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▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ ❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥s ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥s

❨✉r✐ ▲✉❝❤❦♦

❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s✱ P❤②s✐❝s✱ ❛♥❞ ❈❤❡♠✐str② ❇❡✉t❤ ❚❡❝❤♥✐❝❛❧ ❯♥✐✈❡rs✐t② ♦❢ ❆♣♣❧✐❡❞ ❙❝✐❡♥❝❡s ❇❡r❧✐♥ ❇❡r❧✐♥✱ ●❡r♠❛♥② ❋r❛❝t✐♦♥❛❧ P❉❊s✿ ❚❤❡♦r②✱ ❆❧❣♦r✐t❤♠s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ■❈❊❘▼ ❛t ❇r♦✇♥ ❯♥✐✈❡rs✐t② ❏✉♥❡ ✶✽ ✲ ✷✷✱ ✷✵✶✽

❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦✿

• ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r ❛ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡

❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

• ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ ✇❡❛❦ s♦❧✉t✐♦♥s ♦❢ ❛ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥

❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❈❛♣✉t♦ ❞❡r✐✈❛t✐✈❡

• ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r ❛♥ ❛❜str❛❝t s♣❛❝❡✲ ❛♥❞ t✐♠❡✲❢r❛❝t✐♦♥❛❧

❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❍✐❧❜❡rt s♣❛❝❡

• ❙❤♦rt s✉r✈❡② ♦❢ ♦t❤❡r r❡s✉❧ts

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✷ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

❲❤❛t ✐s ❛ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡❄

▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡✿ ❆ ❢✉♥❝t✐♦♥ s❛t✐s✜❡s ❛ ❞✐✛❡r❡♥t✐❛❧ ✐♥❡q✉❛❧✐t② ♦r ❡q✉❛t✐♦♥ ✐♥ ❛ ❞♦♠❛✐♥ D ⇒ ■t ❛❝❤✐❡✈❡s ✐ts ♠❛①✐♠✉♠ ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ D✳ ❆ ✈❡r② ❡❧❡♠❡♥t❛r② ❡①❛♠♣❧❡✿ f ′′(x) > ✵, x ∈]a, b[ ❛♥❞ f ∈ C([a, b]) ⇒ f ❛❝❤✐❡✈❡s ✐ts ♠❛①✐♠✉♠ ✈❛❧✉❡ ❛t ♦♥❡ ♦❢ t❤❡ ❡♥❞♣♦✐♥ts ♦❢ t❤❡ ✐♥t❡r✈❛❧✳ ❖t❤❡r ❡①❛♠♣❧❡s✿ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡s ❢♦r ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛♥❞ ✐♥❡q✉❛❧✐t✐❡s ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡s ❢♦r ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛♥❞ ✐♥❡q✉❛❧✐t✐❡s ❱❡r② r❡❝❡♥t❧②✿ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡s ❢♦r ❢r❛❝t✐♦♥❛❧ P❉❊s

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✸ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

• ❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s

▲❡t k ❜❡ ❛ ♥♦♥♥❡❣❛t✐✈❡ ❧♦❝❛❧❧② ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❈❛♣✉t♦ t②♣❡✿ (DC

(k)f )(t) =

t

✵

k(t − τ)f ′(τ) dτ. ❚❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ t②♣❡✿ (DRL

(k)f )(t) = d

dt t

✵

k(t − τ)f (τ) dτ. ❋♦r ❛♥ ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ f ✇✐t❤ t❤❡ ✐♥❝❧✉s✐♦♥ f ′ ∈ Lloc

✶ (I

R+)✱ ✇❡ ❣❡t (DC

(k)f )(t) = d

dt t

✵

k(t −τ)f (τ) dτ − k(t)f (✵) = (DRL

(k)f )(t)−k(t)f (✵)

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✹ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

P❛rt✐❝✉❧❛r ❝❛s❡s

✶✮ ❚❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ ❈❛♣✉t♦ ❛♥❞ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s✿ k(τ) = τ −α Γ(✶ − α), ✵ < α < ✶. (Dα

∗ f )(t) =

✶ Γ(✶ − α) t

✵

(t − τ)−αf ′(τ) dτ, (Dαf )(t) = d dt

• ✶

Γ(✶ − α) t

✵

(t − τ)−αf (τ) dτ

• .

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✺ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

P❛rt✐❝✉❧❛r ❝❛s❡s

✷✮ ❚❤❡ ♠✉❧t✐✲t❡r♠ ❞❡r✐✈❛t✐✈❡s k(τ) =

n

• k=✶

ak τ −αk Γ(✶ − αk), ✵ < α✶ < · · · < αn < ✶ (DC

(k)f )(t) = n

• k=✶

ak(Dαk

∗ f )(t),

(DRL

(k)f )(t) = n

• k=✶

ak(Dαkf )(t).

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✻ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

P❛rt✐❝✉❧❛r ❝❛s❡s

✸✮ ❉❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❞✐str✐❜✉t❡❞ ♦r❞❡r✿ k(τ) = ✶

✵

τ −α Γ(✶ − α) dρ(α), ✇❤❡r❡ ρ ✐s ❛ ❇♦r❡❧ ♠❡❛s✉r❡ ♦♥ [✵, ✶]✿ (DC

(k)f ) =

✶

✵

(Dα

∗ f )(t) dρ(α),

(DC

(k)f ) =

✶

✵

(Dαf )(t) dρ(α).

