε := Ω\ {λ ∈ C : −d2 − ε < Re λ < 0, | Im λ| < ε} ❡✐❣❡♥✈❛❧✉❡s ✐s ✐s♦❧❛t❡❞ ✐✳❡✳✱ ❤❛✈❡ ♥♦ ♣♦✐♥ts ♦❢ ❛❝❝✉♠✉❧❛t✐♦♥✳

✶✻ ✴ ✹✽

slide-17
SLIDE 17

❙❡❡ ❢♦r ♠♦r❡ ❞❡t❛✐❧ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛rt✐❝❧❡ ❛♥❞ ♠♦♥♦❣r❛♣❤✿ ✶✮ ❱❧❛s♦✈ ❱✳ ❱✳✱ ❘❛✉t✐❛♥ ◆✳ ❆✳✱ ❙♣❡❝tr❛❧ ❆♥❛❧②s✐s ♦❢ ▲✐♥❡❛r ▼♦❞❡❧s ♦❢ ❱✐s❝♦❡❧❛st✐❝✐t②✴✴ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❙❝✐❡♥❝❡✱ ✷✵✶✼✱ ❱✳ ✷✸✵✿✺✱ ♣♣✳ ✻✻✽✕ ✻✼✷✳ ✷✮ ✶✮ ❱❧❛s♦✈✱ ❱✳ ❱✳✱ ❘❛✉t✐❛♥✱ ❲❡❧❧✲♣♦s❡❞ s♦❧✈❛❜✐❧✐t② ♦❢ ✈♦❧t❡rr❛ ✐♥t❡❣r♦✲ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✐♥ ❍✐❧❜❡rt s♣❛❝❡✴✴ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ✷✵✶✻✱ ❱✳ ✺✷✿✾✱ ♣♣✳ ✶✶✷✸â❶✏✲✶✶✸✷❀ ✸✮ ❱❧❛s♦✈ ❱✳ ❱✳✱ ❘❛✉t✐❛♥ ◆✳ ❆✳✱ ❙♣❡❝tr❛❧ ❛♥❛❧②s✐s ♦❢ ❢✉♥❝t✐♦♥❛❧✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ✕ ▼✳ ▼❆❑❙ Pr❡ss✱ ✷✵✶✻✱ ✕ ✹✽✽ ♣✳

✶✼ ✴ ✹✽

slide-18
SLIDE 18

■♥ ♦✉r ♣r❡✈✐♦✉s ✇♦r❦s ✶✮ ❱✳ ❱✳ ❱❧❛s♦✈✱ ◆✳ ❆✳ ❘❛✉t✐❛♥ ❙♣❡❝tr❛❧ ❆♥❛❧②s✐s ♦❢ ❍②♣❡r❜♦❧✐❝ ❱♦❧t❡rr❛ ■♥t❡❣r♦✲❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✴✴ ❉♦❦❧❛❞② ▼❛t❤❡♠❛t✐❝s✱ ✷✵✶✺✱ ❱✳✾✷✿✷✱ ♣♣✳✺✾✵✕✺✾✸✳ ✷✮ ❱✳ ❱✳ ❱❧❛s♦✈✱ ◆✳ ❆✳ ❘❛✉t✐❛♥✱ ❆✳ ❙✳ Pr♦♣❡rt✐❡s ♦❢ s♦❧✉t✐♦♥s ♦❢ ✐♥t❡❣r♦✲ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛r✐s✐♥❣ ✐♥ ❤❡❛t ❛♥❞ ♠❛ss tr❛♥s❢❡r t❤❡♦r② ✴✴ ❚r❛♥s✳ ▼♦s❝♦✇ ▼❛t❤✳ ❙♦❝✳✱ ✷✵✶✹✱ ❱✳✼✺✱ P✳ ✶✽✺✕✷✵✹✳ ✸✮ ❱✳ ❱✳ ❱❧❛s♦✈✱ ◆✳ ❆✳ ❘❛✉t✐❛♥ ❙♣❡❝tr❛❧ ❆♥❛❧②s✐s ❛♥❞ ❘❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❙♦❧✉t✐♦♥s ♦❢ ❆❜str❛❝t ■♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✐♥ ❍✐❧❜❡rt ❙♣❛❝❡ ✴✴ ❖♣❡r❛t♦r ❚❤❡♦r②✿ ❆❞✈❛♥❝❡s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✳ ❙♣r✐♥❣❡r ❇❛s❡❧ ❆●✱ ✷✵✶✹✱ ❱✳✷✸✻✱ ♣♣✳ ✺✶✼✕✺✸✺✳ ✇❡ ❝♦♥s✐❞❡r❡❞ ✐♥ ❞❡t❛✐❧ t❤❡ ❝❛s❡ ✇❤❡♥ B = 0✳ ■♥ t❤✐s ❝❛s❡ t❤❡ ❡q✉❛t✐♦♥ ✭✶✮ ❤❛s t❤❡ ❛❜str❛❝t ❢♦r♠ ♦❢ ●✉rt✐♥✲P✐♣❦✐♥ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ t❤❛t ❞❡s❝r✐❜❡ ❤❡❛t tr❛♥s❢❡r ✐♥ ♠❛t❡r✐❛❧s ✇✐t❤ ♠❡♠♦r② ✇✐t❤ ✜♥✐t❡ s♣❡❡❞ ❛♥❞ ❤❛s ❛ ♥✉♠❜❡r ♦❢ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s✳

