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❙❡♠✐❣r♦✉♣s ❢♦r ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ ❝♦♥✈♦❧✉t✐♦♥ ♠❡♠♦r② t❡r♠s

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

✶ ✴ ✷✺

■♥tr♦❞✉❝t✐♦♥

▼❛♥② ✐♥t❡r❡st✐♥❣ ♣❤②s✐❝❛❧ ♣❤❡♥♦♠❡♥❛ ✭s✉❝❤ ❛s ✈✐s❝♦❡❧❛st✐❝✐t②✱ ♣♦♣✉❧❛t✐♦♥ ❞②♥❛♠✐❝s ♦r ❤❡❛t ✢♦✇ ✐♥ r❡❛❧ ❝♦♥❞✉❝t♦rs✱ t♦ ♥❛♠❡ s♦♠❡✮ ❛r❡ ♠♦❞❡❧❧❡❞ ❜② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇❤✐❝❤ ❛r❡ ✐♥✢✉❡♥❝❡❞ ❜② t❤❡ ♣❛st ✈❛❧✉❡s ♦❢ ♦♥❡ ♦r ♠♦r❡ ✈❛r✐❛❜❧❡s✿ t❤❛t ❛r❡ s♦✲❝❛❧❧❡❞ ❡q✉❛t✐♦♥s ✇✐t❤ ♠❡♠♦r②✳ ❚❤❡ ♠❛✐♥ ♣r♦❜❧❡♠ ✐♥ t❤❡ ❛♥❛❧②s✐s ♦❢ ❡q✉❛t✐♦♥s ♦❢ t❤✐s ❦✐♥❞ ❧✐❡s ✐♥ t❤❡✐r ♥♦♥❧♦❝❛❧ ❝❤❛r❛❝t❡r✱ ❞✉❡ t♦ t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ ♠❡♠♦r② t❡r♠s ✭✐♥ ❣❡♥❡r❛❧✱ t❤❡ t✐♠❡ ❝♦♥✈♦❧✉t✐♦♥ ♦❢ t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ ❛❣❛✐♥st ❛ s✉✐t❛❜❧❡ ♠❡♠♦r② ❦❡r♥❡❧s✮✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❝♦rr❡❝t ♠♦❞❡❧❧✐♥❣ ♦❢ ♠❛t❡r✐❛❧s ✇✐t❤ ♠❡♠♦r② ❤❛s ❛❧✇❛②s r❡♣r❡s❡♥t❡❞ ❛ ♠❛❥♦r ❝❤❛❧❧❡♥❣❡ t♦ ♠❛t❤❡♠❛t✐❝✐❛♥s✳ ❚❤❡ ♦r✐❣✐♥s ♦❢ ♠♦❞❡r♥ ✈✐s❝♦❡❧❛st✐❝✐t② ❛♥❞✱ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ♦❢ t❤❡ s♦✲❝❛❧❧❡❞ ❤❡r❡❞✐t❛r② s②st❡♠s tr❛❞✐t✐♦♥❛❧❧② tr❛❝❡ ❜❛❝❦ t♦ t❤❡ ✇♦r❦s ♦❢ ▲✉❞✇✐❣ ❇♦❧t③♠❛♥♥ ❛♥❞ ❱✐t♦ ❱♦❧t❡rr❛✱ ✇❤♦ ✜rst ✐♥tr♦❞✉❝❡❞ t❤❡ ♥♦t✐♦♥ ♦❢ ♠❡♠♦r② ✐♥ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ t❤❡ ❛♥❛❧②s✐s ♦❢ ❡❧❛st✐❝ ♠❛t❡r✐❛❧s✳ ❚❤❡ ❦❡② ❛ss✉♠♣t✐♦♥ ✐♥ t❤❡ ❤❡r❡❞✐t❛r② t❤❡♦r② ♦❢ ❡❧❛st✐❝✐t② ❝❛♥ ❜❡ st❛t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✳ ❋♦r ❛♥ ✈✐s❝♦❡❧❛st✐❝ ❜♦❞② ♦❝❝✉♣②✐♥❣ ❛ ❝❡rt❛✐♥ r❡❣✐♦♥ B ⊂ RN ❛t r❡st✱ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♠❡❝❤❛♥✐❝❛❧ s②st❡♠ ❛t ❛♥② ♣♦✐♥t x ∈ B ✐s ❛ ❢✉♥❝t✐♦♥ ❜♦t❤ ♦❢ t❤❡ ✐♥st❛♥t❛♥❡♦✉s str❡ss ❛♥❞ ♦❢ ❛❧❧ t❤❡ ♣❛st str❡ss❡s ❛t x✳

