Tập hút toàn cục đối với một lớp phương trình parabolic phi tuyến chứa toán tử caffarelli kohn nirenberg

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ◆●❯❨➍◆ ❚❍➚ ❇❸❈❍ P❍×Đ◆● ❚❾P ❍Ĩ❚ ❚❖⑨◆ ❈Ư❈ ✣➮■ ❱❰■ ▼❐❚ ▲❰P P❍×❒◆● ❚❘➐◆❍ P❆❘❆❇❖▲■❈ P❍■ ❚❯❨➌◆ ❈❍Ù❆ ❚❖⑩◆ ❚Û ❈❆❋❋❆❘❊▲▲■✲❑❖❍◆✲◆■❘❊◆❇❊❘● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙ß ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✹ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❚❾P ❍Ĩ❚ ❚❖⑨◆ ❈Ư❈ ✣➮■ ❱❰■ ▼❐❚ ▲❰P P❍×❒◆● ❚❘➐◆❍ P❆❘❆❇❖▲■❈ P❍■ ❚❯❨➌◆ ❈❍Ù❆ ❚❖⑩◆ ❚Û ❈❆❋❋❆❘❊▲▲■✲❑❖❍◆✲◆■❘❊◆❇❊❘● ❈❤✉②➯♥ ♥❣➔♥❤✿ ●■❷■ ❚➑❈❍ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙ß ❚❖⑩◆ ữớ ữợ ❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✹ ▲í✐ ❝❛♠ ✤♦❛♥ ❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ tæ✐ ữợ sỹ ữợ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔ tr➻♥❤ ❜➔② ❞ü❛ tr➯♥ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❣✐→♦ ữợ t ❣✐↔ ◆❣✉②➵♥ ❚❤à ❇↕❝❤ P❤÷đ♥❣ ❳→❝ ♥❤➟♥ ❝õ❛ ❦❤♦❛ ❝❤✉②➯♥ ổ ữợ ỡ ữủ t ữợ sỹ ữợ ✈➔ ♥❤✐➺t t➻♥❤ ❝❤➾ ❜↔♦ ❝õ❛ ❚✐➳♥ s➽ ◆❣✉②➵♥ ✣➻♥❤ ❇➻♥❤✱ ❇ë ❑❤♦❛ ❤å❝ ✈➔ ❈ỉ♥❣ ♥❣❤➺✳ ❊♠ ①✐♥ ✤÷đ❝ tọ ỏ t ỡ s s ỏ qỵ ♠➳♥ ✤è✐ ✈ỵ✐ t❤➛②✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ ❑❤♦❛ ❙❛✉ ✤↕✐ ❤å❝✱ ❑❤♦❛ ❚♦→♥ tr÷í♥❣ ✣↕✐ ❤å❝ s÷ ♣❤↕♠✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❚r✉♥❣ t➙♠ ❤å❝ ❧✐➺✉ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ t→❝ ❣✐↔ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ❚→❝ ❣✐↔ t ỡ ỗ ♥❣❤✐➺♣ ✈➔ ❝→❝ t❤➔♥❤ ✈✐➯♥ tr♦♥❣ ❧ỵ♣ ❝❛♦ ❤å❝ t♦→♥ ❑✷✵ ✤➣ ❧✉ỉ♥ q✉❛♥ t➙♠✱ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ ✈➔ q✉→ tr➻♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥ ❣✐ó♣ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚✉② ❝â ♥❤✐➲✉ ❝è ❣➢♥❣✱ s♦♥❣ t❤í✐ ❣✐❛♥ ✈➔ ♥➠♥❣ ❧ü❝ ❝õ❛ ❜↔♥ t❤➙♥ ❝â ❤↕♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤â tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❘➜t ♠♦♥❣ ✤÷đ❝ sü õ õ ỵ t ổ ũ t t❤➸ ❜↕♥ ✤å❝✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✽ ♥➠♠ ✷✵✶✹ ❚→❝ ❣✐↔ ◆❣✉②➵♥ ❚❤à ❇↕❝❤ P❤÷đ♥❣ ✐ ▼ư❝ ❧ư❝ ▼Ð ✣❺❯ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ❈→❝ ❦❤æ♥❣ ❣✐❛♥ ❤➔♠✿ ❑❤æ♥❣ ❣✐❛♥ ❤➔♠ ♣❤ư t❤✉ë❝ t❤í✐ ❣✐❛♥✿ ❚➟♣ ❤ót t♦➔♥ ❝ư❝✿ ✶ ✼ ✶✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸✳✷ ❚➟♣ ❤ót t♦➔♥ ❝ư❝✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t t út t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ❚➟♣ ❤ót ✤➲✉ ❝õ❛ q✉→ tr➻♥❤ ✤ì♥ trà✿ ▼ët sè ❜➜t ✤➥♥❣ t❤ù❝ t❤÷í♥❣ ❞ò♥❣✿ ✷ ❙Ü ❚➬◆ ❚❸■ ❈Õ❆ ◆●❍■➏▼ ❨➌❯ ✣à♥❤ ♥❣❤➽❛ t ỹ tỗ t ❝õ❛ ❜➔✐ t♦→♥✿ ✸ ❙Ü ❚➬◆ ❚❸■ ❚❾P ❍Ó❚ ❚❖⑨◆ ❈Ư❈ ❚r÷í♥❣ ❤đ♣ ỉtỉ♥ỉ♠✿ ❚r÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ ỉtỉ♥ỉ♠✿ ❑➌❚ ▲❯❾◆ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✷✷ ✷✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✽ ✸✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✽ ✸✾ ✶ ▼Ð ✣❺❯ sỷ t tr ỵ t➔✐✿ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t✐➳♥ ❤â❛ ♣❤✐ t✉②➳♥ ①✉➜t ❤✐➺♥ ♥❤✐➲✉ tr♦♥❣ ❝→❝ q✉→ tr➻♥❤ ❝õ❛ ✈➟t ❧➼✱ ❤â❛ ❤å❝ ✈➔ s✐♥❤ ❤å❝✱ ❝❤➥♥❣ ❤↕♥ ❝→❝ q✉→ tr➻♥❤ tr✉②➲♥ ♥❤✐➺t ✈➔ ❦❤✉➳❝❤ t→♥✱ q✉→ tr➻♥❤ tr✉②➲♥ sâ♥❣ tr♦♥❣ ❝ì ❤å❝ ❝❤➜t ❧ä♥❣✱ ❝→❝ ♣❤↔♥ ù♥❣ ❤â❛ ❤å❝✱ ❝→❝ ♠æ ❤➻♥❤ q✉➛♥ t❤➸ tr♦♥❣ s✐♥❤ ❤å❝✱✳✳✳❱✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ỳ ợ ữỡ tr õ ỵ q trồ tr♦♥❣ ❦❤♦❛ ❤å❝ ✈➔ ❝æ♥❣ ♥❣❤➺✳ ❈❤➼♥❤ ✈➻ ✈➟② ♥â ✤➣ ✈➔ ✤❛♥❣ t❤✉ ❤ót ✤÷đ❝ sü q✉❛♥ t➙♠ ❝õ❛ ♥❤✐➲✉ ♥❤➔ ❦❤♦❛ ❤å❝ tr➯♥ t❤➳ ❣✐ỵ✐✳ ❈→❝ ✈➜♥ ✤➲ ✤➦t r❛ ❧➔ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ✤➦t ✤ó♥❣ ❝õ❛ ❜➔✐ t sỹ tỗ t t sỹ tở ❧✐➯♥ tö❝ ❝õ❛ ♥❣❤✐➺♠ t❤❡♦ ❞ú ❦✐➺♥ ✤➣ ❝❤♦✮ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ✤à♥❤ t➼♥❤ ❝õ❛ ♥❣❤✐➺♠ ✭t➼♥❤ trì♥✱ ❞→♥❣ ✤✐➺✉ t✐➺♠ ❝➟♥ ❝õ❛ ♥❣❤✐➺♠✱✳✳✳✮✳ ❙❛✉ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ✤➦t ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥✱ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❞→♥❣ ✤✐➺✉ t✐➺♠ ❝➟♥ ❝õ❛ ♥❣❤✐➺♠ ❦❤✐ t❤í✐ ❣✐❛♥ r❛ ✈ỉ ❝ị♥❣ r➜t q✉❛♥ trå♥❣ ✈➻ ♥â ❝❤♦ ♣❤➨♣ t❛ ❤✐➸✉ ✈➔ ❞ü ✤♦→♥ ①✉ t❤➳ ♣❤→t tr✐➸♥ ❝õ❛ ❤➺ ✤ë♥❣ ❧ü❝ tr♦♥❣ t÷ì♥❣ ❧❛✐✱ tø ✤â t❛ ❝â t❤➸ ❝â ♥❤ú♥❣ ✤✐➲✉ ❝❤➾♥❤ t❤➼❝❤ ❤đ♣ ✤➸ ✤↕t ✤÷đ❝ ❦➳t q✉↔ ♠♦♥❣ ♠✉è♥✳ ❱➲ ♠➦t t♦→♥ ❤å❝✱ ✤✐➲✉ ♥➔② ❧➔♠ ♥↔② s✐♥❤ ởt ữợ ự ợ ữủ t tr tr♦♥❣ ❦❤♦↔♥❣ ❜❛ t❤➟♣ ❦➾ ❣➛♥ ✤➙② ❧➔ ▲➼ t❤✉②➳t ❝→❝ ❤➺ ✤ë♥❣ ❧ü❝ ✈æ ❤↕♥ ❝❤✐➲✉✳ ▲➼ t❤✉②➳t ♥➔② ♥➡♠ ð ❣✐❛♦ ❝õ❛ ✸ ❝❤✉②➯♥ ♥❣➔♥❤ ❧➔ ▲➼ t❤✉②➳t ❤➺ ✤ë♥❣ ❧ü❝✱ ▲➼ t❤✉②➳t ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕♦ ❤➔♠ r✐➯♥❣ ✈➔ ▲➼ t❤✉②➳t ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✭①❡♠ ❇↔♥❣ ♣❤➙♥ ❧♦↕✐ t♦→♥ ❤å❝ ♥➠♠ ✷✵✶✵✮✳ ❇➔✐ t♦→♥ ❝ì ❜↔♥ ❝õ❛ ❧➼ t❤✉②➳t ♥➔② ❧➔ ♥❣❤✐➯♥ ❝ù✉ sü tỗ t t t ỡ t ❤ót✱ ❝❤➥♥❣ ❤↕♥ ✤→♥❤ ❣✐→ sè ❝❤✐➲✉ ❢r❛❝t❛❧ ❤♦➦❝ sè ❝❤✐➲✉ ❍❛✉s❞♦r❢❢✱ sü ♣❤ư t❤✉ë❝ ❧✐➯♥ tư❝ ❝õ❛ t➟♣ ❤ót t❤❡♦ t❤❛♠ ❜✐➳♥✱ t➼♥❤ trì♥ ❝õ❛ t➟♣ ❤ót✱ ①→❝ ✤à♥❤ ❝→❝ ♠♦❞❡s✳✳✳ ✷ ❚➟♣ ❤ót t♦➔♥ ❝ư❝ ❝ê ✤✐➸♥ ❧➔ ♠ët t➟♣ ❝♦♠♣❛❝t✱ ❜➜t ❜✐➳♥✱ ❤ót t➜t ❝↔ ❝→❝ q✉ÿ ✤↕♦ ❝õ❛ ❤➺ ✈➔ ❝❤ù❛ ✤ü♥❣ ♥❤✐➲✉ t❤æ♥❣ t✐♥ ✈➲ ❞→♥❣ ✤✐➺✉ t✐➺♠ ❝➟♥ ❝õ❛ ❤➺✳ ❈ư t❤➸ ✈ỵ✐ ♠é✐ q trữợ ởt tớ T tũ ỵ t t ữủ ởt q ✤↕♦ ♥➡♠ tr➯♥ t➟♣ ❤ót t♦➔♥ ❝ư❝ ♠➔ ❞→♥❣ ✤✐➺✉ ❦❤✐ t❤í✐ ❣✐❛♥ ✤õ ❧ỵ♥ ❝õ❛ ❤❛✐ q✉ÿ ✤↕♦ ♥➔② s❛✐ ❦❤→❝ ✤õ ♥❤ä tr➯♥ ♠ët ❦❤♦↔♥❣ ❝â ✤ë ❞➔✐ T✳ ❚✉② ♥❤✐➯♥✱ t➟♣ ❤ót t♦➔♥ ❝ư❝ ❝❤➾ →♣ ❞ư♥❣ ❝❤♦ ❝→❝ tr÷í♥❣ ❤đ♣ ỉtỉ♥ỉ♠✱ tr♦♥❣ ❦❤✐ r➜t ♥❤✐➲✉ q✉→ tr➻♥❤ ❝â ♥❣♦↕✐ ❧ü❝ ♣❤ư t❤✉ë❝ ✈➔♦ t❤í✐ ❣✐❛♥✳ ❉♦ ✤â ❝➛♥ ♣❤↔✐ ♠ð rë♥❣ ❦❤→✐ ♥✐➺♠ t➟♣ ❤ót ❝❤♦ ❝→❝ ❤➺ ✤ë♥❣ ❧ü❝ ❦❤æ♥❣ ætæ♥æ♠✳ ❱✐➺❝ ♠ð rë♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➲ t➟♣ ❤ót ✤➣ ❞➝♥ ✤➳♥ ❦❤→✐ ♥✐➺♠ t➟♣ ❤ót ✤➲✉ ❝❤♦ tr÷í♥❣ ❤đ♣ q✉ÿ ✤↕♦ ♥❣❤✐➺♠ ❜à ❝❤➦♥ ❦❤✐ t❤í✐ ❣✐❛♥ t t✐➳♥ r❛ ✈ỉ ❤↕♥✳ ❚r♦♥❣ ❜❛ t❤➟♣ ❦➾ ❣➛♥ ✤➙②✱ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ✈➔ t❤✉ ✤÷đ❝ ♥❤✐➲✉ ❦➳t q✉↔ ✈➲ ❧➼ t❤✉②➳t t út ố ợ ợ ữỡ tr ✤↕♦ ❤➔♠ r✐➯♥❣ ✭①❡♠✱❝❤➥♥❣ ❤↕♥✱ ❝✉è♥ ❝❤✉②➯♥ ❦❤↔♦ ❬✹❪ ✈➔ tờ q ởt tr ỳ ợ ữỡ tr r ữủ ự t ợ ữỡ tr r ợ ữỡ tr ổ t q✉→ tr➻♥❤ tr♦♥❣ ✈➟t ❧➼✱ ❤â❛ ❤å❝ ✈➔ s✐♥❤ ❤å❝ ♥❤÷ q✉→ tr➻♥❤ tr✉②➲♥ ♥❤✐➺t✱ q✉→ tr➻♥❤ ♣❤↔♥ ù♥❣ ❦❤✉➳❝❤ t→♥✱ ♠æ ❤➻♥❤ t♦→♥ ❤å❝ tr♦♥❣ s✐♥❤ ❤å❝ q✉➛♥ t❤➸✱✳✳✳ ỹ tỗ t t út t ố ợ ữỡ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ♥û❛ t✉②➳♥ t➼♥❤ ❦❤ỉ♥❣ s✉② ❜✐➳♥ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ ♥❤✐➲✉ t→❝ ❣✐↔✱ tr♦♥❣ ❝↔ ♠✐➲♥ ❜à ❝❤➦♥ ✈➔ ❦❤æ♥❣ ❜à ❝❤➦♥ ✭①❡♠ ❬✶✵❪✱ ❬✶✺❪✮✳ ❚➼♥❤ ❧✐➯♥ tư❝ ❝õ❛ t➟♣ ❤ót t♦➔♥ ❝ư❝ ố ợ t r ữủ ự tr ❝→❝ ❝æ♥❣ tr➻♥❤ ❬✸❪✱ ❬✾❪✱ ❬✶✵❪✱ ❬✶✹❪✳ ❙ü ❤✐➸✉ ❜✐➳t ✈➲ ❝→❝ t➼♥❤ ❝❤➜t ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➺ ✤ë♥❣ ❧ü❝ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ✈➜♥ ✤➲ q✉❛♥ trå♥❣ ❝õ❛ ✈➟t ỵ t ố ợ ữỡ tr ỡ tr ỵ tt tố ữ tỡ ỗ q✉❛♥ trå♥❣ tr♦♥❣ ❤➺ t❤è♥❣ ♥â✐ tr➯♥ tø q✉❛♥ ✤✐➸♠ ❝õ❛ ❝→❝ t❤✉②➳t ð ❤➺ t❤è♥❣ ✤ë♥❣ ❧ü❝✱ ♥â ❝➛♥ tt t tr ởt ỵ tt tữỡ ữỡ ♥û❛ q✉→ tr➻♥❤ ✤❛ trà✳ ❚r♦♥❣ ♥❤ú♥❣ ♥➠♠ q✉❛✱ ✤➣ õ ởt số ỵ tt ữớ t õ t ♥❣❤✐➯♥ ❝ù✉ ♥❤✐➲✉ ❣✐→ trà ❜✐➳♥ t❤✐➯♥ ✈➔ ❝→❝ ❤➔♠ ú ỗ ỵ tt tờ qt ỵ tt ỹ út q ✈➔ ❱✐s❤✐❦ ❬✽❪ ✈➔ ❧➼ t❤✉②➳t ✈➲ ♥û❛ q✉→ tr➻♥❤ ✤❛ trà ❝õ❛ ▼❡❧♥✐❦ ✈➔ ❱❛❧❡r♦ ✸ ❬✶✾❪✱❬✷✶❪✳ ◆❤í ♥❤ú♥❣ ỵ tt õ ởt số t q q t➟♣ ❤ót ❣➛♥ ✤➙② ✤÷❛ r❛ sü ❦❤→❝ ❜✐➺t tr♦♥❣ tr÷í♥❣ ❤đ♣ ♣❤÷ì♥❣ tr➻♥❤ t❤ỉ♥❣ t❤÷í♥❣ ❬✷✵❪✱ ❬✷✶❪✱ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ❬✶✻❪✱ ❬✷✼❪✳✳✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ ❣✐↔ t❤✐➳t ❝â ❜✐➯♥ ❧➔ ∂Ω Ω ❧➔ ♠✐➲♥ ❜à ❝❤➦♥ tr♦♥❣ RN , N ≥ ✈➔ ①➨t ❜➔✐ t♦→♥ s❛✉✿ ∂u − div(|x|−pγ |∇u|p−2 ∇u) + f (t, u) = g(x, t), x ∈ Ω, ∂t u|t=τ = uτ (x), x ∈ Ω, u|∂Ω = 0, τ ∈ R, uτ ∈ L2 (Ω)✱ t > τ✱ (1.1) p, γ t❤ä❛ g ∈ L2c (R; L2 (Ω)) tr♦♥❣ ✤â L2c (R; L2 (Ω)) ❧➔ t➟♣ t➜t ❝↔ ❝→❝ 2 ❝♦♠♣❛❝t tr♦♥❣ Lloc (R; L (Ω)) ✤÷đ❝ ✤÷❛ r❛ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ✶✳✹✳✶ ❤➔♠ tr♦♥❣ ✤â ❢ ♣❤✐ t✉②➳♥✱ ♥❣♦↕✐ ❧ü❝ ❣✱ ❝→❝ sè ♠➣♥ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✭❍✶✮✿ ợ f :RìRR tử tọ |f (t, u)| ≤ C1 |u|q−1 + k1 (1.2) uf (t, u) ≥ C2 |u|q − k2 (1.3) q ≥ 2, C1 , C2 , k1 , k2 ❧➔ ❝→❝ sè ữỡ ữợ 2N p2 N +2 N N N − ≤γ+1< p p ❚❛ ✤÷❛ r❛ ❣✐↔ t❤✐➳t ✭❍✶✮✲✭❍✸✮✳ P❤✐ t✉②➳♥ ❢ ❣✐↔ sû ❝â ♠ët sü ♣❤→t tr✐➸♥ ✤❛ t❤ù❝ ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ t✐➯✉ ❝❤✉➞♥✳ ▼ët ✈➼ ❞ö ✤✐➸♥ ❤➻♥❤ ❝õ❛ ❝→❝ ❤➔♠ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭❍✶✮ ❧➔ f (t, u) = |u|q−2 u.arctant, q ≥ ✭①❡♠ ❬✼❪ ❝❤÷ì♥❣ ✺ ♣❤➛♥ ✸✳✸ ✈➔ ✸✳✺✮ ❞à❝❤ ❣å♥ ❝→❝ ✤↕✐ ❧÷đ♥❣ ❦➼ ❤✐➺✉ tr♦♥❣ L2loc (R; L2 (Ω))✳ ✭❍✸✮ ❧➔ ✤✐➲✉ ❦✐➺♥ ✤↔♠ ✹ ❜↔♦ 1,p D0,Ω (Ω) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♥➠♥❣ ❧÷đ♥❣ tü ♥❤✐➯♥ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ✈✐➺❝ ❣✐↔✐ ❜➔✐ t♦→♥ ✭✶✳✶✮✱ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ tr♦♥❣ ♣❤➛♥ ✶✳✶ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ✣➙② ❧➔ ✤✐➲✉ ❦✐➺♥ q✉❛♥ trå♥❣ ✤➸ ❝❤ù♥❣ sỹ tỗ t t ❜➡♥❣ ❝→❝❤ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝♦♠♣❛❝t✳ ❇➔✐ t♦→♥ ✭✶✳✶✮ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛❢❢❛r❡❧❧✐✲❑♦❤♥✲◆✐r❡♥❜❡r❣ ❬✺❪✱ ❝â ❝❤ù❛ ♠ët sè ✈➜♥ ✤➲ q✉❛♥ trå♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝✳ ❈❤➥♥❣ ❤↕♥ ♥❤÷ ♣❤÷ì♥❣ tr➻♥❤ ♥❤✐➺t ♥û❛ t✉②➳♥ t➼♥❤ ✭❦❤✐ s✉② ❜✐➳♥ ✭❦❤✐ p = 2✮✱ γ = 0, p = 