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✼ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

❈♦♥❞✐t✐♦♥s ♦♥ t❤❡ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥

❑✶✮ ❚❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ˜ k ♦❢ k✱ ˜ k(p) = ∞

✵

k(t) e−pt dt, ❡①✐sts ❢♦r ❛❧❧ p > ✵✱ ❑✷✮ ˜ k(p) ✐s ❛ ❙t✐❧t❥❡s ❢✉♥❝t✐♦♥✱ ❑✸✮ ˜ k(p) → ✵ ❛♥❞ p˜ k(p) → ∞ ❛s p → ∞✱ ❑✹✮ ˜ k(p) → ∞ ❛♥❞ p˜ k(p) → ✵ ❛s p → ✵✳

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✽ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❣❡♥❡r❛❧ ❞❡r✐✈❛t✐✈❡s

✭❆✮ ❋♦r ❛♥② λ > ✵✱ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ❢♦r t❤❡ ❢r❛❝t✐♦♥❛❧ r❡❧❛①❛t✐♦♥ ❡q✉❛t✐♦♥ (DC

(k)f )(t) = −λf (t),

t > ✵, u(✵) = ✶ ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ uλ = uλ(t) t❤❛t ❜❡❧♦♥❣s t♦ t❤❡ ❝❧❛ss C ∞(I R+) ❛♥❞ ✐s ❛ ❝♦♠♣❧❡t❡❧② ♠♦♥♦t♦♥❡ ❢✉♥❝t✐♦♥✳ ✭❇✮ ❚❤❡r❡ ❡①✐sts ❛ ❝♦♠♣❧❡t❡❧② ♠♦♥♦t♦♥❡ ❢✉♥❝t✐♦♥ κ = κ(t) ✇✐t❤ t❤❡ ♣r♦♣❡rt② t

✵

k(t − τ)κ(τ) dτ ≡ ✶, t > ✵. ✭❈✮ ❋♦r f ∈ Lloc

✶ (I

R+)✱ t❤❡ r❡❧❛t✐♦♥s (DC

(k)I(k)f )(t) = f (t),

(DRL

(k)I(k)f )(t) = f (t)

❤♦❧❞ tr✉❡✱ ✇❤❡r❡ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ✐♥t❡❣r❛❧ I(k) ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❢♦r♠✉❧❛ (I(k)f )(t) = t

✵

κ(t − τ)f (τ) dτ.

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✾ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

• ❡♥❡r❛❧ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥

▲❡t Ω ❜❡ ❛♥ ♦♣❡♥ ❛♥❞ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✐♥ I Rn ✇✐t❤ ❛ s♠♦♦t❤ ❜♦✉♥❞❛r② ∂Ω ✭❢♦r ❡①❛♠♣❧❡✱ ♦❢ C ✷ ❝❧❛ss✮ ❛♥❞ T > ✵✳ ❚❤❡ ❣❡♥❡r❛❧ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥✿ (DC

(k)u(x, ·))(t)=D✷(u)+D✶(u)−q(x)u(x, t)+F(x, t), (x, t) ∈ Ω×(✵, T],

✇❤❡r❡ q ∈ C(¯ Ω)✱ q(x) ≥ ✵, x ∈ ¯ Ω✱ D✶(u) =

n

• i=✶

bi(x) ∂u ∂xi , D✷(u) =

n

• i,j=✶

ai,j(x) ∂✷u ∂xi∂xj ❛♥❞ D✷ ✐s ❛ ✉♥✐❢♦r♠❧② ❡❧❧✐♣t✐❝ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r✳

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✶✵ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

❈❛✉❝❤② ♣r♦❜❧❡♠

❑♦❝❤✉❜❡✐ ❝♦♥s✐❞❡r❡❞ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ u(x, ✵) = u✵(x), x ∈ I Rn ❢♦r t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❣❡♥❡r❛❧ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ D✷ = ∆✱ D✶ ≡ ✵ ❛♥❞ q ≡ ✵✳ ❍✐s ♠❛✐♥ r❡s✉❧ts✿ ✶✮ ❚❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❤❛s ❛ ✉♥✐q✉❡ ❛♣♣r♦♣r✐❛t❡❧② ❞❡✜♥❡❞ s♦❧✉t✐♦♥ ❢♦r ❛ ❜♦✉♥❞❡❞ ❣❧♦❜❛❧❧② ❍ö❧❞❡r ❝♦♥t✐♥✉♦✉s ✐♥✐t✐❛❧ ✈❛❧✉❡ u✵✳ ✷✮ ❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ s♦❧✉t✐♦♥ t♦ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❛ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ❛♥❞ t❤✉s t❤❡ ❣❡♥❡r❛❧ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ❞❡s❝r✐❜❡s ❛ ❦✐♥❞ ♦❢ ✭❛♥♦♠❛❧♦✉s✮ ❞✐✛✉s✐♦♥✳

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✶✶ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

■♥✐t✐❛❧✲❜♦✉♥❞❛r②✲✈❛❧✉❡ ♣r♦❜❧❡♠

▲✉❝❤❦♦ ❛♥❞ ❨❛♠❛♠♦t♦✿ ❆♥❛❧②s✐s ♦❢ ✉♥✐q✉❡♥❡ss ❛♥❞ ❡①✐st❡♥❝❡ ♦❢ s♦❧✉t✐♦♥ t♦ t❤❡ ✐♥✐t✐❛❧✲❜♦✉♥❞❛r②✲✈❛❧✉❡ ♣r♦❜❧❡♠ ❢♦r t❤❡ ❣❡♥❡r❛❧ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ u(x, t)

• t=✵ = u✵(x), x ∈ ¯

Ω ❛♥❞ t❤❡ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ u(x, t)

• (x,t)∈∂Ω×(✵,T] = v(x, t), (x, t) ∈ ∂Ω × (✵, T].

