❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ◆●❯❨➍◆ ❚❍➚ ❚❍❆◆❍ ▲❖❆◆ ❍➏ P❍×❒◆● ❚❘➐◆❍ ◆❆❱■❊❘✲❙❚❖❑❊❙ ❈➶ ❚❘➍ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❍➔ ◆ë✐✱ ✷✵✶✽ ❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ◆●❯❨➍◆ ❚❍➚ ❚❍❆◆❍ ▲❖❆◆ ❍➏ P❍×❒◆● ❚❘➐◆❍ ◆❆❱■❊❘✲❙❚❖❑❊❙ ❈➶ ❚❘➍ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ▼➣ sè ữớ ữợ ❚❙✳ ✣➔♦ ❚rå♥❣ ◗✉②➳t ❍➔ ◆ë✐✱ ✷✵✶✽ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ✐ ▲í✐ ❝❛♠ ✤♦❛♥ ✐✐ ▼ët sè ❦➼ ❤✐➺✉ t❤÷í♥❣ ❞ò♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥ ▼ð ✤➛✉ ✶ ✷ ỹ tỗ t t t ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ◆❛✈✐❡r✲❙t♦❦❡s ❝â tr➵ ✺ ✶✳✶ ✣➦t ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t ✈➔ t➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✾ ✷ ❙ü tỗ t t ứ ố ợ ữỡ tr rts õ tr ỹ tỗ t t t ứ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❚➼♥❤ ê♥ ✤à♥❤ ♠ô ❝õ❛ ♥❣❤✐➺♠ ❞ø♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ❑➳t ❧✉➟♥ ✷✾ ✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✵ ▲í✐ ❝↔♠ ì♥ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❚❙✳ ✣➔♦ ❚rå♥❣ ◗✉②➳t✱ ♥❣÷í✐ ✤➣ ❝❤➾ ❜↔♦ t➟♥ t➻♥❤ ✈➔ ❝❤♦ tæ✐ ♥❤ú♥❣ ♥❤➟♥ ①➨t q✉➼ ❜→✉ ✤➸ tæ✐ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❜↔♥ ❧✉➟♥ ✈➠♥ ♥➔② ♠ët ❝→❝❤ tèt ♥❤➜t✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ð ❦❤♦❛ ❚♦→♥✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✱ ❣✐ó♣ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♠ët ❝→❝❤ t❤✉➟♥ ❧đ✐✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ỷ ỡ tợ trữớ P ổ ỗ ♥❣❤✐➺♣✱ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥✱ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ ✤ë♥❣ ✈✐➯♥ ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ✤➸ tæ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝ ❝õ❛ ♠➻♥❤✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✵✻ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ◆❣✉②➵♥ ❚❤à ❚❤❛♥❤ ▲♦❛♥ ✐ ▲í✐ ❝❛♠ ổ ữợ sỹ ữợ ❚❙✳ ✣➔♦ ❚rå♥❣ ◗✉②➳t✱ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝❤✉②➯♥ ♥❣➔♥❤ ❚♦→♥ ❣✐↔✐ t➼❝❤ ✈ỵ✐ ✤➲ t➔✐ ❙t♦❦❡s ❝â tr➵✧ ✧❍➺ ♣❤÷ì♥❣ tr➻♥❤ ◆❛✈✐❡r✲ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❜ð✐ ❝❤➼♥❤ ♥❤➟♥ t❤ù❝ ❝õ❛ ❜↔♥ t❤➙♥ tæ✐✳ ❚r♦♥❣ s✉èt q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✱ tæ✐ ✤➣ ❦➳ t❤ø❛ ♥❤ú♥❣ t❤➔♥❤ tü✉ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈ỵ✐ sü tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✻ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ◆❣✉②➵♥ ❚❤à ❚❤❛♥❤ ▲♦❛♥ ✐✐ ▼ët sè ❦➼ ❤✐➺✉ t❤÷í♥❣ ❞ò♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥ ∆ t♦→♥ tû ▲❛♣❧❛❝❡❀ ∇ ✈❡❝t♦r ❣r❛❞✐❡♥t❀ ∇· t♦→♥ tû ❣r❛❞✐❡♥t❀ H, V ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❤➔♠ ❞ò♥❣ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❤➺ ◆❛✈✐❡r✲❙t♦❦❡s❀ V ❦❤ỉ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ (·, ·), | · | t