✷ ✴ ✷✺

❋♦r ❡①❛♠♣❧❡ ✇❡ ❝♦♥s✐❞❡r t❤❡ ●✉rt✐♥✲P✐♣❦✐♥ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ utt(x, t) = uxx(x, t) − t K(t − τ)uxx(x, τ)dτ + f(x, t) ✇❤✐❝❤ ❞❡s❝r✐❜❡s t❤❡ ♣r♦❝❡ss ♦❢ ❤❡❛t ♣r♦♣❛❣❛t✐♦♥ ✐♥ ♠❡❞✐❛ ✇✐t❤ ♠❡♠♦r②✱ ♣r♦❝❡ss ♦❢ ✇❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ t❤❡ ✈✐s❝♦❡❧❛st✐❝ ♠❡❞✐❛ ❛♥❞ ❛❧s♦ ❛r✐s✐♥❣ ✐♥ t❤❡ ♣r♦❜❧❡♠s ♦❢ ♣♦r♦✉s ♠❡❞✐❛ ✭❉❛r❝② ❧❛✇✮✳ ❙❡❡

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

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

✸ ✴ ✷✺

❲❡ ❝♦♥s✐❞❡r ❛❧s♦ t❤❡ ❡①❛♠♣❧❡ ♦❢ ❞②♥❛♠✐❝❛❧ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ♦❢ ♠♦t✐♦♥ ♦❢ ✈✐s❝♦❡❧❛st✐❝ ✐s♦tr♦♣✐❝ ♠❡❞✐❛ utt(x, t) = ρ−1 [µ∆u(x, t) + (µ + λ)/3 · ❣r❛❞(❞✐✈u(x, t))] − − t K(t − τ)ρ−1µ [∆u(x, τ) + 1/3 · ❣r❛❞(❞✐✈u(x, τ))] dτ− − t Q(t − τ)ρ−1λ [1/3 · ❣r❛❞(❞✐✈u(x, τ))] dτ + f(x, t) ✭✶✮ ❤❡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✜❡❞✳ ❙❡❡ ❆✳❆✳ ■❧②✉s❤✐♥✱ ❇✳❊✳ P♦❜❡❞r②❛ ❋♦✉♥❞❛t✐♦♥s ♦❢ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ t❤❡♦r② ♦❢ t❤❡r♠♦✈✐s❝♦❡❧❛st✐❝✐t②✳ ✲ ▼✳ ◆❛✉❦❛✱ ✶✾✼✵✱ ❘✳▼✳ ❈❤r✐st❡♥s❡♥ ❚❤❡♦r② ♦❢ ✈✐s❝♦❡❧❛st✐❝✐t②✳ ❆♥ ✐♥tr♦❞✉❝t✐♦♥✳ ✲ ❆❝❛❞❡♠✐❝ Pr❡ss ◆❡✇ ❨♦r❦ ❛♥❞ ▲♦♥❞♦♥✱ ✶✾✼✶

✹ ✴ ✷✺

❊①❛♠♣❧❡s ♦❢ t❤❡ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥s

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

m

• k=1

akRk (t) , Q (t) =

m

• k=1

bkRk (t) ✭✷✮ ■✮ Rk (t) = e−βkt✱ ✇❤❡r❡ βk+1 > βk > 0✱ l ∈ N✱ βk → +∞ (j → +∞)✳ ■■✮ Rk (t) = tα−1e−βkt Γ(α) ✱ ■■■✮ Rk(t) = tα−1

∞

• n=0

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

✺ ✴ ✷✺

Rk(t) = ∞ e−tτdµk(τ), ✇❤❡r❡ dµj ✐s ❛ ♣♦s✐t✐✈❡ ♠❡❛s✉r❡✳ ❲❡ ✐❞❡♥t✐❢② t❤✐s ♠❡❛s✉r❡ ✇✐t❤ ✐ts ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s µj✱ s♦ µj ✐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❤❡ ❘❛❜♦t♥♦✈ ❢✉♥❝t✐♦♥ ✭■■■✮ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ Rk(t) = 1 2πi lim

R→+∞ γ+iR

• γ−iR

eλtdλ λα + βk = sin πα π

+∞

• e−tτdτ

τ α + 2βk cos πα + β2

kτ −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.