✶✽ ✴ ✹✽

slide-19
SLIDE 19

❙t❛t❡♠❡♥t ♦❢ t❤❡ ♣r♦❜❧❡♠ ✇✐t❤ ❢r❛❝t✐♦♥❛❧✲❡①♣♦♥❡♥t✐❛❧ ❦❡r♥❡❧

❲❡ ❝♦♥s✐❞❡r ♦♥ t❤❡ ♣♦s✐t✐✈❡ s❡♠✐❛①✐s R+ = (0, ∞) t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥✐t✐❛❧ ♣r♦❜❧❡♠ ❢♦r t❤❡ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ d2u(t) dt2 + A2u(t) − t K(t − s)A2u(s)ds = f(t), t ∈ R+, ✭✶✸✮ u(+0) = ϕ0, u(1)(+0) = ϕ1, ✭✶✹✮ ✇❤❡r❡ A ✐s ❛ ♣♦s✐t✐✈❡ s❡❧❢✲❛❞❥♦✐♥t ♦♣❡r❛t♦r ❛❝t✐♥❣ ✐♥ t❤❡ s❡♣❛r❛❜❧❡ ❍✐❧❜❡rt s♣❛❝❡ H✱ ❤❛✈✐♥❣ t❤❡ ❜♦✉♥❞❡❞ ✐♥✈❡rs❡ ♦♣❡r❛t♦r✳ ❚❤❡ ♦♣❡r❛t♦r A ❝❛♥ ❜❡ r❡❛❧✐③❡❞ ❛s A2y = −y′′(x)✱ ✇❤❡r❡ x ∈ (0, π)✱ y(0) = y(π) = 0✱ ♦r A2y = −∆y ✇✐t❤ ❉✐r✐❝❤❧❡t ❝♦♥❞✐t✐♦♥s ✐♥ t❤❡ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ Q ⊂ Rn ✇✐t❤ s♠♦♦t❤ ❜♦✉♥❞❛r② ✭H = L2(Q)✮ ♦r A2y = −∆y − 1/3 · ❣r❛❞(❞✐✈y)✱ y = y(x) ∈ R3 ✇✐t❤ ❉✐r✐❝❤❧❡t ❝♦♥❞✐t✐♦♥s ✐♥ t❤❡ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ Ω ⊂ R3 ✇✐t❤ s♠♦♦t❤ ❜♦✉♥❞❛r② ✭H = L2(Ω)✮✳

✶✾ ✴ ✹✽

slide-20
SLIDE 20

❑❡r♥❡❧s ♦❢ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥s ❛r❡ ✇✐❞❡❧② ✉s❡❞ ✐♥ t❤❡ ♣r♦❜❧❡♠s ♦❢ ❤❡❛t ♣r♦♣❛❣❛t✐♦♥ ✐♥ ♠❡❞✐❛ ✇✐t❤ ♠❡♠♦r② ❛♥❞ ✈✐s❝♦❡❧❛st✐❝✐t②✿ K (t) =

∞

  • j=1

cjRj (t) , ✭✶✺✮ ✇❤❡r❡ cj > 0✱ j ∈ N✱ ❢✉♥❝t✐♦♥s Rj (t) ❛r❡ ❞❡✜♥❡❞ ❜② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②s ■✮ Rj (t) = e−βjt ✇❤❡r❡ βj+1 > βj > 0✱ j ∈ N✱ βj → +∞ (j → +∞)✳ ■■✮ Rj(t) = tα−1

∞

  • n=0

(−βj)ntnα Γ[(n + 1)α], ✕ ❢r❛❝t✐♦♥❛❧✲❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s✱ ✇❤❡r❡ 0 < α < 1, Γ(·) ✕ ❣❛♠♠❛✲❢✉♥❝t✐♦♥✱ 0 < βj < βj+1,✱ j ∈ N✱ βj → +∞✱ j → +∞✳

✷✵ ✴ ✹✽

slide-21
SLIDE 21

K(t) = ∞ e−tτdµ(τ), ✭✶✻✮ ✇❤❡r❡ dµ ✐s ❛ ♣♦s✐t✐✈❡ ♠❡❛s✉r❡✳ ❲❡ ✐❞❡♥t✐❢② t❤✐s ♠❡❛s✉r❡ ✇✐t❤ ✐ts ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ µ✱ s♦ µ ✐s ✐♥❝r❡❛s✐♥❣✱ ❝♦♥t✐♥✉♦✉s ❢r♦♠ t❤❡ r✐❣❤t✱ ❛♥❞ t❤❡ ✐♥t❡❣r❛❧ ✐s ✐♥t❡r♣r❡t❡❞ ❛s ❛ ❙t✐❡❧t❥❡s ✐♥t❡❣r❛❧✳ ■♥ ♣❛rt✐❝✉❧❛r t❤❡ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥ ✭■■✮ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ Rj(t) = 1 2πi lim

R→+∞ γ+iR

  • γ−iR

eλtdλ λα + βj = sin πα π

+∞

  • e−tτdτ

τ α + 2βj cos πα + β2

j τ −α ,

✇❤❡r❡ λα (0 < α < 1) ✐s t❤❡ ♠❛✐♥ ❜r❛♥❝❤ ♦❢ ♠✉❧t✐✈❛❧✉❡ ❢✉♥❝t✐♦♥ f(λ) = λα✱ λ ∈ C ✇✐t❤ ❛ ❝✉t ♦♥ ♥❡❣❛t✐✈❡ r❡❛❧ s❡♠✐❛①✐s t❤❛t ✐s λα = |λ|αeiα arg λ, −π < arg λ < π.

✷✶ ✴ ✹✽

slide-22
SLIDE 22

❚❤❡ r❡s✉❧ts ❛❜♦✉t ❝♦rr❡❝t s♦❧✈❛❜✐❧✐t②

▲❡t ✉s ❞❡♥♦t❡ ❜② W n

2,γ (R+, An) t❤❡ ❙♦❜♦❧❡✈ s♣❛❝❡ ♦❢ t❤❡ ✈❡❝t♦r✲✈❛❧✉❡❞

❢✉♥❝t✐♦♥s ♦♥ t❤❡ ♣♦s✐t✐✈❡ s❡♠✐❛①✐s R+ = (0, ∞) ✇✐t❤ t❤❡ ✈❛❧✉❡s ✐♥ t❤❡ s♣❛❝❡ H ❡q✉✐♣❡❞ ❜② t❤❡ ♥♦r♠ uW n

2,γ(R+,An) ≡

∞ e−2γt

  • u(n)(t)
  • 2

H + Anu(t)2 H

  • dt

1/2 , γ ≥ 0. ❋♦r ♠♦r❡ ❞❡t❛✐❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ s♣❛❝❡ W n

