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❙♣❡❝tr❛❧ ❛♥❛❧②s✐s ♦❢ ✐♥t❡❣r♦❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛r✐s✐♥❣ ✐♥ ✈✐s❝♦❡❧❛st✐❝✐t②

◆✳ ❆✳ ❘❛✉t✐❛♥✱ ❱✳ ❱✳ ❱❧❛s♦✈ ✭▲♦♠♦♥♦s♦✈ ▼♦s❝♦✇ ❙t❛t❡ ❯♥✐✈❡rs✐t②✮ ❖❚■◆❉✲✷✵✶✻✱ ❉❡❝❡♠❜❡r ✶✼✕✷✵✱ ✷✵✶✻

✶ ✴ ✹✷

■♥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❛♣❤② ❝✐t❡❞ ❜❡❧♦✇ ❢♦r ♠♦r❡ ❞❡t❛✐❧s✮ ❞❡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✳ ❲❡ ❛♥❛❧②s❡ s♣❡❝tr❛❧ ♣r♦❜❧❡♠s ❢♦r t❤❡ ♦♣❡r❛t♦r✕✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ✇❤✐❝❤ ❛r❡ t❤❡ s②♠❜♦❧s ♦❢ t❤❡s❡ ❡q✉❛t✐♦♥s✳ ▼♦r❡♦✈❡r ✇❡ st✉❞② t❤❡ s♣❡❝tr✉♠ ♦❢ t❤❡ ❛❜str❛❝t ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ♦❢ ●✉rt✐♥✲P✐♣❦✐♥ t②♣❡✳

✷ ✴ ✹✷

▲❡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. ✭✷✮

✸ ✴ ✹✷

❆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) =

∞

• k=1

ake−γkt, Q(t) =

∞

• k=1

bke−γkt, ✭✸✮ ✇❤❡r❡ ak > 0✱ bk 0✱ γk+1 > γk > 0✱ k ∈ N✱ γk → +∞ (k → +∞)✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ tr✉❡✿

∞

• k=1

ak γk < 1,

∞

• k=1

bk γk < 1. ✭✹✮ ❚❤❡ ❝♦♥❞✐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) =

∞

• k=1

ak < +∞, Q(0) =

∞

• k=1

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

1 (R+)✳

✹ ✴ ✹✷

❚❤❡ ❡q✉❛t✐♦♥ ✭✶✮ ❝❛♥ ❜❡ r❡❣❛r❞❡❞ ❛s ❛♥ ❛❜str❛❝t ❢♦r♠ ♦❢ ❞✐♥❛♠✐❝❛❧ ✈✐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 = −1 3ρ−1λ · ❣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✐♦♥ ✭✻✮✳

✺ ✴ ✹✷

■♥ ❝❛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❝❤✳ ❘❛t✳ ▼❡❝❤✳ ❆♥❛❧✳✱ ✶✾✻✽✱ ❱✳ ✸✶✱ P✳ ✶✶✸✕✶✷✻✳ ✷✮ Pr✉ss ❏✳ ❊✈♦❧✉t✐♦♥❛r② ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✴✴ ▼♦♥♦❣r❛♣❤s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ✶✾✾✸✱ ❱✳✽✼✱ ❇✐r❦❤❛✉s❡r ❱❡r❧❛❣✳ ❇❛s❡❧✲❇❛st♦♥✲❇❡r❧✐♥✳ ✸✮ ❆♠❡♥❞♦❧❛ ●✳✱ ❋❛❜r✐③✐♦ ▼✳✱ ●♦❧❞❡♥✱ ❏✳ ▼✳ ❚❤❡r♠♦❞②♥❛♠✐❝s ♦❢ ♠❛t❡r✐❛❧s ✇✐t❤ ♠❡♠♦r②✿ t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ◆❡✇ ❨♦r❦✿ ❙♣r✐♥❣❡r✱ ✷✵✶✷✳

✻ ✴ ✹✷

❆♣♣❧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) =

∞

• k=1

ak (λ + γk), ˆ Q(λ) =

∞

• k=1

bk (λ + γk), ✭✽✮

❉❡✜♥✐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)✳

✼ ✴ ✹✷

❉❡♥♦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 ✳

✽ ✴ ✹✷

❈♦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♦❜❧❡♠ ✭✶✮✱ ✭✷✮✳

✾ ✴ ✹✷

❚❤❡♦r❡♠

❙✉♣♣♦s❡ t❤❛t f(1)(t) ∈ L2,γ0 (R+, H) ❢♦r s♦♠❡ γ0 0 ❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✹✮ ✐s 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✳

✶✵ ✴ ✹✷

❙♣❡❝tr❛❧ ❆♥❛❧②s✐s

▲❡t ✉s st✉❞② t❤❡ str✉❝t✉r❡ ❛♥❞ ❧♦❝❛❧✐③❛t✐♦♥ ♦❢ t❤❡ s♣❡❝tr❛ ♦❢ ♦♣❡r❛t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ) ✇❤❡♥ t❤❡ ❝♦♥❞✐t✐♦♥s ✭✹✮ ❛♥❞ ✭✺✮ ❛r❡ s❛t✐s✜❡❞✳ ❲❡ s❤❛❧❧ s✉♣♣♦s❡ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss✉♠♣t✐♦♥s ♦♥ t❤❡ s❡q✉❡♥❝❡ {γk}∞