❚❤❡✐r ♠❛✐♥ r❡s✉❧ts✿ ✶✮ ❯♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥ ❜♦t❤ ✐♥ t❤❡ str♦♥❣ ❛♥❞ ✐♥ t❤❡ ✇❡❛❦ s❡♥s❡s ✭❊st✐♠❛t❡s ♦❢ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s ✕❃ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ ❣❡♥❡r❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✕❃ ❆ ♣r✐♦r② ♥♦r♠ ❡st✐♠❛t❡s ♦❢ s♦❧✉t✐♦♥s ✕❃ ❯♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥s✮✳ ✷✮ ❊①✐st❡♥❝❡ ♦❢ s♦❧✉t✐♦♥ ✐♥ t❤❡ ✇❡❛❦ s❡♥s❡ ✭❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✕❃ ❋♦r♠❛❧ s♦❧✉t✐♦♥s ✐♥ ❢♦r♠ ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❋♦✉r✐❡r s❡r✐❡s ✕❃ ❈♦♥✈❡r❣❡♥❝❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ❢♦r♠❛❧ s♦❧✉t✐♦♥s✮✳

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✶✷ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

❊st✐♠❛t❡s ♦❢ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s

▲❡t t❤❡ ❝♦♥❞✐t✐♦♥s ▲✶✮ k ∈ C ✶(I R+) ∩ Lloc

✶ (I

R+)✱ ▲✷✮ k(τ) > ✵ ❛♥❞ k′(τ) < ✵ ❢♦r τ > ✵✱ ▲✸✮ k(τ) = o(τ −✶), τ → ✵✳ ❜❡ ❢✉❧✜❧❧❡❞✳ ▲❡t ❛ ❢✉♥❝t✐♦♥ f ∈ C([✵, T]) ❛tt❛✐♥ ✐ts ♠❛①✐♠✉♠ ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ [✵, T] ❛t t❤❡ ♣♦✐♥t t✵, t✵ ∈ (✵, T]✱ ❛♥❞ f ′ ∈ C(✵, T] ∩ L✶(✵, T)✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❡q✉❛❧✐t✐❡s ❤♦❧❞ tr✉❡✿ (DRL

(k)f )(t✵) ≥ k(t✵)f (t✵),

(DC

(k)f )(t✵) ≥ k(t✵)(f (t✵) − f (✵)) ≥ ✵.

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✶✸ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡

▲❡t ✉s ❞❡✜♥❡ t❤❡ ♦♣❡r❛t♦r P(k)(u) := (DC

(k)f )(t) − D✷(u) − D✶(u) + q(x)u(x, t).

▲❡t t❤❡ ❝♦♥❞✐t✐♦♥s ▲✶✮✲▲✸✮ ❜❡ ❢✉❧✜❧❧❡❞ ❛♥❞ ❛ ❢✉♥❝t✐♦♥ u s❛t✐s❢② t❤❡ ✐♥❡q✉❛❧✐t② P(k)(u) ≤ ✵, (x, t) ∈ Ω × (✵, T]. ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❤♦❧❞s tr✉❡✿ max

(x,t)∈ ¯ Ω×[✵,T] u(x, t) ≤ max{max x∈¯ Ω u(x, ✵),

max

(x,t)∈ ∂Ω×[✵,T] u(x, t), ✵}.

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✶✹ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

❆ ♣r✐♦r✐ ♥♦r♠ ❡st✐♠❛t❡s

▲❡t t❤❡ ❝♦♥❞✐t✐♦♥s ❑✶✮✲❑✹✮ ❛♥❞ ▲✶✮✲▲✸✮ ❜❡ ❢✉❧✜❧❧❡❞ ❛♥❞ u ❜❡ ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧✲❜♦✉♥❞❛r②✲✈❛❧✉❡ ♣r♦❜❧❡♠ ❢♦r t❤❡ ❣❡♥❡r❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡ ♦❢ t❤❡ ✉♥✐❢♦r♠ s♦❧✉t✐♦♥ ♥♦r♠ ❤♦❧❞s tr✉❡✿ uC(¯

Ω×[✵,T]) ≤ max{M✵, M✶} + M f (T),

✇❤❡r❡ M✵ = u✵C(¯

Ω), M✶ = vC(∂Ω×[✵,T]), M = FC(Ω×[✵,T]),

❛♥❞ f (t) = t

✵

κ(τ) dτ, t

✵

k(t − τ)κ(τ) dτ ≡ ✶, t > ✵.