✻ ✴ ✷✺

■♥ t❤✐s t❛❧❦✱ ✇❡ ❞✐s❝✉ss ❛♥ ❛❜str❛❝t ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ ♠❡♠♦r② ❛r✐s✐♥❣ ❢r♦♠ ❧✐♥❡❛r ✈✐s❝♦❡❧❛st✐❝✐t② ✭✶✮✱ ♣r❡s❡♥t✐♥❣ ❛♥ ❛♣♣r♦❛❝❤ ✇❤✐❝❤ ❝❛♥ ❜❡ ❡❛s✐❧② ❡①t❡♥❞❡❞ ❛♥❞ ❛❞❛♣t❡❞ t♦ ♠❛♥② ♦t❤❡r ❞✐✛❡r❡♥t✐❛❧ ♠♦❞❡❧s ❝♦♥t❛✐♥✐♥❣ ♠❡♠♦r② t❡r♠s✳ ❖✉r ❛♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ t❤❡ r❡s✉❧ts ♣r❡s❡♥t❡❞ ✐♥ ❢♦❧❧♦✇✐♥❣ ♠♦♥♦❣r❛♣❤s✿ ❬✶❪ ❑✳❏✳ ❊♥❣❡❧✱ ❘✳ ◆❛❣❡❧ ❖♥❡✲P❛r❛♠❡t❡r ❙❡♠✐❣r♦✉♣s ❢♦r ▲✐♥❡❛r ❊✈♦❧✉t✐♦♥ ❊q✉❛t✐♦♥s✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✱ ✷✵✵✵✳ ❬✷❪ ❆♠❡♥❞♦❧❛ ●✳✱ ❋❛❜r✐③✐♦ ▼✳✱ ●♦❧❞❡♥ ❏✳ ▼✳ ❚❤❡r♠♦❞②♥❛♠✐❝s ♦❢ ▼❛t❡r✐❛❧s ✇✐t❤ ♠❡♠♦r②✳❚❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❙♣r✐♥❣❡r ◆❡✇✲❨♦r❦ ✲ ❉♦r❞r❡❝❤t ✲ ❍❡✐❞❡❧❜❡r❣ ✲ ▲♦♥❞♦♥✱ ✷✵✶✷ ❬✸❪ ❙✳ ●✳ ❑r❡✐♥ ▲✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✱ ❚❤❛♥s❧✳ ▼❛t❤✳ ▼♦♥♦❣r❛♣❤s✱ ✈♦❧✳✷✾✱ ❆▼❙✱ ✶✾✼✶✳ ❛♥❞ t❤❡ ❧✐t❡r❛t✉r❡ ❝✐t❡❞ t❤❡r❡✐♥✳

✼ ✴ ✷✺

❙t❛t❡♠❡♥t ♦❢ t❤❡ ♣r♦❜❧❡♠

▲❡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 + (A + B) u(t) −

m

• k=1

t Rk(t − s) (akA + bkB) u(s)ds = = f(t), ✭✸✮ u(+0) = ϕ0, u(1)(+0) = ϕ1. ✭✹✮ ✇❤❡r❡ ak > 0✱ bk > 0✱ k ∈ N✳

✽ ✴ ✷✺

❙✉♣♣♦s❡ t❤❛t t❤❡ ❦❡r♥❡❧s ♦❢ t❤❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦rs Rk : R+ → R+ ❛r❡ t❤❡ ✭str✐❝t❧② ♣♦s✐t✐✈❡✮ ♥♦♥✐♥❝r❡❛s✐♥❣ s✉♠♠❛❜❧❡ ❢✉♥❝t✐♦♥s✱ Rk(t) ∈ L1(R+) ❛♥❞ lim

t→+∞ Rk(t) = 0 ❢♦r ❛♥② k = 1...m✳

■♥ ❛❞❞✐t✐♦♥✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ s❛t✐s✜❡❞✿

m

• k=1
• ak

+∞ Rk(s)ds

• < 1,

m

• k=1
• bk

+∞ Rk(s)ds

• < 1.

✭✺✮

✾ ✴ ✷✺

❘❡❞✉❝t✐♦♥ t♦ t❤❡ ✜rst✲♦r❞❡r ♦♣❡r❛t♦r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥

❙❡t Mk(t) := +∞

t

Rk(s)ds = +∞ Rk(t + s)ds, k ∈ N, A0 :=

• 1 −

m

• k=1

akMk(0)

• A +
• 1 −

m

• k=1

bkMk(0)

• B,

Ak := akA + bkB ❇② ✈✐rt✉❡ ♦❢ ✇❡❧❧✲❦♥♦✇♥ r❡s✉❧ts ✭s❡❡ ♠♦♥♦❣r❛♣❤ ❚✳ ❑❛t♦ P❡rt✉r❜❛t✐♦♥ ❚❤❡♦r② ❢♦r ▲✐♥❡❛r ❖♣❡r❛t♦rs✴✴ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣ ◆❡✇ ❨♦r❦✱ ✶✾✽✵✱ ♣✳ ✸✻✶❪✮ ❛♥❞ ❝♦♥❞✐t✐♦♥ ✭✺✮✱ A0✱ Ak ❛r❡ t❤❡ s❡❧❢✲❛❞❥♦✐♥t ♣♦s✐t✐✈❡ ♦♣❡r❛t♦rs ❢♦r ❛♥② k = 1...m✳ ❘❡♠❛r❦✳ ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♦♣❡r❛t♦rs A ❛♥❞ B t❤❛t A0✱ Ak ❛r❡ t❤❡ ✐♥✈❡rt✐❜❧❡ ♦♣❡r❛t♦rs✱ Qk := A1/2

k

A−1/2 ❛r❡ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs✱ ❛♥❞ A−1 ✐s ❛ ❝♦♠♣❛❝t ♦♣❡r❛t♦r✳