2,γ (R+, An) s❡❡ t❤❡ ♠♦♥♦❣r❛♣❤

❏✳ ▲✳ ▲✐♦♥s ❛♥❞ ❊✳ ▼❛❣❡♥❡s ◆♦♥❤♦♠♦❣❡♥❡♦✉s ❇♦✉♥❞❛r②✲❱❛❧✉❡ Pr♦❜❧❡♠s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ✴✴ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✲❍❡✐❞❡❧❜❡r❣✲◆❡✇ ❨♦r❦✳ ✶✾✼✷✱ ❝❤❛♣t❡r ✶✳ ❋♦r n = 0 ✇❡ ❤❛✈❡ W 0

2,γ

  • R+, A0

≡ L2,γ (R+, H)✱ ❛♥❞ ❢♦r γ = 0 ✇❡ s❤❛❧❧ ✇r✐t❡ W n

2,0 = W n 2 ✳

✷✷ ✴ ✹✽

slide-23
SLIDE 23

❉❡✜♥✐t✐♦♥

❱❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ u ✐s ❝❛❧❧❡❞ t❤❡ str♦♥❣ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✸✮✱ ✭✶✹✮ ✐❢ ✐t ❜❡❧♦♥❣s t♦ t❤❡ s♣❛❝❡ W 2

2,γ(R+, A2) ❢♦r s♦♠❡ γ 0✱ s❛t✐s✜❡s t❤❡

❡q✉❛t✐♦♥ ✭✶✸✮ ❛❧♠♦st ❡✈❡r②✇❤❡r❡ ♦♥ t❤❡ s❡♠✐❛①✐s R+✱ ❛♥❞ ❛❧s♦ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✭✶✹✮✳

❉❡✜♥✐t✐♦♥

❱❡❝t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ u ✐s ❝❛❧❧❡❞ t❤❡ ✇❡❛❦ ✭❣❡♥❡r❛❧✐③❡❞✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✸✮✱ ✭✶✹✮ ✐❢ ✐t ❜❡❧♦♥❣s t♦ t❤❡ s♣❛❝❡ W 1

2,γ(R+, A) ❢♦r s♦♠❡ γ 0✱

s❛t✐s✜❡s t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ u(+0) = ϕ0 ❛♥❞ ❛❧s♦ s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t② A[u(t) − t K(t − s)u(s)ds], Av(t)L2,γ(R+,H)− −

  • u′(t), v′(t)
  • L2,γ(R+,H) + 2γ
  • u′(t), v(t)
  • L2,γ(R+,H) =

= f(t), v(t)L2,γ(R+,H) + (ϕ1, v(0))H ❢♦r ❡✈❡r② v(t) ∈ W 1

2,γ(R+, A)✱ s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥

lim

t→+∞ v(t)e−γt = 0.

✷✸ ✴ ✹✽

slide-24
SLIDE 24

▲❡t ✉s ❝♦♥✈❡rt t❤❡ ❞♦♠❛✐♥ Dom(Aβ) ♦❢ t❤❡ ♦♣❡r❛t♦r Aβ✱ ✭β > 0✮ ✐♥t♦ t❤❡ ❍✐❧❜❡rt s♣❛❝❡ Hβ✱ ❜② ✐♥tr♦❞✉❝✐♥❣ t❤❡ ♥♦r♠ · β = Aβ · ♦♥ t❤❡ s♣❛❝❡ Dom(Aβ) ✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t t❤❡ ❣r❛♣❤ ♥♦r♠ ♦❢ t❤❡ ♦♣❡r❛t♦r Aβ✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠s ♣r❡s❡♥t t❤❡ r❡s✉❧ts ♦♥ t❤❡ ❝♦rr❡❝t s♦❧✈❛❜✐❧✐t② ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✸✮✱ ✭✶✹✮✳ ❉❡♥♦t❡ ❜② d t❤❡ ❝♦♥st❛♥t ✐♥❞❡♣❡♥❞❡♥t ♦♥ ✈❡❝t♦r✲❢✉♥❝t✐♦♥ f ❛♥❞ ✈❡❝t♦rs ϕ0 ❛♥❞ ϕ1✳

✷✹ ✴ ✹✽

slide-25
SLIDE 25

■♥ t❤❡ ✜rst ❝❛s❡ t❤❡ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥ K(t) ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠ K(t) =

∞

  • j=1

cje−βjt, ✭✶✼✮ ✇❤❡r❡ t❤❡ ❝♦❡✣❝✐❡♥ts cj > 0✱ βj+1 > βj > 0✱ j ∈ N✱ βj → +∞ (j → +∞) ❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥

∞

  • j=1

cj βj < 1. ✭✶✽✮ ✐s s❛t✐s✜❡❞✳ ❆❧♦♥❣ ✇✐t❤ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✶✽✮ ✇❡ s❤❛❧❧ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ K(0) =

∞

  • j=1

cj < +∞. ✭✶✾✮

✷✺ ✴ ✹✽

slide-26
SLIDE 26

❚❤❡♦r❡♠ ✭✺✮

❙✉♣♣♦s❡ t❤❛t Af(t) ∈ L2,γ2 (R+, H) ❢♦r s♦♠❡ γ2 0 ❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✶✽✮ ✐s s❛t✐s✜❡❞✳ ❚❤❡♥ ✶✮ ✐❢ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✶✾✮ ✐s s❛t✐s✜❡❞ ❛♥❞ ϕ0 ∈ H2✱ ϕ1 ∈ H1✱ t❤❡♥ t❤❡ ♣r♦❜❧❡♠ ✭✶✸✮✱ ✭✶✹✮ ❤❛s t❤❡ ✉♥✐q✉❡ str♦♥❣ s♦❧✉t✐♦♥ ✐♥ t❤❡ s♣❛❝❡ W 2

2,γ

  • R+, A2

❢♦r ❛r❜✐tr❛r② γ > γ2✳ ▼♦r❡♦✈❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡ ✐s ✈❛❧✐❞ uW 2