k=1 ❛r❡ ✈❛❧✐❞✿

sup

k∈N

γ2

k(γk+1 − γk) = +∞,

✭✶✵✮ lim

k→∞

γk − γk−1 γk = 0. ✭✶✶✮ ❘❡♠❛r❦ t❤❛t ❝♦♥❞✐t✐♦♥ ✭✶✶✮ ✐s s❛t✐s✜❡❞ ✇❤❡♥ t❤❡ s❡q✉❡♥❝❡ γk ≃ kα✱ α > 0✳ ❘❡❛❧❧② ✐♥ t❤✐s ❝❛s❡ γk − γk−1 γk ∼ α k → 0✱ (k → ∞)✳ ❚❤❡s❡ ♣♦✇❡r ❛s②♠♣t♦t✐❝s ❛r✐s✐♥❣ ✐♥ ❛✈❡r❛❣✐♥❣ t❤❡♦r② ✇❤❡r❡ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡ {γk}∞

k=1 ❛r❡ ♣♦✐♥ts

♦❢ t❤❡ s♣❡❝tr✉♠ ♦❢ ❛ s♣❡❝✐❛❧ ❡❧❧✐♣t✐❝ ❙t♦❦❡s✲t②♣❡ ♣r♦❜❧❡♠ ✇✐t❤ ♣❡r✐♦❞✐❝ ❝♦♥❞✐t✐♦♥s ✭s❡❡ t❤❡ ♠♦♥♦❣r❛♣❤ ❊✳ ❙❛♥❝❡s P❛❧❡♥s✐❛ ◆♦♥❤♦♠♦❣❡♥❡♦✉s ▼❡❞✐❛ ❛♥❞ ❖s❝✐❧❧❛t✐♦♥ ❚❤❡♦r② ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣ ◆❡✇ ❨♦r❦✱ ✶✾✽✵✮✳ ■♥ t✉r♥ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✶✶✮ ✐s ♥♦t s❛t✐s✜❡❞ ✐❢ t❤❡ s❡q✉❡♥❝❡ γk = cqk✱ q > 1✱ c > 0✳ ❙✉❝❤ ❜❡❤❛✈✐♦r ♦❢ t❤❡ s❡q✉❡♥❝❡ {γk}∞

k=1 ✐s r❛t❤❡r s❡❧❞♦♠ ✐♥ ❛♣♣❧✐❝❛t✐♦♥s✳

✶✶ ✴ ✹✷

▲❡t ✉s ❞❡t❡r♠✐♥❡ t❤❡ ❧♦❝❛❧✐③❛t✐♦♥ ♦❢ ♥♦♥r❡❛❧ r♦♦ts ♦❢ t❤❡ ❡q✉❛t✐♦♥ (L(λ)f, f) = λ2 + (Af, f) + (Bf, f) −

∞

• k=1

ak(Af, f) + bk(Bf, f) λ + γk = 0, (f ∈ D(A), ||f|| = 1) ✭✶✷✮ ▲❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥s✿ ω2 = ((A + B)f, f) > 0✱ ck = ((akA + bkB)f, f) ω−2 > 0✱ A(λ) =

∞

• k=1

ak(λ + γk)−1✱ B(λ) =

∞

• k=1

bk(λ + γk)−1✳ ❯♥❞❡r t❤❡s❡ ♥♦t❛t✐♦♥s t❤❡ ❡q✉❛t✐♦♥ ✭✶✷✮ t❛❦❡ ❛ ❢♦r♠✿ λ2 ω2 + 1 =

∞

• k=1

ck λ + γk , λ ∈ C. ✭✶✸✮ ■t ✐s ❡❛s② s❤♦✇ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥s

∞

• k=1

ck < +∞✱

∞

• k=1

ckγ−1

k

< 1 ❛r❡ ✈❛❧✐❞✳

✶✷ ✴ ✹✷

❚❤❡♥ ✇❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛✿

▲❡♠♠❛ ✭✶✮

❙✉♣♣♦s❡ t❤❡ ❝♦♥❞✐t✐♦♥s ✭✶✵✮✱ ✭✶✶✮ ❛r❡ s❛t✐s✜❡❞ ❛♥❞ t❤❡ s❡r✐❡s

∞

• k=1

ck ✐s ❝♦♥✈❡r❣❡♥t✳ ❚❤❡♥ t❤❡ ❡q✉❛t✐♦♥ ✭✶✸✮ ❤❛s t✇♦ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ r♦♦ts λ±

0 = α0 ± iβ0 ∈ C✱ α0, β0 ∈ R✱ α0 < 0 ❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❡q✉❛❧✐t❡s

− 1 2

∞

• k=1

ck α0 −1 2

∞

• k=1

ω2ck ω2 + γ2

k

. ✭✶✹✮ ❛r❡ ✈❛❧✐❞ ❢♦r r❡❛❧ ♣❛rt α0 ♦❢ r♦♦ts λ±

0 ✳

■♥ ♦✉r ♣r❡✈✐♦✉s ♥♦t❛t✐♦♥s t❤❡ ✐♥❡q✉❛❧✐t✐❡s ✭✶✹✮ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠ −1 2

∞

• k=1

((akA + bkB)f, f) ((A + B)f, f) α0 −1 2

∞

• k=1

((akA + bkB)f, f)

• (A + B + γ2

kI)f, f

, ✇❤❡r❡ f ∈ D(A)✱ ||f|| = 1 ❛♥❞ α0 ✐s r❡❛❧ ♣❛rt ♦❢ ♥♦♥r❡❛❧ ③❡r♦❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭✶✷✮✳