✶✵ ✴ ✷✺

❉❡✜♥✐t✐♦♥ ✭✶✮

❆ ✈❡❝t♦r ❢✉♥❝t✐♦♥ u(t) ✐s ❝❛❧❧❡❞ ❛ str♦♥❣ s♦❧✉t✐♦♥ ♦❢ ♣r♦❜❧❡♠ ✭✸✮✱ ✭✹✮ ✐❢ u(t) ∈ C2(R+, H)✱ Au(t), Bu(t) ∈ C(R+, H)✱ u(t) s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥ ✭✸✮ ❢♦r ❡✈❡r② t ∈ R+ ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✭✹✮✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❛r✐❛❜❧❡s✿ v(t) := u′(t)✱ ξ0(t) := A1/2 u(t)✱ ξk(t, τ) := t Rk(t + τ − s)QkA1/2 v(s)ds, τ > 0, k = 1...m. f1(t) := f(t) −

m

• k=1

Mk(t)Akϕ0.

✶✶ ✴ ✷✺

❚❤❡r❡❢♦r❡✱ t❤❡ ♣r♦❜❧❡♠ ✭✸✮✱ ✭✹✮ ✐s ✭❢♦r♠❛❧❧②✮ tr❛♥s❧❛t❡❞ ✐♥t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠ ♦♥ t❤❡ s❡♠✐❛①✐s R+ = (0, ∞)✿                  dv(t) dt + A1/2

• ξ0(t) +

m

• k=1

Q∗

k

+∞ ξk(t, τ)dτ

• = f1(t),

dξ0(t) dt = A1/2 v(t), dξk(t, τ) dt = Rk(τ)QkA1/2 v(t) + ∂ ∂τ ξk(t, τ), τ > 0, k = 1...m, ✭✻✮ v(t)|t=0 = ϕ1, ξ0(t)|t=0 = A1/2 ϕ0, ξk(t, τ)|t=0 = 0, τ > 0, k = 1...m. ✭✼✮ ◆♦✇ ♦✉r ♠❛✐♥ ❣♦❛❧s ❛r❡ ❢♦❧❧♦✇s✿ ❋✐rst❧②✱ ✇❡ ❤❛✈❡ t♦ ✇r✐t❡ ✭✻✮✱ ✭✼✮ ❛s ❛ ♣r♦❜❧❡♠ ✐♥ ❛ s✉✐t❛❜❧❡ ❢✉♥❝t✐♦♥❛❧ s♣❛❝❡✱ ♣r♦✈✐❞✐♥❣ ❛♥ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss r❡s✉❧t✳ ❙❡❝♦♥❞❧②✱ ✇❡ ❤❛✈❡ t♦ ❡st❛❜❧✐s❤ ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡ ✭♥♦t ♦♥❧② ❢♦r♠❛❧✮ ❜❡t✇❡❡♥ t❤❡ s♦❧✉t✐♦♥s t♦ ✭✻✮✱ ✭✼✮ ❛♥❞ t❤❡ s♦❧✉t✐♦♥s t♦ t❤❡ ♦r✐❣✐♥❛❧ ♣r♦❜❧❡♠ ✭✸✮✱ ✭✹✮✳

✶✷ ✴ ✷✺

❚r❛♥s❧❛t✐♦♥ s❡♠✐❣r♦✉♣s ✐♥ L2

rk(R+, H) s♣❛❝❡s

❚❤❡ ✜rst st❡♣ t♦ s❡t t❤❡ ♣r♦❜❧❡♠ ✭✻✮✱ ✭✼✮ ✐♥ ♣r♦♣❡r ❢✉♥❝t✐♦♥❛❧ s♣❛❝❡ ✐s t♦ ✐♥t❡r♣r❡t ✐♥ ❛ ❝♦rr❡❝t ✇❛② t❤❡ ❞❡r✐✈❛t✐✈❡

∂ ∂τ (·) ❛♣♣❡❛r❡❞ ✐♥ t❤❡ t❤✐r❞ ❡q✉❛t✐♦♥

♦❢ ✭✻✮✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ♥❡✇ ♠❡♠♦r② ❦❡r♥❡❧s rk(τ) = 1 Rk(τ) : R+ → R+, ❛♥❞ ✇❡ ♣✉t rk(0) := lim

