Tập hút đều đối với một lớp phương trình parabolic suy biến tựa tuyến tính không ôtônôm

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ NGUYỄN THỊ NGỌC HÂN ❚❾P ❍Ó❚ ✣➋❯ ✣➮■ ❱❰■ ▼❐❚ ▲❰P P❍×❒◆● ❚❘➐◆❍ P❆❘❆❇❖▲■❈ ❙❯❨ ❇■➌◆ ❚Ü❆ ❚❯❨➌◆ ❚➑◆❍ ❑❍➷◆● ➷❚➷◆➷▼ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✽ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ NGUYỄN THỊ NGỌC HÂN ❚❾P ❍Ó❚ ✣➋❯ ✣➮■ ❱❰■ ▼❐❚ ▲❰P P❍×❒◆● ❚❘➐◆❍ P❆❘❆❇❖▲■❈ ❙❯❨ ❇■➌◆ ❚Ü❆ ❚❯❨➌◆ ❚➑◆❍ ❑❍➷◆● ➷❚➷◆➷▼ ◆❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ▼➣ sè✿ ✽ ✹✻ ✵✶ ✵✷ ữớ ữợ ❤å❝ ❚❙✳P❍❸▼ ❚❍➚ ❚❍Õ❨ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✽ ▲í✐ ❝❛♠ ✤♦❛♥ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❦❤ỉ♥❣ trò♥❣ ❧➦♣ ✈ỵ✐ ✤➲ t➔✐ ❦❤→❝✳ ❈→❝ t❤ỉ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr ữủ ró ỗ ố ◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✽ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ ◆❣✉②➵♥ ❚❤à ◆❣å❝ ❍➙♥ ❳→❝ ♥❤➟♥ ❳→❝ ♥❤➟♥ ❝õ❛ tr÷ð♥❣ ❦❤♦❛ ữớ ữợ P ❚❤õ② i ▲í✐ ❝↔♠ ì♥ ❚ỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ❚❙✳ P❤↕♠ ❚❤à ❚❤õ② ✱ ữớ ổ t t ữợ tổ tr sốt q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✤➸ tæ✐ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚ỉ✐ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ❜❛♥ ❧➣♥❤ ✤↕♦ ♣❤á♥❣ s❛✉ ✣↕✐ ❤å❝ ❝ò♥❣ t♦➔♥ t❤➸ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ❑❤♦❛ ❚♦→♥ tr÷í♥❣ ✣❍❙P ❚❤→✐ ◆❣✉②➯♥ ✤➣ tr✉②➲♥ t❤ư ❝❤♦ tỉ✐ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ q✉❛♥ trå♥❣✱ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ✈➔ ❝❤♦ tỉ✐ ♥❤ú♥❣ þ ❦✐➳♥ ✤â♥❣ ❣â♣ q✉þ ❜→✉ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ▲✉➟♥ ✈➠♥ ❝❤➢❝ ❝❤➢♥ s➩ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❦❤✐➳♠ ❦❤✉②➳t ✈➻ ✈➟② rt ữủ sỹ õ õ ỵ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ✤➸ ❧✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ ❝❤➾♥❤ ❤ì♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ◆❣✉②➵♥ ❚❤à ◆❣å❝ ❍➙♥ ii ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ỡ ử ởt số ỵ ✈➔ ✈✐➳t t➢t ▼ð ✤➛✉ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❈→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❤➔♠ ✶✳✸ ❚➟♣ ❤ót t♦➔♥ ❝ư❝ ✶✳✹ ✶i ii✷ ✸ iii v✺ ✶ ✹ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸✳✷ ❚➟♣ út t ỹ tỗ t t út t ❝ö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ❚➟♣ ❤ót ✤➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✹✳✶ ❚➟♣ ❤ót ✤➲✉ ❝õ❛ q✉→ tr➻♥❤ ✤ì♥ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✹✳✷ ❚➟♣ ❤ót ✤➲✉ ❝õ❛ ♥û❛ q✉→ tr➻♥❤ ✤❛ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✺ ▼ët sè ❜➜t ✤➥♥❣ t❤ù❝ t❤÷í♥❣ ❞ò♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✻ ▼ët sè ❜ê ✤➲ q✉❛♥ trå♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ iii ✷ ❚➟♣ ❤ót ✤➲✉ ✤è✐ ✈ỵ✐ ởt ợ ữỡ tr r s tỹ t t ❦❤æ♥❣ ætæ♥æ♠ ✷✸ ✷✳✶ ✣➦t ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t ỹ tỗ t t út tr ❚➼♥❤ trì♥ ❝õ❛ t➟♣ ❤ót ✤➲✉ tr♦♥❣ tr÷í♥❣ ❤đ♣ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ✈➔ p=2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ L2 (Ω) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✶ ❚➟♣ (L2 (Ω), Lq (Ω)) ✷✳✹✳✷ ❚➟♣ (L2 (Ω), D01 (Ω, ρ) ∩ Lq (Ω)) ✲ ❤ót ✤➲✉ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✷✼ ✸✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✲ ❤ót ✤➲✉ ✸✾ ✳ ✳ ✳ iv ởt số ỵ ✈➔ ✈✐➳t t➢t R = (−∞; +∞) : t➟♣ Rn : ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì t✉②➳♥ t➼♥❤ t❤ü❝ ♥ ❝❤✐➲✉ C([a; b], Rn ) : t➟♣ C(Ω) : ❧➔ C k (Ω) : ❝→❝ sè t❤ü❝ t➜t ❝↔ ❝→❝ ❤➔♠ ❧✐➯♥ tư❝ tr➯♥ ❬❛❀ ❜❪ ✈➔ ♥❤➟♥ ❣✐→ trà tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❧✐➯♥ tư❝ tr➯♥ ♠✐➲♥ Ω ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ✤➲✉ ❝➜♣ ❦ tr➯♥ ♠✐➲♥ L2 ([a, b], Rm ) : C ∞ (Ω) : ❧➔ t➟♣ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ ❜➟❝ ❤❛✐ tr➯♥ ❬❛✱ ❜❪ ✈➔ ❧➜② ❣✐→ trà tr♦♥❣ k∈N k Cc (), Cc (), , ỵ tr C(Ω), C k (Ω), , ✈ỵ✐ ❣✐→ ❝♦♠♣❛❝t ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tư❝ ❝➜♣ ✈ỉ ❤↕♥ tr➯♥ ♠✐➲♥ ❱ỵ✐ ❣✐→ ❝♦♠♣❛❝t ❚r♦♥❣ ✤â ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❧ô② t❤ø❛ ❜➟❝ ♣ ❦❤↔ t➼❝❤ ▲❡❜❡s❣✉❡ : (Ω) Lp (Ω) |(Ω)|p dx) p , (1 ≤ p < ∞) =( (Ω) ∞ L (u) = {u : u → R|u ❧➔ ❚r♦♥❣ ✤â : Ω C k (Ω) ✣÷đ❝ ①→❝ ✤à♥❤ ❜➡♥❣ Lp (Ω) : ❧➔ Ω ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tư❝ ❝➜♣ ✈ỉ ❤↕♥ tr➯♥ ♠✐➲♥ C0∞ (Ω) : ▲➔ u Rn L∞ (u) ✤♦ ✤÷đ❝ ▲❡❜❡s❣✉❡, = ❡ss sup |u| u v u L∞ (u) < ∞} Ω Rm L1loc (Ω) : tỗ t L1 () : ỗ õ ✤ë ✤♦ ▲❡❜❡s❣✉❡ Ω ⊂⊂ Ω t❤➻ v(x) ∈ L1 (Ω ) Lploc (u) = {u : u → R|u ∈ Lp (V ), H k (u), Wpk (u)(k = 1, 2, ) ❧➔ ✈ỵ✐ ♠å✐ Ω|v(x)| < +∞ V u} ỵ ổ C k,β (u), C k,β (u), (k = 0, 1, , < β ≤ 1) ❧➔ u = (ux1 , , uxn ) ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❍♦❧❞❡r ✈➨❝tì ❣r❛❞✐❡♥t ❝õ❛ ❤➔♠ ✉ n u= uxi xi ❧➔ t♦→♥ tû ▲❛♣❧❛❝❡ ❝õ❛ ❤➔♠ ✉ i=1 ✷ : ❦➳t t❤ó❝ ❝❤ù♥❣ ♠✐♥❤ vi ▼ð ✤➛✉ ✶✳ ▲➼ ❞♦ ❝❤å♥ ✤➲ t➔✐ ✳ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t✐➳♥ ❤â❛ ①✉➜t tr q tr t ỵ õ s ự ỳ ợ ữỡ tr õ ỵ q trồ tr ✈➔ ❝æ♥❣ ♥❣❤➺✳ ❈❤➼♥❤ ✈➻ ✈➟② ♥â ✤➣ ✈➔ ✤❛♥❣ t❤✉ ❤ót ✤÷đ❝ sü q✉❛♥ t➙♠ ❝õ❛ ♥❤✐➲✉ ♥❤➔ ❦❤♦❛ ❤å❝ tr➯♥ t❤➳ ❣✐ỵ✐✳ ❈→❝ ✈➜♥ ✤➲ ✤➦t r❛ ❧➔ ự sỹ tỗ t sỹ tở tö❝ ❝õ❛ ♥❣❤✐➺♠ t❤❡♦ ❞ú ❦✐➺♥ ✤➣ ❝❤♦ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ✤à♥❤ t➼♥❤ ❝õ❛ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥✳ ❚r♦♥❣ t ỵ tt ❧ü❝ t✐➯✉ ❤❛♦ ✈ỉ ❤↕♥ ❝❤✐➲✉ ✤÷đ❝ ♣❤→t tr✐➸♥ ♠↕♥❤ ỵ tt ỵ tt ỹ ỵ tt ữỡ tr r ỵ tt ữỡ tr tữớ t ỡ ỵ tt ự sỹ tỗ t t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ t➟♣ ❤ót✳ ◆❤✐➲✉ ❦➳t q✉↔ ỵ tt t út ố ợ ợ ữỡ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕♦ ❤➔♠ r✐➯♥❣ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ởt tr ỳ ợ ữỡ tr r ữủ ự t ợ ữỡ tr r ỹ tỗ t t út t ố ợ ợ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ♥û❛ t✉②➳♥ t➼♥❤ ❦❤ỉ♥❣ s✉② ❜✐➳♥ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ ♥❤✐➲✉ t→❝ ❣✐↔ tr♦♥❣ ♠✐➲♥ ❜à ❝❤➦♥✳ ❚➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ t➟♣ út t ố ợ t r ữủ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❝→❝ ❝æ♥❣ tr➻♥❤❬✸❪✱ ❬✻❪✱ ❬✶✷❪✳ ❈❤♦ ✤➳♥ t q ỵ tt t út ố ợ ợ ữỡ tr r ổ s rt ú t ỵ tt t út t ố ợ ữỡ tr r s ❜✐➳♥ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ ❜➔✐ t♦→♥ ❝❤ù❛ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ s✉② ❜✐➳♥ ❝â ♣❤➛♥ ❝❤➼♥❤ ❞↕♥❣ − Φ(u) ❤♦➦❝ −div(Φ(u) (u)) tr♦♥❣ ✤â Φ(0) = 0❀ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ s✉② ❜✐➳♥ ❝❤ù❛ t♦→♥ tû ●r❛s❤✐♥❀ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ s✉② ❜✐➳♥ ❦✐➸✉ ❈❛❧❞✐r♦❧✐ ✲ ▼✉ss✐♥❛✳✳✳❈→❝ ❦➳t q✉↔ ✈➲ sü tỗ t t út ữủ ự tr ự sỹ tỗ t t t t út ố ợ ợ ữỡ tr➻♥❤ ♣❛r❛❜♦❧✐❝ s✉② ❜✐➳♥ ❧➔ ✈➜♥ ✤➲ t❤í✐ sü✱ ❝â þ ♥❣❤➽❛ ❦❤♦❛ ❤å❝ ✈➔ ❤ù❛ ❤➭♥ ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ tr♦♥❣ ❝→❝ ❜➔✐ t♦→♥ t❤ü❝ t➳✳ ❱ỵ✐ ♥❤ú♥❣ ❧➼ ❞♦ tr➯♥✱ ❝❤ó♥❣ tỉ✐ ❧ü❛ ❝❤å♥ ✈➜♥ ✤➲ tr➯♥ ❧➔♠ ♥ë✐ ❞✉♥❣ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❧✉➟♥ ✈➠♥ ✈ỵ✐ t➯♥ ❣å✐ út ố ợ ởt ợ ữỡ tr ♣❛r❛❜♦❧✐❝ s✉② ❜✐➳♥ tü❛ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ ætæ♥æ♠ ✏✳ ✷✳ ▼ö❝ ✤➼❝❤ ✈➔ ♥❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉✳ ✷✳✶✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ♥❣❤✐➯♥ ự sỹ tỗ t ởt số t t t út t ỗ t trỡ số rt ố ợ ởt ợ ữỡ tr s ❦✐➸✉ ❈❛❧❞✐r♦❧✐ ✲ ▼✉ss✐♥❛ tr♦♥❣ ♠✐➲♥ ❜à ❝❤➦♥✳ ✷✳✷✳ ◆❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉ ✳ ❚r➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ổ t út t sỹ tỗ t↕✐ t➟♣ ❤ót t♦➔♥ ❝ư❝✱ sè ❝❤✐➲✉ ❢r❛❝t❛❧✳ ❚r➻♥❤ ❜➔② t q sỹ tỗ t t út ố ợ ởt ợ ữỡ tr r s tỹ t t➼♥❤ ❦❤ỉ♥❣ ỉtỉ♥ỉ♠ tr➯♥ ♠✐➲♥ ❜à ❝❤➦♥ ✸✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✳ Ω ⊂ RN ✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ sü tỗ t t ú tổ sỷ ữỡ r t ủ ợ t ự sỹ tỗ t t ❤ót ✈➔ t➼♥❤ trì♥ ❝õ❛ t➟♣ ❤ót ❝❤ó♥❣ tỉ✐ sû sư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝õ❛ ❧➼ t❤✉②➳t ❤➺ ✤ë♥❣ ❧ü❝ ✈ỉ ❤↕♥ ❝❤✐➲✉✱ ♥â✐ r✐➯♥❣ ❧➔ ✷ ✈➔ t❛ ❝â ❝→❝ ✤→♥❤ ❣✐→ s❛✉ T | τ T | a(un , v)dt| = p−1 |u|p−2 udiv(ρ∇v)|dxdt, dt τ Ω p p T p p ||un ||Lp (Ω) ||v||V dt ≤ C||un ||Lp (Qτ,T ) ||v||Lp (τ,T ;V ) , ≤C τ | f (un ), v | ≤ ||f (un )||Lq Qτ,T ) ||v||Lq (Qτ,T ) , ( | g, v | ≤ ||g||Lq (Qτ,T ) ||v||Lq (Qτ,T ) , ✈ỵ✐ ♠å✐ v ∈ Lp (τ, T ; V )∩Lq (Qτ,T )✱ ❦❤✐ ✤â { Lp (τ, T ; V ∗ ) + Lq (Qτ,T )✳ ❑➳t ❤ñ♣ ✤✐➲✉ ♥➔② ✈ỵ✐ ✭✷✳✶✵✮ ✈➔ sû ❞ư♥❣ ▼➺♥❤ ✤➲ ✷✳✹ tr♦♥❣ ❬✷❪ t❛ t❤✉ ✤÷đ❝ un → u tr♦♥❣ Lp (Qτ,T )✱ ❇➙② ❣✐í t❛ ①➨t dun } ❜à ❝❤➦♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ dt {un } ❦❤✐ ✤â tn → t0 ✱ ✈ỵ✐ ❧➔ t✐➲♥ ❝♦♠♣❛❝t tr♦♥❣ Lp (Qτ,T )✳ ❉♦ ✤â✱ u ∈ L2 (τ, T ; L2 (Ω))✳ tn , t0 ∈ (τ, T ]✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ un (tn ) → u(t0 ) tr♦♥❣ L2 (Ω) ❱➻ un (tn ) u(t0 ) L2 (Ω)✱ tr♦♥❣ ♥➯♥ lim inf ||un (tn )||L2 (Ω) ≥ ||u(t0 )||L2 (Ω) n→∞ ◆➳✉ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ t❤➻ un (tn ) → u(t0 ) tr♦♥❣ lim supn→∞ ||un (tn )||L2 (Ω) ≤ lim inf n→∞ ||u(t0 )||L2 (Ω) L2 (Ω)✳ ❚❛ ❝â t ||un (t)||2L2 (Ω) ≤ ||un (s)||2L2 (Ω) + K(t − s) + ||u(t)||2L2 (Ω) ≤ ||u(s)||2L2 (Ω) + K(t − s) + ✈ỵ✐ ♠å✐ t ≤ s, t, s ∈ [τ, T ]✱ t❤✉ë❝ ✈➔♦ n✳ tr♦♥❣ ✤â (gσn (v), un (v))dv, s t σ = gσ (gσ (v), un (v))dv, s ✈➔ ❤➡♥❣ sè K>0 ❉♦ ✤â✱ ❤➔♠ t Jn (t) = ||un (t)||2L2 (Ω) − Kt − (gσn (v), un (v))dv, τ t J(t) = ||u(t)||2L2 (Ω) − Kt − ✷✾ (gσ (v), un (v))dv, τ ❦❤ỉ♥❣ ♣❤ư [τ, T ]✳ ❧➔ ❧✐➯♥ tư❝ ✈➔ ❦❤ỉ♥❣ t➠♥❣ tr➯♥ L2 (Ω) tr♦♥❣ ❤➛✉ ❦❤➢♣ t ∈ (τ, T ), un → u L2 (τ, T ; L2 (Ω))✱ t❛ ❝â s❛♦ ❝❤♦ tr♦♥❣ Jn (t) → J(t) ❚✐➳♣ t❤❡♦ t❛ ❝❤ù♥❣ ♠✐♥❤ τ < tm < t0 ❍ì♥ ♥ú❛✱ ✈➻ un (t) → u(t) L2 (τ, T ; L2 (Ω)) ❤➛✉ ❦❤➢♣ ❦❤✐ n → ∞✳ gσn gσ t ∈ (τ, T )✳ lim supn→∞ Jn (tn ) ≤ J(t0 )✳ Jn (tm ) → J(tm ) ✈➔ tr♦♥❣ ●✐↔ sû ❚❤➟t ✈➟②✱ ✤➦t tm < t n ✳ ❱➻ Jn ❦❤æ♥❣ t➠♥❣✱ t❛ ❝â Jn (tm ) − J(t0 ) ≤ |Jn (tm ) − J(tm )| + |J(tm ) − J(t0 )| > ợ t tỗ t tm n0 (tm ) s❛♦ ❝❤♦ Jn (tn ) ≤ ✱ ✈ỵ✐ ♠å✐ n ≥ n0 ✳ ❱➻ ✈➟②✱ t lim sup Jn (tn ) = n→∞ lim sup ||un (t)||2L2 (Ω) n→∞ − Kt − (gσ (v), u(v))dv τ t ≤ ❉♦ ✤â✱ ||u(t0 )||2L2 (Ω) ❬✶❪ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❍✶✮ ✲ ✭❍✸✮ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â ❤å ♥û❛ q✉→ tr➻♥❤ ✤❛ trà tr♦♥❣ (gσ (v), u(v))dv τ limn→∞ sup ||un (tn )||L2 (Ω) ≤ ||u(t0 )||L2 () ỵ A Kt {Uσ }σ∈Σ ❝â ♠ët t➟♣ ❤ót t♦➔♥ ❝ư❝ ✤➲✉ ❝♦♠♣❛❝t L2 (Ω)✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ✭✷✳✼✮✱ t❛ ❝â ||un (t)||2L2 (Ω) ≤ ||u(0)||2L2 (Ω) e−(t−τ ) + K1 + K2 ||g||2L2 b = ❉♦ ✤â ❤➻♥❤ ❝➛✉ ❝õ❛ →♥❤ ①↕ T (B) ||u(0)||2L2 (Ω) e−(t−τ ) s❛♦ ❝❤♦ UΣ (t, 0, B) ⊂ B0 ✱ ❚❛ ✤à♥❤ ♥❣❤➽❛ t➟♣ t♦➔♥ ❝ö❝ tr♦♥❣ +R B0 = {u ∈ L2 (Ω) : ||u|| ≤ (t, u) → UΣ (t, 0, u)✱ L2 (Ω)✱ K √ R2 + } ♥❣❤➽❛ ❧➔ ✈ỵ✐ ❜➜t ❦➻ ✈ỵ✐ ♠å✐ K = UΣ (1, 0, B0 ) t õ B B(L2 ()) tỗ t t T (B) ứ sỹ tỗ t↕✐ t➟♣ ❤ót ❧➔ ❝♦♠♣❛❝t✳ ✸✵ ❧➔ t➟♣ ❤➜♣ t❤ư ❍ì♥ ♥ú❛✱ ✈➻ B0 ❧➔ t➟♣ ❤➜♣ t❤ư✱ t❛ ❝â UΣ (t, τ, B) = UΣ (t, t − 1, UΣ (t − 1, τ, B)) = UT (t−1)σ (1, 0, UT (τ )σ (t − − τ, 0, B)) ⊂ UΣ (1, 0, B0 ) ⊂ K ✈ỵ✐ ♠å✐ σ ∈ Σ, B ∈ B(L2 (Σ))✱ ❑❤✐ ✤â ♠å✐ ❞➣② ✈➔ t ≥ T (B, τ )✳ {ξn } ∈ Uσn (tn , τ, B0 ), σn ∈ Σ, tn → +∞, B ∈ B(L2 (Σ))✱ ❧➔ t✐➲♥ ❝♦♠♣❛❝t✳ ❉♦ ✤â✱ →♥❤ ①↕ Uσ ❝â ❣✐→ trà ❝♦♠♣❛❝t ✈ỵ✐ ❜➜t ❦➻ ❈✉è✐ ❝ò♥❣✱ t❛ ❝❤ù♥❣ ♠✐♥❤ →♥❤ ①↕ tr➯♥ ✈ỵ✐ ♠é✐ t↕✐ t≥τ ≥0 (σ, x) → Uσ (t, τ, x) σ ∈ Σ✳ ❧➔ ♥û❛ ❧✐➯♥ tö❝ ❝è ✤à♥❤✳ ●✐↔ sû ✤✐➲✉ ♥➔② ❦❤ỉ♥❣ ✤ó♥❣✱ tù❝ ❧➔✱ tỗ u0 L2 (), t 0, σ0 ∈ Σ, > 0, δn → 0, un ∈ Bδn (u0 ) , σn → σ0 ✱ ξn ∈ Uσn (t, τ, un ) s❛♦ ❝❤♦ {ξn } ∈ / B (U0 (t, , u0 )) sỹ tỗ t t➟♣ ❤ót t♦➔♥ ❝ư❝ tr♦♥❣ L2 (Ω) ✱ t❛ ❝â ✈➔ ◆❤÷♥❣ t❤❡♦ ❇ê ✤➲ ✈➲ ξn → ξ ∈ Uσ0 (t, τ, u0 ) ✭s❛✐ ❦❤→❝ ♠ët ❞➣② ❝♦♥✮✱ t ỹ tỗ t t➟♣ ❤ót t♦➔♥ ❝ư❝ ✤➲✉ ❝♦♠♣❛❝t ❦➳t ❤đ♣ ✈ỵ✐ ✣à♥❤ ❧➼ ✈➲ t➟♣ ❤ót ✤➲✉ ❝õ❛ ♥û❛ q✉→ tr➻♥❤ ✤❛ trà✳ ✷✳✹ ❚➼♥❤ trì♥ ❝õ❛ t➟♣ ❤ót ✤➲✉ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ✈➔ p = ❚❛ ①➨t ❇➔✐ t♦→♥ ✭✷✳✶✮ ❝❤♦ tr÷í♥❣ ❤đ♣ p = 2✿ ∂u − div(ρ(x)∇u) + f (u) = g(t, x), x ∈ Ω, t > τ, ∂t u|t=τ = uτ (x), x ∈ Ω, ✭✷✳✶✶✮ u|∂Ω = 0, tr♦♥❣ ✤â ❧ü❝ g u L2 () trữợ số số ❤↕♥❣ ♣❤✐ t✉②➳♥ f ✈➔ ♥❣♦↕✐ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❍✶✮ ✲ ✭❍✸✮✳ ❈❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ❇➔✐ t♦→♥ ✭✷✳✶✶✮ tr♦♥❣ ✤â sè ❤↕♥❣ ♣❤✐ t✉②➳♥ ✸✶ f t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ (H21 ) f ∈ C (R, R) t❤ä❛ ♠➣♥ ✭✷✳✷✮ ✱ ✈➔ f (u) ≥ −l (l > 0), ✈➔ ♥❣♦↕✐ ❧ü❝ g ✭✷✳✶✷✮ t❤ä❛ ♠➣♥ t❤➯♠ ♠ët sè ✤✐➲✉ ❦✐➺♥ ♥➔♦ ✤â✳ ❚÷ì♥❣ tỹ ữ tr sỹ tỗ t ♥❤➜t ♥❣❤✐➺♠ ②➳✉✱ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✭✷✳✶✷✮ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❇➔✐ t♦→♥ ✭✷✳✶✶✮ ❝â ❞✉② ♥❤➜t ♥❣❤✐➺♠✳ ❱➻ ✈➟②✱ t❛ ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ ♠ët ❤å ❝→❝ q✉→ tr➻♥❤ tr♦♥❣ L2 (Ω) ✈ỵ✐ Uσ (t, τ )uτ = u(t)✱ tr♦♥❣ ✤â ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✶✶✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ u(t) uτ {Uσ (t, τ )}σ∈Σ ❧➔ ♥❣❤✐➺♠ ②➳✉ ❞✉② ♥❤➜t ✈➔ ♥❣♦↕✐ ❧ü❝ σ ∈ Σ✳ ❉♦ t➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠✱ t❛ ❝â Uσ (t + h, τ + h) = UT (h)σ (t, τ ), ∀σ ∈ Σ, t ≥ τ, τ ∈ R, h ∈ R ❚❛ s➩ sû ❞ư♥❣ ❧➼ t❤✉②➳t t➟♣ ❤ót ✤➲✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❦➨♣ ✤è✐ ✈ỵ✐ q✉→ tr➻♥❤ ❧✐➯♥ tư❝ ữỡ ữợ ữủ t t ❬✶✵❪ ❝➟♥ ✤➸ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ trì♥ ❝õ❛ t➟♣ ❤ót ✤➲✉ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ♠➺♥❤ ✤➲ s❛✉✳ ▼➺♥❤ ✤➲ ✷✳✹✳✶✳ ❱ỵ✐ ❝→❝ ❣✐↔ t❤✐➳t (H1), (H21) ✈➔ (H3)✱ ❤å ❝→❝ q✉→ tr➻♥❤ {Uσ (t, τ )}σ∈Σ s✐♥❤ ❜ð✐ ❇➔✐ t♦→♥ ✭✷✳✶✮ tr♦♥❣ tr÷í♥❣ ❤đ♣ (L2 (Ω) × Σ, L2 (Ω)) − ❧✐➯♥ tư❝ ②➳✉ ✈ỵ✐ p=2 ❧➔ t ≥ τ, ✈➔ (L2 (Ω) × Σ, D01 (, ) Lq ()) ự