✶✸ ✴ ✹✷

▲❡t ✉s ❞❡t❡r♠✐♥❡ t❤❡ ❧♦❝❛t✐♦♥ ♦❢ r❡❛❧ ③❡r♦❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭✶✷✮✳ ❉❡♥♦t❡ ❜② τ = (Af, f)ω−2✱ 0 inf τ τ sup τ 1✳ ❘❡✇r✐t❡ t❤❡ ✭✶✷✮ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠ λ2 ω2 +1 = τ

∞

• k=1

ak λ + γk +(1−τ)

∞

• k=1

bk λ + γk =: τA(λ)+(1−τ)B(λ). ✭✶✺✮ ❈♦♥s✐❞❡r t❤❡ ❡q✐❛t✐♦♥ Φτ(p) := τA(p) + (1 − τ)B(p) = 1, ✭✶✻✮ ❚❤❡ ❢✉♥❝t✐♦♥ Φτ(p) → ∞ ❢♦r p → −γk✱ k ∈ N ❛♥❞ ✐t ✐s ♠♦♥♦t♦♥♦✉s❧② ❞❡❝r❡❛s❡s ❢♦r r❡❛❧ p ∈ (−γk, −γk−1)✱ k ∈ N ✭γ0 = 0✮✱ ❤❡♥❝❡ t❤❡ ❡q✉❛t✐♦♥ ✭✶✻✮ ❤❛s t❤❡ ✐♥✜♥✐t② ❝♦♥s❡q✉❡♥❝❡ ♦❢ r❡❛❧ ③❡r♦❡s pk(τ) ∈ (−γk, −γk−1)✱ k ∈ N✳ ■♥ t✉r♥ t❤❡ ❡q✉❛t✐♦♥ ✭✶✺✮ ❛❧s♦ ❤❛s t❤❡ ✐♥✜♥✐t② ❝♦♥s❡q✉❡♥❝❡ ♦❢ r❡❛❧ ③❡r♦❡s λk(τ) ∈ (−γk, pk(τ)) ✭❜② ❝♦♥str❛❝t✐♦♥✮✱ t❤❡r❡❢♦r❡ λk(τ) ∈ (−γk, max

0<τ<1 pk(τ)) ❢♦r ❛♥② τ ∈ (0, 1)✱ k ∈ N✳

✶✹ ✴ ✹✷

▲❡♠♠❛ ✭✷✮

❙✉♣♣♦s❡ t❤❡ ❝♦♥❞✐t✐♦♥s ✭✶✵✮✱ ✭✶✶✮ ❛r❡ s❛t✐s✜❡❞✳ ❚❤❡♥ r❡❛❧ ③❡r♦❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭✶✷✮ ❜❡❧♦♥❣ t♦ ✐♥t❡r✈❛❧s ∆k = (−γk, ˜ pk) ✇❤❡r❡ ˜ pk = max {pk(τ ′), pk(τ ′′)}✱ pk(τ) ❛r❡ r❡❛❧ ③❡r♦❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭✶✻✮ ❜❡❧♦♥❣✐♥❣ t♦ ✐♥t❡r✈❛❧s (−γk, −γk−1)✱ k ∈ N ✭γ0 = 0✮✱ τ ′ :=

• A−1/2A0A−1/2

−1✱ τ ′′ :=

• A−1/2

AA−1/2

• ✳

❘❡♠❛r❦✳ ❆❝❝♦r❞✐♥❣ t♦ ❧❡♠♠❛ ✷✳✶ ❢r♦♠ t❤❡ ❛rt✐❝❧❡ ❆✳❆✳ ❙❤❦❛❧✐❦♦✈ ❙tr♦♥❣❧② ❉❛♠♣❡❞ P❡♥❝✐❧s ♦❢ ❖♣❡r❛t♦rs ❛♥❞ ❙♦❧✈❛❜✐❧✐t② ♦❢ t❤❡ ❈♦rr❡s♣♦♥❞✐♥❣ ❖♣❡r❛t♦r✲ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✭▼❛t❤❡♠❛t✐❝s ♦❢ t❤❡ ❯❙❙❘✲❙❜♦r♥✐❦ ✭✶✾✽✾✮✱ ✻✸✭✶✮✿✾✼✮ ♦♣❡r❛t♦r A−1/2BA−1/2 ❛❞♠✐ts ❜♦✉♥❞❡❞ ❝❧♦s✉r❡ ✐♥ t❤❡ s♣❛❝❡ H✳ ❍❡♥❝❡ ✇❡ ♦❜t❛✐♥ t❤❛t ♦♣❡r❛t♦r A−1/2A0A−1/2 = I + A−1/2BA−1/2 ❛❞♠✐ts ❜♦✉♥❞❡❞ ❝❧♦s✉r❡ ✐♥ t❤❡ s♣❛❝❡ H✳ ■♥ t✉r♥ ♦✇✐♥❣ t♦ ❧❡♠♠❛ ✷✳✶ ❢r♦♠ ❝✐t❡❞ ❛rt✐❝❧❡ ❛♥❞ ❞✉❡ t♦ s❡❧❢❛❞❥♦✐♥t♥❡ss ♦❢ ♦♣❡r❛t♦r A0 = A + B ♦♣❡r❛t♦r A−1/2 AA−1/2 ❛❧s♦ ❛❞♠✐ts ❜♦✉♥❞❡❞ ❝❧♦s✉r❡ ✐♥ t❤❡ s♣❛❝❡ H✳ ❚❤✉s ✈❛❧✉❡s τ ′ ❛♥❞ τ ′′ ❛r❡ ❞❡✜♥❡❞ ❝♦rr❡❝t❧②✳