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✶✺ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

❯♥✐q✉❡♥❡ss ♦❢ t❤❡ s♦❧✉t✐♦♥

❚❤❡ ✐♥✐t✐❛❧✲❜♦✉♥❞❛r②✲✈❛❧✉❡ ♣r♦❜❧❡♠ ❢♦r t❤❡ ❣❡♥❡r❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ❡q✉❛t✐♦♥ ♣♦ss❡ss❡s ❛t ♠♦st ♦♥❡ s♦❧✉t✐♦♥✳ ❚❤✐s s♦❧✉t✐♦♥ ❝♦♥t✐♥✉♦✉s❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ♣r♦❜❧❡♠ ❞❛t❛ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐❢ u ❛♥❞ ˜ u s♦❧✉t✐♦♥s t♦ t❤❡ ♣r♦❜❧❡♠s ✇✐t❤ t❤❡ s♦✉r❝❡s ❢✉♥❝t✐♦♥s F ❛♥❞ ˜ F ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ❛♥❞ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s u✵ ❛♥❞ ˜ u✵ ❛♥❞ v ❛♥❞ ˜ v✱ r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ F − ˜ FC(¯

Ω×[✵,T]) ≤ ǫ,

u✵ − ˜ u✵C(¯

Ω) ≤ ǫ✵, v − ˜

vC(∂Ω×[✵,T]) ≤ ǫ✶, t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦r♠ ❡st✐♠❛t❡ ❤♦❧❞s tr✉❡✿ u − ˜ uC(¯

Ω×[✵,T]) ≤ max{ǫ✵, ǫ✶} + ǫ f (T)

✇✐t❤ f (t) = t

✵

κ(τ) dτ, t

✵

k(t − τ)κ(τ) dτ ≡ ✶, t > ✵.

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✶✻ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡

▲✐t❡r❛t✉r❡

❆✳◆✳ ❑♦❝❤✉❜❡✐✱ ●❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❝❛❧❝✉❧✉s✱ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s✱ ❛♥❞ r❡♥❡✇❛❧ ♣r♦❝❡ss❡s✳ ■♥t❡❣r✳ ❊q✉❛✳ ❖♣❡r❛t♦r ❚❤❡♦r② ✼✶ ✭✷✵✶✶✮✱ ✺✽✸✕✻✵✵✳ ❨✉✳ ▲✉❝❤❦♦ ❛♥❞ ▼✳ ❨❛♠❛♠♦t♦✱ ●❡♥❡r❛❧ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥✿ ❙♦♠❡ ✉♥✐q✉❡♥❡ss ❛♥❞ ❡①✐st❡♥❝❡ r❡s✉❧ts ❢♦r t❤❡ ✐♥✐t✐❛❧✲❜♦✉♥❞❛r②✲✈❛❧✉❡ ♣r♦❜❧❡♠s✳ ❋r❛❝t✳ ❈❛❧❝✳ ❆♣♣❧✳ ❆♥❛❧✳ ✶✾ ✭✷✵✶✻✮✱ ✻✼✻✕✻✾✺✳

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✶✼ ✴ ✸✸

✷✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❈❛♣✉t♦ ❞❡r✐✈❛t✐✈❡

❙✐♥❣❧❡✲t❡r♠ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥

■♥✐t✐❛❧✲❜♦✉♥❞❛r②✲✈❛❧✉❡ ♣r♦❜❧❡♠✿ ∂α

t u(x, t) = n

• i,j=✶

∂i(aij(x)∂ju(x, t))+c(x)u(x, t)+F(x, t), x ∈ Ω ⊂ Rn, t > ✵ u(x, t) = ✵, x ∈ ∂Ω, t > ✵, u(x, ✵) = a(x), x ∈ Ω ✇✐t❤ ✵ < α < ✶ ❛♥❞ ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ Ω ✇✐t❤ ❛ s♠♦♦t❤ ❜♦✉♥❞❛r② ∂Ω✳ ■♥ ✇❤❛t ❢♦❧❧♦✇s✱ ✇❡ ❛❧✇❛②s s✉♣♣♦s❡ t❤❛t t❤❡ s♣❛t✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ✐s ❛ ✉♥✐❢♦r♠❧② ❡❧❧✐♣t✐❝ ♦♥❡✳

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✶✽ ✴ ✸✸

✷✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❈❛♣✉t♦ ❞❡r✐✈❛t✐✈❡

❲❡❛❦ s♦❧✉t✐♦♥ ✐♥ t❤❡ ❢r❛❝t✐♦♥❛❧ ❙♦❜♦❧❡✈ s♣❛❝❡

❋♦r u ∈ C ✶[✵, T]✱ t❤❡ ❈❛♣✉t♦ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ✐s ❞❡✜♥❡❞ ❜② ∂α

t u(x, t) =

✶ Γ(✶ − α) t

✵

(t − s)−α∂su(x, s)ds, x ∈ Ω. ❘❡❝❡♥t❧②✱ t❤❡ ❈❛♣✉t♦ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ∂α

t ✇❛s ❡①t❡♥❞❡❞ t♦ ❛♥ ♦♣❡r❛t♦r

❞❡✜♥❡❞ ♦♥ t❤❡ ❝❧♦s✉r❡ Hα(✵, T) ♦❢ ✵C ✶[✵, T] := {u ∈ C ✶[✵, T]; u(✵) = ✵} ✐♥ t❤❡ ❢r❛❝t✐♦♥❛❧ ❙♦❜♦❧❡✈ s♣❛❝❡ Hα(Ω)✳ ■♥ ✇❤❛t ❢♦❧❧♦✇s✱ ✇❡ ✐♥t❡r♣r❡t ∂α

t u ❛s t❤✐s ❡①t❡♥s✐♦♥ ✇✐t❤ t❤❡ ❞♦♠❛✐♥

Hα(✵, T)✳ ❚❤✉s ✇❡ ✐♥t❡r♣r❡t t❤❡ ♣r♦❜❧❡♠ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ❛s t❤❡ ❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ s✉❜❥❡❝t t♦ t❤❡ ✐♥❝❧✉s✐♦♥s u(·, t) ∈ H✶

✵(Ω),

t > ✵, u(x, ·) − a(x) ∈ Hα(✵, T), x ∈ Ω.