2✮✱ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ♣✲▲❛♣❧❛❝✐❛♥✭❦❤✐ = 0, p = ỹ tỗ t t➼♥❤ ❝❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✶✮ ✤➣ t❤✉ ❤ót ✤÷đ❝ sü q✉❛♥ t➙♠ ❝õ❛ ❝→❝ ♥❤➔ ❚♦→♥ ❤å❝ tr♦♥❣ ♥❤ú♥❣ ♥➠♠ ❣➛♥ ✤➙② ❬✷❪✱ ❬✶✷❪✳ ❚✉② ♥❤✐➯♥ sü ❤✐➸✉ ❜✐➳t tèt ♥❤➜t ❝õ❛ ❝❤ó♥❣ tỉ✐✱ ❞÷í♥❣ ♥❤÷ ➼t ✤÷đ❝ ✤÷❛ r❛ tr♦♥❣ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ❞➔✐ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ✭✶✳✶✳✮ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ✈ỵ✐ ♠ư❝ t✐➯✉ ❞ü❛ tr➯♥ ❦➳t q✉↔ ❝õ❛ ❝ỉ♥❣ tr➻♥❤ ❬✷✸❪✱ ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ✈➔ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ❝→❝❤ ❣✐↔✐ ❜➔✐ t♦→♥ ✭✶✳✶✮ q ữợ tổ q t út t♦➔♥ ❝ö❝ t❤è♥❣ ♥❤➜t ❝❤♦ ♥û❛ q✉→ tr➻♥❤ ✤❛ trà✳ Ð ✤➙② ❦❤æ♥❣ ❝â ❤↕♥ ❝❤➳ sü ①→❝ ✤à♥❤ ❝õ❛ ❤➔♠ ♣❤✐ t✉②➳♥ ❢ ✈➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤è✐ ✈ỵ✐ ự sỹ tỗ t ❜➔✐ t♦→♥ ✭✶✳✶✮✱ ♥❤÷♥❣ ❦❤ỉ♥❣ tè✐ ÷✉✳ ❱➻ ✈➟②✱ ✤➸ ✤÷❛ r❛ ❝→❝ ❣✐↔✐ ♣❤→♣ ❝❤ó♥❣ t❛ ❝➛♥ sû ❞ư♥❣ ❝→❝ ❧➼ t❤✉②➳t ✈➲ t➟♣ ❤ót ❝❤♦ q✉→ tr➻♥❤ ✤❛ trà✳ ✣✐ t❤❡♦ ✤÷í♥❣ ❧è✐ ❝❤✉♥❣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❬✶✻❪✱ ❬✶✼❪✱ ❬✷✷❪✱ ❬✷✼❪ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ❦❤ỉ♥❣ s✉② ❜✐➳♥✱ ❝❤ó♥❣ tỉ✐ ❝❤ù♥❣ ♠✐♥❤ sü tỗ t t út t tr trữớ ủ ổtổổ ổ ổtổổ ữ ỵ r t ổ tở tớ t sỹ tỗ t↕✐ ➼t ♥❤➜t ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ ♥❣÷đ❝ ❧↕✐ tr÷í♥❣ ❤đ♣ s✉② ❜✐➳♥ ♥û❛ t✉②➳♥ t➼♥❤✱ ❝ư t❤➸ ❦❤✐ g=0 ✈➔ p = 2✱ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr ú ỵ õ ✤✐➲✉ ❦✐➺♥ ❜ê s✉♥❣ fu (t, u) ≥ −C3 ❝❤♦ t➜t ❝↔ t > τ ✱ u ∈ R ❤♦➦❝ ♠ët ❣✐↔ t❤✐➳t (f (t, u) − f (t, v)).(u − v) ≥ −C |u − v|2 ✈ỵ✐ t > τ ✱ u, v ∈ R✱ t❛ tr➯♥ ❢✱ ✈➼ ỡ õ t ự r sỹ tỗ t↕✐ ✈➔ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✶✮✳ ❱ỵ✐ ♥❤ú♥❣ ❧➼ ❞♦ ð tr➯♥✱ ❝❤ó♥❣ tỉ✐ ❧ü❛ ❝❤å♥ ự sỹ tỗ t t ự sỹ tỗ t t út t♦➔♥ ❝ư❝ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✶✮ tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣ ætæ♥æ♠ ✈➔ ❦❤æ♥❣ ætæ♥æ♠ ✺ ❧➔♠ ♥ë✐ ❞✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ▲✉➟♥ ✈➠♥ ✈ỵ✐ t➯♥ ❣å✐ ✿ ✧❚➟♣ ❤ót t♦➔♥ ố ợ ởt ợ ữỡ tr r t ❝❤ù❛ t♦→♥ tû ❈❛❢❢❛r❡❧❧✐✲ ❑♦❤♥✲◆✐r❡♥❜❡r❣✧✳ ✷✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ự sỹ tỗ t t sỹ tỗ t t út t ố ợ ởt ợ ữỡ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ♣❤✐ t✉②➳♥ ❝❤ù❛ t♦→♥ tû ❈❛❢❢❛r❡❧❧✐✲❑♦❤♥✲◆✐r❡♥❜❡r❣✳ ✸✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤✿ ✲ ❈❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ❜➔✐ t♦→♥ ✭✶✳✶✮ tr ợ s ã ự sỹ tỗ t t ã ự sỹ tỗ t t út t ố ợ t tr trữớ ủ ổtổổ ổ ổtổổ Pữỡ ự ã ỷ ữỡ ♣❤→♣ ①➜♣ ①➾ ●❛❧❡r❦✐♥ ✈➔ ✤→♥❤ ❣✐→ ①➜♣ ①➾ ♥❣❤✐➺♠ ự sỹ tỗ t t ã ỷ ữỡ t ♣❤÷ì♥❣ ♣❤→♣ tr♦♥❣ ❧➼ t❤✉②➳t ❝õ❛ ❤➺ ✤ë♥❣ ❧ü❝ ✈ỉ ự sỹ tỗ t t út t♦➔♥ ❝ư❝ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✶✮ tr♦♥❣ tr÷í♥❣ ❤đ♣ ỉtỉ♥ỉ♠ ✈➔ ❦❤ỉ♥❣ ỉtỉ♥ỉ♠✳ ✺✳ ❇è ❝ư❝ ❝õ❛ ▲✉➟♥ ✈➠♥✿ ▲✉➟♥ ỗ ữỡ ❑➳t ❧✉➟♥ ✈➔ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ✷✼ ∆p,γ ✤➸ ❝❤➾ r❛ r➡♥❣ ψ = −∆p,γ u 1,p w ∈ V := Lp (τ, T ; D0,γ (Ω)), ❚❛ sû ❞ö♥❣ t➼♥❤ ♥û❛ ❧✐➯♥ tö❝ ❝õ❛ t♦→♥ tû ✣➦t v = u − λw✱ tr♦♥❣ ✤â λ>0 ✈➔ T λ (ψ + ∆p,γ (u − λw), w)dt ≥ 0, ❚❛ ❝â✿ τ T (ψ + ∆p,γ (u − λw), w)dt ≥ ❉♦ ✤â✿ (2.17) τ ❈❤♦ λ→0 (2.17)✱ tr♦♥❣ t❛ ❝â✿ T (ψ + ∆p,γ u, w)dt ≥ 0, ∀w ∈ V τ ψ = −∆p,γ u ❉♦ ✤â u = ∆p,γ u − f (t, u) + g(t) tr♦♥❣ V 1,p q ❚❛ t❤➜② u(τ ) = uτ ✳ ❈❤å♥ ✈➔✐ ϕ ∈ C ([τ, T ]; D0,γ (Ω) ∩ L (Ω)) ✈ỵ✐ φ(T ) = 0, V, sỷ ỵ trở s t ❝â t❤➸ ❱➟② ❦✐➸♠ tr❛ r➡♥❣✿ T T −pγ − (u, ϕ )dt + τ |x| T p−2 |∇u| ∇u∇ϕdxdt + τ Ω f (t, u)ϕdxdt τ Ω T = (u(τ ), ϕ(τ )) + gϕdxdt τ Ω ❙û ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ●❛❧❡r❦✐♥ t❛ ❝â✿ T T −pγ − (un , ϕ )dt + τ |x| T p−2 |∇un | ∇un ∇ϕdxdt + τ Ω f (t, un )ϕdxdt τ Ω T = (un (τ ), ϕ(τ )) + gϕdxdt τ Ω ▲➜② ❣✐ỵ✐ ❤↕♥ ❦❤✐ T n → ∞✱ T −pγ − (u, ϕ )dt + τ |x| t❛ ❝â✿ T p−2 |∇u| ∇u∇ϕdxdt + τ Ω f (t, u)ϕdxdt τ Ω T = (uτ , ϕ(τ )) + gϕdxdt τ Ω u ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ (1.1) tr➯♥ (τ, T )✳ ❈✉è✐ ❝ò♥❣ t❛ ❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ ♥❣❤✐➺♠ u t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ (2.3) t t u tỗ t↕✐ tr➯♥ t♦➔♥ (τ, +∞) ❉♦ ✤â u(τ ) = u ú t t tỗ t t ♥❤➜t ♠ët ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✶✮✳ ✷✽ ❈❤÷ì♥❣ ✸ ❙Ü ❚➬◆ ❚❸■ ❚❾P ❍Ĩ❚ ❚❖⑨◆ ❈Ư❈ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② t❛ ♥❣❤✐➯♥ ❝ù✉ ❜➔✐ t♦→♥ ✭✶✳✶✮ tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣ ỉtỉ♥ỉ♠ ✈➔ ❦❤ỉ♥❣ ỉtỉ♥ỉ♠✳ • ❇➔✐ t♦→♥ ✭✶✳✶✮ tr♦♥❣ trữớ ủ ổtổổ tớ ã f g t ❇➔✐ t♦→♥ ✭✶✳✶✮ tr♦♥❣ tr÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ ỉtỉ♥ỉ♠✿ t❤í✐ ❣✐❛♥ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ f ✈➔ g ♣❤ư t❤✉ë❝ ✈➔♦ t✳ ✸✳✶ ❚r÷í♥❣ ❤đ♣ ỉtỉ♥ỉ♠✿ ❳➨t tr÷í♥❣ ❤đ♣ f ✈➔ g ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ t❤í✐ ❣✐❛♥ t ✈➔ ❝❤ó♥❣ t❛ ♥❤➢❝ ❧↕✐ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❝→❝ ❞á♥❣ ✤❛ trà✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✶✳✶✳ [20] ●✐↔ sû E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ⑩♥❤ ①↕ G ✿❬ 0; +∞ ✮ ×E → 2E ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❞á♥❣ ✤❛ trà ♥➳✉ t❤ä❛ ♠➣♥ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ s❛✉✿ G(0, w) = w, ∀w ∈ E; + ✭✷✮ G(t1 + t2 , w) ⊂ G(t1 , G(t2 , w)), ∀w ∈ E, t1 , t2 ∈ R ✱ tr♦♥❣ ✤â G(t, B) = G(t, x), B ⊂ E ✭✶✮ x∈B ◆â ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❞á♥❣ ✤❛ trà ♥❣➦t ♥➳✉✿ G(t1 + t2 , w) = G(t1 , G(t2 , w)), ∀w ∈ E, t1 , t2 ∈ R+ ✷✾ ❚❛ ①➨t ❜➔✐ t (1.1) ợ = ỷ ỵ ✷✳✷✳✶ t❛ ①➙② ❞ü♥❣ ♠ët →♥❤ ①↕ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿ G(t, u0 ) = u(t)|u(.) ❧➔ ♥❣❤✐➺♠ ②➳✉ t♦➔♥ ❝ö❝ ❝õ❛ ✭✶✳✶✮ t❤ä❛ ♠➣♥ u(0) = u0 ❇ê ✤➲ ✸✳✶✳✷✳ G ❧➔ ♥û❛ ❞á♥❣ ✤❛ trà ♥❣➦t t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ✸✳✶✳✶ ❈❤ù♥❣ ♠✐♥❤ ξ ∈ G(t1 + t2 , u0 ), t❤➻ ξ = u(t1 + t2 ) tr♦♥❣ ✤â u(t) ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ (1.1) ❑➼ ❤✐➺✉ v(t) = u(t1 + t2 ), t❛ t❤➜② r➡♥❣ v(.) ❝ơ♥❣ t❤✉ë❝ t➟♣ ♥❣❤✐➺♠ ❝õ❛ (1.1) ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ v(0) = u(t2 ) ❉♦ ✤â ξ = v(t1 ) ∈ G(t1 , u(t2 )) ⊂ G(t2 , G(t2 , u0 )) ❱➟② G(t1 , G(t2 , u0 )) ⊂ G(t1 +t2 , u0 ) ◆➳✉ ξ ∈ G(t1 , G(t2 , u0 )) t❤➻ ξ = v(t1 ), tr♦♥❣ ✤â v(0) ∈ G(t2 , u0 ) ❚❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣  v(0) = u(t2 ), tr♦♥❣ ✤â v(0) ∈ G(t2 , u0 ), u(0) = u0 u(τ ), ≤ τ < t2 ❚➟♣ ❤ñ♣ w(τ ) ❂ v(τ − t ), τ ≥ t2 ❚ø u ✈➔ v ❧➔ ♥❣❤✐➺♠ ❝õ❛ (1.1) t❛ t❤➜② w ❝ơ♥❣ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ (1.1) ✈ỵ✐ w(0) = u(0) = u0 ❉♦ ξ = v(t1 ) = w(t1 + t2 )✱ t❛ ❝â ξ ∈ G(t1 + t2 , u0 ) ●✐↔ sû r➡♥❣ ✣à♥❤ ♥❣❤➽❛ ✸✳✶✳✸✳ [20] ❚➟♣ A ❧➔ t➟♣ ❤ót t♦➔♥ ❝ư❝ ❝õ❛ ❞á♥❣ ✤❛ trà G ♥➳✉ t❤ä❛ ♠➣♥ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✯ A ❧➔ t➟♣ ❤➜♣ t❤ö✱ tù❝ ❧➔ dist(G(t, B), A) → ❦❤✐ t → ∞, ∀B ⊂ E, ❜à ❝❤➦♥✳ A ⊂ G(t, A), ∀t ≥ ¯ ✯ ◆➳✉ ❇ ❧➔ ♠ët t➟♣ ❤➜♣ t❤ö ❝õ❛ G ✱ t❤➻ A ⊂ B tr♦♥❣ ✤â dist(C, A) = sup inf ||c − a|| ❧➔ ♥û❛ ❦❤♦↔♥❣ ✯ A ❧➔ ♥û❛ ❜➜t ❜✐➳♥ ➙♠ c∈C a∈A ❍❛✉s❞♦r❢❢✳ s ự sỹ tỗ t t ❤ót t♦➔♥ ❝ư❝ ❝❤♦ ♥û❛ ❞á♥❣ ✤❛ trà G ✣à♥❤ ỵ [[19], [20]] sỷ ỷ ỏ tr ♥❣➦t G t❤ä❛ ♠➣♥✿ ✭✶✮ G ❧➔ t✐➯✉ ❤❛♦ ✤✐➸♠✱ tự tỗ t K > s u0 ∈ E, u(t) ∈ ✸✵ G(t, u0 ) t❛ ❝â✿ ||u(t)||E ≤ K ♥➳✉ G(t, ) ❧➔ ♠ët →♥❤ ①↕ ✤â♥❣ ✈ỵ✐ ❜➜t η; ξn ∈ G(t, ηn ) t❤➻ ξ ∈ G(t, η); ✭✸✮ G ❧➔ ♥û❛ ❝♦♠♣❛❝t tr➯♥ t✐➺♠ ❝➟♥✱ + tr♦♥❣ ❊ s❛♦ ❝❤♦✿ T (B), γT (B) (B) := ✭✷✮ t ≥ t0 (||u0 ||E ); ❦➻ t ≥ 0✱ tù❝ ❧➔ ♥➳✉ ξn → ξ, ηn → tù❝ ❧➔ ♥➳✉ ❇ ❧➔ ♠ët t➟♣ ❜à ❝❤➦♥ G(t, B) ❜à ❝❤➦♥✱ ✈ỵ✐ ♠å✐ ❞➣② t≥T (B) ξn ∈ G(tn , B) ✈ỵ✐ tn → ∞ ❧➔ t✐➲♥ ❝♦♠♣❛❝t tr♦♥❣ ❊✳ ❑❤✐ ✤â G ❝â ♠ët t➟♣ ❤ót ❝♦♠♣❛❝t t♦➔♥ ❝ư❝ ❆ tr♦♥❣ ❜✐➳♥✱ tù❝ G(t, A) = A, ✈ỵ✐ ∀t ≥ 0✳ ❊✳ ❍ì♥ ♥ú❛ ❆ ❧➔ ❜➜t ❇ê ✤➲ ✸✳✶✳✺✳ G(t∗, )❧➔ →♥❤ ①↕ ❝♦♠♣❛❝t ✈ỵ✐ ♠é✐ t∗ ∈ (0, T ]✳ ❈❤ù♥❣ ♠✐♥❤✿ ❇ê ✤➲ ♥➔② ❧➔ ♠ët tr÷í♥❣ ❤đ♣ ❝õ❛ ❜ê ✤➲ ✸✳✷✳✻ ữợ t ự sỹ tỗ t t út t ỵ ữợ t❤✐➳t ✭❍✶✮ ✲ ✭❍✸✮✱ tr♦♥❣ ✤â f ✈➔ g ❦❤æ♥❣ ♣❤ö t❤✉ë❝ ✈➔♦ t✱ ♥û❛ ❞á♥❣ ✤❛ trà ❝❤➦t G ✤÷đ❝ s✐♥❤ ❜ð✐ ❜➔✐ t♦→♥ ✭✶✳✶✮ ❝â ♠ët t➟♣ ❤ót t♦➔♥ ❝ư❝ ❝♦♠♣❛❝t ❜➜t ❜✐➳♥ tr♦♥❣ L (Ω)✳ ❈❤ù♥❣ ♠✐♥❤✿ ❚❛ s➩ ❦✐➸♠ tr❛ ❝→❝ ❣✐↔ t❤✐➳t ❝õ❛ ✤à♥❤ rữợ t sỷ u(t) G(t, u0 )✱ t❛ ❝â✿ 1d ||u(t)||2L2 (Ω) + |x|−pγ |∇u|p dx + C2 ||u||qLq (Ω) ≤ k2 |Ω| + ugdx dt Ω Ω ≤ k2 |Ω| + ε ||u||2L2 (Ω) + C ||g||2L2 () ú ỵ r C2 ||u||qLq () ≥ λ ||u||2L2 (Ω) − C, C = C(q, |Ω|) > t❛ ❝â 1d ||u||2L2 (Ω) + λ||u||2L2 (Ω) ≤ C(q, |Ω|, ||g||L2 (Ω) ) dt ❚ø ✤â t❛ ❝â✿ ||u(t)||2L2 (Ω) ≤ ||u(0)||2L2 (Ω) e−2λt + C(q, |Ω|, ||g||L2 (Ω) ) ❉♦ ✤â t❛ ❝â G ❧➔ t✐➯✉ ❤❛♦ ✤✐➸♠✳ ❇➙② ❣✐í t❛ ❦✐➸♠ tr❛ ❣✐↔ t❤✐➳t ỵ (3.1) n G(t, ηn ), ξn → ξ, ηn → η tr♦♥❣ L2 (Ω)✳ {un } s❛♦ ❝❤♦ un (t) = ξn , un (0) = ηn ●✐↔ sû ❞➣② ❑❤✐ ✤â tỗ t ởt ỵ ữ tr ự ỵ t õ un u tr L2 (Q0,T )✱ ✯ un (t) u(t) tr♦♥❣ L2 (Ω) ✈ỵ✐ t ∈ [0, T ] ✈➔ u(0) = η ✱ ✯ f (un ) f (u) tr♦♥❣ Lp (Q0,T ), dun du ✯ tr♦♥❣ ❱✱ dt dt −1,p ✯−∆p,γ un −∆p,γ u tr♦♥❣ Lp (0, T ; D−γ (Ω)), ✯ ❉♦ ✤â ❝❤✉②➸♥ q✉❛ ❣✐ỵ✐ ❤↕♥ tr♦♥❣ ✤➥♥❣ t❤ù❝ T T T < un , v > + 0 dt Ω u(t) ❧➔ ♥❣❤✐➺♠ ξ ∈ G(t, η) t❛ ❝â ✤â |x|−pγ |∇un |p−2 ∇un ∇v+ dt T f (un )v = Ω ②➳✉ ❝õ❛ ✭✶✳✶✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ dt u(0) = η ✳ gv Ω ❉♦ ❱ỵ✐ ❣✐↔ t❤✐➳t ✭✸✮ t❛ t❤➜② ✈ỵ✐ ♥ ✤õ ❧ỵ♥✱ G(tn , B) = G(t∗ + tn − t∗ , B) ⊂ G(t∗ , G(tn − t∗ B)) ⊂ G(t∗ , B ∗ ), t∗ > ✈➔ B ∗ ❧➔ ♠ët t➟♣ ❜à ❝❤➦♥ tr♦♥❣ L2 (Ω)✳ ❙û ❞ö♥❣ ❜ê ✤➲ ✸✳✶✳✺✱ ♥➳✉ ξn ∈ G(tn , B) t❤➻ {ξn } ❧➔ t✐➲♥ ❝♦♠♣❛❝t tr♦♥❣ L (Ω) ❚❤❡♦ ✤à♥❤ ❧➼ ✸✳✶✳✹ ❝❤ó♥❣ t❛ ❦➳t ❧✉➟♥ ♥û❛ ❞á♥❣ ✤❛ trà ♥❣➦t G ✤÷đ❝ s✐♥❤ tr♦♥❣ ✤â ❜ð✐ ❜➔✐ t♦→♥ ✭✶✳✶✮ tr♦♥❣ tr÷í♥❣ ❤đ♣ ỉtỉ♥ỉ♠ ❝â ♠ët t➟♣ ❤ót t♦➔♥ ❝ư❝ ❝♦♠♣❛❝t ❜➜t ❜✐➳♥ tr♦♥❣ L2 (Ω) ✸✳✷ ❚r÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ ỉtỉ♥ỉ♠✿ ❈❤ó♥❣ t❛ ♥❤➢❝ ❧↕✐ ♠ët sè ✤à♥❤ ♥❣❤➽❛ ✈➔ ♠ët sè ❦➳t q✉↔ ❧➦♣ ❝õ❛ ❝→❝ ❤➔♠ (f (s, ), g(., s)) = σ(s) ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤➦❝ tr÷♥❣ ❝õ❛ ✭✶✳✶✮✳ ❚❛ σ(s + h) = (f (s + h, ), g(., s + h)) ✈➔ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❞➣② σ(s + hn ) tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ Σ✳ ❍å ❝õ❛ ♠é✐ n∈N ❞↕♥❣ ♥❤÷ ✈➟② ✤÷đ❝ ❣å✐ ❧➔ ❜❛♦ ❝õ❛ σ tr♦♥❣ Σ ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ H(σ), tù❝ ❧➔✿ ①➨t ❜➔✐ t♦→♥ ✭✶✳✶✮ ✈ỵ✐ ♠ët ❤å ✸✷ H(σ) = clΣ {σ( + h)|h ∈ R} ◆➳✉ ❜❛♦ H(σ) ❧➔ ♠ët t➟♣ ❝♦♠♣❛❝t tr♦♥❣ Σ Rd = (t, τ ) ∈ R2 |τ ≤ t Σ✱ t❛ ♥â✐ r➡♥❣ σ ❧➔ ❝♦♠♣❛❝t ❞à❝❤ ❝❤✉②➸♥ tr♦♥❣ ❑➼ ❤✐➺✉✿ ●å✐ ❳ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ✱ P✭❳✮ ✈➔ ❇✭❳✮ ❧➔ t➟♣ t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ✈➔ t➟♣ t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❜à ❝❤➦♥ ❝õ❛ ❳✳ ●✐↔ sû ❩ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ Z = ϕ ∈ C(R; R) : |ϕ(u)| ≤ Cϕ (1 + |u|q−1 ), |ϕ(u)| ||ϕ||Z = sup q−1 u∈R + |u| ❑❤✐ ✤â ❩ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❚❛ ❝â ✈ỵ✐ fn → f Σ ❦➼ ❤✐➺✉✿ Cϕ > , tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ C(R; Z) ♥➳✉✿ lim sup ||fn (s, ) − f (s, )||Z = (3.2) n→∞ s∈[t,t+r] t ∈ R, r > f0 ∈ C(R; Z), g0 ∈ L2,w loc (R; L (Ω)) ✈ỵ✐ ♠å✐ ●✐↔ sû ✈➔ H(f0 ) = clC(R;Z) {f0 ( + h)|h ∈ R} , H(g0 ) = clL2,w (R;L2 (Ω)) {g0 ( + h)|h ∈ R} , loc L2,w loc (R; L (Ω)) ✤÷đ❝ tr❛♥❣ ❜à gn → g tr♦♥❣ L2,w loc (R; L (Ω)) ♥➳✉ tr♦♥❣ ✤â tỉ♣ỉ tr♦♥❣ t♦➔♥ ❝ư❝✱ tù❝ ❧➔ ✿ ❜ð✐ tà♥❤ ❤ë✐ tö ②➳✉ t+r (gn (s, x) − g(s, x))φ(x, s)dsdx = lim n→∞ t ✈ỵ✐ ♠å✐ t ∈ R, r > Ω ✈➔ ϕ ∈ L2 (Qt,t+r ) ❚❛ ❦➼ ❤✐➺✉ Σ = H(f0 ) × H(g0 ) ▼➺♥❤ ✤➲ ✸✳✷✳✶✳ [8] ❍➔♠ f ∈ C(R; Z) ❧➔ ❝♦♠♣❛❝t ❞à❝❤ ❝❤✉②➸♥ ♥➳✉ ✈➔ ∀R > t❛ ❝â✿ ✭✶✮ |f (t, v)| ≤ C(R), ∀t ∈ R, v ∈ [−R, R]✱ ✭✷✮ |f (t1 , v1 ) − f (t2 , v2 )| ≤ α(|t1 − t2 | + |v1 − v2 |, R), ∀t1 , t2 ∈ R, v1 , v2 ∈ [−R, R], ✈ỵ✐ C(R) > ✈➔ α(., ) ❧➔ ♠ët ❤➔♠ t❤ä❛ ♠➣♥ α(s, R) → ❦❤✐ s → 0+ ❝❤➾ ♥➳✉ ✸✸ ❚ø ❜➙② ❣✐í t❛ ❧✉ỉ♥ ❣✐↔ sû g f ❧➔ ❤➔♠ ❝♦♠♣❛❝t ❞à❝❤ ❝❤✉②➸♥✳ ❈ị♥❣ ✈ỵ✐ ❧➔ ❤➔♠ ❝♦♠♣❛❝t ❞à❝❤ ❝❤✉②➸♥ tr♦♥❣ t➟♣ ❝♦♠♣❛❝t tr♦♥❣ L2,w loc (R; L (Ω))✱ T (h) : Σ → Σ L2,w loc (R; L (Ω)) ✱ t❛ t❤➜② r➡♥❣ Σ ❧➔ ❦❤✐ ✤â tø ❬✽❪ t❛ ❝â✿ ❧➔ ❧✐➯♥ tö❝ ✈➔ T (h)Σ ⊂ Σ, ∀h ∈ R ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳✷✳ [21] ⑩♥❤ ①↕ U : Rd × X → P(X)✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❞á♥❣ ✤❛ trà ✭▼❙P✮ ♥➳✉ (1)U (, , ) = Id ỗ ♥❤➜t ✮ (2)U (t, τ, x) ⊂ U (t, s, U (s, τ, x)), ∀x ∈ X, t, sτ ∈ R, τ ≤ s ≤ t ❯ ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ q✉→ tr➻♥❤ ✤❛ trà ♥❣➦t ♥➳✉ U (t, τ, x) = U (t, s, U (s, τ, x)) Dτ,σ (uτ )❧➔ t➟♣ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ②➳✉ t♦➔♥ ❝ö❝ ữủ ợ t t♦→♥ ✭✶✳✶✮ ✈ỵ✐ (fσ , gσ ) t❤❛② ❝❤♦ (f, g) s❛♦ ❝❤♦ u(τ ) = uτ ✳ ❱ỵ✐ ♠é✐ σ = (f, g) ∈ Σ t❛ ①➨t ❤å ▼❙P {Uσ : σ ∈ Σ} ❚❛ ❦➼ ❤✐➺✉ t➟♣ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ Uσ (t, τ, uτ ) = {u = u(t)|u(.) ∈ Dτ,σ (uτ )} ❇ê ✤➲ ✸✳✷✳✸✳ Uσ (t, τ, uτ ) ❧➔ ♠ët ♥û❛ q✉→ tr➻♥❤ ✤❛ trà✳ ❍ì♥ ♥ú❛ Uσ (t + s, τ + s, uτ ) = UT (s)σ (t, τ, uτ ), ∀uτ ∈ L2 (Ω), (t, τ ) ∈ Rd , s ∈ R z ∈ Uσ (t, τ, uτ ) t❛ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣✿ z ∈ Uσ (t, s, Uσ (s, τ, uτ ))✳ ●å✐ y(.) ∈ Dτ,σ (uτ ) s❛♦ ❝❤♦ y(τ ) = uτ ✈➔ y(t) = z ✳ ❘ã r➔♥❣ y(s) ∈ Uσ (s, τ, uτ )✳ ❑❤✐ ✤â ♥➳✉ t❛ ✤à♥❤ ♥❣❤➽❛ z(t) = y(t) ✈ỵ✐ t ≥ s t❛ t❤➜② r➡♥❣ z(s) = y(s) ✈➔ ❤✐➸♥ ♥❤✐➯♥ z(.) ∈ Ds,σ (y(s))✳ ❉♦ ❞â z(t) ∈ Uσ (t, s, Uσ (s, τ, uτ )) ✣➦t z ∈ Uσ (t + s, τ + s, uτ ) õ tỗ t u(.) D +s, (u ) s❛♦ ❝❤♦ z = u(t+s) ✈➔ v(.) = u(.+s) ∈ Dτ,T (s)σ (uτ )✱ ❞♦ ✤â z = v(t) ∈ Uτ,T (s)σ (uτ )✳ ◆❣÷đ❝ ❧↕✐ ✱ z ∈ Uτ,T (s)σ (uτ ) tø ✤â z ∈ Dτ,T (s)σ (uτ ) s❛♦ ❝❤♦ z = u(t) ✈➔ v(.) = u(−s + ) ∈ Dτ +s,σ (uτ ) ✈➻ ✈➟② z = v(t + s) ∈ Uσ (t + s, τ + s, uτ ) ❑➼ ❤✐➺✉ UΣ (t, τ, x) = Uσ (t, τ, x) ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ σ∈Σ ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳✹✳ [21] ❚➟♣ ❤đ♣ A ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ❤ót t♦➔♥ ❝ư❝ ✤➲✉ ✤è✐ ✈ỵ✐ ❝→❝ ♥û❛ q✉→ tr➻♥❤ ✤❛ trà UΣ ♥➳✉✿ (1)A ❧➔ ♥û❛ ❜➜t ❜✐➳♥ ➙♠✱ tù❝ ❧➔ A ⊂ UΣ (t, τ, A), ∀t ≥ τ ; (2)A ❧➔ ❤ót ✤➲✉✱ tù❝ ❧➔ ✿ dist(UΣ (t, τ, B), A) → ❦❤✐ t → ∞, ∀B ∈ B(X) ✸✹ τ ∈ R❀ ✈➔ (3) ✈ỵ✐ ♠å✐ t➟♣ ❤ót ✤â♥❣ ✤➲✉ ❨✱ t❛ ❝â A⊂Y t ỹ t ỵ [21] sỷ r ❤å ❝→❝ ♥û❛ q✉→ tr➻♥❤ ✤❛ trà UΣ t❤ä❛ ♠➣♥ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ s❛✉✿ (1) ❚r➯♥ Σ ❝â ♠ët t♦→♥ tû ❞à❝❤ ❝❤✉②➸♥ ❧✐➯♥ tö❝ T (s)σ(t) = σ(t + s), ∀s ∈ R s❛♦ ❝❤♦ T (h)Σ ⊂ Σ ✈➔ ✈ỵ✐ ♠é✐ (t, τ ) ∈ Rd , σ ∈ Σ, s ∈ R, x ∈ X t❛ ❝â✿ Uσ (t + s, τ + s, x) = UT (s)σ (t, τ, x)❀ (2)Uσ ❧➔ ♥û❛ ❝♦♠♣❛❝t t✐➺♠ ❝➟♥ tr➯♥ ✤➲✉❀ (3)UΣ ❧➔ t✐➯✉ ❤❛♦ ✤✐➸♠❀ (4) ⑩♥❤ ①↕ (x, σ) → Uσ (t, 0, x) ❝â ❣✐→ trà ✤â♥❣ ❧➔ ✇ ✲ ♥û❛ ❧✐➯♥ tö❝ tr➯♥✳ ❑❤✐ ✤â✱ ❤å ❝→❝ ♥û❛ q✉→ tr➻♥❤ ✤❛ trà UΣ ❝â t➟♣ ❤ót ❝♦♠♣❛❝t t♦➔♥ ❝ư❝ ✤➲✉ A✳ ❇ê ✤➲ ✸✳✷✳✻✳ ◆➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❍✶✮ ✲ ✭❍✸✮ ✤÷đ❝ t❤ä❛ ♠➣♥ ✈➔ {un}n∈ N ❧➔ ♠ët ❞➣② ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ✭✶✳✶✮ ✈ỵ✐ ♥❤ú♥❣ ✤➦❝ tr÷♥❣ ❧✐➯♥ q✉❛♥ ❞➣② {σn } ⊂ Σ s❛♦ ❝❤♦✿ (1)un (τ ) → uτ tr♦♥❣ L2 (Ω)) (2)n tr õ tỗ t ởt t ợ trữ u(τ ) = uT ✈➔ un (t∗ ) → u(t∗ ) tr♦♥❣ L2 (Ω)) ✈ỵ✐ ♠é✐ σ s❛♦ t∗ > τ σn = (fn , gn )✳ ❱➻ f t❤ä❛ ♠➣♥ ✭❍✶✮✱ ∀t ∈ R ✈➔ fn ∈ H(f ) t❛ t❤➜② r➡♥❣ fn ❝ô♥❣ t❤ä❛ ♠➣♥ ✭❍✶✮✳ ▼➦t ❦❤→❝✱ ú ỵ r {un ( )} tr L2 ()) ||gn ||L2b |g||L2b õ ỵ ❧✉➟♥ t÷ì♥❣ ❈❤ù♥❣ ♠✐♥❤✳ ✿ ●✐↔ sû tü ♥❤÷ tr♦♥❣ ✤à♥❤ ❧➼ ✷✳✷✳✶✱ t❛ ❝â✿ 1,p {un } ❜à ❝❤➦♥ tr♦♥❣ V = Lp (τ, T, D0,γ (Ω)) ∩ Lq (τ, T ; Lq (Ω)), −1,p {un } ❜à ❝❤➦♥ tr♦♥❣ V = Lp (τ, T, D−γ (Ω)) + Lq (τ, T ; Lq (Ω)), {un } ❜à ❝❤➦♥ tr♦♥❣ C([τ, T ]; L2 (Ω)), {fn (t, un )} ❜à ❝❤➦♥ tr♦♥❣ Lq (Qτ,T ), −1,p {−∆p,γ un } ❜à ❝❤➦♥ tr♦♥❣ Lp (τ, T ; D−γ (Ω)) ❚ø ✤â t❛ ❝â✿ un (t) u(t) tr♦♥❣ L2 (Ω), ∀t ∈ [τ, T ], ✣➦t σn → σ = (f¯, g ¯) tr♦♥❣ Σ✱ t❛ t❤➜② r➡♥❣ ✉ ❧➔ ♠ët ♥❣❤✐➺♠ t ợ trữ s u(τ ) = uT ✱ ❧➜② ❣✐ỵ✐ ❤↕♥ ✤➥♥❣ t❤ù❝ s❛✉ t❛ ✸✺ ✤÷đ❝✿ T T (un v + |x|−pγ |∇un |p−2 ∇un ∇v + fn (t, un )v)dxdt = τ gn vdxdt τ Ω Ω ∀v ∈ V ✳ ❉♦ gn g¯ tr♦♥❣ L2 (τ, T ; L2 (Ω)) ♥➯♥ fn (t, un ) f¯(t, u) tr♦♥❣ Lq (Qτ,T )✳ ✣➛✉ t✐➯♥ t❛ ❝❤➾ r❛ r➡♥❣ fn (t, un ) → f¯(t, u) tr♦♥❣ Lq (Qτ,T )✳ ❚❤➟t ✈➟②✿ T |fn (t, un ) − f¯(t, un )|q dxdt τ Ω T = τ Ω |fn (t, un ) − f¯(t, un )|q (1 + |un |q−1 )q dxdt q−1 q (1 + |un | ) sup ||fn − f¯||Z ≤ q T (1 + |un |q )dxdt → τ Ω [τ,T ] fn → f¯ tr♦♥❣ ❩ ✈➔ {un } ❜à ❝❤➦♥ tr♦♥❣ Lq (Qτ,T )✳ ▼➦t ❦❤→❝✱ ❞♦ f¯(t, un ) ❜à ❝❤➦♥ tr♦♥❣ Lq (Qτ,T )✱ sû ❞ư♥❣ ❜ê ✤➲ ✶✳✸ tr♦♥❣ ❬❬✶✽❪ ❝❤÷ì♥❣ ✶❪ ✈➔ t➼♥❤ ❧✐➯♥ tư❝ ❝õ❛ f¯ ♥❤÷ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ỵ t t r f(t, un ) → f¯(t, u) Lq (Qτ,T )✳ ②➳✉ tr♦♥❣ ❉♦ ✤â t❛ ❝â✿ fn (t, un )−f¯(t, u) = (fn (t, un )−f¯(t, un ))+(f¯(t, un )−f¯(t, u)) → u∗n → u(t∗ ) ❇➙② ❣✐í t❛ ❝❤➾ r❛ r➡♥❣ tr♦♥❣ L2 (Ω) ✈ỵ✐ ♠é✐ ②➳✉ tr♦♥❣ t∗ > τ ✳ ✣➸ ❝â ✤÷đ❝ un (t) t❛ ♣❤↔✐ ❦✐➸♠ tr❛ r➡♥❣ u(t) tr♦♥❣ L2 (Ω), ∀t ∈ [τ, T ], ||un (t∗ )||L2 (Ω) → ||u(t∗ )||L2 (Ω) ✣➦t✿ t Jn (t) = ||un (t)||2L2 (Ω) − (gn (s), un (s))ds − (2k2 |Ω| + 2λ)(t − τ ), τ t J(t) = ||u(t)||2L2 (Ω) − (g(s), u(s))ds − (2k2 |Ω| + 2λ)(t − τ ) τ ❉➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝ ❝→❝ ❤➔♠ Jn (t), J(t) ❧➔ ❧✐➯♥ tư❝ ✈➔ ❦❤ỉ♥❣ t➠♥❣ tr➯♥ Lq (Qτ,T ✸✻ [τ, T ]✳ ✣➛✉ t✐➯♥ t❛ ❝❤➾ r❛ r➡♥❣✿Jn (t) → J(t) ❤➛✉ ❦❤➢♣ ♥ì✐ ✈ỵ✐ t ∈ [τ, T ], ❚❤➟t ✈➟②✱ |Jn (t) − J(t)| ≤ ||un (t)||2L2 (Ω) − ||u(y)||2L2 (Ω) t [(gn (s), un (s)) − (g(s), u(s))]ds +2 τ ≤ ||un (t) − u(t)||L2 (Ω) (||un (t)||L2 (Ω) − ||u(t)||L2 (Ω) ) t t [gn (s), un (s) − u(s)]ds + +2 τ [gn (s) − g(s), u(s)]ds τ ✈➔ t [gn (s), un (s) − u(s)]ds ≤ ||gn ||L2 (Qτ,t ) ||un (t) − u(t)||L2 (Ω) → τ ❦❤✐ n → ∞ ✈➔ un → u ♠↕♥❤ tr♦♥❣ L2 (Qτ,t ) ✈➔ {gn } ❦❤✐ n→∞ ❜à ❝❤➦♥ tr♦♥❣ t L2 (Qτ,t ) [gn (s) − g(s), u(s)]ds → ◆❣♦➔✐ r❛✱ ✈➔ gn g τ L2 (Qτ,t )✳ ❉♦ ✤â Jn (t) → J(t) ❤➛✉ ❦❤➢♣ ♥ì✐ ✈ỵ✐ t ∈ [τ, T ]✳ ❚❤ü❝ t➳ ❧➔ un (t) → u(t) tr♦♥❣ L (Ω) ❤➛✉ ❦❤➢♣ ♥ì✐ ✈ỵ✐ t ∈ [τ, T ]✳ ∗ ❚❛ ❝❤å♥ ♠ët ❞➣② ❦❤æ♥❣ t➠♥❣ {tm } ⊂ [τ, T ], tm → t s❛♦ ❝❤♦ Jn (tm ) J(tm ) ❦❤✐ n → ∞✳ ❑❤✐ ✤â✱ ❞♦ t➼♥❤ ❧✐➯♥ tö❝ Jn (tm ) Jn (t∗ ) ❦❤✐ m → ∞ tr♦♥❣ ❱➟②✿ Jn (t∗ ) − J(t∗ ) ≤ Jn (tm ) − J(t∗ ) = Jn (tm ) − J(tm ) + J(tm ) − J(t∗ ) < ε ✈ỵ✐ n ≥ n0 (ε) ✈➔ ε>0 ❜➜t ❦➻✳ ❉♦ ✤â lim sup Jn (t∗ ) ≤ J(t∗ ) ✈➔ ❞♦ ✤â lim sup ||un (t∗ )|| ≤ ||u(t∗ )|| ❚ø sü ❤ë✐ tö ②➳✉ ∗ ∗ un (t ) → u(t ) un (t∗ ) ♠↕♥❤ tr♦♥❣ u(t∗ ) t❛ ❝â ||un (t∗ )|| → ||u(t∗ )|| L2 (Ω) ❦❤✐ n → ∞ ❞♦ ✈➟② ỵ sỷ ✲ ✭❍✸✮ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â ❤å ❝→❝ ♥û❛ q✉→ tr➻♥❤ ✤❛ trà {Uσ (t, τ )} ❝â ♠ët t➟♣ ❤ót t♦➔♥ ❝ư❝ ✤➲✉ A ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❜✐➳t r➡♥❣ ✈ỵ✐ ♠é✐ σn = (fn , gn ) ∈ Σ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❍✶✮ ✲ ✭❍✷✮✳ gn ∈ H(g), ❚ø ||gn ||L2b ≤ ||g||L2b ✈ỵ✐ σn t❛ ❝â✿ t❛ ❝â ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✶✮ ||un (t)||2L2Ω ≤ ||un (τ )||2L2 (Ω) e−λ(t−τ ) + ❉♦ ✤â ♥➳✉ un ❧➔ ♠ët ♥❣❤✐➺♠ ②➳✉ 2k2 |Ω| ||g|| + +2 Lb λ(1 − e−λ ) λ R0 s❛♦ ❝❤♦ ♥➳✉ un (τ ) ∈ BR , ❤➻♥❤ ❝➛✉ tr♦♥❣ LΩ t➙♠ ❖ ❜→♥ ❦➼♥❤ ❘ t tỗ t T0 = T0 (, R) s ứ t tự tr t t r tỗ t ởt sè un (t) ∈ BR0 , ∀t ≥ T0 , ❝â UΣ (t, τ, BR ) ⊂ BR0 , ∀t ≥ T0 (τ, R) ❝❤♦ t❛ ❉♦ ✤â {Uσ (t, )} tọ tr ỵ ✸✳✷✳✺ ❚❛ ✤à♥❤ ♥❣❤➽❛ t➟♣ ❤ñ♣ K = UΣ (1, 0, BR0 )✳ ❚ø ❜ê ✤➲ ✸✳✷✳✻ t❛ ❝â ❑ ❧➔ ❝♦♠♣❛❝t✳ BR0 ❧➔ t➟♣ ❤➜♣ t❤ö ✱ t❛ ❝â✿ Uσn (t, τ, BR ) = Uσn (t, t − 1, Uσn (t − 1, τ, BR )) = UT (t−1)σn (1, 0, UT (τ )σn (t − − τ, 0, BR )) ⊂ UΣ (1, 0, BR0 ) ⊂ K, ∀σn ∈ Σ, BR ∈ B(L2 (Ω)) ✈➔ t ≥ T0 (τ, BR ) ❚ø ✤â ✈ỵ✐ ❜➜t ❦➻ ❞➣② {ξn } t❤ä❛ ♠➣♥ {ξn } ∈ Uσn (tn , τ, BR0 ), σn ∈ Σ, tn → +∞, BR ∈ B(L2 (Ω)) ❧➔ t✐➲♥ ❝♦♠♣❛❝t tr♦♥❣ L2 (Ω) ◆â ❧➔ ♠ët ❤➺ q✉↔ ❝õ❛ ❜ê ✤➲ ✸✳✷✳✻ →♥❤ ①↕ Uσ ❝â ❣✐→ trà ❝♦♠♣❛❝t ✈ỵ✐ ❜➜t ❦➻ σ ∈ Σ ❈✉è✐ ❝ò♥❣✱ t❛ ❝❤ù♥❣ ♠✐♥❤ →♥❤ ①↕ (σ, x) → Uσ (t, τ, x) ❧➔ ♥û❛ ❧✐➯♥ tư❝ tr➯♥ ✈ỵ✐ ♠é✐ t ≥ τ ●✐↔ sû ✤✐➲✉ õ ổ ú tỗ t u ∈ L2 (Ω), t ≥ τ, σ ¯ ∈ Σ, ε > 0, δn → 0, un ∈ Bδn (¯ u), σn → σ ¯ , ✈➔ ξn ∈ Uσn (t, τ, un ) s❛♦ ❝❤♦ {ξn } ∈ / Bε (Uσ¯ (t, τ, u¯)) ❚✉② ♥❤✐➯♥ ❜ê ✤➲ ✸✳✷✳✻ ❧↕✐ ❧➔ ξn → ξ ∈ Uσ ¯) ✤✐➲✉ ✤â ♠➙✉ t❤✉➝♥✳ ¯ (t, τ, u ❍ì♥ ♥ú❛✱ ✈➻ ❉♦ ✤â✱ t❤❡♦ ✤à♥❤ ❧➼ ✸✳✷✳✺ ✈➔ ❜ê ✤➲ ✸✳✷✳✻ t❛ ❦➳t ❧✉➟♥ ❤å ❝→❝ ♥û❛ q✉→ tr➻♥❤ ✤❛ trà {Uσ (t, τ )} ❝â ♠ët t➟♣ ❤ót t♦➔♥ ❝ư❝ ✤➲✉ A ✸✽ ❑➌❚ ▲❯❾◆ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tæ✐ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ✈➔ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ❝❤ù♥❣ sỹ tỗ t t sỹ tỗ t t út t ố ợ ởt ợ ữỡ tr r t ự t tỷ r ✲ ❑♦❤♥ ✲ ◆✐r❡♥❜❡r❣✳ ❑➳t q✉↔ ❝❤➼♥❤ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr ự sỹ tỗ t ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✶✮✳ ✷✳ ❈❤ù♥❣ ♠✐♥❤ sü tỗ t t út t ố ợ t tr tr trữớ ủ ổtổổ ổ ổtổổ ữợ ❝ù✉ t✐➳♣ t❤❡♦✱ ✤è✐ ✈ỵ✐ ❜➔✐ t♦→♥ ♥➔② ❝❤ó♥❣ t❛ ❝â t❤➸ ♥❣❤✐➯♥ ❝ù✉ t✐➳♣ t➼♥❤ ❧✐➯♥ t❤æ♥❣ ✈➔ sè ❝❤✐➲✉ ❝õ❛ t➟♣ ❤ót ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ tr♦♥❣ ❝↔ ❤❛✐ tr÷í♥❣ ❤đ♣ ỉtỉ♥ỉ♠ ✈➔ ❦❤ỉ♥❣ ỉtỉ♥ỉ♠✳ ✸✾ ❚➔✐ ❧✐➺✉ t ỡ s ỵ t❤✉②➳t ❤➺ ✤ë♥❣ ❧ü❝ ✈æ ❤↕♥ ❝❤✐➲✉✑ ✱ ✣↕✐ ❤å❝ s÷ ♣❤↕♠✱ ❍➔ ◆ë✐✳ ❬✷❪ ❆❜❞❡❧❧❛♦✉✐✱ ❇✱ ❈♦❧♦r❛❞♦✱ ❊✱ P❡r❛❧✱ ■✿ ✏❊①✐st❡♥❝❡ ❛♥❞ ♥♦♥❡①✐st❡♥❝❡ r❡s✉❧ts ❢♦r ❛ ❝❧❛ss ♦❢ ❧✐♥❡❛r ❛♥❞ s❡♠✐✲❧✐♥❡❛r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s r❡✲ ❧❛t❡❞ t♦ s♦♠❡ ❈❛❢❢❛r❡❧❧✐✲❑♦❤♥✲◆✐r❡♥❜❡r❣ ✐♥❡q✉❛❧✐t✐❡s✑✳ ❏ ❊✉r ▼❛t❤ ❙♦❝✳✻✱ ✶✶✾✕✶✹✽ ✭✷✵✵✹✮✳ ❬✸❪ ❆✳ ❱✳ ❇❛❜✐♥ ✭✷✵✵✻✮✱ ✏●❧♦❜❛❧ ❆ttr❛❝t♦rs ✐♥ ❆❉❊✑✱ ❍❛ss❡❧❜❧❛tt✱ ❇✳✭❡❞✳✮ ❡t ❛❧✳✱ ❍❛♥❞❜♦♦❦ ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠✳ ❱♦❧✉♠❡ ✶❇✳ ❆♠st❡r❞❛♠✿ ❊❧❡✲ s❡✈✐❡r✳ ✾✸✽✲✶✵✽✺✳ ❬✹❪ ❆✳❱✳ ❇❛❜✐♥ ❛♥❞ ▼✳■✳ ❱✐s❤✐❦ ✭✶✾✾✷✮✱ ✏❆ttr❛❝t♦rs ♦❢ ❊✈♦❧✉t✐♦♥ ❊q✉❛✲ t✐♦♥s✑ ❚r❛♥s❧✳ ❢r♦♠ t❤❡ ❘✉ss✐❛♥ ❜② ❆✳❱✳ ❇❛❜✐❧✱❙t✉❞✐❡s ✐♥ ▼❛t❤❡♠❛t✲ ✐❝s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✳ ✷✺ ❆♠st❡r❞❛♠ ❡t❝✳ ◆♦rt❤✲ ❍♦❧❧❛♥❞✳ ✺✸✷ ♣✳ ❬✺❪ ❈❛❢❢❛r❡❧❧✐✱ ▲✱ ❑♦❤♥✱ ❘✱ ◆✐r❡♥❜❡r❣✱ ▲✿ ✏❋✐rst ♦r❞❡r ✐♥t❡r♣♦❧❛t✐♦♥ ✐♥✲ ❡q✉❛❧✐t✐❡s ✇✐t❤ ✇❡✐❣❤ts✑✳ ❈♦♠♣♦s✐t✐♦ ▼❛t❤✳✺✸✱ ✷✺✾✕✷✼✺ ✭✶✾✽✹✮✳ ❬✻❪ ❈✳✲❑✳ ❩❤♦♥❣✱ ▼✳✲❍✳ ❨❛♥❣✱ ❛♥❞ ❈✳✲❨✳ ❙✉♥✭✷✵✵✻✮✱ ✏❚❤❡ ❡①✐st❡♥❝❡ ♦❢ ❣❧♦❜❛❧ ❛ttr❛❝t♦rs ❢♦r t❤❡ ♥♦r♠✲t♦✲✇❡❛❦ ❝♦♥t✐♥✉♦✉s s❡♠✐❣r♦✉♣ ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥ t♦ t❤❡ ♥♦♥❧✐♥❡❛r r❡❛❝t✐♦♥✲❞✐❢❢✉s✐♦♥ ❡q✉❛t✐♦♥s✑✱ ❏♦✉r♥❛❧ ♦❢ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ✈♦❧✳ ✷✷✸✱ ♥♦✳ ✷✱ ♣♣✳ ✸✻✼✕✸✾✾✳ ❬✼❪ ❈❤❡♣②③❤♦✈✱ ❱❱✱ ❱✐s❤✐❦✱ ▼■✿ ✏❆ttr❛❝t♦rs ❢♦r ❊q✉❛t✐♦♥s ♦❢ ▼❛t❤❡✲ ♠❛t✐❝❛❧ P❤②s✐❝s✑✳ ■♥ ❆♠ ▼❛t❤ ❙♦❝ ❈♦❧❧♦q P✉❜❧ ❆♠ ▼❛t❤ ❙♦❝✱ ✈♦❧✳ ✹✾✱Pr♦✈✐❞❡♥❝❡✱ ❘■ ✭✷✵✵✷✮✳ ✹✵ ❬✽❪ ❈❤❡♣②③❤♦✈✱ ❱❱✱ ❱✐s❤✐❦✱ ▼■✿ ✏❊✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s ❛♥❞ t❤❡✐r tr❛❥❡❝✲ t♦r② ❛ttr❛❝t♦r✑✳ ❏ ▼❛t❤ P✉r❡ ❆♣♣❧✳✼✻✱ ✾✶✸✕✾✻✹ ✭✶✾✾✼✮✳ ❬✾❪ ❈✳ ❚✳ ❆♥❤✱ P✳ ◗✳ ❍✉♥❣✱ ❚✳ ❉✳ ❑❡✱ ❛♥❞ ❚ ✳❚✳ P❤♦♥❣✭✷✵✵✽✮✱ ✏●❧♦❜❛❧ ❛ttr❛❝t♦rs ❢♦r ❛ s❡♠✐❧✐♥❡❛r ♣❛r❛❜♦❧✐❝❡q✉❛t✐♦♥ ✐♥✈♦❧✈✐♥❣ ●r✉s❤✐♥ ♦♣✲ ❡r❛t♦r✑✱ ❊❧❡❝tr♦♥✐❝ ❏♦✉r♥❛❧ ♦❢ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ♥♦✳ ✸✷✱ ♣♣✳ ✶✲ ✶✶✳ ❬✶✵❪ ❈✳ ❚✳ ❆♥❤ ❛♥❞ ❚✳❚ P❤♦♥❣ ✭✷✵✵✾✮✱ ✏●❧♦❜❛❧ ❛ttr❛❝t♦rs ❢♦r ❛ s❡♠✐❧✐♥❡❛r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ ✇❡✐❣❤t❡❞ ♣✲▲❛♣❧❛❝✐❛♥ ♦♣❡r❛t♦rs✑✱ ❆♥♥✳ P♦❧✳▼❛t❤✳ ✾✽✱ ✷✺✶✲✷✼✶✳ ❬✶✶❪ ❈✳ ❚✳ ❆♥❤ ❛♥❞ ◆✳❱✳ ◗✉❛♥❣ ✭✷✵✶✶✮✱ ✏❯♥✐❢♦r♠ ❛ttr❛❝t♦rs ❢♦r ♥♦♥❛✉✲ t♦♥♦♠♦✉s ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥ ✐♥✈♦❧✈✐♥❣ ●r✉s❤✐♥ ♦♣❡r❛t♦r✑✱ ❆❝t❛ ▼❛t❤✳ ❱✐❡t♥❛♠✳✸✻✱ ♥♦✳ ✶✱✶✾✲✸✸✳ ❬✶✷❪ ❉❛❧❧✬❛❣❧✐♦✱ ❆✱ ●✐❛❝❤❡tt✐✱ ❉✱ P❡r❛❧✱ ■✿ ✏❘❡s✉❧ts ♦♥ ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s r❡❧❛t❡❞ t♦ s♦♠❡ ❈❛❢❢❛r❡❧❧✐✲❑♦❤♥✲◆✐r❡♥❜❡r❣ ✐♥❡q✉❛❧✐t✐❡s✑✳ ❙■❆▼ ▼❛t❤ ❆♥❛❧✳✸✻✱ ✻✾✶✕✼✶✻ ✭✷✵✵✹✮✳ ❬✶✸❪ ❏✳ ❈✳ ❘♦❜✐♥s♦♥✭✷✵✵✶✮✱ ✏■♥❢✐♥✐t❡✲❉✐♠❡♥s✐♦♥❛❧ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s✑✱ ❈❛♠❜r✐❞❣❡ ❚❡①ts ✐♥ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✱❯❑✳ ❬✶✹❪ ❏✳ ▼✳ ❆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 ✶✻✽✱ ✺✸✸✲✺✺✾✳ ❬✶✺❪ ❏✳ ▼✳ ❆rr✐❡t❛✱ ❏✳❲✳ ❈❤♦❧❡✇❛✱ ❚✳ ❉❧♦t❦♦ ❛♥❞ ❆✳ ❘♦❞r✐❣✉❡③✲❇❡r♥❛❧ ✭✷✵✵✹✮✱ ✏❆s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ❛♥❞ ❛ttr❛❝t♦rs ❢♦r r❡❛❝t✐♦♥ ❞✐❢❢✉s✐♦♥ ❡q✉❛t✐♦♥s ✐♥ ✉♥❜♦✉♥❞❡❞ ❞♦♠❛✐♥s✑✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✺✻✱✺✶✺❀✺✺✹✳ ❬✶✻❪ ❑❛♣✉st②❛♥✱ ❆❱✿ ✏●❧♦❜❛❧ ❛ttr❛❝t♦rs ♦❢ ❛ ♥♦♥❛✉t♦♥♦♠♦✉s r❡❛❝t✐♦♥✲ ❞✐❢❢✉s✐♦♥ ❡q✉❛t✐♦♥✑✳ ❉✐❢❢ ❊q✉✸✽✭✶✵✮✿✶✹✻✼✕✶✹✼✶ ✭✷✵✵✷✮✳ ❬❚r❛♥s❧❛t✐♦♥ ❢r♦♠ ❉✐❢❢❡r❡♥s✐❛❧ ❯r❛✈♥❡♥✐②❛ ✸✽✭✶✵✮✱ ✶✸✼✽✲✶✸✽✶ ✭✷✵✵✷✮❪✳ ❬✶✼❪ ❑❛♣✉st②❛♥✱ ❆❱✱ ❙❤❦✉♥❞✐♥✱ ❉❱✿ ✏●❧♦❜❛❧ ❛ttr❛❝t♦r ♦❢ ♦♥❡ ♥♦♥❧✐♥❡❛r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥✑✳ ❯❦r❛✐♥ ▼❛t❤ ❩❤✳✺✺✱ ✹✹✻✕✹✺✺ ✭✷✵✵✸✮✳ ✹✶ ❬✶✽❪ ▲✐♦♥s✱ ❏▲✿ ✏◗✉❡❧q✉❡s ▼➨t❤♦❞❡s ❞❡ ❘➨s♦❧✉t✐♦♥ ❞❡s Pr♦❜❧➧♠❡s ❛✉① ▲✐♠✐t❡s ◆♦♥ ▲✐♥➨❛✐r❡s✑✳ ❉✉♥♦❞✱ P❛r✐s ✭✶✾✻✾✮✳ ❬✶✾❪ ▼❡❧♥✐❦✱ ❱❙✱ ❱❛❧❡r♦✱ ❏✿ ✏❆❞❞❡♥❞✉♠ t♦ ❖♥ ❛ttr❛❝t♦rs ♦❢ ♠✉❧t✐✲✈❛❧✉❡❞ s❡♠✐❢❧♦✇s ❛♥❞ ❞✐❢❢❡r❡♥t✐❛❧ ✐♥❝❧✉s✐♦♥s✑✳ ❙❡t ❱❛❧✉❡❞ ❆♥❛❧✳ ✶✻✱ ✺✵✼✕✺✵✾ ✭✷✵✵✽✮✳ ❬✷✵❪ ▼❡❧♥✐❦✱ ❱❙✱ ❱❛❧❡r♦✱ ❏✿ ✏❖♥ ❛ttr❛❝t♦rs ♦❢ ♠✉❧t✐✲✈❛❧✉❡❞ s❡♠✐❢❧♦✇s ❛♥❞ ❞✐❢❢❡r❡♥t✐❛❧ ✐♥❝❧✉s✐♦♥s✑✳ ❙❡t ❱❛❧✉❡❞ ❆♥❛❧✳✻✱✽✸✕✶✶✶ ✭✶✾✾✽✮✳ ❬✷✶❪ ▼❡❧♥✐❦✱ ❱❙✱ ❱❛❧❡r♦✱ ❏✿ ✏❖♥ ❣❧♦❜❛❧ ❛ttr❛❝t♦rs ♦❢ ♠✉❧t✐✲✈❛❧✉❡❞ s❡♠✐♣r♦✲ ❝❡ss❡s ❛♥❞ ♥♦♥❛✉t♦♥♦♠♦✉s ❡✈❛❧✉t✐♦♥ ✐♥❝❧✉s✐♦♥s✑✳ ❙❡t ❱❛❧✉❡❞ ❆♥❛❧✳✽✱ ✸✼✺✕✹✵✸ ✭✷✵✵✵✮✳ ❬✷✷❪ ▼♦r✐❧❧❛s✱ ❋✱ ❱❛❧❡r♦✱ ❏✿ ✏❆ttr❛❝t♦rs ❢♦r r❡❛❝t✐♦♥✲❞✐❢❢✉s✐♦♥ ❡q✉❛t✐♦♥s ✐♥ ✇✐t❤ ❝♦♥t✐♥✉♦✉s ♥♦♥❧✐♥❡❛r✐t②✑✳ ❆s②♠♣t♦t ❆♥❛❧✳✹✹✱ ✶✶✶✕✶✸✵ ✭✷✵✵✺✮✳ ❬✷✸❪ ◆✳❉✳❇✐♥❤ ❛♥❞ ❈✳❚✳❆♥❤✿ ❆ttr❛❝t♦rs ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s r❡❧❛t❡❞ t♦ ❈❛❢❢❛r❡❧❧✐✲❑♦❤♥✲◆✐r❡♥❜❡r❣ ✐♥❡q✉❛❧✐t✐❡s✳ ❇♦✉♥❞❛r② ❱❛❧✉❡ Pr♦❜✲ ❧❡♠s✳ ✶✲✶✾ ✭✷✵✶✷✮✳ ❬✷✹❪ ❘♦s❛✱ ❘✿ ✏❚❤❡ ❣❧♦❜❛❧ ❛ttr❛❝t♦r ❢♦r t❤❡ ✷❉ ◆❛✈✐❡r✲❙t♦❦❡s ❢❧♦✇ ♦♥ s♦♠❡ ✉♥❜♦✉♥❞❡❞ ❞♦♠❛✐♥s✑✳ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✸✷✱✼✶✕✽✺ ✭✶✾✾✽✮✳ ❬✷✺❪ ❘✳ ❚❡♠❛♠ ✭✶✾✾✼✮✱ ✏■♥❢✐♥✐t❡✲❉✐♠❡♥s✐♦♥❛❧ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ✐♥ ▼❡✲ ❝❤❛♥✐❝s ❛♥❞ P❤②s✐❝s✑✱ ✷♥❞ ❡❞✐t✐♦♥✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✳ ❬✷✻❪ ❚❡♠❛♠✱ ❘✿ ✏◆❛✈✐❡r✲❙t♦❦❡s ❊q✉❛t✐♦♥s ❛♥❞ ◆♦♥❧✐♥❡❛r ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s✑✳❙■❆▼ ✭s❡r✐❡s ❧❡❝t✉r❡s✮✱ P❤✐❧❛❞❡❧♣❤✐❛✱ ✷ ✭✶✾✾✺✮✳ ❬✷✼❪ ❱❛❧❡r♦✱ ❏✱ ❑❛♣✉st②❛♥✱ ❆✿ ✏❖♥ t❤❡ ❝♦♥♥❡❝t❡❞♥❡ss ❛♥❞ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦✉r ♦❢ s♦❧✉t✐♦♥s ♦❢ r❡❛❝t✐♦♥✲❞✐❢❢✉s✐♦♥ s②st❡♠s✑✳ ❏ ▼❛t❤ ❆♥❛❧ ❆♣♣❧✳✸✷✸✱ ✻✶✹✕✻✸✸ ✭✷✵✵✻✮✳ ❬✷✽❪ ❱✳❱✳ ❈❤❡♣②③❤♦✈ ❛♥❞ ▼✳■✳ ❱✐s❤✐❦ ✭✷✵✵✷✮✱ ✏❆ttr❛❝t♦rs ❢♦r ❊q✉❛t✐♦♥s ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✑✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ❈♦❧❧♦q✳ P✉❜❧✳✱ ❱♦❧✳ ✹✾✱ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✳ ... ❤å❝ tr➯♥ t❤➳ ❣✐ỵ✐✳ ❈→❝ ✈➜♥ ✤➲ ✤➦t r❛ ❧➔ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ✤➦t ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭sü tỗ t t sỹ tở tử ❝õ❛ ♥❣❤✐➺♠ t❤❡♦ ❞ú ❦✐➺♥ ✤➣ ❝❤♦✮ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ✤à♥❤ t➼♥❤ ❝õ❛ ♥❣❤✐➺♠ ✭t➼♥❤ trì♥✱ ❞→♥❣ ✤✐➺✉ t✐➺♠... ②➯✉ ❝➛✉ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ❉♦ ✤â ♥➳✉ ✤➦t tr♦♥❣ ✤à♥❤ ❝õ❛ t➼♥❤ ❝♦♠♣❛❝t t ú ỵ X ổ ỗ ỷ õ ởt t tử ❝❤➦♥ B✱ S(t) ❝â t❤➻ ❜❛ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ S(t) ❧➔ ❝♦♠♣❛❝t t✐➺♠ ❝➟♥✱ ❜✮ ◆û❛ ♥❤â♠ S(t)... ❧➔ ❝♦♠♣❛❝t ②➳✉✳ ❚❛ s➩ sû ❞ư♥❣ ❧➼ t❤✉②➳t t➟♣ ❤ót ✤➲✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❦➨♣ ✤è✐ ✈ỵ✐ q✉→ tr➻♥❤ ❧✐➯♥ tử ữỡ ữợ ữủ t t ❝➟♥ ✤➸ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ trì♥ ❝õ❛ t➟♣ ❤ót ✤➲✉ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✳ ❇ê ✤➲ ✶✳✹✳✸✳