✶✺ ✴ ✹✷

❚❤❡♦r❡♠ ✭❚❤❡ ♠❛✐♥ s♣❡❝tr❛❧ t❤❡♦r❡♠✮

❙✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥s ✭✹✮✱ ✭✺✮✱ ✭✶✵✮✱ ✭✶✶✮ ❛r❡ s❛t✐s✜❡❞✳ ❚❤❡♥ t❤❡ s♣❡❝tr❛ ♦❢ ♦♣❡r❛t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ) ❜❡❧♦♥❣s t♦ t❤❡ ✉♥✐t ✐♥t❡r✈❛❧s ∆k = (−γk, ˜ pk] ⊂ (−γk, −γk−1)✱ k ∈ N ✭γ0 = 0✮ ❛♥❞ t❤❡ str✐♣ {λ ∈ C|α1 Re λ α2}✱ ✇❤❡r❡ ˜ pk = max {pk(τ ′), pk(τ ′′)}✱ pk(τ) ❛r❡ r❡❛❧ ③❡r♦❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ Φτ(p) := τ

∞

• k=1

ak(p + γk)−1+(1−τ)

∞

• k=1

bk(p + γk)−1 = 1, (0 ≤ τ ≤ 1). ❜❡❧♦♥❣✐♥❣ t❤❡ ✐♥t❡r✈❛❧s (−γk, −γk−1)✱ k ∈ N ✭γ0 = 0✮✱ τ ′ :=

• A−1/2A0A−1/2

−1✱ τ ′′ :=

• A−1/2

AA−1/2

• ✱ (0 < τ ′ < τ ′′ 1)✱

α1 = −1 2 sup

f=1 ∞

• k=1

((akA + bkB)f, f) ((A + B)f, f) , f ∈ D(A), α2 = −1 2 inf

f=1 ∞

• k=1

((akA + bkB)f, f)

• (A + B + γ2

kI)f, f

, f ∈ D(A).

✶✻ ✴ ✹✷

❘❡♠❛r❦

❈♦♥st❛♥ts α1 ❛♥❞ α2 ❢r♦♠ t❤❡ ♠❛✐♥ s♣❡❝tr❛❧ t❤❡♦r❡♠ s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡s α1 −1 2

• A−1/2

∞

• k=1

akA +

∞

• k=1

bkB

• A−1/2
• ,

α2 < −1 2

• (a1A + b1B)−1/2

A0 + γ2

1I

• (a1A + b1B)−1/2
• −1

.

❚❤❡♦r❡♠

❯♥r❡❛❧ ♣❛rt ♦❢ t❤❡ s♣❡❝tr❛ ♦❢ ♦♣❡r❛t♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ) ✐s s②♠♠❡tr✐❝ ❝♦♥❝❡r♥✐♥❣ t❤❡ r❡❛❧ ❛①✐s ❛♥❞ ❝♦♥s✐sts ♦❢ ❡✐❣❡♥✈❛❧✉❡s ♦❢ ✜♥✐t❡ ❛❧❣❡❜r❛✐❝ ♠✉❧t✐♣❧✐❝✐②✳ ▼♦r❡♦✈❡r ❢♦r ❛♥② ε > 0 ✐♥ t❤❡ ❞♦♠❛✐♥ Ωε := C\ {λ : α1 ≤ Re λ ≤ α2, | Im λ| < ε}✱ ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ✐s♦❧❛t❡❞ t❤❛t ✐s ❤❛✈❡ ♥♦ ❛❝❝✉♠✉❧❛t✐♦♥ ♣♦✐♥ts✳

✶✼ ✴ ✹✷

• ❡♥❡r❛❧✐③❛t✐♦♥s

❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠ ✭✶✮✱ ✭✷✮ 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, ✐♥ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ s❝❛❧❛r ❢✉♥❝t✐♦♥s K(t) ❛♥❞ Q(t) ❛❞♠✐t 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❡✳

✶✽ ✴ ✹✷

❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ tr✉❡✿ 0 < ∞ dµ(τ) τ < 1, 0 < ∞ dη(τ) τ < 1, ✭✶✽✮ K(0) = ∞ dµ(τ) ≡ ❱❛r µ|∞

0 < +∞,

Q(0) = ∞ dη(τ) ≡ ❱❛r η|∞

0 < +∞.

✭✶✾✮ ❍❡r❡ t❤❡ s✉♣♣♦rts µ ❛♥❞ η ❜❡❧♦♥❣ t♦ t❤❡ ✐♥t❡r✈❛❧ (d0, +∞)✱ d0 > 0✳

▲❡♠♠❛

❙✉♣♣✉s❡ t❤❛t ❝♦♥❞✐t✐♦♥s ✭✶✽✮✱ ✭✶✾✮ ❤♦❧❞s✳ ❚❤❡♥ t❤❡ ♦♣❡r❛t♦r✲❢✉♥❝t✐♦♥ L(λ) ✐s ✐♥✈❡rt✐❜❧❡ ✐♥ ❝❧♦s❡❞ r✐❣❤t ❤❛❧❢✲♣❧❛♥❡ ❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡

• A1/2L−1(λ)A1/2
• ❝♦♥st,

Re λ > γ > 0. ✐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✳

❚❤❡♦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 ❛r❜✐tr❛r② θ0 > 0 t❤❡r❡ ❡①✐st ♥✉♠❜❡r R0 > 0✱ s✉❝❤ 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=1