τ→0 rk(τ)✱ k = 1...m✳ ■♥ ✈✐❡✇ ♦❢ t❤❡ ❛ss✉♠♣t✐♦♥s ♦♥

Rk(τ) t❤❡ ❢✉♥❝t✐♦♥s rk(τ) ❛r❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ♥♦♥❞❡❝r❡❛s✐♥❣ ♦♥ R+ ✇✐t❤ ♥♦♥♥❡❣❛t✐✈❡ ❞❡r✐✈❛t✐✈❡✳ ▼♦r❡♦✈❡r✱ lim

τ→+∞ rk(τ) = +∞✳

▲❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡ s♣❛❝❡s Ωk = L2

rk(R+, H)✱ k = 1...m✱ ❡q✉✐♣♣❡❞ ✇✐t❤

t❤❡ ♥♦r♠ ||u||Ωk = +∞ rk(s)||u(s)||2

Hds

1/2 ,

✶✸ ✴ ✷✺

❲❡ ❝♦♥s✐❞❡r t❤❡ str♦♥❣❧② ❝♦♥t✐♥✉♦✉s s❡♠✐❣r♦✉♣ Lk(t) ♦❢ ❧❡❢t tr❛♥s❧❛t✐♦♥s ♦♥ Ωk✱ ❞❡✜♥❡❞ ❜② (Lk(t)ξ) (τ) = ξ(t + τ). ❚❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ❣❡♥❡r❛t♦r ♦❢ Lk(t) ✐s t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r Tkξ(τ) = ξ′(τ) ♦♥ Ωk ✇✐t❤ ❞♦♠❛✐♥ D(Tk) =

• ξ ∈ Ωk : ξ′(τ) ∈ Ωk, lim

τ→+∞ ||ξ(τ)||H = 0

• .

▲❡♠♠❛ ✭✶✮

❋♦r ❡✈❡r② ξ ∈ D(Tk)✿ +∞ r′k(s)||ξ(s)||2

Hds < ∞ ❛♥❞ t❤❡r❡ ❡①✐sts ❧✐♠✐t

lim

τ→0 rk(τ)||ξ(τ)||2 H ✭❡q✉❛❧ t♦ ③❡r♦ ✐❢ lim τ→∞ rk(τ) = 0✮✳ ▼♦r❡♦✈❡r✱

2 Re ∂τξ(τ), ξ(τ)Ωk = − lim

τ→0 rk(τ)||ξ(τ)||2 H−

+∞ r′k(s)||ξ(s)||2

Hds 0.

✶✹ ✴ ✷✺

❚❤❡ s❡♠✐❣r♦✉♣ ✐♥ t❤❡ ❡①t❡♥❞❡❞ ❍✐❧❜❡rt s♣❛❝❡

❲❡ ♠❛❦❡ t❤❡ ❞♦♠❛✐♥ Dom(Aβ

0) ♦❢ t❤❡ ♦♣❡r❛t♦r Aβ 0✱ β > 0 ❛ ❍✐❧❜❡rt s♣❛❝❡

Hβ✱ ❜② ✐♥tr♦❞✉❝✐♥❣ t❤❡ ♥♦r♠ ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❣r❛♣❤ ♥♦r♠ ♦❢ t❤❡ ♦♣❡r❛t♦r Aβ

0 ♦♥ Dom(Aβ 0) ✇✐t❤ st❛♥❞❛r❞ ✐♥♥❡r ♣r♦❞✉❝t u1, u2β =

• Aβ

0u1, Aβ 0u2

• H✳

❲❡ ❢♦r♠✉❧❛t❡ ✭✻✮✱ ✭✼✮ ❛s ❛♥ ❛❜str❛❝t ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥ ♦♥ ❛ s✉✐t❛❜❧❡ ❍✐❧❜❡rt s♣❛❝❡✳ ❚♦ t❤✐s ❡♥❞✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❡①t❡♥❞❡❞ ❍✐❧❜❡rt s♣❛❝❡ H = H ⊕ H ⊕ (⊕m

k=1Ωk)

♥♦r♠❡❞ ❜② (v, ξ0, ξ1(τ), ..., ξm(τ))2

H = ||v||2 H + ||ξ0||2 H + m

• k=1

||ξk||2

Ωk,

τ > 0.