2,γ(R+,A2) ≤ d

  • Af(t)L2,γ(R+,H) +
  • A2ϕ0
  • H + Aϕ1H
  • ;

✭✷✵✮ ✷✮ ✐❢ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✶✾✮ ✐s ♥♦t s❛t✐s✜❡❞ ❛♥❞ ϕ0 ∈ H3✱ ϕ1 ∈ H2✱ t❤❡♥ t❤❡ ♣r♦❜❧❡♠ ✭✶✸✮✱ ✭✶✹✮ ❤❛s t❤❡ ✉♥✐q✉❡ str♦♥❣ s♦❧✉t✐♦♥ ✐♥ t❤❡ s♣❛❝❡ W 2

2,γ

  • R+, A2

❢♦r ❛r❜✐tr❛r② γ > γ2✳ ▼♦r❡♦✈❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡ ✐s ✈❛❧✐❞ uW 2

2,γ(R+,A2) ≤ d

  • Af(t)L2,γ(R+,H) +
  • A3ϕ0
  • H +
  • A2ϕ1
  • H
  • . ✭✷✶✮

✷✻ ✴ ✹✽

slide-27
SLIDE 27

◆♦✇ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥ K(t) ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t②♣❡ K (t) =

∞

  • j=1

cjRj (t) , ✭✷✷✮ ✇❤❡r❡ cj > 0✱ j ∈ N✱ Rj(t) = tα−1

∞

  • n=0

(−βj)ntnα Γ[(n + 1)α], ✭✷✸✮ ✇❤❡r❡ 0 < α < 1, Γ(·) ✕ ❣❛♠♠❛✲❢✉♥❝t✐♦♥✱ 0 < βj < βj+1,✱ j ∈ N✱ βj → +∞✱ j → +∞✳

∞

  • j=1

cj βj < 1, ✭✷✹✮ ❆❧♦♥❣ ✇✐t❤ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✷✹✮ ✇❡ s❤❛❧❧ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥

∞

  • j=1

cj < +∞. ✭✷✺✮

✷✼ ✴ ✹✽

slide-28
SLIDE 28

❚❤❡♦r❡♠ ✭✻✮

❙✉♣♣♦s❡ t❤❛t Af (t) ∈ L2,γ0 (R+, H) ❢♦r s♦♠❡ γ0 > 0✱ t❤❡ ❦❡r♥❡❧ K (t) ❤❛s t❤❡ ❢♦r♠ ✭✷✷✮✱ ✭✷✸✮ ✇✐t❤ ❝♦♥st❛♥t α (0 < α < 1)✱ ❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✷✹✮ ✐s s❛t✐s✜❡❞✳ ▼♦r❡♦✈❡r ϕ0 ∈ H3✱ ϕ1 ∈ H2✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts γ1 > γ0 t❤❛t t❤❡ ♣r♦❜❧❡♠ ✭✶✸✮✱ ✭✶✹✮ ❤❛s t❤❡ ✉♥✐q✉❡ str♦♥❣ s♦❧✉t✐♦♥ ✐♥ t❤❡ s♣❛❝❡ W 2

2,γ

  • R+, A2

❢♦r ❛r❜✐tr❛r② γ γ1 ❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡ ✐s ✈❛❧✐❞ uW 2

2,γ(R+,A2) d

  • AfL2,γ(R+,H) +
  • A3ϕ0
  • H +
  • A2ϕ1
  • H
  • .

❚❤❡♦r❡♠ ✭✼✮

❙✉♣♣♦s❡ t❤❛t f (t) ∈ L2,γ0 (R+, H) ❢♦r s♦♠❡ γ0 > 0✱ t❤❡ ❦❡r♥❡❧ K (t) ❤❛s t❤❡ ❢♦r♠ ✭✷✷✮✱ ✭✷✸✮ ✇✐t❤ ❝♦♥st❛♥t α (0 < α < 1)✱ ❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✷✹✮ ✐s s❛t✐s✜❡❞✳ ▼♦r❡♦✈❡r ϕ0 ∈ H2✱ ϕ1 ∈ H✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts γ1 > γ0 t❤❛t t❤❡ ♣r♦❜❧❡♠ ✭✶✸✮✱ ✭✶✹✮ ❤❛s t❤❡ ✉♥✐q✉❡ ✇❡❛❦ s♦❧✉t✐♦♥ ✐♥ t❤❡ s♣❛❝❡ W 1

2,γ (R+, A) ❢♦r ❛r❜✐tr❛r② γ γ1 ❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡ ✐s ✈❛❧✐❞

uW 1

2,γ(R+,A) d

  • fL2,γ(R+,H) +
  • A2ϕ0
  • H + Aϕ1H
  • .

✷✽ ✴ ✹✽

slide-29
SLIDE 29

❙♣❡❝tr❛❧ ❆♥❛❧②s✐s

❲❡ s❤❛❧❧ s✉♣♣♦s❡ ✐♥ ✇❤❛t ❢♦❧❧♦✇s t❤❛t ♦♣❡r❛t♦r A ❤❛✈❡ ❛ ❝♦♠♣❛❝t ✐♥✈❡rs❡✳ ▲❡t ✉s ❞❡♥♦t❡ ❜② {ej}∞

j=1 t❤❡ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ❝♦♥s✐st✐♥❣ ♦❢ ❡✐❣❡♥✈❡❝t♦rs

♦❢ ♦♣❡r❛t♦r A ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❡✐❣❡♥✈❛❧✉❡s aj✿ Aej = ajej✱ j ∈ N✳ ❚❤❡ ❡✐❣❡♥✈❛❧✉❡s aj ❛r❡ ♥✉♠❡r❛t❡❞ ✐♥ ✐♥❝r❡❛s✐♥❣ ♦r❞❡r 0 < a1 < a2 < ...❀ an → +∞ ❢♦r n → +∞✳