ổ ữợ tr ổ H; ((ã, ã)), ã t ổ ữợ tr ổ ❣✐❛♥ V; · ∗ ❝❤✉➞♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ V; V ; C0∞ (Ω) ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ✈æ ❤↕♥ ❝â ❣✐→ ❝♦♠♣❛❝t tr♦♥❣ Lp (Ω) ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❜➟❝ C([0, T ]; X) ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ [0, T ]❀ Lp (0, T ; X) ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ ❜➟❝ p → ❤ë✐ tư ♠↕♥❤❀ ❤ë✐ tư ②➳✉❀ → ♣❤➨♣ ♥❤ó♥❣ ❧✐➯♥ tư❝❀ →→ ♣❤➨♣ ♥❤ó♥❣ ❝♦♠♣❛❝t❀ ✶ p ❦❤↔ t➼❝❤ ▲❡❜❡s❣✉❡ tr♦♥❣ tr➯♥ [0, T ]❀ Ω; Ω; ▼ð ✤➛✉ ✶✳ ỵ t ữỡ tr ♣❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ❝ì ❤å❝ ❝❤➜t ❧ä♥❣ ①✉➜t ❤✐➺♥ ❦❤✐ ♠æ t↔ ❝❤✉②➸♥ ✤ë♥❣ ❝õ❛ ❝→❝ ❝❤➜t ❧ä♥❣ ✈➔ ❦❤➼ ữ ữợ ổ ọ ữợ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ t÷ì♥❣ ✤è✐ tê♥❣ q✉→t✱ ✈➔ ❝❤ó♥❣ ①✉➜t ❤✐➺♥ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ♥❤✐➲✉ ❤✐➺♥ t÷đ♥❣ q✉❛♥ trå♥❣ tr♦♥❣ ❦❤♦❛ ❤å❝ ❤➔♥❣ ❦❤ỉ♥❣✱ ❦❤➼ t÷đ♥❣ ❤å❝✱ ❝ỉ♥❣ ♥❣❤✐➺♣ ❞➛✉ ♠ä✱ ✈➟t ❧➼ ♣❧❛s♠❛✱ ✳ ✳ ✳ ✳ ▼ët tr♦♥❣ ỳ ợ ữỡ tr q trồ tr ỡ ❝❤➜t ❧ä♥❣ ❧➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ◆❛✈✐❡r✲❙t♦❦❡s ❝â ❞↕♥❣✿     ∂u − ν∆u + (u · ∇)u + ∇p = f (x, t), ∂t    ð ✤â u = u(x, t), p = p(x, t) t➻♠✱ ν = ❝♦♥st > div u = 0, t÷ì♥❣ ù♥❣ ❧➔ ❤➔♠ ✈➨❝tì ✈➟♥ tè❝ ✈➔ ❤➔♠ →♣ s✉➜t ❝➛♥ ❧➔ ❤➺ sè ♥❤ỵt ✈➔ f ❧➔ ♥❣♦↕✐ ❧ü❝✳ ▼➦❝ ❞ò ✤÷đ❝ ✤÷❛ r❛ ❧➛♥ ✤➛✉ t✐➯♥ ✈➔♦ ♥➠♠ 1822✱ ❝❤♦ ✤➳♥ ♥❛② ✤➣ ❝â ❤➔♥❣ ✈↕♥ ❜➔✐ ❜→♦ ✈➔ s→❝❤ ✈✐➳t ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ◆❛✈✐❡r✲❙t♦❦❡s✱ t✉② ♥❤✐➯♥ ♥❤ú♥❣ ❤✐➸✉ ❜✐➳t ❝õ❛ t❛ ✈➲ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❝á♥ ❦❤→ ❦❤✐➯♠ tè♥✳ ❉♦ ♥❤✉ ❝➛✉ ❝õ❛ ❑❤♦❛ ❤å❝ ✈➔ ❈æ♥❣ ♥❣❤➺ ♠➔ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ◆❛✈✐❡r✲❙t♦❦❡s ♥â✐ r✐➯♥❣ ✈➔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ❝ì ❤å❝ ❝❤➜t ❧ä♥❣ ♥â✐ ❝❤✉♥❣ ♥❣➔② ❝➔♥❣ trð ♥➯♥ t❤í✐ sü ✈➔ ❝➜♣ t❤✐➳t✳ ◆❤ú♥❣ ✈➜♥ ✤➲ ❝ì ❜↔♥ ✤➦t r❛ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ ✷ tr➻♥❤ tr♦♥❣ ❝ì t ọ ỹ tỗ t t ♥❤➜t ✈➔ t➼♥❤ ❝❤➼♥❤ q✉✐ ❝õ❛ ♥❣❤✐➺♠✿ ◆❣❤✐➺♠ ð ✤➙② ❝â t❤➸ ❧➔ ♥❣❤✐➺♠ ②➳✉ ❤♦➦❝ ♥❣❤✐➺♠ ♠↕♥❤✳ ❚➼♥❤ ❝❤➼♥❤ q✉✐ ð ✤➙② ❝â t❤➸ ❧➔ t➼♥❤ ❝❤➼♥❤ q✉✐ t❤❡♦ ❜✐➳♥ t❤í✐ ❣✐❛♥✱ ❤♦➦❝ t➼♥❤ ❝❤➼♥❤ q✉✐ t❤❡♦ ❜✐➳♥ ❦❤ỉ♥❣ ❣✐❛♥✳ • ❉→♥❣ ✤✐➺✉ t✐➺♠ ❝➟♥ ❝õ❛ ♥❣❤✐➺♠✿ ◆❣❤✐➯♥ ❝ù✉ ❞→♥❣ ✤✐➺✉ ❝õ❛ ♥❣❤✐➺♠ ❦❤✐ t❤í✐ ❣✐❛♥ t r❛ ✈ỉ ❝ò♥❣✳ ❑❤✐ ♥❣♦↕✐ ❧ü❝ f ✏❧ỵ♥✑✱ t❛ ♥❣❤✐➯♥ ❝ù✉ sü tỗ t t t t út õ ♠ët t➟♣ ❝♦♠♣❛❝t✱ ❜➜t ❜✐➳♥✱ ❤ót ❝õ❛ ❝→❝ t➟♣ ❜à ❝❤➦♥ ✈➔ ❝❤ù❛ ✤ü♥❣ ♥❤✐➲✉ t❤æ♥❣ t✐♥ ✈➲ ❞→♥❣ ✤✐➺✉ t✐➺♠ ❝➟♥ ❝õ❛ ♥❣❤✐➺♠❀ ❝á♥ ❦❤✐ ♥❣♦↕✐ ❧ü❝ f ✏♥❤ä✑ ✈➔ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ t❤í✐ ❣✐❛♥✱ t❛ ♥❣❤✐➯♥ ❝ù✉ sü tỗ t t t ứ tự ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❞ø♥❣ t÷ì♥❣ ù♥❣✱ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✤❛♥❣ ①➨t ❞➛♥ ✤➳♥ ♥❣❤✐➺♠ ❞ø♥❣ ♥➔② ❦❤✐ t❤í✐ ❣✐❛♥ t r❛ ✈ỉ ❝ò♥❣✳ ✣➦❝ ❜✐➺t✱ ❦❤✐ tr↕♥❣ t❤→✐ ❝õ❛ ❤➺ ♣❤ö t❤✉ë❝ ✈➔♦ ❝↔ q✉→ ❦❤ù ❝õ❛ ♥❣❤✐➺♠ t❤➻ ♥❣♦↕✐ ❧ü❝ s➩ ①✉➜t ❤✐➺♥ t❤➯♠ sè ❤↕♥❣ ❝❤ù❛ tr➵ ✭①❡♠ ❬✶✱ ✷✱ ✸❪✮✳ ❑❤✐ ✤â✱ sỹ tỗ t t t t ❝➟♥ ❝õ❛ ♥❣❤✐➺♠ ❝ơ♥❣ ❧➔ ❝→❝ ✈➜♥ ✤➲ t❤í✐ sü ❝➛♥ ✤÷đ❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉✳ ❱➻ ✈➟②✱ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ✤➲ t➔✐ ◆❛✈✐❡r✲❙t♦❦❡s ❝â tr➵✑ ✏❍➺ ♣❤÷ì♥❣ tr➻♥❤ ❧➔♠ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ✷✳ ▼ö❝ ✤➼❝❤ ự ự t q sỹ tỗ t↕✐✱ ❞✉② ♥❤➜t ✈➔ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ♥❣❤✐➺♠ ❞ø♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ◆❛✈✐❡r✲❙t♦❦❡s ❦❤✐ ♥❣♦↕✐ ❧ü❝ ❝â tr➵✳ ự ự sỹ tỗ t↕✐✱ t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ♥❣❤✐➺♠❀ • ❈❤ù♥❣ ♠✐♥❤ t➼♥❤ ứ ố ợ ữỡ tr ◆❛✈✐❡r✲ ✸ ❙t♦❦❡s ❝â tr➵✳ ✹✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ự ố tữủ ự ữỡ tr➻♥❤ ◆❛✈✐❡r✲❙t♦❦❡s ❝â tr➵✳ • P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿ ❙ü tỗ t t t Pữỡ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❙û ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❝õ❛ ❧➼ t❤✉②➳t ❤➺ ✤ë♥❣ ❧ü❝ t✐➯✉ ❤❛♦ ✈æ ❤↕♥ ❝❤✐➲✉✱ ❧➼ t❤✉②➳t ❤➺ ◆❛✈✐❡r✲❙t♦❦❡s✳ ✻✳ ✣â♥❣ ❣â♣ ❝õ❛ ❧✉➟♥ ✈➠♥ ❉ü❛ t❤❡♦ t➔✐ ❧✐➺✉ ❬✺❪✱ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ õ tố t q sỹ tỗ t ♥❣❤✐➺♠ ②➳✉ ❦❤✐ N = 2, 3✱ t➼♥❤ ❞✉② ♥❤➜t tỗ t t t ♥❣❤✐➺♠ ❞ø♥❣ ②➳✉ ❦❤✐ ♥❣❤✐➺♠ ❞ø♥❣ ②➳✉ ❦❤✐ N =2 N = 2, N = 2, ✈➔ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ◆❛✈✐❡r✲❙t♦❦❡s ❝â tr➵✳ ✹ sỹ t O ữ tr ợ R > um,R (x) = um (x)χ(|x|2 /R2 )✳ ˜ N E = (L2 (O)) tr➯♥ s❛♦ ❝❤♦ O ⊂ B(0, R) ✈➔ ˜ = Ω ∩ B(0, 2R)✱ O ❉♦ ✤â t➼♥❤ ❝♦♠♣❛❝t ❝á♥ ✤ó♥❣ ✈ỵ✐ ✈➔ ˜ N ⊂ X = (H01 (O)) ✈ỵ✐ ♣❤➨♣ ♥❤ó♥❣ ❝♦♠♣❛❝t✱ ✈➔ t❛ ✈➝♥ ❜↔♦ t♦➔♥ ❝→❝ ❤➔♠ ❜❛♥ ✤➛✉ um Ω ∩ B(0, R)✳ ◆❤÷ ✈➟②✱ t❛ ❝â t❤➸ t✐➳♣ tư❝ ❝❤ù♥❣ ♠✐♥❤ trü❝ t✐➳♣ ✈ỵ✐ um,R ✳ um t❤❛② ✈➻ ✈ỵ✐ ❤➔♠ ❱➻ ✤✐➲✉ ❦✐➺♥ ✐✐✮ tr♦♥❣ ✣à♥❤ ❧➼ ✶✳✷ ✤÷đ❝ t❤ä❛ ♠➣♥ ✈➻ ✤✐➲✉ ❦✐➺♥ ✭✶✳✶✵✮✱ t❛ ❝➛♥ ❝❤ù♥❣ tä ✤✐➲✉ ❦✐➺♥ ✐✮ ❝ơ♥❣ ✤ó♥❣✳ ❚❤➟t ✈➟②✱ t❛ s➩ ❝❤ù♥❣ tä t➼♥❤ ❝❤➜t s❛✉ ✤ó♥❣ ✈ỵ✐ t♦➔♥ ❜ë ♠✐➲♥ Ω, sup ||τh um − um ||L2 (0,T −h;(L2 (Ω))N ) → ❦❤✐ h → ✭✶✳✶✹✮ m∈N ❳➨t h>0 (t, t + h) (0, T ) ọ tũ ỵ tứ ✭✶✳✾✮ t❛ s✉② r❛ ✈ỵ✐ t+h m ∇um (s).∇wj dxds (u (t + h) − u(t))wj dx + ν Ω t Ω t+h b(um (s), um (s)wj ds + t t+h t+h = g1 (s, um s )wj dxds f (s), wj ds + t t Ω t+h g2 (s, um s ), wj ds + t ◆❤➙♥ ❝↔ ❤❛✐ ✈➳ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✈ỵ✐ γmj (t + h) − γmj (t) ✈➔ ❧➜② tê♥❣ t❤❡♦ ❝â |um (t + h) − u(t)|2 dx Ω t+h ∇um (s).(∇um (t + h)−∇um (t))dxds = −ν t t+h Ω b(um (s), um (s), um (t + h) − um (t))ds + t t+h m m g1 (s, um s )(u (t + h) − u (t))dxdt + t Ω t+h m m f (s) + g2 (s, um s ), u (t + h) − u (t) ds + t ✶✻ j, t❛ ❚❛ t❤➜② ✈➳ ♣❤↔✐ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② ❧➔ ❜à ❝❤➦♥ ❜ð✐ t+h ν |∇um (t + h) − ∇um (t)| |∇um (s)|ds t t+h GN (|um (s)| , um (s) , um (t + h) − um )ds + t t+h m m |g1 (s, um s )| |u (t + h) − u (t)| ds + t t+h + ( f (s) ∗ m m + g2 (s, um s ) ∗ ) u (t + h) − u (t) ds, t tr♦♥❣ ✤â ❞↕♥❣ ✸✲t✉②➳♥ t➼♥❤ b GN : R3 → R ❧➔ ❜à ❝❤➦♥ ❜ð✐ ❤➔♠ GN (x, y, z) =    