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

f=1 ∞

• k=1

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

• (A + B + d2

2I)f, f

• ,

f ∈ D(A). ✭✷✵✮

✷✵ ✴ ✹✷

❘❡♠❛r❦

▲❡t ✉s ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ❤♦❧❞✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts s✉❝❤ γ0 > 0✱ t❤❛t ♦♣❡r❛t♦r✲❢✉♥❝t✐♦♥ L−1(λ) s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡

• L−1(λ)
• const

|λ|| Re λ| ✭✷✶✮ ♦♥ t❤❡ s❡t {λ : Re λ < −R0} ∪ {λ : Re λ > γ0}✳

❘❡♠❛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 ❝♦♥❞✐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✐♦♥✳ ❚❤❡s❡ r❡s✉❧ts ✇❛s ♣r♦✈❡❞ ✐♥ t❤❡ ❛rt✐❝❧❡s ✶✮ ❱✳❱✳ ❱❧❛s♦✈ ❛♥❞ ◆✳❆✳ ❘❛✉t✐❛♥✱ ❈♦rr❡❝t s♦❧✈❛❜✐❧✐t② ❛♥❞ s♣❡❝tr❛❧ ❛♥❛❧②s✐s ♦❢ ✐♥t❡❣r♦❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛r✐s✐♥❣ ✐♥ t❤❡ ✈✐s❝♦❡❧❛st✐❝✐t② t❤❡♦r② ✴✴ ❙♦✈r❡♠✳ ♠❛t✳ ❋✉♥❞❛♠✳ ♥❛♣r❛✈❧✳ ❬❈♦♥t❡♠♣✳ ▼❛t❤✳ ❋✉♥❞❛♠✳ ❉✐r❡❝t✐♦♥s❪✱ ✷✵✶✺✱ ✺✽✱ ✷✷✕✹✷✳ ✭✐♥ ❘✉ss✐❛♥✮ ✷✮ ❱✳❱✳ ❱❧❛s♦✈ ❛♥❞ ◆✳❆✳ ❘❛✉t✐❛♥✱ ❙♣❡❝tr❛❧ ❆♥❛❧②s✐s ♦❢ ■♥t❡❣r♦❞✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✐♥ ❛ ❍✐❧❜❡rt ❙♣❛❝❡ ✴✴ ❙♦✈r❡♠✳ ♠❛t✳ ❋✉♥❞❛♠✳ ♥❛♣r❛✈❧✳ ❬❈♦♥t❡♠♣✳ ▼❛t❤✳ ❋✉♥❞❛♠✳ ❉✐r❡❝t✐♦♥s❪✱ ✷✵✶✻✱ ✻✷✱ ✺✸✕✼✶✳ ✭✐♥ ❘✉ss✐❛♥✮

✷✷ ✴ ✹✷

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

✷✸ ✴ ✹✷

Pr❡✈✐♦✉s r❡s✉❧ts

■♥ ♦✉r ♣r❡✈✐♦✉s ✇♦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✳ ❚❤❡♥ t❤❡ ♣r♦❜❧❡♠ ✭✶✮✱ ✭✷✮ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠ d2u(t) dt2 + A2u(t) − t K(t − s)A2u(s)ds = f(t), t ∈ R+, u(+0) = ϕ0, u(1)(+0) = ϕ1.

✷✹ ✴ ✹✷

▲❡t ✉s ❞❡♥♦t❡ ❜② {en}∞

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

♦❢ ♦♣❡r❛t♦r A2 ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❡✐❣❡♥✈❛❧✉❡s aj✿ A2en = a2

nen✱ n ∈ N✳

❚❤❡ ❡✐❣❡♥✈❛❧✉❡s a2

n ❛r❡ ♥✉♠❡r❛t❡❞ ✐♥ ✐♥❝r❡❛s✐♥❣ ♦r❞❡r 0 < a2 1 < a2 2 < ...❀

a2

n → +∞ ❢♦r n → +∞✳

❈♦♥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✳ ❚❤✉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✳

✷✺ ✴ ✹✷

▲❡t ✉s ❞❡♥♦t❡ ❜② {en}∞

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

♦❢ ♦♣❡r❛t♦r A2 ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❡✐❣❡♥✈❛❧✉❡s aj✿ A2en = a2

nen✱ n ∈ N✳

❚❤❡ ❡✐❣❡♥✈❛❧✉❡s a2

n ❛r❡ ♥✉♠❡r❛t❡❞ ✐♥ ✐♥❝r❡❛s✐♥❣ ♦r❞❡r 0 < a2 1 < a2 2 < ...❀

a2

n → +∞ ❢♦r n → +∞✳

❈♦♥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✳ ❚❤✉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✳

✷✻ ✴ ✹✷

❚❤❡♦r❡♠

▲❡t ✉s s✉♣♣♦s❡ B = 0 ❛♥❞ ❝♦♥❞✐t✐♦♥s ✭✹✮✱ ✭✶✵✮ ❛r❡ s❛t✐s✜❡❞✳ ❚❤❡♥ 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 t❤❛t ✐s

σ(L) := {λk,n ∈ R|k ∈ N, n ∈ N} ∪

• λ±

n |n ∈ N

• ,

✭✷✷✮ ✇❤❡r❡ λ±

n ❛r❡ ♥♦♥r❡❛❧ ❝♦♥❥✉❣❛t❡ ❝♦♠♣❧❡① ③❡r♦❡s λ+ n = λ− n ♦❢ ❢✉♥❝t✐♦♥s

ln(λ)✳ ▼♦r❡♦✈❡r t❤❡ r❡❛❧ ③❡r♦❡s s❛t✐s❢② t❤❡ ✐♥❡q✉❛❧✐t✐❡s ... − γk+1 < xk+1 < λk+1,n < −γk < ... < −γ1 < x1 < λ1,n < 0, k ∈ N, ✭✷✸✮ ✇❤❡r❡ xk ❛r❡ t❤❡ r❡❛❧ ③❡r♦❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ 1 − ˆ K(λ) ❛♥❞ λk,n = xk + O