✷✾ ✴ ✹✽

slide-30
SLIDE 30

❆♣♣❧②✐♥❣ t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ t♦ t❤❡ ❡q✉❛t✐♦♥ ✭✶✸✮ ✇✐t❤ ③❡r♦ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣❡r❛t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L (λ) = λ2I + A2 − ˆ K(λ)A2, ✇❤✐❝❤ ❛r❡ t❤❡ s②♠❜♦❧ ✭❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ q✉❛s✐✲♣♦❧②♥♦♠✐❛❧✮ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭✶✸✮ ✇❤❡r❡ ˆ K(λ) t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ K(t)✱ I ✐s ✐❞❡♥t✐t② ♦♣❡r❛t♦r ❛❝t✐♥❣ ✐♥ t❤❡ s♣❛❝❡ H✳

❉❡✜♥✐t✐♦♥

❚❤❡ s❡t ♦❢ ✈❛❧✉❡s λ ∈ C ✐s ❝❛❧❧❡❞ t❤❡ r❡s♦❧✈❡♥t s❡t R(L) ♦❢ ♦♣❡r❛t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ) ✐❢ t❤❡r❡ ❡①✐sts L−1(λ) ✐s ❜♦✉♥❞❡❞ ❢♦r ❛♥② λ ∈ R(L)✳ ❚❤❡ s❡t σ(L) = {λ ∈ C\R(L) | L(λ) ❡①✐sts} ✐s ❝❛❧❧❡❞ t❤❡ s♣❡❝tr❛ ♦❢ ♦♣❡r❛t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ)✳

✸✵ ✴ ✹✽

slide-31
SLIDE 31

▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ♣r♦❥❡❝t✐♦♥ ln (λ) = (L (λ) en, en) = λ2 + a2

n

  • 1 − ˆ

K(λ)

  • .

♦❢ t❤❡ ♦♣❡r❛t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ) ♦♥ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ❢♦r♠❡❞ ❜② t❤❡ ✈❡❝t♦r en✱ ✇❤❡r❡ Aen = anen✱ n ∈ N✱ an → +∞ ❢♦r n → +∞✳ ❚❤✉s ✇❡ ♦❜t❛✐♥ t❤❡ ❝♦✉♥t❛❜❧❡ s❡t ♦❢ t❤❡ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s ln(λ)✱ n ∈ N✳ ❚❤❡♥ t❤❡ s♣❡❝tr✉♠ ♦❢ t❤❡ ♦♣❡r❛t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ) ✐s t❤❡ ❝❧♦s✉r❡ ♦❢ t❤❡ ③❡r♦❡s s❡t ♦❢ t❤❡ ❢✉♥❝t✐♦♥s {ln(λ)}∞

n=1✳

✸✶ ✴ ✹✽

slide-32
SLIDE 32

❲❡ s✉♣♣♦s❡ t❤❡ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥ K(t) ❤❛s t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭✶✼✮✳ ❚❤❡♥ ˆ K(λ) =

∞

  • j=1

cj λ + βj t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ K(t)✳

❚❤❡♦r❡♠ ✭✽✮

▲❡t ✉s s✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥s ✭✶✽✮ ❛♥❞ ✭✶✾✮ ❛r❡ s❛t✐s✜❡❞✳ ❚❤❡♥ t❤❡ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s λ±

n ✱ λ+ n = λ− n ♦❢ t❤❡ ✈❡❝t♦r ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ)

❛s②♠♣t♦t✐❝❛❧❧② r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦r♠ λ±

n = ±i

  • an + O

1 an

  • − 1

2K(0) + O 1 a2

n

  • , an → +∞.

✭✷✻✮

✸✷ ✴ ✹✽

slide-33
SLIDE 33

❚❤❡♦r❡♠ ✭✾✮

▲❡t ✉s s✉♣♣♦s❡ t❤❡ ❝♦♥❞✐t✐♦♥s ✭✶✽✮ ❛♥❞ ✭✶✾✮✳ ❚❤❡♥ t❤❡ s♣❡❝tr✉♠ ♦❢ t❤❡ ♦♣❡r❛t♦r ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ) ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❢♦❧❧♦✇s σ(L) := σR ∪ σI ✇❤❡r❡ σR ❛♥❞ σI ❛r❡ t❤❡ r❡❛❧ ❛♥❞ ♥♦♥r❡❛❧ ♣❛rt ♦❢ s♣❡❝tr✉♠ ♦❢ ♦♣❡r❛t♦r ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ)✱ ❝♦rr❡s♣♦♥❞❡♥t❧②✳ ▼♦r❡♦✈❡r t❤❡ r❡❛❧ ♣❛rt ♦❢ s♣❡❝tr✉♠ ♦❢ t❤❡ ♦♣❡r❛t♦r ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ) ✐s t❤❡ ❝❧♦s✉r❡ ♦❢ t❤❡ r❡❛❧ ③❡r♦❡s s❡t {λk,n|k ∈ N, n ∈ N} ♦❢ t❤❡ ❢✉♥❝t✐♦♥s {ln(λ)}∞

n=1 t❤❛t

s❛t✐s❢② t❤❡ ✐♥❡q✉❛❧✐t✐❡s ... − βk+1 < xk+1 < λk+1,n < −βk < ... < −β1 < λ1,n, k ∈ N, ✭✷✼✮ ✇❤❡r❡ xk ❛r❡ t❤❡ r❡❛❧ ③❡r♦❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ˆ K(λ)✱ ❛♥❞ λk,n = xk + O

  • 1/a2

n

  • .

σI =

  • λ±

n ∈ C\R, λ− n = λ+ n |n ∈ N

  • ,

✇❤❡r❡ λ±

n ✲ ♥♦♥r❡❛❧ ❡✐❣❡♥✈❛❧✉❡s ♦❢ ♦♣❡r❛t♦r ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ) ❤❛s t❤❡

r❡♣r❡s❡♥t❛t✐♦♥ ✭✷✻✮✳

✸✸ ✴ ✹✽

slide-34
SLIDE 34

◆♦✇ ✇❡ s✉♣♣♦s❡ t❤❡ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥ K(t) ❤❛s t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭✷✷✮✱ ✭✷✸✮✳ ❚❤❡♥ t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ K(t) ✐s ❢♦❧❧♦✇s ˆ K(λ) =

∞

  • j=1

cj λα + βj , 0 < α < 1. ❍❡r❡ λα (0 < α < 1) ✐s t❤❡ ♠❛✐♥ ❜r❛♥❝❤ ♦❢ ♠✉❧t✐✈❛❧✉❡ ❢✉♥❝t✐♦♥ f(λ) = λα✱ λ ∈ C ✇✐t❤ ❛ ❝✉t ♦♥ ♥❡❣❛t✐✈❡ r❡❛❧ s❡♠✐❛①✐s t❤❛t ✐s λα = |λ|αeiα arg λ✱ −π < arg λ < π.