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✶✾ ✴ ✸✸

✷✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❈❛♣✉t♦ ❞❡r✐✈❛t✐✈❡

❘❡s✉❧ts r❡❣❛r❞✐♥❣ t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡

▲✉❝❤❦♦✿ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ str♦♥❣ s♦❧✉t✐♦♥ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ c(x) ≤ ✵, x ∈ Ω. ▲✉❝❤❦♦ ❛♥❞ ❨❛♠❛♠♦t♦✿ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ ✇❡❛❦ s♦❧✉t✐♦♥ ✐♥ t❤❡ ❝❛s❡ c ∈ C(Ω) ✇✐t❤♦✉t t❤❡ ♥♦♥✲♥❡❣❛t✐✈✐t② ❝♦♥❞✐t✐♦♥ c(x) ≤ ✵, x ∈ Ω✳

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✷✵ ✴ ✸✸

✷✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❈❛♣✉t♦ ❞❡r✐✈❛t✐✈❡

❈♦♥s❡q✉❡♥❝❡s ❢r♦♠ t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡

▲❡t ✉s ♥♦✇ ❞❡♥♦t❡ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ✐♥✐t✐❛❧✲❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠ ❢♦r t❤❡ ❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ❜② ua,F✳ ✶✮ ◆♦♥✲♥❡❣❛t✐✈✐t② ♣r♦♣❡rt②✿ ▲❡t a ∈ L✷(Ω) ❛♥❞ F ∈ L✷(Ω × (✵, T))✳ ■❢ F(x, t) ≥ ✵ ❛✳❡✳ ✭❛❧♠♦st ❡✈❡r②✇❤❡r❡✮ ✐♥ Ω × (✵, T) ❛♥❞ a(x) ≥ ✵ ❛✳❡✳ ✐♥ Ω✱ t❤❡♥ ua,F(x, t) ≥ ✵ ❛✳❡✳ ✐♥ Ω × (✵, T)✳ ✷✮ ❈♦♠♣❛r✐s♦♥ ♣r♦♣❡rt②✿ ▲❡t a✶, a✷ ∈ L✷(Ω) ❛♥❞ F✶, F✷ ∈ L✷(Ω × (✵, T)) s❛t✐s❢② t❤❡ ✐♥❡q✉❛❧✐t✐❡s a✶(x) ≥ a✷(x) ❛✳❡✳ ✐♥ Ω ❛♥❞ F✶(x, t) ≥ F✷(x, t) ❛✳❡✳ ✐♥ Ω × (✵, T)✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡♥ ua✶,F✶(x, t) ≥ ua✷,F✷(x, t) ❛✳❡✳ ✐♥ Ω × (✵, T)✳

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✷✶ ✴ ✸✸

✷✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❈❛♣✉t♦ ❞❡r✐✈❛t✐✈❡

❈♦♠♣❛r✐s♦♥ ♣r♦♣❡rt② r❡❣❛r❞✐♥❣ t❤❡ ❝♦❡✣❝✐❡♥t c = c(x)

▲❡t ✉s ♥♦✇ ✜① ❛ s♦✉r❝❡ ❢✉♥❝t✐♦♥ F = F(x, t) ≥ ✵ ❛♥❞ ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ a = a(x) ≥ ✵ ❛♥❞ ❞❡♥♦t❡ ❜② uc = uc(x, t) t❤❡ ✇❡❛❦ s♦❧✉t✐♦♥ t♦ t❤❡ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❝♦❡✣❝✐❡♥t c = c(x)✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣❛r✐s♦♥ ♣r♦♣❡rt② ✐s ✈❛❧✐❞✿ ▲❡t c✶, c✷ ∈ C(Ω) s❛t✐s❢② t❤❡ ✐♥❡q✉❛❧✐t② c✶(x) ≥ c✷(x) ✐♥ Ω✳ ❚❤❡♥ uc✶(x, t) ≥ uc✷(x, t) ✐♥ Ω × (✵, T)✳

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✷✷ ✴ ✸✸

✷✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❈❛♣✉t♦ ❞❡r✐✈❛t✐✈❡

P♦s✐t✐✈✐t② ♣r♦♣❡rt②

❈♦♥❞✐t✐♦♥s✿ ✶✮ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ a ∈ L✷(Ω)✱ a ≥ ✵✱ a ≡ ✵ ❛✳❡✳ ✐♥ Ω✱ ✷✮ t❤❡ ✇❡❛❦ s♦❧✉t✐♦♥ u ❜❡❧♦♥❣s t♦ C((✵, T]; C(Ω))✱ ✸✮ t❤❡ s♦✉r❝❡ ❢✉♥❝t✐♦♥ ✐s ✐❞❡♥t✐❝❛❧❧② ❡q✉❛❧ t♦ ③❡r♦✱ ✐✳❡✳✱ F(x, t) ≡ ✵, x ∈ Ω, t > ✵✳ ❚❤❡♥ t❤❡ ✇❡❛❦ s♦❧✉t✐♦♥ u ✐s str✐❝t❧② ♣♦s✐t✐✈❡✿ u(x, t) > ✵, x ∈ Ω, ✵ < t ≤ T.

