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❙♣❡❝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 ✶✾✲✷✷✱ ✷✵✶✾ ✶ ✴ ✹✽

■♥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✳ ✷ ✴ ✹✽

▲❡t ✉s H ❜❡ ❛ s❡♣❛r❛❜❧❡ ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ A ❜❡ ❛ s❡❧❢✲❛❞❥♦✐♥t ♣♦s✐t✐✈❡ ♦♣❡r❛t♦r A ∗ = A � κ 0 ✭ κ 0 > 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 , ∞ ) ✿ � t � t d 2 u ( t ) + Au ( t )+ Bu ( t ) − K ( t − s ) Au ( s ) ds − Q ( t − s ) Bu ( s ) ds = dt 2 0 0 = f ( t ) , t ∈ R + , ✭✶✮ u (1) (+0) = ϕ 1 . u (+0) = ϕ 0 , ✭✷✮ ✸ ✴ ✹✽

❆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✿ � ∞ � ∞ e − tτ dµ ( τ ) , e − tτ dη ( τ ) , K ( t ) = Q ( t ) = ✭✸✮ 0 0 ✇❤❡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❡ s❛t✐s✜❡❞✿ � ∞ � ∞ dµ ( τ ) dη ( τ ) < 1 , < 1 , ✭✹✮ τ τ 0 0 ❍❡r❡ t❤❡ s✉♣♣♦rts µ ❛♥❞ η ❜❡❧♦♥❣ t♦ t❤❡ ✐♥t❡r✈❛❧ ( d 0 , + ∞ ) ✱ d 0 > 0 ✳ ❚❤❡ ❝♦♥❞✐t✐♦♥s ✭✹✮ ♠❡❛♥s t❤❛t K ( t ) , Q ( t ) ∈ L 1 ( R + ) ✱ � K � L 1 < 1 ✱ � Q � L 1 < 1 ✳ ■❢ ❝♦♥❞✐t✐♦♥s ✭✹✮ ❛r❡ s✉♣♣❧❡♠❡♥t❡❞ ✇✐t❤ t❤❡ ❝♦♥❞✐t✐♦♥s � ∞ dµ ( τ ) ≡ ❱❛r µ | ∞ K (0) = 0 < + ∞ , 0 � ∞ dη ( τ ) ≡ ❱❛r η | ∞ Q (0) = 0 < + ∞ . ✭✺✮ 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 (( A + B ) x, x ) > 1 . ✭✻✮ || x || =1 , x ∈ Dom ( A ) ✺ ✴ ✹✽

❚❤❡ ❡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 − 1 µ (∆ u + (1 / 3) ❣r❛❞ ( ❞✐✈ u )) , B = − ρ − 1 λ (1 / 3) ❣r❛❞ ( ❞✐✈ u ) , u ( x, t ) ∈ R 3 ✐s ❞✐s♣❧❛❝❡♠❡♥t ✈❡❝t♦r ♦❢ ✈✐s❝♦❡❧❛st✐❝ ❤❡r❡❞✐t❛r② ❤❡r❡ u = � ✐s♦tr♦♣✐❝ ♠❡❞✐❛ t❤❛t ✜❧❧ t❤❡ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ Ω ⊂ R 3 ✇✐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 L 2 (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❦ ❛♥❞ ▲♦♥❞♦♥✱ ✶✾✼✶ ✻ ✴ ✹✽