❚❤❡♦r❡♠ ✭✶✵✮

❙✉♣♣♦s❡ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✷✹✮ ✐s s❛t✐s✜❡❞✳ ❚❤❡♥ t❤❡ s♣❡❝tr❛ ♦❢ ♦♣❡r❛t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ) ❧✐❡s ♦♥ t❤❡ ❧❡❢t ❝♦♠♣❧❡① ❤❛❧❢♣❧❛♥❡✳

✸✹ ✴ ✹✽

slide-35
SLIDE 35

❚❤❡♦r❡♠ ✭✶✶✮

▲❡t ✉s s✉♣♣♦s❡ t❤❡ ❝♦♥❞✐t✐♦♥s ✭✷✹✮ ❛♥❞ cj = 0 ❢♦r ❛r❜✐tr❛r② j ♠♦r❡ t❤❛♥ s♦♠❡ N ∈ N ❛r❡ s❛t✐s✜❡❞✳ ❚❤❡♥ t❤❡ s♣❡❝tr✉♠ ♦❢ t❤❡ ♦♣❡r❛t♦r ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ) ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❢♦❧❧♦✇s σ(L) :=

  • λ±

n ∈ C\R, λ− n = λ+ n |n ∈ N

  • ,

✇❤❡r❡ t❤❡ ❝♦♠♣❧❡① ❡✐❣❡♥✈❛❧✉❡s λ±

n ✱ λn+ = ¯

λ−

n ♦❢ t❤❡ ✈❡❝t♦r ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥

L(λ) ❛s②♠♣t♦t✐❝❛❧❧② r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦r♠ λ±

n = − sin

πα 2

  • a1−α

n

Q 2 ± ian

  • 1 − cos

πα 2

  • a−α

n

Q 2

  • + o
  • a1−α

n

  • ,

n → +∞, ✭✷✽✮ ✇❤❡r❡ Q =

N

  • j=1

cj✳

✸✺ ✴ ✹✽

slide-36
SLIDE 36

❘❡♠❛r❦

❋♦r α = 1 t❤❡ ❛s②♠♣t♦t✐❝ ❢♦r♠✉❧❛ ✭✷✽✮ ❝♦♠❡s t♦ ❛s②♠♣t♦t✐❝ ❢♦r♠✉❧❛ ✭✷✻✮ ✇❤❡r❡ t❤❡ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥ K(t) ❤❛s t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭✶✼✮✳

✸✻ ✴ ✹✽

slide-37
SLIDE 37

❙❡❡ ❢♦r ♠♦r❡ ❞❡t❛✐❧ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛rt✐❝❧❡s✿ ✶✮ ❱✳ ❱✳ ❱❧❛s♦✈✱ ◆✳ ❆✳ ❘❛✉t✐❛♥ ❲❡❧❧✲P♦s❡❞♥❡ss ❛♥❞ ❙♣❡❝tr❛❧ ❆♥❛❧②s✐s ♦❢ ❱♦❧t❡rr❛ ■♥t❡❣r♦✲❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✇✐t❤ ❙✐♥❣✉❧❛r ❑❡r♥❡❧s✴✴ ❉♦❦❧❛❞② ▼❛t❤❡♠❛t✐❝s✱ ✷✵✶✽✱ ❱✳✾✽✿✷✱ ♣♣✳✺✵✷✕✺✵✺✳ ✷✮ ❱❧❛s♦✈✱ ❱✳ ❱✳✱ ❘❛✉t✐❛♥✱ ❘❡s❡❛r❝❤ ♦❢ ♦♣❡r❛t♦r ♠♦❞❡❧s ❛r✐s✐♥❣ ✐♥ ✈✐s❝♦❡❧❛st✐❝✐t② t❤❡♦r②✳ ✭❘✉ss✐❛♥✮✴✴ ❙♦✈r❡♠✳ ▼❛t✳ ❋✉♥❞❛♠✳ ◆❛♣r❛✈❧✳✱ ✷✵✶✽✱ ❱✳ ✻✹✿✶✱ ♣♣✳ ✻✵✕✼✸✳

✸✼ ✴ ✹✽

slide-38
SLIDE 38

❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ s♦❧✉t✐♦♥s

❖♥ t❤❡ ❜❛s❡ ♦❢ t❤❡ s♣❡❝tr❛❧ t❤❡♦r❡♠s ✇❡ ♦❜t❛✐♥ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✸✮✱ ✭✶✹✮✳

✸✽ ✴ ✹✽

slide-39
SLIDE 39

▲❡t ✉s s✉♣♣♦s❡ t❤❡ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥ K(t) ❤❛s t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭✶✼✮✳

❚❤❡♦r❡♠ ✭✶✷✮

▲❡t ✉s s✉♣♣♦s❡ t❤❛t f(t) = 0 ❢♦r t ∈ R+✱ ✈❡❝t♦r✲❢✉♥❝t✐♦♥ u(t) ∈ W 2

2,γ

  • R+, A2

✱ γ > 0 ✐s ❛ str♦♥❣ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✸✮✱ ✭✶✹✮ ❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✶✽✮ ✐s s❛t✐s✜❡❞✳ ❚❤❡♥✱ ❢♦r ❛r❜✐tr❛r② t ∈ R+ t❤❡ s♦❧✉t✐♦♥ u(t) ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✸✮✱ ✭✶✹✮ ✐s r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡r✐❡s u(t) =

∞

  • n=1
  • ωn(t, λ+

n ) + ωn(t, λ− n ) + ∞

  • k=1

ωn(t, λkn)

  • en,

✭✷✾✮ t❤❛t ✐s ❝♦♥✈❡r❣❡♥t ❜② t❤❡ ♥♦r♠ ♦❢ t❤❡ s♣❛❝❡ H✱ ✇❤❡r❡ ωn(t, λ) = (ϕ1n + λϕ0n) eλt l(1)

n (λ)

.