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✷✸ ✴ ✸✸

✷✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❈❛♣✉t♦ ❞❡r✐✈❛t✐✈❡

▲✐t❡r❛t✉r❡

❨✉✳ ▲✉❝❤❦♦✱ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ ❣❡♥❡r❛❧✐③❡❞ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥✳ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✺✶ ✭✷✵✵✾✮✱ ✷✶✽✕✷✷✸✳ ❘✳ ●♦r❡♥✢♦✱ ❨✉✳ ▲✉❝❤❦♦ ❛♥❞ ▼✳ ❨❛♠❛♠♦t♦✱ ❚✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❢r❛❝t✐♦♥❛❧ ❙♦❜♦❧❡✈ s♣❛❝❡s✳ ❋r❛❝t✳ ❈❛❧❝✳ ❆♣♣❧✳ ❆♥❛❧✳ ✶✽ ✭✷✵✶✺✮✱ ✼✾✾✕✽✷✵✳ ❨✳ ▲✐✉✱ ❲✳ ❘✉♥❞❡❧❧✱ ▼✳ ❨❛♠❛♠♦t♦✱ ❙tr♦♥❣ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r ❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥s ❛♥❞ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ t♦ ❛♥ ✐♥✈❡rs❡ s♦✉r❝❡ ♣r♦❜❧❡♠✳ ❋r❛❝t✳ ❈❛❧❝✳ ❆♣♣❧✳ ❆♥❛❧✳ ✶✾ ✭✷✵✶✻✮✱ ✽✽✽✕✾✵✻✳ ❨✉✳ ▲✉❝❤❦♦✱ ▼✳ ❨❛♠❛♠♦t♦✱ ❖♥ t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r ❛ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥✳ ❋r❛❝t✳ ❈❛❧❝✳ ❆♣♣❧✳ ❆♥❛❧✳ ✷✵ ✭✷✵✶✼✮✱ ✶✶✸✶✕✶✶✹✺✳ ❨✉✳ ▲✉❝❤❦♦✱ ▼✳ ❨❛♠❛♠♦t♦✱ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ t✐♠❡✲❢r❛❝t✐♦♥❛❧ P❉❊s✳ ❈❤❛♣t❡r ✐♥ ❍❛♥❞❜♦♦❦ ♦❢ ❋r❛❝t✐♦♥❛❧ ❈❛❧❝✉❧✉s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ t♦ ❜❡ ♣✉❜❧✐s❤❡❞ ✐♥ ✷✵✶✽✳

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✷✹ ✴ ✸✸

✸✳ ❆❜str❛❝t t✐♠❡✲ ❛♥❞ s♣❛❝❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥

❆❜str❛❝t t✐♠❡✲ ❛♥❞ s♣❛❝❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥

▲❡t X ❜❡ ❛ ❍✐❧❜❡rt s♣❛❝❡ ♦✈❡r I R ✇✐t❤ t❤❡ s❝❛❧❛r ♣r♦❞✉❝t (·, ·)✳ ❋♦r ✵ < α, β < ✶✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❍✐❧❜❡rt s♣❛❝❡ X✿ Dα

t u(t) = −(−A)βu

✐♥ X✱ t > ✵ ❛❧♦♥❣ ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ u(✵) = a ∈ X. ❆ss✉♠♣t✐♦♥s✿ t❤❡ ♦♣❡r❛t♦r A ✐s s❡❧❢✲❛❞❥♦✐♥t✱ ❤❛s ❝♦♠♣❛❝t r❡s♦❧✈❡♥t✱ ❛♥❞ (−∞, ✵] ⊂ ρ(−A)✱ ρ(−A) ❜❡✐♥❣ t❤❡ r❡s♦❧✈❡♥t ♦❢ −A✳ ❲❡ ♥♦t❡ t❤❛t u(·, t) := u(t) ∈ D((−A)β) ❢♦r t > ✵ ❛♥❞ s♦ ❛ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✐s ✐♥❝♦r♣♦r❛t❡❞ ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥✳

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✷✺ ✴ ✸✸

✸✳ ❆❜str❛❝t t✐♠❡✲ ❛♥❞ s♣❛❝❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥

◆♦♥✲♥❡❣❛t✐✈✐t② ♣r♦♣❡rt②

❋♦r ✵ < α, β < ✶✱ ❧❡t ✉s ❞❡♥♦t❡ ❛ s♦❧✉t✐♦♥ t♦ t❤❡ ❛❜str❛❝t t✐♠❡✲ ❛♥❞ s♣❛❝❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ❜② uα,β(t)✳ ■❢ a ≥ ✵ ✐♥ Ω✱ t❤❡♥ uα,β(·, t) ≥ ✵ ✐♥ Ω ❢♦r t ≥ ✵✳