• 1/a2

n

• ✳ ▼♦r❡♦✈❡r ✐❢ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✺✮ ✐s s❛t✐s✜❡❞ t❤❡♥ t❤❡

❝♦♥❥✉❣❛t❡ ❝♦♠♣❧❡① ③❡r♦❡s λ±

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

λ±

n = ±i

• an + O

1 an

• − 1

2

∞

• k=1

ak + O 1 a2

n

• ,

an → +∞. ✭✷✹✮

✷✼ ✴ ✹✷

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣✐❝t✉r❡ r❡♣r❡s❡♥ts t❤❡ s♣❡❝tr❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ♦♣❡r❛t♦r✲ ❢✉♥❝t✐♦♥ L(λ) ✐♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ λ✱ ✇❤❡r❡ β = −K(0)/2✳

✷✽ ✴ ✹✷

❚❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠ ♣r❡s❡♥ts t❤❡ ❛s②♠♣t♦t✐❝ ♦❢ t❤❡ ❝♦♠♣❧❡① ③❡r♦❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ln(λ) ✇❤❡♥ an → +∞✱ ✐♥ t❤❡ ❝❛s❡ B = 0 ✇❤❡♥ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✺✮ ✐s ♥♦t s❛t✐s✜❡❞✱ ❛♥❞ t❤❡ s❡q✉❡♥❝❡s {ck}∞

k=1 ❛♥❞ {γk}∞ k=1 ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣

❛s②♠♣t♦t✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ck = A kα + O

• 1

kα+1

• ,

✭✷✺✮ γk = Bkβ + O

• kβ−1

, ❢♦r k → +∞✱ ✇❤❡r❡ ck > 0✱ γk+1 > γk > 0✱ k ∈ N ❛♥❞ t❤❡ ❝♦♥st❛♥ts A > 0✱ B > 0✱ 0 < α 1✱ α + β > 1✳

✷✾ ✴ ✹✷

▲❡t ✉s ❞❡♥♦t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥st❛♥t ❜② D := − i 2 ∞ dt tr(i + t) = π 2 sin(πr) · eiπ 2 (1 − r), ✇❤❡r❡ 0 < r < 1✳

✸✵ ✴ ✹✷

❚❤❡♦r❡♠

▲❡t ✉s s✉♣♣♦s❡ B = 0 ❛♥❞ ❝♦♥❞✐t✐♦♥s ✭✷✺✮ ❛r❡ s❛t✐s✜❡❞✳ ❚❤❡♥ ❝♦♥❥✉❣❛t❡ ❝♦♠♣❧❡① ③❡r♦❡s λ±

n ✱ λ+ n = λ− n ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ln(λ) ❛r❡ ❛s②♠♣t♦t✐❝❛❧❧②

r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠ λ±

n = ±ian −

DA βB1−r a1−r

n

+ O(a1−2r

n

), ❢♦r 0 < r < 1 2 λ±

n = ±ian −

DA βB1−r a1−r

n

+ O(1), ❢♦r 1 2 r < 1, λ±

n = ±ian − 1

2 A β ln an + O (1) , ❢♦r r = 1, ❢♦r n → +∞✱ ✇❤❡r❡ r := α + β − 1 β ✱ t❤❡ ❝♦♥st❛♥t D ❞❡♣❡♥❞s ♦♥ r✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣✐❝t✉r❡ r❡♣r❡s❡♥ts t❤❡ s♣❡❝tr❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ♦♣❡r❛t♦r ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ) ✐♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ λ ✐♥ ❝❛s❡ B = 0 ❛♥❞ K(t) ∈ L1(R+)✱ K(t) / ∈ W 1

1 (R+)✳

✸✶ ✴ ✹✷

Im λ Re λ λ±

n

x2 x3 −∞

Ðèñ✳✿ ❙♣❡❝tr❛❧ str✉❝t✉r❡ ✐♥ ❝❛s❡ B = 0✱ K(t) ∈ L1(R+)✱ K(t) / ∈ W 1

1 (R+)✳

✸✷ ✴ ✹✷

❚❤❡♦r❡♠

▲❡t ✉s s✉♣♣♦s❡ B = 0✱ t❤❡ ❝♦♥❞✐t✐♦♥s ✭✹✮✱ ✭✶✵✮ ❛r❡ s❛t✐s✜❡❞✱ ❜✉t t❤❡ ❝♦♥❞✐t✐♦♥ ✭✺✮ ✐s ♥♦t s❛t✐s✜❡❞✳ ❚❤❡♥ t❤❡ ♣❛r❡ ♦❢ t❤❡ ❝♦♥❥✉❣❛t❡ ❝♦♠♣❧❡① ③❡r♦❡s λ±

n ✱ λ+ n = λ− n ♦❢ t❤❡ ♠❡r♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥ ln(λ) ❛s②♠♣t♦t✐❝❛❧❧②

r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦r♠ λ±

n = ±iΘ · an + Φ(an, {ck}∞ k=1, {γk}∞ k=1),

k ∈ N ✭✷✻✮ ✇❤❡r❡ Θ = Θ({ck}∞

k=1, {γk}∞ k=1) ✐s ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t✱ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡

s❡q✉❡♥❝❡s {ck}∞

k=1✱ {γk}∞ k=1✱ Re Φ = O(an)✱ Im Φ = o(an) ✇❤✐❧❡

an → +∞ ❛♥❞ lim

an→∞ Re Φ(an, {ck}∞ k=1, {γk}∞ k=1) = −∞✳

✸✸ ✴ ✹✷

❘❡♣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♦❜❧❡♠ ✭✶✮✱ ✭✷✮ ✇❤❡♥ ♦♣❡r❛t♦r B = 0 ✐♥ t❤❡ ❢♦r♠ ♦❢ t❤❡ s❡r✐❡s✳