✸✾ ✴ ✹✽

slide-40
SLIDE 40

❚❤❡♦r❡♠ ✭✶✸✮

▲❡t ✉s s✉♣♣♦s❡ ✈❡❝t♦r✲❢✉♥❝t✐♦♥ f(t) ∈ C ([0, T], H) ❢♦r ❛r❜✐tr❛r② T > 0✱ ✈❡❝t♦r✲❢✉♥❝t✐♦♥ u(t) ∈ W 2

2,γ

  • R+, A2

✱ γ > 0 ✐s ❛ str♦♥❣ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✸✮✱ ✭✶✹✮ ❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥s ✭✶✽✮✱ ϕ0 = ϕ1 = 0 ❛r❡ s❛t✐s✜❡❞✳ ❚❤❡♥✱ ❢♦r ❛r❜✐tr❛r② t ∈ R+ t❤❡ s♦❧✉t✐♦♥ u(t) ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✸✮✱ ✭✶✹✮ ✐s r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡r✐❡s u(t) =

∞

  • n=1
  • ωn(t, λ+

n ) + ωn(t, λ− n ) + ∞

  • k=1

ωn(t, λkn)

  • en,

✭✸✵✮ t❤❛t ✐s ❝♦♥✈❡r❣❡♥t ❜② t❤❡ ♥♦r♠ ♦❢ t❤❡ s♣❛❝❡ H✱ ✇❤❡r❡ ωn(t, λ) =

t

  • fn(τ)eλ(t−τ)dτ

l(1)

n (λ)

.

✹✵ ✴ ✹✽

slide-41
SLIDE 41

▲❡t ✉s s✉♣♣♦s❡ t❤❡ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥ K(t) ❤❛s t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭✷✷✮✳ ❉❡♥♦t❡ ❜② Kn(τ) = a2

n

  • ˆ

K− (−τ) − ˆ K+ (−τ)

  • τ 2 + a2

n

  • 1 − ˆ

K+ (−τ) τ 2 + a2

n

  • 1 − ˆ

K− (−τ) , ˆ K± (−τ) =

N

  • k=1

ck τ αe±iπα + βk

✹✶ ✴ ✹✽

slide-42
SLIDE 42

❚❤❡♦r❡♠ ✭✶✹✮

▲❡t ✉s s✉♣♣♦s❡ t❤❛t ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ✶✶ ❛r❡ s❛t✐s✜❡❞✱ α ∈ (0, 1/2)✱ f(t) ≡ 0✱ ϕ0 ∈ H3✱ ϕ1 ∈ H2✳ ❚❤❡♥ t❤❡ str♦♥❣ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✸✮✱ ✭✶✹✮ ✐s r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉♠ u(t) = uI(t) + uR(t), t > 0, uI(t) =

∞

  • n=1
  • ωn(t, λ+

n ) + ωn(t, λ− n )

  • en,

ωn(t, λ) = (ϕ1n + λϕ0n) eλt l(1)

n (λ)

, ✭✸✶✮ uR(t) =

∞

  • n=1

wn (t)en, wn (t) = ∞ e−tτKn(τ) (−τϕ0n + ϕ1n)dτ, ✭✸✷✮ ❚❤❡ s❡r✐❡s ❛r❡ ❝♦♥✈❡r❣❡♥t ❜② t❤❡ ♥♦r♠ ♦❢ t❤❡ s♣❛❝❡ H ❛♥❞ λ±

n ❛r❡ ♥♦♥r❡❛❧

❡✐❣❡♥✈❛❧✉❡s ♦❢ ♦♣❡r❛t♦r ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ)✱ ϕkn = (ϕk, en)✱ n ∈ N✱ k = 1, 2✳

✹✷ ✴ ✹✽

slide-43
SLIDE 43

❚❤❡♦r❡♠ ✭✶✺✮

▲❡t ✉s s✉♣♣♦s❡ t❤❛t ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ✶✶ ❛r❡ s❛t✐s✜❡❞✱ α ∈

  • 0, 1

2

  • ✱

ϕ0 = ϕ1 ≡ 0. ❚❤❡♥ t❤❡ str♦♥❣ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✸✮✱ ✭✶✹✮ ✐s r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉♠ u(t) = wI(t) + wR(t), t > 0, wI(t) =

∞

  • n=1

 

t

  • exp(λ+

n (t − τ))

l(1)

n (λ+ n )

+ exp(λ−

n (t − τ))

l(1)

n (λ− n )

  • fn(τ)dτ

  en, wR(t) =

∞

  • n=1

wn (t)en, wn (t) =

t

∞

  • exp(−p(t − τ))Kn(p)dp

  fn(τ)dτ ❚❤❡ s❡r✐❡s ❛r❡ ❝♦♥✈❡r❣❡♥t ❜② t❤❡ ♥♦r♠ ♦❢ t❤❡ s♣❛❝❡ H✱ λ±

n ✲ ❛r❡ ♥♦♥r❡❛❧

❡✐❣❡♥✈❛❧✉❡s ♦❢ ♦♣❡r❛t♦r ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ)✱ fn = (f, en)✳

✹✸ ✴ ✹✽

slide-44
SLIDE 44

❉❡♥♦t❡ ❜② Pn t❤❡ ♦rt❤♦♣r♦❥❡❝t♦r ♦♥ t❤❡ s✉❜s♣❛❝❡✱ ✇❤✐❝❤ ✐s t❤❡ ❧✐♥❡❛r ❝♦✈❡r ♦❢ t❤❡ ❡✐❣❡♥✈❡❝t♦rs {ej}n

j=1✱ ❛♥❞ ❞❡♥♦t❡ ❜② Qn t❤❡ ♦rt❤♦♣r♦❥❡❝t♦r ♦♥ t❤❡

s✉❜s♣❛❝❡ ✇❤✐❝❤ ✐s ♦rt❤♦❣♦♥❛❧ t♦ t❤❡ s✉❜s♣❛❝❡ PnH✳ ❚❤❛t ✐s Qn = I − Pn ❛♥❞ t❤❡ s♣❛❝❡ H ✐s r❡♣r❡s❡♥t❡❞ ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s✉♠ H = PnH ⊕ QnH.