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✷✻ ✴ ✸✸

✸✳ ❆❜str❛❝t t✐♠❡✲ ❛♥❞ s♣❛❝❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥

❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢

❚❤❡ ♠❛✐♥ ✐❞❡❛ ✐s ✜rst t♦ ♣r♦✈❡ ♥♦♥✲♥❡❣❛t✐✈✐t② ♦❢ uα,β ✐♥ t❤❡ ❝❛s❡ α = ✶ ✭s♣❛❝❡✲❢r❛❝t✐♦♥❛❧ ❡q✉❛t✐♦♥✮ ❛♥❞ t❤❡♥ t♦ ❡①t❡♥❞ t❤✐s r❡s✉❧t ❢♦r t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣♦rt❛♥t r❡s✉❧t✿ u✶,β(·, t) ≥ ✵ ✐♥ Ω ❢♦r t ≥ ✵ ✐❢ a ≥ ✵ ✐♥ Ω✳

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✷✼ ✴ ✸✸

✸✳ ❆❜str❛❝t t✐♠❡✲ ❛♥❞ s♣❛❝❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥

❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢

■♥❣r❡❞✐❡♥ts ❢♦r t❤❡ ♣r♦♦❢✿ ✶✮ ■♥t❡❣r❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ((−A)β + ✶)−✶a = sin πβ π ∞

✵

µβ(−A + µ)−✶a µ✷β + ✷µβ cos πβ + ✶dµ, a ∈ X ✷✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r A ❂❃ (−A + µ)−✶a ≥ ✵ ❢♦r µ ≥ ✵ ❛♥❞ a ≥ ✵ ✐♥ Ω✳ ✸✮µ✷β + ✷µβ cos πβ + ✶ > ✵ ❢♦r µ ≥ ✵ ❛♥❞ ✵ < β < ✶✳ ❍❡♥❝❡ (✶ + (−A)β)−✶a ≥ ✵ ✐❢ a(x) ≥ ✵, x ∈ Ω. ❚❤❡♥ u✶,β(·, t) = e−(−A)βta = lim

ℓ→∞

• ✶ + t

ℓ(−A)β−ℓ a ≥ ✵.

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✷✽ ✴ ✸✸

✸✳ ❆❜str❛❝t t✐♠❡✲ ❛♥❞ s♣❛❝❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥

❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢

▲❡t ✵ < α, β < ✶✳ ❚❤❡♥ uα,β(x, t) = ∞

✵

Φα(η)u✶,β(x, tαη)dη, x ∈ Ω, t > ✵ ✇✐t❤ Φα(η) =

∞

• ℓ=✵

(−η)ℓ ℓ!Γ(−αℓ + ✶ − α) ❜❡✐♥❣ ❛ ♣❛rt✐❝✉❧❛r ❝❛s❡ ♦❢ t❤❡ ❲r✐❣❤t ❢✉♥❝t✐♦♥ ✭❛❧s♦ ❦♥♦✇♥ ❛s t❤❡ ▼❛✐♥❛r❞✐ ❢✉♥❝t✐♦♥✮✳ ❇❡❝❛✉s❡ u✶,β(x, t) ≥ ✵ ❢♦r x ∈ Ω ❛♥❞ t ≥ ✵ ❢♦r a ≥ ✵ ❛♥❞ Φα(η) ≥ ✵, η > ✵ ✇❡ ❣❡t t❤❡ ✐♥❡q✉❛❧✐t② uα,β(x, t) ≥ ✵, x ∈ Ω, t ≥ ✵.

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✷✾ ✴ ✸✸

✹✳ ❙❤♦rt s✉r✈❡② ♦❢ ♦t❤❡r r❡s✉❧ts

❙♦♠❡ ♦❢ ♦t❤❡r r❡s✉❧ts

❆✳ ❆❧s❛❡❞✐✱ ❇✳ ❆❤♠❛❞ ❛♥❞ ▼✳ ❑✐r❛♥❡✱ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r ❝❡rt❛✐♥ ❣❡♥❡r❛❧✐③❡❞ t✐♠❡ ❛♥❞ s♣❛❝❡ ❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥s✳ ◗✉❛rt✳ ❆♣♣❧✳ ▼❛t❤✳ ✼✸ ✭✷✵✶✺✮✱ ✶✻✸✕✶✼✺✳ ▼✳ ❆❧✲❘❡❢❛✐✱ ❨✉✳ ▲✉❝❤❦♦✱ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡s ❢♦r t❤❡ ❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥s ✇✐t❤ t❤❡ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s✳ ❋r❛❝t✳ ❈❛❧❝✳ ❆♣♣❧✳ ❆♥❛❧✳ ✶✼ ✭✷✵✶✹✮✱ ✹✽✸✕✹✾✽✳ ▼✳ ❆❧✲❘❡❢❛✐✱ ❨✉✳ ▲✉❝❤❦♦✱ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ ♠✉❧t✐✲t❡r♠ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥s ✇✐t❤ t❤❡ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s✳ ❆♣♣❧✳ ▼❛t❤✳ ❈♦♠♣✉t✳ ✷✺✼ ✭✷✵✶✺✮✱ ✹✵✕✺✶✳ ▼✳ ❆❧✲❘❡❢❛✐✱ ❨✉✳ ▲✉❝❤❦♦✱ ❆♥❛❧②s✐s ♦❢ ❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥s ♦❢ ❞✐str✐❜✉t❡❞ ♦r❞❡r✿ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s✳ ❆♥❛❧②s✐s ✸✻ ✭✷✵✶✻✮✱ ✶✷✸✕✶✸✸✳