❚❤❡♦r❡♠

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

2,γ (R+, A)✱ γ > 0 ✐s ❛ str♦♥❣ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✮✱ ✭✷✮ ❛♥❞

t❤❡ ❝♦♥❞✐t✐♦♥s ✭✹✮✱ ✭✶✵✮ ❛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) = (ϕ1n + λϕ0n) eλt l(1)

n (λ)

, ϕ = (ϕ , e )✱ ϕ = (ϕ , e )✳

✸✹ ✴ ✹✷

❚❤❡♦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)

.

✸✺ ✴ ✹✷

❊①❛♠♣❧❡s ♦❢ ✈❡r② ✉♥st❛❜❧❡ ❧✐♥❡❛r ♣❛rt✐❛❧ ❢✉♥❝t✐♦♥❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

❲❡ ❝♦♥s✐❞❡r ♦♥ t❤❡ ♣♦s✐t✐✈❡ s❡♠✐❛①✐s R+ = (0, ∞) t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥✐t✐❛❧ ♣r♦❜❧❡♠ ❢♦r t❤❡ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ✜rst ♦r❞❡r dv dt +

t

• −∞

A2v(t − s)dσ(s) = f(t), t ∈ R+, ✭✷✾✮ v(t) = 0, t ∈ (−∞, 0), ✭✸✵✮ ✇❤❡r❡ dσ ✐s ❛ ♣♦s✐t✐✈❡ ♠❡❛s✉r❡✳ ❲❡ ✐❞❡♥t✐❢② t❤✐s ♠❡❛s✉r❡ ✇✐t❤ ✐ts ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ σ(s)✱ s♦ σ(s) ✐s ✐♥❝r❡❛s✐♥❣✱ ❝♦♥t✐♥✉♦✉s ❢r♦♠ t❤❡ r✐❣❤t✱ ❛♥❞ t❤❡ ✐♥t❡❣r❛❧ ✐s ✐♥t❡r♣r❡t❡❞ ❛s ❛ ❙t✐❡❧t❥❡s ✐♥t❡❣r❛❧✳

✸✻ ✴ ✹✷

❚❤❡r❡ ✐s ❛♥ ✐♠♣♦rt❛♥t ♣❛rt✐❝✉❧❛r ❝❛s❡ ✇❤❡♥ t❤❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ σ(t) ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠ σ(t) = θ(t − h)✱ ✇❤❡r❡ h > 0✳ ■♥ t❤✐s ♠♦❞❡❧ ❝❛s❡ t❤❡ ✐♥t❡❣r❛❧ t❡r♠ ✐♥ t❤❡ ❡q✉❛t✐♦♥ ✭✷✾✮ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠

t

• −∞

A2u(t − s)dσ(s) = A2u(t − h). ❚❤✉s ❡q✉❛t✐♦♥ ✭✷✾✮ ✐s t❤❡ ❞❡❧❛② ❡q✉❛t✐♦♥✳ ❚❤❡ ♦♣❡r❛t♦r ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ L(λ) = λI + A2e−λh ✐s t❤❡ s②♠❜♦❧ ♦❢ t❤✐s ❡q✉❛t✐♦♥✳

✸✼ ✴ ✹✷

■♥ ❛ ❝❛s❡ ✇❤❡♥ H s♣❛❝❡ ❤❛s t❤❡ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥✱ t❤❡ ❝♦♥s✐❞❡r❡❞ ❞❡❧❛② ❡q✉❛t✐♦♥ ♦❜t❛✐♥❡❞ ✇❛s ✐♥✈❡st✐❣❛t❡❞ ❜② ♠❛♥② ❛✉t❤♦rs✳ ❍♦✇❡✈❡r✱ ✐♥ ❛ ❝❛s❡ ✇❤❡♥ H ❤❛s ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥ ❛♥❞ A ✐s t❤❡ ✉♥❧✐♠✐t❡❞ s❡❧❢✲❝♦♥❥✉❣❛t❡ ♣♦s✐t✐✈❡ ♦♣❡r❛t♦r ✐♥ ❍✐❧❜❡rt s♣❛❝❡ ♦❢ H✱ t❤❡ s♣❡❝t✉♠ ♦❢ t❤❡ ♦♣❡r❛t♦r✲❢✉♥❝t✐♦♥ L(λ)✱ ❛♣♣❛r❡♥t❧②✱ ✇❛s♥✬t st✉❞✐❡❞ ❡❛r❧✐❡r✳

✸✽ ✴ ✹✷

▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦❞❡❧ ❡①❛♠♣❧❡✳ ▲❡t✬s ❛ss✉♠❡ t❤❛t ♦♣❡rt♦r A2 ✐s r❡❛❧✐③❡❞ ❛s ❢♦❧❧♦✇s✿ A2y(x) = −y(2)

xx (x)✱ y(0) = y(π) = 0✱ H = L2(0, π)

❛♥❞ h = 1✳ ■♥ t❤✐s ❝❛s❡ an = n✱ ❛♥❞ s♣❡❝tr✉♠ ♦❢ ♦♣❡r❛t♦r✲❢✉♥❝t✐♦♥ L(λ) ✐s t❤❡ ❝❧♦s✉r❡ ♦❢ t❤❡ ③❡r♦❡s s❡t ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ln(λ) = λ + n2e−λ✱ n ∈ N✱ t❤❛t ✐s σ(L) =

• n∈N
• k∈Z

λnk, ln(λnk) = 0.