❚❤❡♦r❡♠ ✭✶✻✮

▲❡t ✉s s✉♣♣♦s❡ t❤❛t ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ✶✹ ❛r❡ s❛t✐s✜❡❞✳ ❚❤❡♥ ❢♦r ❛♥② ε > 0 t❤❡r❡ ❡①✐sts n0 ∈ N ❛♥❞ δ > 0✱ s♦ t❤❛t ❢♦r ✈❡❝t♦r✲❢✉♥❝t✐♦♥ uI (t)✱ ❞❡✜♥❡❞ ❜② ✭✸✶✮✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡s ❛r❡ ✈❛❧✐❞ Qn0uI (t) θ1

  • Qn0e−kA1−αtA2ϕ0
  • + θ2
  • Qn0e−kA1−αtAϕ1
  • ,

t > 0, ✭✸✸✮ 0 < k < 1 2 πα 2 N

  • j=1

cj − ε, Pn0uI (t) θ3e−δt Pn0ϕ0 +

  • Pn0A−1ϕ1
  • ,

t > 0, ✭✸✹✮ ✇✐t❤ s♦♠❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts δ✱ θ1, θ2, θ3 ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ✈❡❝t♦rs ϕ0, ϕ1.

✹✹ ✴ ✹✽

slide-45
SLIDE 45

❈♦r♦❧❧❛r②

▲❡t ✉s s✉♣♣♦s❡ t❤❛t ✈❡❝t♦r✲❢✉♥❝t✐♦♥ uI(t) ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❢♦r♠✉❧❛ ✭✸✶✮✱ ✇❤❡r❡ λ±

n ❢♦r ❛♥② s✉✣❝✐❡♥t❧② ❧❛r❣❡ n ∈ N✱ ❤❛s t❤❡ ❛s②♠♣t♦t✐❝ ✭✷✽✮✱ ✈❡❝t♦rs

ϕ0 ∈ Hp✱ ϕ1 ∈ Hp−1✱ p ∈ N✳ ❚❤❡♥ ❢♦r ❛♥② ε > 0 t❤❡r❡ ❡①✐sts s✉❝❤ n0 ∈ N✱ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡s ❛r❡ ✈❛❧✐❞ ApQn0uI (t) θ4

  • Qn0e−kA1−αtApϕ0
  • + θ5
  • Qn0e−kA1−αtAp−1ϕ1
  • ,

t > 0, ✭✸✺✮ 0 < k = 1 2 sin πα 2 N

  • j=1

cj − ε, ApPn0uI (t) θ6e−δt Pn0Apϕ0 +

  • Pn0Ap−1ϕ1
  • ,

t > 0, ✭✸✻✮ ✇✐t❤ s♦♠❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts δ✱ θ4, θ5, θ6✱ ✐♥❞❡♣❡♥❞❡♥t ♦♥ t❤❡ ✈❡❝t♦rs ϕ0, ϕ1.

✹✺ ✴ ✹✽

slide-46
SLIDE 46

❚❤❡♦r❡♠ ✭✶✼✮

▲❡t ✉s s✉♣♣♦s❡ t❤❛t ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ✶✹ ❛r❡ s❛t✐s✜❡❞✳ ❚❤❡♥ ❢♦r ❛♥② ε > 0 ✈❡❝t♦r✲❢✉♥❝t✐♦♥ uR (t)✱ ❞❡✜♥❡❞ ❜② t❤❡ ❢♦r♠✉❧❛ ✭✸✷✮✱ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡ uR (t)2 e−2εt k1

  • A−αϕ0
  • 2 + k2
  • A−1−αϕ1
  • 2

+ + k3

  • ε2(2+α)

A−2ϕ0

  • 2 + ε2(1+α)

A−2ϕ1

  • 2

, t > 0, ✭✸✼✮ ✇✐t❤ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts k1, k2, k3✱ ✐♥❞❡♣❡♥❞❡♥t ♦♥ ✈❡❝t♦rs ϕ0, ϕ1✳

✹✻ ✴ ✹✽

slide-47
SLIDE 47

❚❤❡♦r❡♠ ✭✶✽✮

▲❡t ✉s s✉♣♣♦s❡ t❤❛t ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ✶✶ ❛r❡ s❛t✐s✜❡❞ ❛♥❞ Af (t) ∈ L2,γ0 (R+, H) ❢♦r s♦♠❡ γ0 > 0✳ ❚❤❡♥ ❢♦r ❛♥② ε > 0 t❤❡r❡ ❡①✐sts n0 ∈ N✱ t❤❛t ❢♦r s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✸✮✱ ✭✶✹✮ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡ ✐s ✈❛❧✐❞ Amu (t) d1t t

  • Qn0e−kA1−α(t−τ)Am−1f(τ)
  • 2

dτ+ + d2t t e−2δ(t−τ) Pn0Am−1f(τ)

  • 2dτ+

+t

  • k1

t e−2ε(t−τ)

  • Am−(1+α)f(τ)
  • 2

dτ + k2ε2(α+1) t

  • Am−2f(τ)
  • 2dτ
  • ,

✭✸✽✮ ✇❤❡r❡ m = 0, 1, 2, ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts d1✱ d2, k1, k2 ✐♥❞❡♣❡♥❞❡♥t ♦♥ ✈❡❝t♦r✲❢✉♥❝t✐♦♥ f(t)✳

✹✼ ✴ ✹✽

slide-48
SLIDE 48

❚❤❛♥❦ ②♦✉ ✈❡r② ♠✉❝❤ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳

✹✽ ✴ ✹✽