✶✺ ✴ ✷✺

❛♥❞ t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r A ♦♥ H ✇✐t❤ ❞♦♠❛✐♥ D(A) =

• (v, ξ0, ξ1(τ), ..., ξm(τ)) ∈ H : v ∈ H1/2,

ξ0 +

m

• k=1

Q∗

k

+∞ ξk(τ)dτ ∈ H1/2, ξk(τ) ∈ D(Tk), k = 1, ..., m

• ❛❝t✐♥❣ ❛s

A(v, ξ0, ξ1(τ), ..., ξm(τ)) =

• −A1/2
• ξ0 +

m

• k=1

Q∗

k

+∞ ξk(τ)dτ

• ,

A1/2 v, Rk(τ)QkA1/2 v + Tkξk(τ), k = 1, ..., m

• ✶✻ ✴ ✷✺

■♥tr♦❞✉❝✐♥❣ t❤❡ (2 + m)− ❝♦♠♣♦♥❡♥t ✈❡❝t♦rs Z(t) = (v(t), ξ0(t), ξ1(t, τ), ..., ξm(t, τ)) ∈ H ❛♥❞ z = (v0, ξ00, ξ10(τ), ..., ξm0(τ)) ∈ H. ✇❡ ❝♦♥s✐❞❡r t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✐♥ H    d dtZ(t) = AZ(t) (8.1) Z(0) = z (8.2) ✭✽✮

❉❡✜♥✐t✐♦♥ ✭✷✮

❆ ✈❡❝t♦r Z(t) = (v(t), ξ0(t), ξ1(t, τ), ..., ξm(t, τ)) ∈ H ✐s ❝❛❧❧❡❞ ❛ s♦❧✉t✐♦♥ ♦❢ ♣r♦❜❧❡♠ ✭✽✮ ✐❢ ✶✮ v(t) ∈ C([0, +∞), H1/2) ∩ C1([0, +∞), H)✱ ξ0(t) ∈ C1([0, +∞), H) ✷✮ ξk(t, τ) ∈ C1([0, +∞), H) ❢♦r ❛♥② τ > 0✱ k = 1, ..., m❀ ✸✮ Z(t) ∈ C([0, +∞), D(A))✳ ✈❡❝t♦r Z(t) s❛t✐s✜❡s ❡q✉❛t✐♦♥ ✭✽✳✶✮ ❢♦r ❡✈❡r② t ∈ R+ ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✭✽✳✷✮✳

✶✼ ✴ ✷✺

▲❡♠♠❛ ✭✷✮

▲✐♥❡❛r ♦♣❡r❛t♦r A ♦♥ H ✐s ♠❛①✐♠❛❧ ❞✐ss✐♣❛t✐✈❡ t❤❛t ✐s t❤❡ ✐♥❡q✉❛❧✐t② Re Az, zH 0 ❤♦❧❞s ❢♦r ❡✈❡r② z ∈ D(A) ❛♥❞ range (I − A) = H✳ ▼♦r❡♦✈❡r✱ t❤❡ ❡q✉❛❧✐t② Re Az, zH = Re

m

• k=1
• ξ′

k(τ), ξk(τ)

• Ωk

❤♦❧❞s ❢♦r ❡✈❡r② z ∈ D(A)✳

❚❤❡♦r❡♠ ✭✶✮

Pr♦❜❧❡♠ ✭✽✮ ❣❡♥❡r❛t❡s ❛ ❝♦♥tr❛❝t✐♦♥ s❡♠✐❣r♦✉♣ S(t) = etA ♦♥ H s✉❝❤ t❤❛t Z(t) = S(t)z✱ t > 0. ▼♦r❡♦✈❡r✱ t❤❡ ❡♥❡r❣② ❡q✉❛❧✐t② d dt||S(t)z||2

H = − m

• k=1

  lim

τ→0+ rk(τ)||ξk(t, τ)||2 H + +∞

• r′

k(τ)||ξk(t, τ)||2 Hdτ

  ❤♦❧❞s ❢♦r ❡✈❡r② z ∈ D(A)✳

✶✽ ✴ ✷✺

❊①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t②

❲❡ ❢♦r♠✉❧❛t❡ t❤❡ t❤❡♦r❡♠ ❛❜♦✉t ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ♦❢ t❤❡ s❡♠✐❣r♦✉♣ S(t) ♦♥ H✱ ❛ss✉♠✐♥❣ t❤❛t ❦❡r♥❡❧ ❢✉♥❝t✐♦♥s Rk(τ) s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ R′