2−1/2 xyz ♥➳✉   2−1 x1/2 y 3/2 z ♥➳✉ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ N = 2, ✭✶✳✶✺✮ N = ❉♦ ✤â✱ sû ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ P♦✐♥❝❛r➨ ✭✶✳✶✮ ✈➔ ✭✶✳✶✵✮✱ t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ t+h m m m m |u (t + h) − u (t)| dx ≤ u (t + h) − u (t) Gm (s)ds, Ω ✭✶✳✶✻✮ t tr♦♥❣ ✤â ❤➔♠ Gm (s) = Gm : R → R ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉     ν um (s) + (2−1 K1 )1/2 um (s)        −1/2  + f (s) ∗ + g2 (s, um |g1 (s, um s ) ∗ + λ1 s | ♥➳✉ N = 2,  1/4   ν um (s) + 2−1 K1 um (s) 3/2       −1/2   + f (s) ∗ + g2 (s, um |g1 (s, um s | s ) ∗ + λ1 ♥➳✉ N = ✣➸ ❤♦➔♥ t❤➔♥❤ ❝❤ù♥❣ ♠✐♥❤✱ t❛ ❝➛♥ ✤→♥❤ ❣✐→ T −h τh um − um L2 (0,T,−h;(L2 (Ω))N ) |τh um − um |2 dxdt = Ω T −h t+h um (t + h) − um (t) ≤ Gm (s)dsdt t ✶✼ ❱ỵ✐ ✈➳ ♣❤↔✐✱ →♣ ❞ư♥❣ ✤à♥❤ ❧➼ ❋✉❜✐♥✐✱ ✈➔ sû ❞ö♥❣ ❤➔♠         s¯ = s       T − h ♥➳✉ s ≤ 0, ♥➳✉ < s ≤ T − h, ♥➳✉ s > T − h, t❛ ❝â T −h t+h um (t + h) − um (t) Gm (s)dsdt t T ≤ s um (t + h) − um (t) dtds Gm (s) s−h T ≤ 2(hK2 )1/2 Gm (s)ds, ð ✤â t❛ ✤➣ sû ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❨♦✉♥❣✱ ✭✶✳✶✵✮ ✈➔ ❞♦ ≤ s−s−h ≤ h ✈ỵ✐ t➼❝❤ s um (t + h) − um (t) dt ♣❤➙♥ ❚❤➟t ✈➟②✱ s−h s um (t + h) − um (t) dt s−h 1/2 s ≤ 1/2 s m m u (t + h) − u (t) dt dt s−h s−h 1/2 T −h m 1/2 ≤ 2h |∇u | dxdt Ω 1/2 ≤ 2h1/2 K2 T Gm (s)ds ❍ì♥ ♥ú❛✱ t❛ t❤➜② r➡♥❣ ❧➔ ❜à ❝❤➦♥✱ ✈ỵ✐ ✶✽ N =2 ❤♦➦❝ N = 3✳ ❈❤➥♥❣ ❤↕♥✱ ✈ỵ✐ N = 2, t❛ ❝â T (ν + 2−1 K1 1/2 um (s) + f (s) ∗ + g2 (s, um s ) ∗ −1/2 +λ1 |g1 (s, um s )| ds 1/2 T √ ≤ (ν + (2−1 K1 )1/2 ) T um (s) ds + T f (s) + T g2 (s, us ) ∗ ds N =3 gi T um (s) 3/2 Gm ❧➔ ❜à ❝❤➦♥✳ 3/4 um (s) ds ds ≤ T 1/4 0 h0 , tữỡ tỹ ợ sỹ ❜✐➺t ❞✉② ♥❤➜t ❧➔ T ❝❤➦♥ ✈➔ ❝❤♦ ds g1 (s, um s ✈➔ tø ❣✐↔ t❤✐➳t ✭✐✐✮ ✈➔ ✭✐✈✮ ❝❤♦ t❛ t➼❝❤ ♣❤➙♥ ◆❤÷ ✈➟②✱ ❤➔♠ 1/2 T √ + T λ−1/2 ❚r÷í♥❣ ❤đ♣ ∗ ds 1/2 T √ 1/2 T √ ❧➔ ❜à ❝❤➦♥✱ ❞♦ ✤â tø ✭✶✳✶✻✮ s✉② r❛ t❛ t❤✉ ✤÷đ❝ ✭✶✳✶✹✮✳ ✶✾ |um (t + h) − um (t)|2 ❧➔ ữỡ ỹ tỗ t t ứ ố ợ ữỡ tr rts õ tr➵ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❞ü❛ t❤❡♦ t➔✐ ❧✐➺✉ ❬✺❪ ❝❤ó♥❣ tổ tr sỹ tỗ t t ứ ữỡ tr rts ợ số ❝❤✐➲✉ ❤♦➦❝ N = 3✱ r❛✱ ✈ỵ✐ N =2 ❦❤✐ sè ❤↕♥❣ tr➵ ❝â ❞↕♥❣ ✤➦❝ ❜✐➺t ✈➔ ❦❤✐ ❤➺ sè ♥❤ỵt ✤õ ❧ỵ♥✳ ◆❣♦➔✐ N = 2✱ t❛ t❤➜② r➡♥❣ ♠å✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t✐➳♥ ❤â❛ ❤ë✐ tư ✤➳♥ ♥❣❤✐➺♠ ❞ø♥❣ t❤❡♦ tè❝ ✤ë ♠ô✳ ●✐↔ sû f ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ t❤í✐ ❣✐❛♥✱ ✈➔ sè ❤↕♥❣ tr➵ ❝â ❞↕♥❣ G(u(t − ρ(t)) G : RN → RN ợ G tọ ữ tr trữớ ủ G(0) = g(t, ut ) = ữ ợ tớ tự tỗ t số L1 > |G(u) − G(v)|RN ≤ L1 |u − v|RN s❛♦ ❝❤♦ ∀u, v ∈ RN ❈ư t❤➸✱ ❝❤ó♥❣ tỉ✐ ①➨t ❜➔✐ t♦→♥            ❚➻♠ u ∈ L2 (−h, T ; V ) ∩ L∞ (0, T ; H), s❛♦ ❝❤♦ ∀v ∈ V d (u(t), v) + νa(u(t), v)) + b(u(t), u(t), v) = f, v + (G(ut ), v) dt u(0) = u0 , u(t) = φ(t), t ∈ (−h, 0) ✷✵ ✭✷✳✶✮ ❚❛ ❝â ❦❤→✐ ♥✐➺♠ ♥❣❤✐➺♠ ❞ø♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✮ ♥❤÷ s❛✉✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶ u∗ ∈ V ✳ ✭❬✺❪✮ ▼ët ♥❣❤✐➺♠ ❞ø♥❣ ②➳✉ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✮ ❧➔ ♠ët ❤➔♠ s❛♦ ❝❤♦ νa(u∗ , v) + b(u∗ , u∗ , v) = f, v + (G(u∗ ), v), ∀v ∈ V ✭✷✳✷✮ ✷✳✶ ❙ü tỗ t t t ứ r ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❦➳t q✉↔ ✈➲ sü tỗ t t t ứ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ◆❛✈✐❡r✲❙t♦❦❡s ❝â tr➵✳ ❚❛ ❝â ❦➳t q✉↔ sỹ tỗ t t ứ q✉❛ ✤à♥❤ ❧➼ s❛✉✳ ✣à♥❤ ❧➼ ✷✳✶ ✤â✿ ✭❬✺❪✮ ✳ ●✐↔ sû r➡♥❣ G t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ tr➯♥ ✈➔ ν > Lλ11 ❑❤✐ ❛✮ ❱ỵ✐ ♠å✐ f V tỗ t t t ởt ❜✮ ◆➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ ν − λ−1 L1 > (2λ1 )−1/2 f ✈ỵ✐ N = 2, ∗, ✈➔ ν − λ−1 L1 ✤÷đ❝ t❤ä❛ ♠➣♥✱ t❤➻ ✭✷✳✷✮ > 2−1 (λ1 )−1/4 f ∗ ✈ỵ✐ N = 3, ❝â ❞✉② ♥❤➜t ♥❣❤✐➺♠✳ ❈❤ù♥❣ ♠✐♥❤✳ ❛✮ ❙ü tỗ t ữỡ tỹ ữ trữợ t B = {w1, w2 } ❧➔ ♠ët ❝ì sð trü❝ ❝❤✉➞♥ tr♦♥❣ H ✤÷đ❝ t↕♦ tø ❝→❝ ♣❤➛♥ tû ❝õ❛ t➼♥❤ ❝õ❛ ❝❤ó♥❣ ❧➔ trò ♠➟t tr♦♥❣ V✳ ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ữợ V s tờ ủ t ữợ um Vm Vm = t✉②➳♥ t➼♥❤[w1 , w2 , , wm ] ✈➔ ①➨t ❜➔✐ t♦→♥ t➻♠ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ν((um , v m )) + b(z m , um , v m ) = f, v m + (G(z m ), v m ) ð ✤â z m ∈ Vm z m ∈ Vm ♣❤✐➳♠ ❤➔♠ (u, v) → ν((u, v)) + b(z m , u, v) ✸✲t✉②➳♥ t➼♥❤✱ ❧✐➯♥ tö❝ ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❜ù❝ tr➯♥ v → f, v + (G(z m ), v) Vm ①→❝ ✤à♥❤ tr➯♥ ❞✉② ♥❤➜t ❝õ❛ ✭✷✳✸✮✱ t❛ ❦➼ ❤✐➺✉ ♥❣❤✐➺♠ ✤â ❧➔ t♦→♥ tỷ ữợ Vm ì Vm t z m Vm , tỗ t ♠ët ♥❣❤✐➺♠ um ✣à♥❤ ♥❣❤➽❛ Tm : Vm → Vm ❧➔ Tm (z m ) = um ❚❛ s➩ ❝❤ù♥❣ tä r➡♥❣ ✈ỵ✐ ♠é✐ ❜➜t ✤ë♥❣ ❝❤♦ →♥❤ ①↕ um ∈ V m Tm m, t❛ ❝â t❤➸ →♣ ❞ö♥❣ ✤à♥❤ ❧➼ ✤✐➸♠ ❤↕♥ ❝❤➳ tr➯♥ ♠ët t➟♣ ❝♦♥ Km ⊂ Vm , tø ✤â ❝â ✤÷đ❝ sü t❤ä❛ ♠➣♥ ν((um , v m )) + b(um , um , v m ) = f, v m + (G(um ), v m ) ❚✐➳♣ t❤❡♦✱ t❛ ❧➜② ❧➔ ❧➔ t✉②➳♥ t➼♥❤ ✈➔ ❧✐➯♥ tö❝✳ ❉♦ ✤â✱ t❤❡♦ ✣à♥❤ ỵ r ợ ộ tỷ ố tỗ t tỷ ố trữợ ú þ r➡♥❣ ✈ỵ✐ ♠é✐ ❤➔♠ ∀v m ∈ Vm , um = v m ν um ∀v m ∈ Vm tr ởt ữợ ữủ t ❝â ≤ f ∗ m um + λ−1 L1 z um , ❤❛② ν um ≤ f ❱➻ ν > λ−1 L1 , t❛ t❤✉ ✤÷đ❝ ♥➯♥ t❛ ❝â t❤➸ ❧➜② ν um ∗ m + λ−1 L1 z k > s❛♦ ❝❤♦ k(ν − λ−1 L1 ) ≥ f −1 m ≤ kν − kλ−1 L1 + λ1 L1 z Km = {z ∈ Vm : z ≤ k} ∗ , ✈➻ ✈➟② ❇➙② ❣✐í t❛ ✤à♥❤ ♥❣❤➽❛ t➟♣ ❧➔ ởt t ỗ t V ữ Tm ①↕ Km ✈➔♦ ❝❤➼♥❤ ♥â✳ ❚✐➳♣ t❤❡♦✱ t❛ s➩ →♣ ❞ö♥❣ ✤à♥❤ ❧➼ ✤✐➸♠ ❜➜t ✤ë♥❣ ❇r♦✉✇❡r ✈➔♦ →♥❤ ①↕ ❧➔♠ ✤✐➲✉ ♥➔②✱ t❛ ❝➛♥ ❦✐➸♠ tr❛ ✤✐➲✉ ❦✐➺♥ Tm ✷✷ ❧➔ ❧✐➯♥ tö❝✳ ❚❤➟t ✈➟②✱ ❧➜② Tm |Km ✳ ✣➸ z1m , z2m ∈ Vm ✈➔ ❦➼ ❤✐➺✉ m um i = T (zi ) t÷ì♥❣ ù♥❣ ❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✮✳ ❱➻ ✈➟②✱ ①➨t ❤✐➺✉ m m m m m m m m m m m ν((um − u2 , v )) + b(z1 , u1 , v ) − b(z2 , u2 , v ) = (G(z1 ) − G(z2 ), v ) ✈ỵ✐ ♠å✐ v m ∈ Vm ✳ m ν um − u2 ✣➦❝ ❜✐➺t✱ ♥➳✉ t❛ ✤➦t m v m = um − u2 ✱ ✈➔ ✈➻ b(u, v, v) = ♥➯♥ m m m m m m = b(z2m , um , u1 − u2 ) − b(z1 , u1 , u1 − u2 ) m + (G(z1m ) − G(z2m ), um − u2 ) m m m m m m = b(z2m , um , u1 − u2 ) ± b(z2 , u1 , u1 − u2 ) m m m m m m − b(z1m , um , u1 − u2 ) + (G(z1 ) − G(z2 ), u1 − u2 ) m m m m m m = b(z2m − z1m , um , u1 − u2 ) + (G(z1 ) − G(z2 ), u1 − u2 ) ❉♦ um ∈ Km ✱ ♥➯♥ ❤❛✐ sè ❤↕♥❣ ❝✉è✐ ❝õ❛ ✤➥♥❣ t❤ù❝ tr➯♥ ❧➔ ❜à ❝❤➦♥✱ ❦❤✐ ✤â t❛ ❝â   ((2λ1 )−1/2 k + λ−1 L1 ) z m − z m um − um , N = 2 1 2 ≤  (2−1 λ−1/4 k + λ−1 L ) z m − z m um − um , N = 1 1 m ν um − u2 ❚ø ✤✐➲✉ ♥➔② t❛ t❤✉ ✤÷đ❝ t tử ữợ Tm |Km s q ợ t ữủ tr ữợ t ữủ sỹ tỗ t ởt ❝õ❛ ✭✷✳✷✮✳ ❚❤➟t ✈➟②✱ ❧➜② ν um vm = um tr♦♥❣ ✭✷✳✹✮✱ t❛ ❝â = f, um + (G(um ), um ) ≤ f ∗ m um + λ−1 L1 u ✣✐➲✉ ♥➔② s✉② r❛ m (ν − λ−1 ≤ f L1 ) u ❉♦ ✤â✱ ❝→❝ ♥❣❤✐➺♠ ❝❤➦♥ ❝õ❛ ❞➣② s❛♦ ❝❤♦ um um {um }, u t ữủ tr ữợ ❜à ❝❤➦♥ ✤➲✉ tr♦♥❣ V ❚ø t➼♥❤ ❜à t❛ ❝â t❤➸ ❧➜② ♠ët ❞➣② ❤ë✐ tö ②➳✉ ✭♠➔ ✈➝♥ ❦➼ ❤✐➺✉ ❧➔ tr♦♥❣ V✳ ❍ì♥ ♥ú❛✱ ✈ỵ✐ ❜➜t ❦ý t➟♣ ❜à ❝❤➦♥✱ ❝❤➼♥❤ q✉② ✷✸ {um }✮ O ⊂ Ω✱ t❛ ❝ô♥❣ ❝â t➼♥❤ ❜à ❝❤➦♥ ✤➲✉ ❝õ❛ um |O → u|O tr♦♥❣ um |O ✱ sû ❞ư♥❣ ♣❤➨♣ ♥❤ó♥❣ ❝♦♠♣❛❝t✱ t❛ t❤✉ ✤÷đ❝ (L2 (O))N ✳ ❚✐➳♣ t❤❡♦✱ t❛ ❝è ✤à♥❤ ♣❤➛♥ tû ❜➜t ❦➻ tr➻♥❤ ✭✷✳✹✮ ✈ỵ✐ ♠å✐ m > j, wj ∈ B ✳ ❱➻ t❛ ❝â ♠ët ❞➣② ❝♦♥ ❝→❝ ♣❤÷ì♥❣ ❞♦ ✤â t❛ ❝â t❤➸ q ợ ộ tỷ õ ữủ ν((u, wj )) + b(u, u, wj ) = f, wj + (G(u), wj ) ❙è ❤↕♥❣ ✤➛✉ t✐➯♥ t❤✉ ✤÷đ❝ ❜ð✐ sü ❤ë✐ tư ②➳✉ tr♦♥❣ t➼♥❤ ❧➔ ❤ë✐ tö ♥➳✉ ❣✐→ ❝õ❛ wj ❧➔ ❝♦♠♣❛❝t ✭ð ✤➙② ❦➼ ❤✐➺✉ wj ✮✱ ❜à ❝❤➦♥ ✈ỵ✐ ❜✐➯♥ trì♥ ✈➔ ❝❤ù❛ ❣✐→ ❝õ❛ tr♦♥❣ V ✱ um u✳ ❙è ❤↕♥❣ ✸✲t✉②➳♥ Oj ⊂ Ω ❧➔ ♠ët t➟♣ ♠ð ❞♦ ✤â t❛ ❝â sü ❤ë✐ tö ♠↕♥❤ um → u (L2 (Oj ))N ✳ ❈✉è✐ ❝ò♥❣✱ ✈ỵ✐ sè ❤↕♥❣ (G(um ), wj ), t❛ ❝â t÷ì♥❣ tü |(G(um ), wj ) − (G(u), wj )| ≤ G(um ) − G(u) ≤ L1 um − u ❞♦ sü ❤ë✐ tö ♠↕♥❤ tr♦♥❣ ♣❤➛♥ tû tr♦♥❣ (L2 (Oj ))N |wj | |wj | → 0, wj ✳ ❱➻ ❝→❝ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ B = {w1 , w2 } ❧➔ trò ♠➟t tr♦♥❣ V ✱ ♥➯♥ ✭✷✳✷✮ t❤ä❛ ♠➣♥ ✈ỵ✐ u∗ = u✳ ❜✮ ❚➼♥❤ ❞✉② ♥❤➜t✿ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ t t ữợ tt ũ ủ rữợ t t t r ♥❣❤✐➺♠ v = u∗ (L2 (Oj ))N (L2 (Oj ))N ữ t s r ú ợ ộ ❧➜② ✭✷✳✺✮ N =2 u∗ ✈➔ N = ❝õ❛ ✭✷✳✷✮ ❧➔ ❜à ❝❤➦♥✱ t❤➟t ✈➟②✱ tø ✭✷✳✷✮ t❛ t❤✉ ✤÷đ❝ u∗ ≤ ❇➙② ❣✐í✱ ❣✐↔ sû ✭✷✳✷✮ ❝â ❤❛✐ ♥❣❤✐➺♠ u1 f ∗ ν − λ−1 L1 ✈➔ u2 ✳ ❑❤✐ ✤â✱ ∀v ∈ V t❛ ❝â ν((u1 − u2 , v)) + b(u1 , u1 , v) − b(u2 , u2 , v) = (G(u1 ) − G(u2 ), v) ✷✹ ▲➜② v = u1 − u2 ✈➔ ❞♦ ν u1 − u2 b(u, v, v) = 0✱ ✭✈ỵ✐ N = 2✮ ✤✐➲✉ ♥➔② ❞➝♥ ✤➳♥ ≤ b(u2 − u1 , u1 , u1 − u2 ) + (G(u1 ) − G(u2 ), u1 − u2 ) u2 − u1 + λ−1 L1 u2 − u1 ≤ 2−1/2 |u2 − u1 | u1 ỷ ữợ ữủ tr ợ u1 ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ P♦✐♥❝❛r➨ ✭✶✳✶✮✱ t❛ t❤✉ ✤÷đ❝ ν u1 − u2 ≤ (2λ1 )−1/2 f ν − λ−1 L1 ∗ u2 − u1 2 + λ−1 L1 u2 − u1 ✭✷✳✻✮ ❚ø ✤â s✉② r❛✱ (ν − λ−1 L1 ) u1 − u2 ≤ (2λ1 )−1/2 f ∗ u2 − u1 ❱➻ ❣✐↔ t❤✐➳t −1/2 (ν − λ−1 f L1 ) > (2λ1 ) ♥➯♥ tø ✭✷✳✻✮ s✉② r❛ ❤ñ♣ N =2 u1 − u2 = 0, ∗, ❤❛② t➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❞ø♥❣ tr♦♥❣ tr÷í♥❣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❚r÷í♥❣ ❤đ♣ N =3 t÷ì♥❣ tü ♥❤÷ s❛✉✿ |b(u2 − u1 , u1 , u1 − u2 )| ≤ 2−1 u1 |u2 − u1 |1/2 u2 − u1 −1/4 ≤ 2−1 λ1 3/2 f ∗ u2 − u1 −1 ν − λ1 L ❚ø ✤â (ν − λ−1 L1 ) u1 − u2 ❇➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ♥➔② ❝❤♦ t❛ −1/4 ≤ 2−1 λ1 f ∗ u1 − u2 u1 = u2 ✣à♥❤ ❧➼ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✷✳✷ ❚➼♥❤ ê♥ ✤à♥❤ ♠ơ ❝õ❛ ♥❣❤✐➺♠ ❞ø♥❣ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ♥❣❤✐➺♠ ❞ø♥❣ ②➳✉ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✷✮✱ ♥❣❤➽❛ ❧➔ ♠å✐ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✷✮ ①➨t ✈ỵ✐ sè ❝❤✐➲✉ ✷✺ N =2 ✤➲✉ ❤ë✐ tö ✈➲ u∗ ✣à♥❤ ❧➼ ✷✳✷ ✭❬✺❪✮ ✈ỵ✐ tè❝ ✤ë ♠ơ ❦❤✐ t❤í✐ ❣✐❛♥ t r❛ ✈ỉ ❝ò♥❣✳ ✳ ❈❤♦ G : R2 → R2 ❧➔ →♥❤ ①↕ ▲✐♣s❝❤✐t③ ✈ỵ✐ ❤➡♥❣ sè ▲✐♣s❝❤✐t③ ✈➔ t❤ä❛ ♠➣♥ G(0) = 0✳ ●✐↔ sû r➡♥❣ sè ❤↕♥❣ g(t, u1) tr♦♥❣ ✭✶✳✷✮ ✤÷đ❝ ❝❤♦ ❜ð✐ g(t, ut) = G(u(t − ρ(t)) ✈ỵ✐ ρ ∈ C 1((R)+; [0, h]) s❛♦ ❝❤♦ ρ (t) ≤ ρ∗ < ✈ỵ✐ ♠å✐ t ≥ 0✳ ●✐↔ sû f ∈ V , ✈➔ ν > λ−1 L1 ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ t❤ä❛ ♠➣♥ L1 > νλ1 > 1/2 L1 1/2 (1 − ρ∗ ) (2−1 λ1 ) f + −1 ν − λ1 L ∗ ✭✷✳✼✮ ❑❤✐ ✤â ✭✷✳✷✮ ❝â ♠ët ♥❣❤✐➺♠ ❞ø♥❣ ❞✉② ♥❤➜t u∗, ✈➔ ♠å✐ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✷✮ ❤ë✐ tư tỵ✐ u∗ t❤❡♦ tè❝ ✤ë ♠ơ ❦❤✐ t , tự tỗ t số ữỡ C ✈➔ λ✱ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ u0 ∈ H ✈➔ φ ∈ L2(−h, 0, V )✱ ♥❣❤✐➺♠ u ❝õ❛ ✭✶✳✷✮ t❛ ❝â |u(t) − u∗ |2 ≤ Ce−λt ( u0 − u∗ + φ − u∗ L2 (−h,0;V ) ) ✭✷✳✽✮ ✈ỵ✐ f (t) ≡ f, ✈➔ ✈ỵ✐ ♠å✐ t ≥ ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t u ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✷✮ ✈ỵ✐ f (t) ≡ f ✱ ✈➔ u V sỹ tỗ t↕✐ ✈➔ ❞✉② ♥❤➜t ❝õ❛ u∗ ✤÷đ❝ ✤↔♠ ❜↔♦ ❜ð✐ ✣à♥❤ ❧➼ ✷✳✶✳ ✣➸ ❝❤ù♥❣ tä t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ♥❣❤✐➺♠ ❞ø♥❣✱ t❛ ✤➦t w(t) = u(t) − u∗ , ❦❤✐ ✤â d (w(t), v) + ν((w(t), v)) + b(u(t), u(t), v) − b(u∗ , u∗ , v) dt = (G(u(t − ρ(t))), v) − (G(u∗ ), v) ❱➻ t❛ ❝â b(u(t), u(t), w(t)) − b(u∗ , u∗ , w(t)) = b(u∗ , w(t), u∗ ) − b(u(t), w(t), u(t)) = b(u∗ , w(t), u∗ ) ± b(u(t), w(t), u∗ ) − b(u(t), w(t), u(t)) = −b(w(t), w(t), u∗ ) − b(u(t), w(t), w(t)) = −b(w(t), w(t), u∗ ), ✷✻ õ t õ ữủ ữợ ữủ s ✤➙② ✭ð ✤➙② λ δ ✈➔ ✤÷đ❝ ①→❝ ✤à♥❤ ❧➔ ❝→❝ ❣✐→ trà ❞÷ì♥❣ ✤➸ ✤÷đ❝ ①→❝ ✤à♥❤ s❛✉✮✿ d d λt (e |w(t)|2 ) = eλt |w(t)|2 + eλt |w(t)| dt dt = λeλt |w(t)|2 + 2eλt (−ν w(t) − b(u(t), u(t), w(t)) + b(u∗ , u∗ , w(t)) + (G(u(t − ρ(t))) − G(u∗ ), w(t))) ≤ eλt (λ|w(t)|2 − 2ν w(t) + 2b(w(t), w(t), u∗ ) ✭✷✳✾✮ + 2L1 |(w(t) − ρ(t)) w(t) )| λt ≤ λ−1 e (λ + δL1 − 2νλ1 ) w(t) + 2eλt |b(w(t), w(t), u∗ | + (δλ1 )−1 L1 eλt w(t − ρ(t)) ❉♦ |b(w(t), w(t), u∗ )| ≤ (2λ)−1/2 w(t) u∗ , ✈➔ u∗ ≤ f ∗ ν − λ−1 L1 ❚❛ ❝â✱ (2λ1 )−1/2 f |b(w(t), w(t), u | ≤ ν − λ−1 L1 ∗ ∗ w(t) ❚❤❛② ❜➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ♥➔② ✈➔♦ ✭✷✳✾✮✱ t❛ ✤÷đ❝ (2λ1 )1/2 f ∗ λ + δL1 − 2νλ1 + ν − λ−1 L1 d λt (e |w(t)|2 ) ≤ λ−1 eλt dt w(t) + (δλ1 )−1 L1 eλt w(t − ρ(t) , ✈➟② ✈ỵ✐ ♠å✐ t ∈ [0, T ], t λt 2 −1 e |w(t)| ≤ |w(0)| + (δλ1 ) eλt w(s − ρ(s)) ds L1 +λ−1 λ + δL1 − 2νλ1 + ✷✼ (2λ1 )1/2 f ∗ ν − λ−1 L1 t eλs w(s) ds ❳➨t ❤↕♥❣ tû ❝❤ù❛ tr➵ ❜➯♥ ✈➳ ♣❤↔✐✱ t❛ t❤➜② ❤➔♠ φ(t) := t − ρ(t) t➠♥❣ ♥❣➦t✱ ♠➔ ρ ❧➜② ❣✐→ trà tr➯♥ [0, h]✱ ✈➻ t❤➳ φ−1 (η) ≤ η + h, ✈➔ t❛ ❝â t❤➸ ✤ê✐ ❜✐➳♥ η = s − ρ(s) = φ(s), ✤➸ t❤✉ ✤÷đ❝ t t−ρ(0) λt e eλφ w(s − ρ(s)) ds = −1 (η) w(η) −ρ(0) ≤ eλh − ρ∗ dη − ρ (φ−1 (η)) t eλη w(η) dη ❑➳t ❤ñ♣ ❤❛✐ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ❝â ✤÷đ❝✱ eλt |w(t)|2 ≤ |w(0)|2 + (δλ1 )1 L1 + 1 ữ ỵ r + δL1 − 2νλ1 + δ∗ = (1 − ρ∗ )−1/2 õ tỗ t eh >0 t eλs w(s) ds −h (2λ1 )1/2 f ∗ ν − λ−1 L1 ❧➔ ❝ü❝ t✐➸✉ ❝õ❛ →♥❤ ①↕ t eλs w(s) δ →δ+ ✳ δ(1 − ρ∗ ) (2λ1 )1/2 f ∗ L1 eλh + ≤ δ∗ (1 − ρ∗ ) ν − λ−1 L1 ◆❤÷ ✈➟②✱ t❛ ❝â λt ∗ e |u(t) − u | ≤ u − u ∗ λ−1 L1 eλh + 1 − ρ∗ eλη ω(η) dη −h ❚ø ✤â t❛ t❤✉ ✤÷đ❝ ✭✷✳✽✮ ✈ỵ✐ λ−1 L1 eλh C = max 1, 1 − ρ∗ ✣à♥❤ ❧➼ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✷✽ ds ✤õ ♥❤ä ✤➸ λ + δ∗ L1 − 2νλ1 + ∗ ❑❤✐ ✤â✱ ❑➳t ❧✉➟♥ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ◆❛✈✐❡r✲❙t♦❦❡s ❦❤✐ ♥❣♦↕✐ ❧ü❝ ❝❤ù❛ tr➵✳ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ỗ r ữủ t q sỹ tỗ t t ữỡ tr rts ❝â tr➵✳ ✷✳ ❚r➻♥❤ ❜➔② ✤÷đ❝ ❦➳t q✉↔ ✈➲ sü tỗ t t ữỡ tr ◆❛✈✐❡r✲❙t♦❦❡s ❝â tr➵✳ ✸✳ ❚r➻♥❤ ❜➔② ✤÷đ❝ ❦➳t q✉↔ ✈➲ t ứ ố ợ ữỡ tr➻♥❤ ◆❛✈✐❡r✲❙t♦❦❡s ❤❛✐ ❝❤✐➲✉ ❦❤✐ ♥❣♦↕✐ ❧ü❝ ❝❤ù❛ tr➵✳ ▲✉➟♥ ✈➠♥ ❝❤➾ ❞ø♥❣ ❧↕✐ ð ✈✐➺❝ tr➻♥❤ ❜➔②✱ s➢♣ ①➳♣ ❧↕✐ ♠ët ❝→❝❤ ❝â ❤➺ t❤è♥❣ ♠ët sè ❦➳t q✉↔ ❣➛♥ ✤➙② tr♦♥❣ ❬✺❪ ✈➲ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ◆❛✈✐❡r✲❙t♦❦❡s ❝â tr➵✳ ❍✐ ✈å♥❣ ❧✉➟♥ ✈➠♥ ❧➔ ❝✉è♥ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ tèt ❝❤♦ ❜↕♥ ✤å❝ ❦❤✐ t➻♠ ❤✐➸✉ ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ◆❛✈✐❡r✲❙t♦❦❡s ❝â tr➵✳ ▲✉➟♥ ✈➠♥ ❦❤â tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✱ t→❝ ❣✐↔ r➜t ♠♦♥❣ ữủ sỹ õ ỵ qỵ t ổ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ✦✳ ✷✾ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❚✳ ❈❛r❛❜❛❧❧♦✱ ❏✳ ❘❡❛❧✱ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤ ❞❡❧❛②s✱ Pr♦❝✳ ❘✳ ❙♦❝✳ ▲♦♥❞✳ ❆ ✹✺✼ ✭✷✵✵✶✮ ✷✹✹✶✲✷✹✺✸✳ ❬✷❪ ❚✳ ❈❛r❛❜❛❧❧♦✱ ❏✳ ❘❡❛❧✱ ❆s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦✉r ♦❢ ✷❉✲◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤ ❞❡❧❛②s✱ Pr♦❝✳ ❘✳ ❙♦❝✳ ▲♦♥❞✳ ❆ ✹✺✾ ✭✷✵✵✸✮ ✸✶✽✶✲✸✶✾✹✳ ❬✸❪ ❚✳ ❈❛r❛❜❛❧❧♦✱ ❏✳ ❘❡❛❧✱ ❆ttr❛❝t♦rs ❢♦r ✷❉✲◆❛✈✐❡r✲❙t♦❦❡s ♠♦❞❡❧s ✇✐t❤ ❞❡❧❛②s✱ ❏✳ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✷✵✺ ✭✷✵✵✹✮ ✷✼✶✕✷✾✼✳ ❬✹❪ ❈✳ ❋♦✐❛s✱ ❖✳ ▼❛♥❧❡②✱ ❘✳ ❘♦s❛✱ ❘✳ ❚❡♠❛♥✱ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❛♥❞ t✉r✲ ❜✉❧❡♥❝❡✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✭✷✵✵✹✮✳ ❬✺❪ ▼✳❏✳ ●❛rr✐❞♦✲❆t✐❡♥③❛ ❛♥❞ P✳ ▼❛r➼♥✲❘✉❜✐♦✱ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤ ❞❡✲ ❧❛②s ♦♥ ✉♥❜♦✉❞❡❞ ❞♦♠❛✐♥s✱ ❬✻❪ ❏✳ ❙✐♠♦♥✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✱ ✻✹ ✭✷✵✵✻✮✱ ✶✶✵✵✲✶✶✶✽✳ ❊q✉❛t✐♦♥s ❞❡ ◆❛✈✐❡r❙t♦❦❡s✱ ❈♦✉rs ❞❡ ❉❊❆ ✷✵✵✷✲✷✵✵✸✱ ❯♥✐✈❡rs✐t➨ ❇❧❛✐s❡ P❛s❝❛❧✳ ❬✼❪ ❘✳ ❚❡♠❛♥✱ ◆❛✈✐❡r✲❙t♦❦❡s ❊q✉❛t✐♦♥s✿ ❚❤❡♦r② ❛♥❞ ◆✉♠❡r✐❝❛❧ ❆♥❛❧②s✐s✱ ✷♥❞ ❡❞✳✱ ◆♦rt❤✲❍♦❧❧❛♥❞✱ ❆♠st❡r❞❛♠✱ ✶✾✼✾✳ ✸✵