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✸✵ ✴ ✸✸

✹✳ ❙❤♦rt s✉r✈❡② ♦❢ ♦t❤❡r r❡s✉❧ts

❙♦♠❡ ♦❢ ♦t❤❡r r❡s✉❧ts

❆✳ ❇❡r♥❛r❞✐s✱ ❋✳❏✳ ▼❛rtí♥✲❘❡②❡s✱ P✳❘✳ ❙t✐♥❣❛✱ ❏✳▲✳ ❚♦rr❡❛✱ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡s✱ ❡①t❡♥s✐♦♥ ♣r♦❜❧❡♠ ❛♥❞ ✐♥✈❡rs✐♦♥ ❢♦r ♥♦♥❧♦❝❛❧ ♦♥❡✲s✐❞❡❞ ❡q✉❛t✐♦♥s✳ ❏♦✉r♥❛❧ ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✷✻✵ ✭✼✮ ✭✷✵✶✻✮✱ ✻✸✸✸✕✻✸✻✷ ❏✳ ❏✐❛✱ ❑✳ ▲✐✱ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡s ❢♦r ❛ t✐♠❡✲s♣❛❝❡ ❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥✳ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ▲❡tt❡rs ✻✷ ✭✷✵✶✻✮✱ ✷✸✕✷✽✳ ❨✳ ▲✐✉✱ ❙tr♦♥❣ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r ♠✉❧t✐✲t❡r♠ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥s ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥ t♦ ❛♥ ✐♥✈❡rs❡ s♦✉r❝❡ ♣r♦❜❧❡♠✳ ❈♦♠♣✉t❡rs ❛♥❞ ▼❛t❤❡♠❛t✐❝s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s ✼✸ ✭✷✵✶✼✮✱ ✾✻✕✶✵✽✳ ❩✳ ▲✐✉✱ ❙❤✳ ❩❡♥❣✱ ❨✳ ❇❛✐✱ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡s ❢♦r ♠✉❧t✐✲t❡r♠ s♣❛❝❡✲t✐♠❡ ✈❛r✐❛❜❧❡✲♦r❞❡r ❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s✳ ❋r❛❝t✳ ❈❛❧❝✳ ❆♣♣❧✳ ❆♥❛❧✳ ✶✾ ✭✷✵✶✻✮✱ ✶✽✽✕✷✶✶✳

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✸✶ ✴ ✸✸

✹✳ ❙❤♦rt s✉r✈❡② ♦❢ ♦t❤❡r r❡s✉❧ts

❙♦♠❡ ♦❢ ♦t❤❡r r❡s✉❧ts

❨✉✳ ▲✉❝❤❦♦✱ ■♥✐t✐❛❧✲❜♦✉♥❞❛r②✲✈❛❧✉❡ ♣r♦❜❧❡♠s ❢♦r t❤❡ ❣❡♥❡r❛❧✐③❡❞ ♠✉❧t✐✲t❡r♠ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥✳ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✼✹ ✭✷✵✶✶✮✱ ✺✸✽✕✺✹✽✳ ❨✉✳ ▲✉❝❤❦♦✱ ❇♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ❢♦r t❤❡ ❣❡♥❡r❛❧✐③❡❞ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ♦❢ ❞✐str✐❜✉t❡❞ ♦r❞❡r✳ ❋r❛❝t✳ ❈❛❧❝✳ ❆♣♣❧✳ ❆♥❛❧✳ ✶✷ ✭✷✵✵✾✮✱ ✹✵✾✕✹✷✷✳ ❍✳ ❨❡✱ ❋✳ ▲✐✉✱ ❱✳ ❆♥❤✱ ■✳ ❚✉r♥❡r✱ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❛♥❞ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞ ❢♦r t❤❡ ♠✉❧t✐✲t❡r♠ t✐♠❡✲s♣❛❝❡ ❘✐❡s③✲❈❛♣✉t♦ ❢r❛❝t✐♦♥❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❆♣♣❧✳ ▼❛t❤✳ ❈♦♠♣✉t✳ ✷✷✼ ✭✷✵✶✹✮✱ ✺✸✶✕✺✹✵✳ ❘✳ ❩❛❝❤❡r✱ ❇♦✉♥❞❡❞♥❡ss ♦❢ ✇❡❛❦ s♦❧✉t✐♦♥s t♦ ❡✈♦❧✉t✐♦♥❛r② ♣❛rt✐❛❧ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ ❞✐s❝♦♥t✐♥✉♦✉s ❝♦❡✣❝✐❡♥ts✳ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✹✽ ✭✷✵✵✽✮ ✶✸✼✕✶✹✾✳

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✸✷ ✴ ✸✸

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

(DC

(k)f )(t) =

t

✵

k(t − τ)f ′(τ) dτ Dα

t u(t) = −(−A)βu

◗✉❡st✐♦♥s ❛♥❞ ❝♦♠♠❡♥ts ❛r❡ ✇❡❧❝♦♠❡✦

❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✸✸ ✴ ✸✸