k(τ) + γRk(τ) 0,

✭✾✮ ❢♦r s♦♠❡ γ > 0 ❛♥❞ ❛❧♠♦st ❡✈❡r② τ > 0✳ ❈♦♥❞✐t✐♦♥ ✭✾✮ ✐s q✉✐t❡ ♣♦♣✉❧❛r ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳ ■t ❤❛s ❜❡❡♥ ✉s❡❞ ❜② s❡✈❡r❛❧ ❛✉t❤♦rs t♦ ♣r♦✈❡ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ♦❢ s❡♠✐❣r♦✉♣s r❡❧❛t❡❞ t♦ ✈❛r✐♦✉s ❡q✉❛t✐♦♥s ✇✐t❤ ♠❡♠♦r②✳ ❙❡❡✱ ❢♦r ❡①❛♠♣❧❡ ❬✷❪ ❆♠❡♥❞♦❧❛ ●✳✱ ❋❛❜r✐③✐♦ ▼✳✱ ●♦❧❞❡♥ ❏✳ ▼✳ ❚❤❡r♠♦❞②♥❛♠✐❝s ♦❢ ▼❛t❡r✐❛❧s ✇✐t❤ ♠❡♠♦r②✳ ❚❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❙♣r✐♥❣❡r ◆❡✇✲❨♦r❦ ✲ ❉♦r❞r❡❝❤t ✲ ❍❡✐❞❡❧❜❡r❣ ✲ ▲♦♥❞♦♥✱ ✷✵✶✷ ❛♥❞ t❤❡ ❧✐t❡r❛t✉r❡ ❝✐t❡❞ t❤❡r❡✐♥✳ ❘❡♠❛r❦✳ ❊①♣♦♥❡♥t✐❛❧ ❛♥❞ ❢r❛❝t✐♦♥❛❧ ❡①♣♦♥❡♥t✐❛❧ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥s ✭■✮✲✭■■■✮ s❛t✐s❢② t❤❡ ❝♦♥❞✐t✐♦♥ ✭✾✮✳

✶✾ ✴ ✷✺

❚❤❡♦r❡♠ ✭✷✮

▲❡t t❤❡ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥s Rk(τ) s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥ ✭✾✮ ❢♦r s♦♠❡ γ > 0✱ ❛❧♠♦st ❡✈❡r② τ > 0 ❛♥❞ k = 1, ..., m✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ω > 0 s✉❝❤ t❤❛t S(t)zH √ 3 zH e−ωt ❢♦r ❡✈❡r② z ∈ H✳ ▼♦r❡♦✈❡r ω = max

β>0 ωβ✱ ωβ = 1 6 min

• γ

γ1(β); 1 γ2(β)

• ✱

γ1(β) := 3 Mβ max

1km

• Mk(0)
• 1

Mk(β)

• 3

||Qk|| + 1 2β2ak1

• +

+Mβ 3

• 1 +

1 2Mk(0)

• ,

γ2(β) := 1 2 3 Mβ + 1

• max

1km {Mk(0); 1}

Mk(β) := +∞

β

Rk(s)ds, Mβ :=

m

• k=1

Mk(β), Aken = aknen, n ∈ N.

✷✵ ✴ ✷✺

❈♦rr❡❝t s♦❧✈❛❜✐❧✐t②

❲❡ ✇✐❧❧ ❢♦❧❧♦✇ ❜② ❞❡✜♥✐t✐♦♥s ❢r♦♠ t❤❡ ♠♦♥♦❣r❛♣❤ ❬✸❪ ❙✳ ●✳ ❑r❡✐♥ ▲✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✱ ❚❤❛♥s❧✳ ▼❛t❤✳ ▼♦♥♦❣r❛♣❤s✱ ✈♦❧✳✷✾✱ ❆▼❙✱ ✶✾✼✶✳

❉❡✜♥✐t✐♦♥ ✭✸✮

❚❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠    d dtZ(t) = AZ(t) Z(0) = z ✭✶✵✮ ✐s ✉♥✐❢♦r♠❧② ❝♦rr❡❝t ♣♦s❡❞ ✐❢ ✶✮ ❢♦r ❛♥② z ∈ D(A) ✐t ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥✱ ✷✮ t❤✐s s♦❧✉t✐♦♥ ❞❡♣❡♥❞s ❝♦♥t✐♥✉♦✉s❧② ♦♥ t❤❡ ✐♥✐t✐❛❧ ❞❛t❛ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐❢ Zn(0) → 0 (Zn(0) ∈ D(A)) ✐♠♣❧✐❡s t❤❛t Z(t) → 0 ✉♥✐❢♦r♠❧② ✐♥ t ♦♥ ❡❛❝❤ ✐♥t❡r✈❛❧ ✜♥✐t❡ ✐♥t❡r✈❛❧ [0, T]✳

✷✶ ✴ ✷✺

❘❡♠❛r❦

■❢ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✭✶✵✮ ❣❡♥❡r❛t❡s ❛ ❝♦♥tr❛❝t✐♦♥ s❡♠✐❣r♦✉♣ ♦♥ H t❤❡♥ t❤✐s ♣r♦❜❧❡♠ ✐s ✉♥✐❢♦r♠❧② ❝♦rr❡❝t✳