✸✾ ✴ ✹✷

❚❤❡ ❛s②♠♣t♦t✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ③❡r♦❡s λnk ❢♦r k → +∞ ❛♥❞ ✜①❡❞ n ✐s ✇❡❧❧✲❦♥♦✇♥✱ ❤♦✇❡✈❡r ✐t ✐s ✐♥t❡r❡st✐♥❣ ❛♥❞ ✉♥❡①♣❡❝t❡❞ t❤❛t t❤❡r❡ ❡①✐sts t❤❡ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ❡✐❣❡♥✈❛❧✉❡s ♦❢ λnk(n)✱ s✉❝❤ t❤❛t Re λnk(n) → +∞ ❢♦r n → +∞✳ ■♥❞❡❡❞✱ ❧❡t ✉s ✜① n ❛♥❞ ❡①tr❛❝t t❤❡ r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rt ❢r♦♠ ❡①♣r❡ss✐♦♥ λeλ✱ λ = x + iy✳ ❚❤❡♥ t❤❡ ❡q✉❛t✐♦♥ λeλ = −n2 ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠✿

• ex(x cos y − y sin y) = −n2,

x sin y + y cos y = 0. ✭✸✶✮ ❚❤❡ ❛♥❛❧②s✐s ♦❢ s②st❡♠ ✭✸✶✮ s❤♦✇s t❤❛t ✐t ❤❛s s♦❧✉t✐♦♥s✱ ❛s②♠♣t♦t✐❝❛❧❧② r❡♣r❡s❡♥t❛❜❧❡ ❛s ❢♦❧❧♦✇s xn ≈ 2 ln n−ln ln n → +∞, yn ≈ π

• 1 −

1 2 ln n − ln ln n

• ,

n → +∞.

✹✵ ✴ ✹✷

❚❤✉s✱ t❤❡ ♣r♦❜❧❡♠ ✭✷✾✮✱ ✭✸✵✮ ❤❛s t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t②♣❡ e(xn+iyn)ten✱ ✐♥❝r❡❛s✐♥❣ ❢♦r t → +∞ q✉✐❝❦❡r t❤❛♥ ❛♥② ✜①❡❞ ❡①♣♦♥❡♥t eγt✱ ✇❤❡r❡ γ > 0 ✐s ❛ ❝♦♥st❛♥t✳ ❙♦ t❤❡ ♣r♦❜❧❡♠ ✭✷✾✮✱ ✭✸✵✮ ✐s ✉♥st❛❜❧❡ ❛♥❞ ✐t ✐s♥✬t ❝♦rr❡❝t❧② s♦❧✈❛❜❧❡ ✐♥ ❙♦❜♦❧❡✈ s♣❛❝❡ W m

2,γ(R+, A2) ❢♦r ❛♥② γ 0

Ð➻ m ∈ N✳ ▲❡t✬s ♥♦t✐❝❡ t❤❛t ❢♦r h = 0 ✇❡ ❤❛✈❡ t❤❡ ❝❧❛ss✐❝ ♠✐①❡❞ ♣r♦❜❧❡♠ ❢♦r ❤❡❛t ❡q✉❛t✐♦♥ t❤❛t ✐s st❛❜❧❡ ❛♥❞ ❝♦rr❡❝t❧② s♦❧✈❛❜❧❡ ✐♥ t❤❡ ❙♦❜♦❧❡✈ s♣❛❝❡ W 1

2,0(R+, A2)✳ ■t s❤♦✉❧❞ ❜❡ ♥♦t❡❞ ❛❧s♦ t❤❛t t❤❡ s②st❡♠ ✭✸✶✮ ✇❛s

✐♥✈❡st✐❣❛t❡❞ ❡❛r❧✐❡r ❛♥❞ ✐t ✇❛s ❡st❛❜❧✐s❤❡❞ t❤❛t t❤❡ s②st❡♠ ✭✸✶✮ ❤❛s s✉❝❤ s♦❧✉t✐♦♥ λnk(n) = xn + iyn t❤❛t xn → 0✱ n → +∞✳ ❊①❛♠♣❧❡s ♦❢ ✈❡r② ✉♥st❛❜❧❡ ❧✐♥❡❛r ♣❛rt✐❛❧ ❢✉♥❝t✐♦♥❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛r❡ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ✇♦r❦ ❘✳ ❙✳ ■s♠❛❣✐❧♦✈✱ ◆✳ ❆✳ ❘❛✉t✐❛♥✱ ❱✳ ❱✳ ❱❧❛s♦✈✱ ❊①❛♠♣❧❡s ♦❢ ✈❡r② ✉♥st❛❜❧❡ ❧✐♥❡❛r ♣❛rt✐❛❧ ❢✉♥❝t✐♦♥❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✴✴ ❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴✶✹✵✷✳✹✶✵✼

✹✶ ✴ ✹✷

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

✹✷ ✴ ✹✷