trữợ ✈➔ σn σ0 ②➳✉ tr♦♥❣ Σ✳ ❧✐➯♥ tư❝ ②➳✉ ✈ỵ✐ t ≥ τ, τ ∈ R✱ ✈➔ ❣✐↔ sû uτn ❑➼ ❤✐➺✉ uτ t > τ ②➳✉ tr♦♥❣ L2 (Ω) un (t) = Uσn (t, τ )uτn ✳ ❚÷ì♥❣ tü ự sỹ tỗ t t út t tr L2 () ữợ ữủ un ✤ó♥❣ ❝❤♦ un (t)✳ ❈ư t❤➸ ❧➔✱ {un } L∞ (τ, T ; L2 (Ω)) ∪ L2 (τ, T ; D01 (Ω, ρ)) ∩ Lq (τ, T ; Lq (Ω))✳ ✸✷ ❜à ❝❤➦♥ tr♦♥❣ ◆❤➙♥ ❝↔ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➛✉ tr♦♥❣ ✭✷✳✶✶✮ ✈ỵ✐ ✤â ❧➜② t➼❝❤ ♣❤➙♥ tr➯♥ Ω✱ (t − τ ) dun ✱ dt s❛✉ t❛ ❝â d d dun ||L2 (Ω) + (t − τ ) ||un ||2D01 (Ω,ρ) + (t − τ ) dt dt dt dun = (t − τ ) g(t) dx, dt Ω (t − τ )|| F (un )dx Ω u tr♦♥❣ ✤â F (u) = f (s)ds✳ ❉ò♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤② ✤➸ ✤→♥❤ ❣✐→ ✈➳ ♣❤↔✐✱ t❛ ❝â d d (t − τ ) ||un ||2D01 (Ω,ρ) + (t − τ ) dt dt ▲➜② t➼❝❤ ♣❤➙♥ tr➯♥ F (un )dx ≤ (t − τ )||g(t)||2L2 (Ω) Ω [τ, t], τ < t ≤ T ✱ t 1 (t − τ )||un (t)||2D01 (Ω,ρ) − 2 τ ||un (s)||2D01 (Ω,ρ) ds + (t − τ ) τ t − t❛ ❝â F (un )dx ≤ (t − τ ) Ω F (un )dx Ω t τ ||g(s)||2L2 (Ω) ds ❚ø C1 |u|q − C0 ≤ F (u) ≤ C3 |u|q + C0 t❛ ❝â (t − τ )||un (t)||2D01 (Ω,ρ) + C1 (t − τ )||un (t)||qLq (Ω) t t ≤ ||un (s)||D01 (Ω,ρ) + τ τ t + (t − τ ) ||g(s)||2L2 (Ω) ds τ ≤ C, ✈ỵ✐ ♠å✐ {un } ❧➔ ❜à ❝❤➦♥ tr♦♥❣ t > τ, {un (t)} ❞➣② ❝♦♥ um (t) ❝õ❛ Ω L2 (τ, T ; D01 (Ω, ρ)) ∩ Lq (τ, T ; Lq (Ω))✳ ❧➔ ❜à ❝❤➦♥ tr♦♥❣ un (t) (C2 |u|q + C0 )dxds s❛♦ ❝❤♦ D01 (Ω, ρ) ∩ Lq (Ω)✳ um (t) ✸✸ ω(t) ❉♦ ✤â ✈ỵ✐ ❱➻ ✈➟②✱ t❛ ❧➜② ♠ët ②➳✉ tr♦♥❣ L2 (Ω) ✈ỵ✐ t ≥ τ✱ ✈➔ tr♦♥❣ D01 (Ω, ρ) ∩ Lq (Ω) ✈ỵ✐ ✭✷✳✶✶✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ t > τ✱ ✈➔ tr♦♥❣ {Uσn (t, τ )uτn } ω ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ uτ ✳ ❚❤❡♦ t➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠✱ t❛ ❝â L2 (Ω) tr♦♥❣ ✤â D01 (Ω, ρ) ∩ Lq (Ω)✳ ✈➔ ❞♦ ✤â ✤ó♥❣ ✈ỵ✐ Uσm (t, τ )uτm uσ0 (t, τ )uτ ②➳✉ tr♦♥❣ ✣✐➲✉ ♥➔② ✤ó♥❣ ❝❤♦ ♠å✐ ❞➣② ❝♦♥ ❝õ❛ {Uσn (t, τ )uτn }✳ ▼➺♥❤ ✤➲ ✷✳✹✳✷✳ ❱ỵ✐ ❝→❝ ❣✐↔ t❤✐➳t (H1)✱ (H2) ✈➔ (H3)✱ ❤å ❝→❝ q✉→ tr➻♥❤ {Uσ (t, τ )}σ∈Σ ❝â ♠ët t➟♣ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû (L2 (Ω), D01 (Ω, ρ) ∩ Lq (Ω)) B ⊂ L2 (Ω) ❧➔ ❜à ❝❤➦♥✱ ✲ ❤➜♣ t❤ö ✤➲✉ uτ ∈ B, σ ∈ Σ✱ B0 ✳ ✈➔ u = Uσ (t, τ )uτ ❉♦ ✤â✱ t÷ì♥❣ tü ✭✷✳✼✮ ✱t❛ ❝â ||u(t)||2L2 (Ω) ≤ ||u(τ )||2L2 (Ω) e−(t−τ ) + C(1 − e−(t−τ ) ) + tr♦♥❣ ✤â t↕✐ ||σ||L2b ≤ ||g||L2b , ∀σ ∈ Σ = Hω (g)✳ T1 = T1 (B, τ ) C ||g||2L2 , −1 b 1−e ❚ø t tự ố tỗ s ||u(t)||2L2 () ρ0 , ✈ỵ✐ ♠å✐t ≥ T1 , uτ ∈ B ✭✷✳✶✸✮ ❚ø ✭✷✳✼✮ ✈➔ ✭✷✳✶✸✮ ✱ t❛ ❝â t+1 t (||u||2D01 (Ω,ρ) + ||u||qLq (Ω) ) ≤ C4 ✈ỵ✐ ♠å✐ t ≥ T1 ✭✷✳✶✹✮ s ✣➦t F (s) = f (ξ)dξ ✱ ❦❤✐ ✤â tø ✭❍✷✮✱ t❛ ❝â C1 |u|q − C0 ≤ F (u) ≤ C2 |u|q + C0 , C1 ||u||qLq (Ω) − C0 |Ω| ≤ Ω F (u) ≤ C2 ||u||qLq (Ω) + C0 |Ω| ❙♦ s→♥❤ ✈ỵ✐ ✭✷✳✶✹✮ ✱ t❛ ❝â t+1 t (||u||2D01 (Ω,ρ) + F (u) ≤ C5 Ω ✸✹ ✈ỵ✐ ❜➜t ❦➻ t ≥ T1 ✭✷✳✶✺✮ ▼➦t ❦❤→❝✱ ♥❤➙♥ ✭✷✳✶✮ ✈ỵ✐ ||u||2L2 (Ω) + ut ✱ d (||u||2D01 (Ω,ρ) + dt t❛ t❤✉ ✤÷đ❝ 1 F (u)) ≤ ||σ(t)||2L2 (Ω) + ||u(t)||2L2 (Ω) ), 2 Ω ✭✷✳✶✻✮ ✈➻ ✈➟② d (||u||2D01 (Ω,ρ) + dt F (u)) ≤ ||σ(t)||2L2 (Ω) Ω ✭✷✳✶✼✮ ❚ø ✭✷✳✶✺✮ ✈➔ ✭✷✳✶✼✮ ✱ →♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ●r♦♥✇❛❧❧ ✤➲✉✱ t❛ ❝â ||u||2D01 (Ω,ρ) + F (u) ≤ C6 ✈ỵ✐ ♠å✐ t ≥ T1 Ω ❉♦ ✤â ||u(t)||2D01 (Ω,ρ) + ||u(t)||qDq () C L õ tỗ t↕✐ ♠ët t➟♣ t❤ư ❜à ❝❤➦♥ ✤➲✉ ❚➟♣ B0 ✈ỵ✐ ♠å✐ t ≥ T1 (L2 (Ω), D01 (Ω, ρ) ∩ Lq (Ω)) ✲ ❤➜♣ B0 ✳ ❝ô♥❣ ❧➔ t➟♣ (L2 (Ω), L2 (Ω)) ❝❤➦♥ ✤➲✉ ✤è✐ ✈ỵ✐ ❤å ❝→❝ q✉→ tr➻♥❤ ✈➔ (L2 (Ω), Lq (Ω)) {Uσ (t, τ )}σ∈Σ ✳ ✲ ❤➜♣ t❤ö ❜à ❚❤❡♦ ✣à♥❤ ❧➼ ✶✳✹✳✺ ự sỹ tỗ t ởt t ❤ót ✤➲✉✱ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ {Uσ (t, τ )}σ∈Σ ❧➔ ❝♦♠♣❛❝t t✐➺♠ ❝➟♥ ✤➲✉✳ ✷✳✹✳✶ ❚➟♣ (L2(Ω), Lq (Ω)) ✲ ❤ót ✤➲✉ ❚❛ ❣✐↔ sû ♥❣♦↕✐ ❧ü❝ g t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥✿ (H31 )g ∈ L2n (R; L2 (Ω))✱ t➢❝ tà♥❤ t✐➳♥ tr♦♥❣ ❇ê ✤➲ ✷✳✹✳✸✳ ◆➳✉ tr♦♥❣ ✤â ❧➔ t➟♣ ❝→❝ ❤➔♠ ❝❤✉➞♥ L2loc (R; L2 (Ω))✳ ❬✶✵❪ g ∈ L2n (R; L2 (Ω))✱ t❤➻ ✈ỵ✐ ❜➜t ❦➻ t lim sup γ→+∞ t≥τ ✈ỵ✐ ♠å✐ L2n (R; L2 (Ω)) τ τ ∈ R✱ e−γ(t−τ ) ||ϕ||2L2 (Ω) ds = 0, ϕ ∈ Σ✳ ✸✺ {Uσ (t, τ )}σ∈Σ ✣➸ ❝❤➾ r❛ ❤å ❝→❝ q✉→ tr➻♥❤ ❧➔ (L2 (Ω), Lq (Ω)) ✲ ❝♦♠♣❛❝t t✐➺♠ ❝➟♥ ✤➲✉✱ t❛ sû ❞ö♥❣ ❦➳t q✉↔ s❛✉✳ ❇ê ✤➲ ✷✳✹✳✹✳ ❬✻❪ ●å✐ {Uσ (t, τ )}σ∈Σ ❧➔ ❤å ❝→❝ q✉→ tr➻♥❤ tr➯♥ L2 (Ω) ✈➔ ❧➔ (L2 (Ω), L2 (Ω)) ✲ ❝♦♠♣❛❝t t✐➺♠ ❝➟♥ ✤➲✉ ✭✤è✐ ✈ỵ✐ σ ∈ Σ✮✳ ❑❤✐ ✤â {Uσ (t, τ )}σ∈Σ ❧➔ (L2 (Ω), Lq (Ω)) ✲ ❝♦♠♣❛❝t t✐➺♠ ❝➟♥ ✤➲✉✱ {Uσ (t, τ )}σ∈Σ ✶✳ ✷✳ ❱ỵ✐ ❜➜t ❦➻ (L2 (Ω), Lq (Ω)) ❝â ♠ët t➟♣ > 0, τ ∈ R ✲ ❤➜♣ t❤ö ✤➲✉ ✈➔ |Uσ (t, τ )uτ |q < T = T ( , B, τ )✱ ✈ỵ✐ ❜➜t ❦➻ B0 ❀ B ⊂ L2 (Ω)✱ ✈➔ ❜➜t ❦➻ t➟♣ ❝♦♥ ❜à ❝❤➦♥ M = M ( , B) ❝→❝ ❤➡♥❣ sè ❞÷ì♥❣ q < tỗ t s u ∈ B, t ≥ T, σ ∈ Σ Ω(|Uσ (t,τ )ur |M ) t ự ỵ ✷✳✹✳✺✳ tr➻♥❤ ❬✶❪ ❱ỵ✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ {Uσ (t, τ )}σ∈Σ Lq (Ω) ❝â ♠ët t➟♣ (H1)✱ (H21 ) ✈➔ (H31 )✱ ❤å ❝→❝ q✉→ (L2 (Ω), Lq (Ω)) ✲ ❤ót ✤➲✉ Aq ✱ ❝♦♠♣❛❝t tr♦♥❣ ✈➔ ❤ót ♠å✐ t➟♣ ❝♦♥ ❝õ❛ L2 (Ω) tr♦♥❣ tỉ♣ỉ ❝õ❛ Lq (Ω)✳ ❍ì♥ ♥ú❛ Aq = ωτ,Σ (B0 ), tr♦♥❣ ✤â B0 ❧➔ t➟♣ (L2 (Ω), Lq (Ω)) ✲ ❤➜♣ t❤ö ✤➲✉✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✹✳✹ ✈➔ ✣à♥❤ ❧➼ ✸✳✾ tr♦♥❣ ❬✼❪✱ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✿ ✈ỵ✐ ♠å✐ > 0, τ ∈ R ❤❛✐ ❤➡♥❣ sè ❞÷ì♥❣ ✈➔ ❜➜t ❦➻ t➟♣ ❝♦♥ ❜à ❝❤➦♥ T = T (B, , τ ) ✈➔ M = M (B, )✱ |Uσ (t, τ )uτ |q dx < ✈ỵ✐ ❜➜t ❦➻ B ⊂ L2 () tỗ t s u B, t T, σ ∈ Σ Ω(|Uσ (t,τ )ur |≥M ) ▲➜② M ✤õ ❧ỵ♥ s❛♦ ❝❤♦ C1 |u|q−1 ≤ f (u) tr♦♥❣ Ω1 = Ω(u(t) ≥ M ) = {x ∈ Ω : u(x, t) ≥ M } ✈➔ ❦➼ ❤✐➺✉ (u − M )+ = u−M 0, ✸✻ ♥➳✉ ♥➳✉ u≥M u ≤ M ❚r♦♥❣ Ω1 t❛ ❝â C1 (u − M )2q−2 + |σ(t)|2 + 2C1 C1 q−1 ≤ (u − M )q−1 + |σ(t)|2 , + |u| 2C1 σ(t)(u − M )q−1 + ≤ ✈➔ q−1 f (u)(u − M )q−1 (u − M )q−1 + ≥ C1 |u| + C1 C1 q−1 ≥ (u − M )q−1 + |u|q−2 (u − M )q+ + |u| 2 C1 C1 M q−2 q−1 q−1 ≥ (u − M )+ |u| + (u − M )q+ 2 ữỡ tr ợ |(u − M )+ |q−1 ✈➔ sû ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥✱ t❛ t❤✉ ✤÷đ❝ 2d ||(u − M )+ ||qLq (Ω) + 2(q − 1) q dt ρ(x)|∇(u − M )+ |2 |(u − M )+ |q−2 Ω1 + C1 M q−2 |(u − M )+ |q ≤ Ω1 Ω1 |σ|2 C1 ❉♦ ✤â d C1 M q−2 q ||(u − M )+ ||qLq (Ω) + ||(u − M )+ ||qLq (Ω) ≤ dt ❚❤❡♦ ▼➺♥❤ tỗ t ||U (t, )u ||qLq () ≤ ρq ✣➦t k = TB ✱ ρq > ✈ỵ✐ ♠å✐ ✈➔ TB > τ Ω1 q |σ|2 2C1 s❛♦ ❝❤♦ ≥ TB , uτ ∈ B t❛ ❝â ||(u − M )+ (t)||qLq (Ω) ≤ ||(u − M )+ (k)||qLq (Ω) e−λ(t−k) t + ≤ ||(u − M )+ (k)||qLq (Ω) e−λ(t−k) + ✸✼ (e−λ(t−s) k q 2C1 t k q 2C1 |σ|2 ) Ω1 e−λ(t−s) ||σ||2L2 (Ω) , ✭✷✳✶✽✮ tr♦♥❣ ✤â t q 2C1 k ✣➦t C1 M q−2 q λ= ✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✹✳✸✱ t❛ ❝â e−λ(t−s) ||σ||2L2 (Ω) ≤ 2q+2 2q+3 ρq T! = ln( ) + k✱ λ , ✈ỵ✐ σ ∈ Σ, M ≥ M1 ✈ỵ✐ ♠å✐M1 ✭✷✳✶✾✮ ❦❤✐ ✤â ||(u − M )+ (k)||qLq (Ω) e−λ(t−k) ≤ 2q+2 , ✈ỵ✐ ♠å✐ t > T1 ✭✷✳✷✵✮ ❚ø ✭✷✳✶✽✮ ✲ ✭✷✳✷✵✮ ✱ t❛ t❤✉ ✤÷đ❝ |(u − M )+ |q dx ≤ 2q+1 Ω(u(t)≥M ) ▲➦♣ ❧↕✐ ❝→❝ ữợ tr tỗ t M2 T2 , ợ t > T1 , σ ∈ Σ, M ≥ M1 |(u+M )− |q−2 (u+M )− t❤❛② ❝❤♦ |(u−M )+ |q−1 ✱ s❛♦ ❝❤♦ |(u + M )− |q dx ≤ Ω(u(t)≤−M ) 2q+1 , ✈ỵ✐ t > T2 , σ ∈ Σ, M ≥ M2 , tr♦♥❣ ✤â (u + M )− = ▲➜② M3 = max(M1 , M2 )✱ u+M 0, ♥➳✉ ♥➳✉ u ≤ −M, u ≥ −M t❛ ❝â |(|u(t)| − M3 )|q dx ≤ Ω(|u(t)|≥M3 ) 2q+1 , ✈ỵ✐ t > max(T1 , T2 ), σ ∈ Σ ❉♦ ✤â |u(t)|q = Ω(|u(t)|≥2M3 ) ((|u(t)| − M3 ) + M3 )q Ω(|u(t)|≥2M3 ) ≤ 2q ( (|u| − M3 )q + Ω(|u(t)|≥2M3 ) ≤ 2q ( Ω(|u(t)|≥2M3 ) (|u| − M3 )q + Ω(|u(t)|≥2M3 ) ≤ 2q+1 2q+1 = ✸✽ M3q ) (|u| − M3 )q ) Ω(|u(t)|≥2M3 ) ✷✳✹✳✷ ❚➟♣ (L2(Ω), D01(Ω, ρ) ∩ Lq (Ω)) ✲ ❤ót ✤➲✉ ✣➸ ❝❤ù♥❣ ♠✐♥❤ sü tỗ t t sỷ ỹ g (L2 (Ω), D01 (Ω, ρ) ∩ Lq (Ω)) ✲ ❤ót ✤➲✉✱ t❛ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ♠↕♥❤ ❤ì♥ s❛✉✿ (H32 ) ||g(t)||L2 (Ω) ≤ K ✈ỵ✐ ♠å✐ t ∈ R, ✈➔ g L2b (R; L2 ()) rữợ t t ự s ữợ ❦✐➺♥ (H1)✱ (H21) ✈➔ (H32)✱✈ỵ✐ ♠å✐ t➟♣ ❝♦♥ B ⊂ L2 () R tỗ t ❤➡♥❣ sè ❞÷ì♥❣ T = T (B, τ ) ≥ τ s❛♦ ❝❤♦ || d (Uσ (t, τ )uτ )|t=s ||2L2 (Ω) ≤ C dt tr♦♥❣ ✤â C ✤ë❝ ❧➟♣ ✈ỵ✐ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t ❧ü❝ σ(t) ❉♦ ✤â ✈➔ t uτ ∈ B, s ≥ T, σ ∈ Σ, σ✳ u(t) = Uσ (t, τ )uτ t❤❡♦ t❤í✐ ❣✐❛♥ 1d ||v||2L2 (Ω) + dt B ✈ỵ✐ ❜➜t ❦➻ ✈➔ ✤➦t s❛✉ ✤â ❧➜② ✤↕♦ ❤➔♠ ✭✷✳✶✶✮ ✈ỵ✐ ♥❣♦↕✐ v = ut ✱ t❛ ✤÷đ❝ 1 ρ(x)|∇v|2 ≤ l||v||2L2 (Ω) + ||σ (t)||2L2 (Ω) + ||v||2L2 (Ω) 2 Ω 1d 1 ||v||2L2 (Ω) ≤ (l + )||v||2L2 (Ω) + ||σ (t)||2L2 (Ω) dt 2 ❚ø ✭✷✳✶✺✮ ✈➔ ✭✷✳✶✻✮ ✱ t❛ ❝â t+1 ||ut ||2L2 (Ω) ≤ C, t ✈ỵ✐ t ✤õ ⑩♣ ❞ư♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ●r♦♥✇❛❧❧ ✤➲✉✱ t❛ ✤÷đ❝ |ut |2 dx ≤ C, Ω ❦❤✐ t ✤õ ❧ỵ♥✱ tr♦♥❣ ✤â C ✤ë❝ ❧➟♣ ✈ỵ✐ B ✸✾ ✈➔ σ✳ ❧ỵ♥ ✣à♥❤ ỵ tr ợ {U (t, τ )}σ∈Σ Lq (Ω)) ✲ ❤ót ✤➲✉ ❜à ❝❤➦♥ ❝õ❛ ✈➔ s✐♥❤ ❜ð✐ ❜➔✐ t♦→♥ ✭✷✳✶✶✮ ❝â ♠ët t➟♣ A✱ L2 (Ω) (H1)✱ (H21 ) ❝♦♠♣❛❝t tr♦♥❣ t❤❡♦ tæ♣æ ❝õ❛ D01 (Ω, ρ) ∩ Lq (Ω) D01 (Ω, ρ) ∩ Lq (Ω)✳ (H32 )✱ ❤å ❝→❝ q✉→ (L2 (Ω), D01 (Ω, ρ) ∩ ✈➔ ❤ót ♠å✐ t➟♣ ❝♦♥ ❍ì♥ ♥ú❛✱ A = ωτ,Σ (B0 ), tr♦♥❣ ✤â B0 ❧➔ ♠ët t➟♣ ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ B0 (L2 (Ω), D01 (Ω, ρ) ∩ Lq (Ω)) (L2 (Ω), D01 (Ω, ρ) ∩ Lq (Ω)) ❧➔ t➟♣ ❝❤➾ ❝➛♥ ❝❤➾ r❛ r➡♥❣ ✈ỵ✐ ❜➜t ❦➻ ❧➔ t✐➲♥ ❝♦♠♣❛❝t tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ♠✐♥❤ ❝❤♦ ✲ ❤➜♣ t❤ö ✤➲✉✳ uτn ∈ B0 , σn ∈ Σ, tn → ∞, {Uσn (tn , τn )Uτn } D01 (Ω, ρ) ∩ Lq (Ω)✳ {Uσn (tn , τn )uτn } ✲ ❤➜♣ t❤ö ✤➲✉✱ t❛ ❚❤❡♦ ✣à♥❤ ❧➼ ✭✷✳✹✳✺✮ ✱ t❛ ❝❤➾ ❝➛♥ ❧➔ t✐➲♥ ❝♦♠♣❛❝t tr♦♥❣ D01 (Ω, ρ)✳ ❚❛ ❝❤ù♥❣ {Uσn (tn , τn )uτn } ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ D01 (Ω, ρ)✳ ●✐↔ sû {Uσn (tn , τn )uτn } ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ L2 (Ω)✳ ❑➼ ❤✐➺✉ un (tn ) = Uσn (tn , τn )uτn ✱ t❛ ❝â ||un (tn ) − um (tm )||2D01 (Ω,ρ) = Aun (tn ) − Aum (tm ), un (tn ) − um (tm ) = − ∂t un (tn ) − ∂t um (tm ), un (tn ) − um (tm ) − f (un (tn )) − f (um (tm )), un (tn ) − um (tm ) + σn (tn ) − σm (tm ), un (tn ) − um (tm ) ≤ ||∂t un (tn ) − ∂t um (tm )||L2 (Ω) ||un (tn ) − um (tm )||L2 (Ω) + l||un (tn ) − um (tm )||2L2 (Ω) + ||σn (tn ) − σm (tm )||L2 (Ω) ||un (tn ) − um (tm )||L2 (Ω) ❚❤❡♦ ❇ê ✤➲ ✭✷✳✹✳✻✮ ✱ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✹✵ ❑➳t ❧✉➟♥ ❝õ❛ ❧✉➟♥ ✈➠♥ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ s❛✉ ✤➙②✿ ✶✳ ❚r➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ❤➔♠✱ t➟♣ ❤ót t♦➔♥ sỹ tỗ t t út t số ❝❤✐➲✉ ❢r❛❝t❛❧ ❝õ❛ t➟♣ ❤ót t♦➔♥ ❝ư❝✱ t➟♣ ❤ót ✤➲✉ tr♦♥❣ q✉→ tr➻♥❤ ✤ì♥ trà ✈➔ t➟♣ ❤ót ✤➲✉ tr♦♥❣ ỷ q tr tr r sỹ tỗ t sỹ tỗ t t út ♠ët sè ❦➳t q✉↔ ✈➲ t➟♣ ❤ót ✤➲✉ ✤è✐ ✈ỵ✐ ởt ợ ữỡ tr r s tỹ t t ❦❤æ♥❣ ætæ♥æ♠✳ ✹✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❈✳❚✳❆♥❤✱ ◆✳❉✳❇✐♥❤ ❛♥❞ ▲✳❚✳❚❤✉② ✭✷✵✶✷✮✱ ✏ ❖♥ ✉♥✐❢♦r♠ ❣❧♦❜❛❧ ❛t✲ tr❛❝t♦rs ❢♦r ❛ ❝❧❛ss ♦❢ ♥♦♥✲❛✉t♦♥♦♠♦✉s ❞❡❣❡♥❡r❛t❡ ♣❛r❛❜♦❧✐❝ ❡q✉❛✲ t✐♦♥s✑✱ ■♥t✳❏✳❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥✱ ❱♦❧✳ ✹✱ ◆♦s✳ ✶✴✷✱ ✸✺✲✺✺❀ ✐♥✈✐t❡❞ ♣❛♣❡r ♦♥ t❤❡ s♣❡❝✐❛❧ ✐ss✉❡ ✏❉❡❣❡♥❡r❛t❡ ❛♥❤ ❙✐♥❣✉❧❛r P❛r❛❜♦❧✐❝ ❛♥❞ ❊❧❧✐♣t✐❝ ❊q✉❛t✐♦♥s✑✳ ❬✷❪ ❈✳❚✳❆♥❤✱ ◆✳▼✳❈❤✉♦♥❣ ❛♥❞ ❚✳❉✳❑❡ ✭✷✵✶✵✮✱✏●❧♦❜❛❧ ❛ttr❛❝t♦rs ❢♦r t❤❡ ♠✲s❡♠✐❢❧♦✇ ❣❡♥❡r❛t❡❞ ❜② ❛ q✉❛s✐❧✐♥❡❛r ❞❡❣❡♥❡r❛t❡ ♣❛r❛❜♦❧✐❝ ❡q✉❛✲ t✐♦♥✑✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✻✸✱ ✹✹✹✲✹✺✸ ❬✸❪ ❏✳▼✳ ❆rr✐❡t❛✱ ❆✳◆✳ ❈❛r✈❛❧❤♦ ❛♥❞ ❆✳ ❘♦❞✐r✐❣✉❡③✲❇❡r♥❛❧ ✭✷✵✵✵✮✱ ✏ ❯♣✲ ♣❡r s❡♠✐❝♦♥t✐♥✉✐t② ❢♦r ❛ttr❛❝t♦rs ♦❢ ♣❛r❛❜♦❧✐❝ ♣r♦❜❧❡♠s ✇✐t❤ ❧♦❝❛❧✐③❡❞ ❧❛r❣❡ ❞✐❢❢✉s✐♦♥ ❛♥❞ ♥♦♥❧✐♥❡❛r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✑✱ ❏✳ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✳ ✶✻✽✱ ✺✸✸✲✺✺✾ ❬✹❪ ❈✳ ❚✳ ❆♥❤ ❛♥❞ ▲✳ ❚✳ ❚❤✉② ✭✷✵✶✷✮✱ ✏●❧♦❜❛❧ ❛ttr❛❝t♦rs ❢♦r ❛ ❝❧❛ss ♦❢ s❡♠✐✲❧✐♥❡❛r ❞❡❣❡♥❡r❛t❡ ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ♦♥ RN ✑ ✱ ❇✉❧❧✳ P♦❧✳ ❆❝❛❞✳ ▼❛t❤✳ ❙❝✐✳✱ ❛❝❝❡♣t❡❞ ❢♦r ♣✉❜❧✐❝❛t✐♦♥✳ ❬✺❪ ❈✳❚✳ ❆♥❤ ❛♥❞ ◆✳❱✳ ◗✉❛♥❣ ✭✷✵✶✶✮ ✱ ✏❯♥✐❢♦r♠ ❛ttr❛❝t♦rs ❢♦r ♥♦♥✲ ❛✉t♦♥♦♠♦✉s ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥ ✐♥✈♦❧✈✐♥❣ ●r✉s❤✐♥ ♦♣❡r❛t♦r✑✱ ❆❝t❛ ▼❛t❤✳❱✐❡t♥♠✳ ✸✻✱ ♥♦✳ ✶✱ ✶✾✲✸✸✳ ✹✷ ❬✻❪ ❱✳▲✳ ❈❛r❜♦♥❡✱ ❆✳◆✳ ❈❛r✈❛❧❤♦ ❛♥❞ ❑✳ ❙❝❤✐❛❜❡❧✲❙✐❧✈❛ ✭✷✵✵✽✮✱ ✏❈♦♥t✐♥✉✲ ✐t② ♦❢ ❛ttr❛❝t♦rs ❢♦r ♣❛r❛❜♦❧✐❝ ♣r♦❜❧❡♠s ✇✐t❤ ❧♦❝❛❧✐③❡❞ ❧❛r❣❡ ❞✐❢❢✉s✐♦♥✑✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✻✽✱ ✺✶✺✲✺✸✺ ❬✼❪ ●✳❳✳ ❈❤❡♥ ❛♥❞ ❈✳❑✳ ❩❤♦♥❣ ✭✷✵✵✽✮✱ ✏❯♥✐❢♦r♠ ❛ttr❛❝t♦rs ❢♦r ♥♦♥✲ ❛✉t♦♥♦♠♦✉s ♣✲▲❛♣❧❛❝✐❛♥ ❡q✉❛t✐♦♥s✑✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✻✽✱ ✸✸✹✾✲✸✸✻✸ ❬✽❪ ❱✳❱✳ ❈❤❡♣②③❤♦✈ ❛♥❞ ▼✳■✳ ❱✐s❤✐❦ ✭✷✵✵✷✮✱ ✏ ●❧♦❜❛❧ ❆ttr❛❝t♦rs ❢♦r ❊q✉❛t✐♦♥s ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✑✱ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ❈♦❧❧♦q✳ P✉❜❧✳✱ ❱♦❧✳ ✹✾✱ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✳ ❬✾❪ ❚✳❉✳ ❑❡ ❛♥❞ ◆✳✲❈✳ ❲♦♥❣ ✭✷✵✶✶✮✱ ✏ ▲♦♥❣✲t✐♠❡ ❜❡❤❛✈✐♦✉r ❢♦r ❛ ♠♦❞❡❧ ♦❢ ♣♦r♦✉s✲♠❡❞✐✉♠ ❡q✉❛t✐♦♥s ✇✐t❤ ✈❛r✐❛❜❧❡ ❝♦❡❢❢✐❝✐❡♥ts✑✱ ❖♣t✐♠✐③❛t✐♦♥ ✻✵✱ ◆♦✳ ✹✲✻✱ ✼✵✾✲✼✷✹✳ ❬✶✵❪ ❏✳✲▲✳ ▲✐♦♥s ✭✶✾✻✾✮✱ ✏ ◗✉❡❧q✉❡s ▼➨t❤♦❞❡s ❞❡ ❘❡s♦❧✉t✐♦♥ ❞❡s Pr♦❜❧➧♠❡s ❛✉① ▲✐♠✐t❡s ◆♦♥ ▲✐♥➨❛✐r❡s✑✱ ❉✉♥♦❞✱ P❛r✐s✳ ❬✶✶❪ ❙✳❙✳ ▲✉✱ ❍✳◗✳ ❲✉ ❛♥❞ ❈✳❑✳ ❩❤♦♥❣ ✭✷✵✵✺✮✱ ✏ ❆ttr❛❝t♦rs ❢♦r ♥♦♥❛✉✲ t♦♥♦♠♦✉s ✷❉ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤ ♥♦r♠❛❧ ❡①t❡r♥❛❧ ❢♦r❝❡✑✱ ❉✐s✲❝r❡t❡ ❈♦♥t✐♥✳ ❉②♥✳ ❙②st✳ ✷✸✱ ✼✵✶✲✼✶✾✳ ❬✶✷❪ ▲✳❆✳❋ ❞❡ ❖❧✐✈❡r✐❛✱ ❆✳▲✳ P❡r❡✐❛ ❛♥❞ ▼✳❈✳ P❡r❡✐❛ ✭✷✵✵✺✮✱ ✏ ❈♦♥t✐♥✉✐t② ♦❢ ❛ttr❛❝t♦rs ❢♦r ❛ r❡❛❝t✐♦♥✲❞✐❢❢✉s✐♦♥ ♣r♦❜❧❡♠ ✇✐t❤ r❡s♣❡❝t t♦ ✈❛r✐❛✲ t✐♦♥s ♦❢ t❤❡ ❞♦♠❛✐♥✑✱ ❊❧❡❝✳ ❏✳ ❉✐❢❢✳ ❊q✉❛✳ ✶✵✵✱ ✶✲✶✽✳ ❬✶✸❪ ❏✳❈✳ ❘♦❜✐♥s♦♥ ✭✷✵✶✶✮✱ ✏ ■♥❢✐♥✐t❡✲❉✐♠❡♥s✐♦♥❛❧ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s✑✱ ❈❛♠✲❜r✐❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬✶✹❪ ❘✳ ❚❡♠❛♠ ✭✶✾✾✼✮✱ ✏ ■♥❢✐♥✐t❡ ❉✐♠❡♥s✐♦♥❛❧ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ✐♥ ▼❡✲ ❝❤❛♥✐❝s ❛♥❞ P❤②s✐❝s✑✱✷♥❞ ❡❞✐t✐♦♥✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✳ ❬✶✺❪ ❈✳❑✳ ❩❤♦♥❣✱ ▼✳❍✳ ❨❛♥❣ ❛♥❞ ❈✳ ❙✉♥ ✭✷✵✵✻✮✱ ✏❚❤❡ ❡①✐st❡♥❝❡ ♦❢ ❣❧♦❜❛❧ ❛ttr❛❝t♦rs ❢♦r t❤❡ ♥♦r♠✲t♦✲✇❡❛❦ ❝♦♥t✐♥✉♦✉s s❡♠✐❣r♦✉♣ ❛♥❞ ❛♣♣❧✐❝t✐♦♥ t♦ t❤❡ ♥♦♥❧✐♥❡❛r r❡❛❝t✐♦♥✲❞✐❢✉s✐♦♥ ❡q✉❛t✐♦♥s✑✱ ❏✳ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛✲ t✐♦♥s ✶✺✱ ✸✻✼✲✸✾✾✳ ✹✸