❚❤❡♦r❡♠ ✭✻✳✺✱ ❙✳●✳❑r❡✐♥✮

■❢ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✭✶✵✮ ✐s ✉♥✐❢♦r♠❧② ❝♦rr❡❝t t❤❡♥ t❤❡ ❢♦r♠✉❧❛ Z(t) = S(t)z + t S(t − p)F(p)dp ②✐❡❧❞s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❢♦r t❤❡ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥✿ d dtZ(t) = AZ(t) + F(t) Z(0) = z ❢♦r z ∈ D(A) ❛♥❞ ❛ ❢✉♥❝t✐♦♥ F(t) s❛t✐s❢②✐♥❣ ♦♥❡ ♦❢ t❤❡ t✇♦ ❝♦♥❞✐t✐♦♥s ✇❤✐❝❤ ❢♦❧❧♦✇✿ ✶✮ ❚❤❡ ✈❛❧✉❡ F(t) ∈ D(A) ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ AF(t) ∈ C(R+, H)❀ ✷✮ ❚❤❡ ❢✉♥❝t✐♦♥ F(t) ∈ C1(R+, H).

✷✷ ✴ ✷✺

❚❤❡♦r❡♠ ✭✸✮

▲❡t ✉s ❝♦♥s✐❞❡r ♣r♦❜❧❡♠ ✭✽✮ ✇✐t❤ ♥♦♥❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥    d dtZ(t) = AZ(t) + F(t) Z(0) = z ✇❤❡r❡ F(t) := (f1(t), 0, ...0

m+1

)✱ f1(t) = f(t) −

m

• k=1

Mk(t)Akϕ0✱ z =  ϕ1, A1/2 ϕ0, 0...0

• m

  ❛♥❞ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ✻✳✺ ❛r❡ s❛t✐s✜❡❞✳ ❚❤❡♥ t❤❡ ♣r♦❜❧❡♠ ✭✽✮ ❤❛s t❤❡ ✉♥✐q✉❡ str♦♥❣ s♦❧✉t✐♦♥ Z(t) = (v(t), ξ0(t), ξ1(t, τ), ..., ξm(t, τ)) ✇❤❡r❡ v(t) := u′(t)✱ ξ0(t) := A1/2 u(t) ❛♥❞ u(t) ✐s t❤❡ str♦♥❣ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✸✮✱ ✭✹✮✳ ▼♦r❡♦✈❡r✱ ✐❢ t❤❡ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥s Rk(τ) s❛t✐s❢② t❤❡ ❝♦♥❞✐t✐♦♥ ✭✾✮✱ t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡s ✐s ✈❛❧✐❞ Z(t)H d

• zHe−ωt +

t e−ω(t−s)F(s)Hds

• ✇✐t❤ t❤❡ ❝♦♥st❛♥t d ✐♥❞❡♣❡♥❞❡♥t ♦♥ ✈❡❝t♦r✲❢✉♥❝t✐♦♥ F ❛♥❞ ✈❡❝t♦rs ϕ0 ✱ ϕ1 ❛♥❞ ω

❞❡✜♥❡❞ ✐♥ t❤❡ ❚❤❡♦r❡♠ ✻✳

✷✸ ✴ ✷✺

❈♦r♦❧❧❛r②

▲❡t ✉s A1/2 f(t) ∈ C (R+, H) , Mk(t) ∈ C (R+)✱ k = 1, ..., m, ϕ0 ∈ H3/2, ϕ1 ∈ H1/2, t❤❡ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥s Rk(τ) s❛t✐s❢② t❤❡ ❝♦♥❞✐t✐♦♥ ✭✾✮✳ ❚❤❡♥ t❤❡ ♣r♦❜❧❡♠ ✭✶✮✱ ✭✷✮ ❤❛s t❤❡ ✉♥✐q✉❡ str♦♥❣ s♦❧✉t✐♦♥✳ ▼♦r❡♦✈❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡ ✐s ✈❛❧✐❞ E(t) := 1 2

• u′(t)
• 2

H +

• A1/2

u(t)

• 2

H

• d
• ϕ12

H +

• A1/2

ϕ0

• 2

H

• e−2ωt+

+

m

• k=1

t e−ω(t−s) +∞

s

Rk(p)dp

• ds

2 Akϕ02

H+

+ t e−ω(t−s)||f(s)||ds 2 ✇✐t❤ t❤❡ ❝♦♥st❛♥t d ✐♥❞❡♣❡♥❞❡♥t ♦♥ ✈❡❝t♦r✲❢✉♥❝t✐♦♥ f ❛♥❞ ✈❡❝t♦rs ϕ0✱ ϕ1 ❛♥❞ ω ❞❡✜♥❡❞ ✐♥ t❤❡ ❚❤❡♦r❡♠ ✻✳

✷✹ ✴ ✷✺

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

✷✺ ✴ ✷✺