Về các bài toán ngược của phương trình truyền nhiệt với nguồn nhiệt chưa biết

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ◆●❯❨➍◆ ❚❍➚ ❚➐◆❍ ❱➋ ❈⑩❈ ❇⑨■ ❚❖⑩◆ ◆●×Đ❈ ❈Õ❆ P❍×❒◆● ❚❘➐◆❍ ❚❘❯❨➋◆ ◆❍■➏❚ ❱❰■ ◆●❯➬◆ ◆❍■➏❚ ❈❍×❆ ❇■➌❚ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❱✐♥❤ ✲ ✷✵✶✻ ❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ◆●❯❨➍◆ ❚❍➚ ❚➐◆❍ ❱➋ ❈⑩❈ ❇⑨■ ❚❖⑩◆ ◆●×Đ❈ ❈Õ❆ P❍×❒◆● ❚❘➐◆❍ ❚❘❯❨➋◆ ◆❍■➏❚ ❱❰■ ◆●❯➬◆ ◆❍■➏❚ ❈❍×❆ ❇■➌❚ ❚❖⑩◆ ●■❷■ ❚➑❈❍ ▼➣ sè✿ ✻✵✳ ✹✻✳ ✵✶✳ ✵✷ ❈❤✉②➯♥ ♥❣➔♥❤ ✿ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ữớ ữợ ❱❿◆ ✣Ù❈ ❱■◆❍ ✲ ✷✵✶✻ ▲❮■ ◆➶■ ✣❺❯ ❇➔✐ t♦→♥ ♥❣÷đ❝ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t❤÷í♥❣ ①✉②➯♥ ①✉➜t ❤✐➺♥ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ❝õ❛ ❝æ♥❣ ♥❣❤➺✱ t ỵ từ ỷ ỵ ↔♥❤✱✳✳✳ ✣â ❧➔ ♥❤ú♥❣ ❜➔✐ t♦→♥ ❦❤✐ ❝→❝ ❞ú ❦✐➺♥ q tr t ỵ ổ ữủ trỹ t✐➳♣ ♠➔ t❛ ♣❤↔✐ ①→❝ ✤à♥❤ ❝❤ó♥❣ tø ♥❤ú♥❣ ❞ú ❦✐➺♥ ✤♦ ✤↕❝ ❣✐→♥ t✐➳♣✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tổ tợ t ỗ ữỡ tr r t ỗ ữỡ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ❧➔ ♠æ ❤➻♥❤ t♦→♥ ❤å❝ ❝õ❛ ❝→❝ ❜➔✐ t♦→♥ t❤ü❝ t✐➵♥✱ ❝❤➥♥❣ ❤↕♥ ♥❤÷ ❜➔✐ t♦→♥ ①→❝ ✤à♥❤ ỗ t tr ởt q tr tr t t ỗ ổ tr ổ ổ trữớ ữợ t ữủ rt t tr ữợ ự ữ ❈❛♥♥♦♥✱ ❊✇✐♥❣✱ ❨❛♠❛♠♦t♦✱ ❍❛s❛♥♦✈✱ P❡❦t❛s✱ ❈❤✉✲▲✐✲❋✉✱ ✣✐♥❤ ◆❤♦ ❍➔♦✱ ✣➦♥❣ ✣ù❝ ❚rå♥❣✱✳✳✳ ❇➔✐ t♦→♥ ❦➸ tr➯♥ t❤÷í♥❣ ✤➦t ❦❤ỉ♥❣ ❝❤➾♥❤ t❤❡♦ ♥❣❤➽❛ ❍❛❞❛♠❛r❞✱ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❦❤æ♥❣ ♣❤↔✐ ❜❛♦ ụ tỗ t ổ tở tử ✈➔♦ ❞ú ❦✐➺♥ ❝õ❛ ❜➔✐ t♦→♥✳ ✣✐➲✉ ♥➔② ❧➔♠ ❝❤♦ ❜➔✐ t♦→♥ ✤➦t ❦❤ỉ♥❣ ❝❤➾♥❤ ❦❤â ❣✐↔✐ ❤ì♥ ♥❤✐➲✉ s♦ ợ t t ổ tữớ t♦→♥ ❤å❝ ♣❤↔✐ ✤➲ ①✉➜t ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❝❤➾♥❤ ❤â❛ ✤➸ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ✤➦t ❦❤æ♥❣ ❝❤➾♥❤✳ ❚✉② ♥❤✐➯♥✱ ❝❤♦ ✤➳♥ ♥❛② ❝→❝ ❦➳t q✉↔ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❝❤➾♥❤ õ ố ợ t ỗ ữỡ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ✈➝♥ ❝á♥ ❤↕♥ ❝❤➳✳ ❈→❝ ❦➳t q✉↔ ❝❤õ ②➳✉ ✤↕t ✤÷đ❝ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❝➜✉ tró❝ ✤ì♥ ❣✐↔♥ ✈➔ ❝→❝ ✤→♥❤ ❣✐→ s❛✐ sè t❤÷í♥❣ ✤↕t ✤÷đ❝ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ✣➸ t➟♣ ❞÷đt ♥❣❤✐➯♥ ❝ù✉ ❝ơ♥❣ ♥❤÷ ✤➸ ❧➔♠ ♣❤♦♥❣ ♣❤ó t❤➯♠ ❝→❝ t➔✐ ❧✐➺✉ ✈➲ t ỗ ữỡ tr➻♥❤ ♣❛r❛❜♦❧✐❝✱ tr➯♥ ❝ì sð ✶ ❝→❝ ❜➔✐ ❜→♦ ❬✸❪ ✈➔ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪✱ ❬✷❪✱ ❬✹❪✱ ❬✺❪✱ ❝❤ó♥❣ tỉ✐ ❧ü❛ ❝❤å♥ ✧❱➲ ❝→❝ ❜➔✐ t♦→♥ ♥❣÷đ❝ ❝õ❛ ữỡ tr tr t ợ ỗ t ữ t ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ✤➲ t➔✐ ❝❤♦ ▲✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤ ❧➔ ✿ ❧✉➟♥ ✈➠♥ t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❝→❝ t ữủ ữỡ tr tr t ợ ỗ t ữ t tr ỡ s t ợ ♠ư❝ ✤➼❝❤ ✤â ❧✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ✷ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❜ê trđ ❈❤÷ì♥❣ ♥➔② ♥❤➡♠ ♠ö❝ ✤➼❝❤ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ♥ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ✷✱ ❝❤õ ②➳✉ ❝❤ó♥❣ tæ✐ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✱ ❬✹❪✱ ❬✺❪ ✳ ❈❤÷ì♥❣ ✷✳ ❱➲ ❝→❝ ❜➔✐ t♦→♥ ♥❣÷đ❝ ❝õ❛ ữỡ tr tr t ợ ỗ t ữ t ữỡ ♥➔② ♥❤➡♠ ♠ö❝ ✤➼❝❤ tr➻♥❤ ❜➔② ✈➲ ❝→❝ ❜➔✐ t♦→♥ ữủ ữỡ tr tr t ợ ỗ t ữ ❜✐➳t tr➯♥ ❝ì sð t❤❛♠ ❦❤↔♦ ❜➔✐ ❜→♦ ❬✸❪✳ ▲✉➟♥ ữủ tỹ t ữợ sỹ ữợ t ự ✣➦❝ ❜✐➺t✱ tæ✐ ①✐♥ ❜➔② tä sü ❦➼♥❤ trå♥❣ ✈➔ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❚❤➛② ❣✐→♦ ❚❙✳ ◆❣✉②➵♥ ự ữớ t t ữợ ú ù ❝❤➾ ❜↔♦ tæ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✳ ◆❤➙♥ ❞à♣ ♥➔② tæ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✱ P❤á♥❣ ✣➔♦ t↕♦ ❙❛✉ ✣↕✐ ❤å❝ tr÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✱ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❑❤♦❛ ❙÷ ♣❤↕♠ ❚♦→♥ ❤å❝ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❝❤♦ tỉ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✳ ❚ỉ✐ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝ỉ ❣✐→♦ tr♦♥❣ tê ●✐↔✐ t➼❝❤ ð ❑❤♦❛ ❙÷ ♣❤↕♠ ❚♦→♥ ❤å❝ ✲ ✣↕✐ ❤å❝ ❱✐♥❤✱ ✤➣ ❣✐↔♥❣ ❞↕② ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ❝ơ♥❣ ♥❤÷ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✳ ❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❝↔♠ ì♥ ❜è ♠➭✱ ❛♥❤ ❝❤à ❡♠ ✈➔ ♥❤ú♥❣ ♥❣÷í✐ t❤➙♥ ❝ơ♥❣ ♥❤÷ ❜↕♥ ❜➧ ✤➣ ❣✐ó♣ ✤ï✱ ✤ë♥❣ ✈✐➯♥ tỉ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✳ ▼➦❝ ❞ị ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳✱ t❤✐➳✉ sât✳ ❈❤ó♥❣ tỉ✐ r➜t ữủ ỳ ỵ õ õ t❤➛②✱ ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ ❜➧ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✷ ◆❣❤➺ ❆♥✱t❤→♥❣ ✽ ♥➠♠ ✷✵✶✻ ❚→❝ ❣✐↔ ◆❣✉②➵♥ ❚❤à ❚➻♥❤ ✸ ▼ư❝ ❧ư❝ ▲í✐ ♥â✐ ✤➛✉ ❈❤÷ì♥❣✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❜ê trđ ✶✳✶ ❇➔✐ t♦→♥ ✤➦t ❦❤ỉ♥❣ ❝❤➾♥❤ ✶✳✷ P❤÷ì♥❣ ♣❤→♣ ❝❤➾♥❤ ❤â❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✹ ✹ ✻ ❈❤÷ì♥❣✷ ❱➲ ❝→❝ ❜➔✐ t♦→♥ ♥❣÷đ❝ ữỡ tr tr t ợ ỗ t ữ t ✶✵ ✷✳✶ ●✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✷ ❇➔✐ t♦→♥ ■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✸ ❇➔✐ t♦→♥ ■■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✹ ỗ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✵ ✸✶ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸ ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❜ê trđ ✶✳✶ ❇➔✐ t♦→♥ ✤➦t ❦❤ỉ♥❣ ❝❤➾♥❤ ❈→❝ ❦✐➳♥ t❤ù❝ tr♦♥❣ ♣❤➛♥ ♥➔② ✤÷đ❝ ❝❤ó♥❣ tỉ✐ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✺❪✳ ❑❤→✐ ♥✐➺♠ ❜➔✐ t♦→♥ ✤➦t ❦❤ỉ♥❣ ❝❤➾♥❤ ✤÷đ❝ tr➻♥❤ ❜➔② tr➯♥ ❝ì sð ①➨t ♠ët ❜➔✐ t♦→♥ ❞↕♥❣ ♣❤÷ì♥❣ tr➻♥❤ t♦→♥ tû A(u) = f, ð ✤➙② A : E → F ❧➔ t♦→♥ tû tø ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ E ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ F ✱ f ❧➔ ♣❤➛♥ tû t❤✉ë❝ F ✳ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ A ❧➔ t♦→♥ tû tø ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ E ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ F ✳ ❇➔✐ t♦→♥ A(u) = f ✤÷đ❝ ❣å✐ ❧➔ ✤➦t ❝❤➾♥❤ ♥➳✉ (i) P❤÷ì♥❣ tr➻♥❤ A(u) = f ❝â ♥❣❤✐➺♠ ✈ỵ✐ ♠å✐ f ∈ F, (ii) ◆❣❤✐➺♠ ♥➔② ❧➔ ❞✉② ♥❤➜t✱ (iii) ◆❣❤✐➺♠ ♣❤ö t❤✉ë❝ ❧✐➯♥ tö❝ ✈➔♦ ❞ú ❦✐➺♥ ❜➔✐ t♦→♥✳ ◆➳✉ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tr➯♥ ❦❤ỉ♥❣ t❤ä❛ ♠➣♥ t❤➻ ❜➔✐ t♦→♥ ✤÷đ❝ ❣å✐ ❧➔ ✤➦t ❦❤ỉ♥❣ ❝❤➾♥❤✳ ❇➔✐ t♦→♥ t➻♠ ♥❣❤✐➺♠ u ♣❤ư t❤✉ë❝ ✈➔♦ ❞ú ❦✐➺♥ f ✭❦➼ ❤✐➺✉ u = R(f )✮ ✤÷đ❝ ❣å✐ ❧➔ ê♥ ✤à♥❤ tr➯♥ ❝➦♣ ❦❤ỉ♥❣ ❣✐❛♥ (E, F ) ợ ộ > tỗ t↕✐ ♠ët sè δ(ε) > s❛♦ ❝❤♦ ♥➳✉ f1 − f2 ≤ δ(ε) t❤➻ u1 − u2 ≤ ε tr♦♥❣ ✤â ui = R(fi ), ui ∈ E, fi ∈ F, i = 1, 2, ✹ ❚r♦♥❣ ♥❤✐➲✉ ù♥❣ ❞ö♥❣ t❤➻ ✈➳ ♣❤↔✐ ❝õ❛ ❜➔✐ t♦→♥ A(u) = f t❤÷í♥❣ ✤÷đ❝ ❝❤♦ ❜ð✐ ✤♦ ✤↕❝✱ ♥❣❤➽❛ ❧➔ t❤❛② ❝❤♦ ❣✐→ trà ❝❤➼♥❤ ①→❝ f t❛ ❝❤➾ ❜✐➳t ①➜♣ ①➾ fδ ❝õ❛ ♥â t❤ä❛ ♠➣♥ fδ − f ≤ δ ✳ ●✐↔ sû uδ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ A(u) = f ✈ỵ✐ f t❤❛② ❜ð✐ fδ ✭❣✐↔ t❤✐➳t ♥❣❤✐➺♠ tỗ t t f f ữ ợ t t ổ t u õ ❝❤✉♥❣ ❦❤ỉ♥❣ ❤ë✐ tư ✤➳♥ u✳ ✶✳✶✳✷ ❱➼ ❞ư✳ ❳➨t ❜➔✐ t♦→♥ t➻♠ ❤➔♠ u tø ❤➺  ∂ u ∂ 2u   + = 0, ∂x2 ∂y  u(x, 0) = f (x), ∂u |y=0 = ϕ(x), −∞ < x < +∞, ∂y ✭✶✳✶✮ tr♦♥❣ ✤â f (x) (x) trữợ (i) ◆➳✉ ❧➜② f (x) = f1 (x) ≡ ✈➔ ϕ(x) = ϕ1 (x) = sin(ax) (a > 0) t❤➻ a ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ tr➯♥ ❧➔ u1 (x, y) = sin(ax)s❤(ay) a (ii) ◆➳✉ ❧➜② f (x) = f2 (x) = ϕ(x) = ϕ2 (x) ≡ t❤➻ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❧➔ u2 (x, y) ợ ỳ trữợ ✈➔ ♥❣❤✐➺♠ ✤÷đ❝ ①➨t ❧➔ ❦❤♦↔♥❣ ❝→❝❤ s✐♥❤ r❛ tø ❝❤✉➞♥ sup✱ t❛ ❝â f1 − f2 = sup |f1 (x) − f2 (x)| = 0, x∈R ϕ1 − ϕ2 = sup |ϕ1 (x) − ϕ2 (x)| = , a x∈R ✈ỵ✐ a ❦❤→ ❧ỵ♥ t❤➻ ❦❤♦↔♥❣ ❝→❝❤ ϕ1 − ϕ2 ❧↕✐ ❦❤→ ♥❤ä✳ ❚r♦♥❣ ❦❤✐ ✤â✱ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ❝→❝ ♥❣❤✐➺♠ u1 − u2 = sup |u1 (x, y) − u2 (x, y)| x∈R = sup | x∈R = sin(ax)s❤(ay)| a2 s❤(ay), a2 ✈ỵ✐ y > ố ợ tũ ỵ õ t♦→♥ ❦❤ỉ♥❣ ê♥ ✤à♥❤✳ ✺ ✶✳✶✳✸ ❱➼ ❞ư✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ♥❣÷đ❝ t❤í✐ ❣✐❛♥ ut + Au = 0, < t < T, u(T ) − f ε, ợ t tỷ ữỡ ổ A õ ởt ỡ s ỗ tỡ r trỹ {i }i tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✈ỵ✐ ❝❤✉➞♥ ❣✐→ trà r✐➯♥❣ {λi }i s❛♦ ❝❤♦ < λ1 ✈➔ ε > ✤➣ ❝❤♦✳ ❚❛ t❤➜② r➡♥❣ (t) = tr➻♥❤ λ2 v(T ) t ., lim λi = +∞ i→+∞ −λn (t−T ) e φn , λn vt + Av ✈➔ v(t) = 0, à tữỡ ự ợ T ♣❤÷ì♥❣ t = 0, < t < T, = φn , λn T ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ vt + Av v(T ) ❘ã r➔♥❣ (T ) − v(T ) = λ n φn = 0, = = < t < T, λn ✈ỵ✐ ♠å✐ t ∈ [0, T ) t❛ ❝â (t)−v(t) = φn = λn → ❦❤✐ n → +∞ ♥❤÷♥❣ −λn (t−T ) φn λn e = λn (T −t) λn e → +∞ ❦❤✐ n → +∞ ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä ❧í✐ ❣✐↔✐ ❝õ❛ ❜➔✐ t♦→♥ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ❧✐➯♥ tư❝ ✈➔♦ ❞ú ❦✐➺♥ t↕✐ t❤í✐ ✤✐➸♠ ❝✉è✐ t = T ✳ ❉♦ ✤â ❜➔✐ t♦→♥ ♥➔② ❧➔ ❜➔✐ t♦→♥ ✤➦t ❦❤æ♥❣ ❝❤➾♥❤✳ ✶✳✷ P❤÷ì♥❣ ♣❤→♣ ❝❤➾♥❤ ❤â❛ ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ ♣❤÷ì♥❣ tr➻♥❤ A(x) = f0✱ ✈ỵ✐ A ❧➔ ♠ët t♦→♥ tû tø ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ E ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ F ✳ ●å✐ x0 ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ A(x) = f0 ✳ ❚♦→♥ tû R(f, α)✱ ♣❤ö t❤✉ë❝ t❤❛♠ sè α✱ t→❝ ✤ë♥❣ E ✈➔♦ F ✤÷đ❝ ❣å✐ ❧➔ ♠ët t♦→♥ tû ❝❤➾♥❤ ❤â❛ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ A(x) = f0 ✱ i) ỗ t số ữỡ α1 s❛♦ ❝❤♦ t♦→♥ tû R(f, α) ①→❝ ✤à♥❤ ✈ỵ✐ ♠å✐ α ∈ (0, α1 ) ✈➔ ✈ỵ✐ ♠å✐ f ∈ F, dF (f, f0 ) ≤ δ, δ ∈ (0, ); ii) ỗ t ởt sỹ tở α = α(f, δ) s❛♦ ❝❤♦ ∀ε > 0, ∃δ(ε) ≤ δ1 ∀f ∈ F, dF (f, f0 ) ≤ δ ≤ δ1 , dE (xα , x0 ) ≤ ε✱ ð ✤➙② xα ∈ R(f, α(f, δ)) ❚r♦♥❣ ✤à♥❤ ♥❣❤➽❛ tr➯♥✱ ♥➳✉ α ✤÷đ❝ ❝❤å♥ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ f t❤➻ t❛ ❣å✐ ❧➔ ❝→❝❤ ❝❤å♥ t✐➯♥ ♥❣❤✐➺♠✳ ◆➳✉ α ✤÷đ❝ ❝❤å♥ ♣❤ư t❤✉ë❝ f ✈➔ δ t❤➻ t❛ ❣å✐ ❧➔ ❝→❝❤ ❝❤å♥ ❤➟✉ ♥❣❤✐➺♠✳ ✶✳✷✳✷ ◆❤➟♥ ①➨t✳ ❚r♦♥❣ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶ ❦❤ỉ♥❣ ✤á✐ ❤ä✐ t➼♥❤ ✤ì♥ trà ❝õ❛ t♦→♥ tû R(f, α)✳ P❤➛♥ ①➜♣ ①➾ xα ∈ R(fδ , α) ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❝❤➾♥❤ ❤â❛ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ A(x) = f0 ✱ ð ✤➙② α = α(fδ , δ) = α(δ) ✤÷đ❝ ❣å✐ ❧➔ t❤❛♠ sè ❝❤➾♥❤ ❤â❛✳ ❉➵ ❞➔♥❣ ♥❤➟♥ t❤➜② tø ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶ ✱ ♥❣❤✐➺♠ ❤✐➺✉ ❝❤➾♥❤ ❤â❛ ê♥ ✤à♥❤ ✈ỵ✐ ❞ú ❦✐➺♥ ❜❛♥ ✤➛✉✳ ◆❤÷ ✈➟②✱ ✈✐➺❝ t➻♠ ♥❣❤✐➺♠ ①➜♣ ①➾ ♣❤ư t❤✉ë❝ ❧✐➯♥ tư❝ ✈➔♦ ✈➳ ♣❤↔✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ A(x) = f0 ỗ ữợ i) t tỷ õ R(f, α)✱ ii) ❳→❝ ✤à♥❤ ❣✐→ trà ❝õ❛ t❤❛♠ sè ❤✐➺✉ ❝❤➾♥❤ α ❞ü❛ ✈➔♦ t❤æ♥❣ t✐♥ ❝õ❛ ❜➔✐ t♦→♥ ✈➲ ♣❤➛♥ tû fδ ✈➔ s❛✐ sè δ ✳ P❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ①➜♣ ①➾ t❤❡♦ q✉② t➢❝ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❝❤➾♥❤ ❤â❛✳ df (t) ✶✮ ❚➼♥❤ ✤↕♦ ❤➔♠ z = ✱ ❦❤✐ f (t) ❝❤♦ ①➜♣ ①➾✳ ✣↕♦ ❤➔♠ z dt t➼♥❤ ✤÷đ❝ ❞ü❛ ✈➔♦ t✛ s❛✐ ♣❤➙♥ ✶✳✷✳✸ ❱➼ ❞ö✳ R(f, α) = f (t + α) − f (t) α ◆➳✉ t❤❛② ❝❤♦ f (t) t❛ ❜✐➳t ①➜♣ ①➾ ❝õ❛ ♥â ❧➔ fδ (t) = f (t) + g(t)✱ ð ✤➙② |g(t)| ≤ δ ✈ỵ✐ ♠å✐ t✱ ❦❤✐ ✤â R(fδ , α) = ❈❤♦ α → 0✱ f (t + α) − f (t) g(t + α) − g(t) + α α f (t + α) − f (t) → z α ✼ ♥➯♥ t❛ ❝â ∞ {Nξ (x, ξ + h; t − τ ) − Nξ (x, ξ; t − τ )}F1 (φ(u(ξ, τ )))h−1 dξ ∞ Nξ (x, ξ; t − τ ){F1 (φ(u(ξ − h, τ ))) − F1 (φ(u(ξ, τ )))}h−1 dξ = h h Nξ (x, ξ; t − τ )F1 (φ(u(ξ, τ )))h−1 dξ − ❚❛ ♥❤➟♥ t❤➜② r➡♥❣ lim |h−1 {F1 (φ(u(ξ − h, τ ))) − F1 (φ(u(ξ, τ )))}| h↓0 ≤ C2 M lim |h−1 (u(ξ − h, τ ) − u(ξ, τ ))| h↓0 ≤ C2 M sup |∂x u(x, t)|, QT ∗ ♥❣♦➔✐ r❛ h Nξ (0, ξ; t − τ )F1 (φ(u(ξ, τ )))h−1 dξ = lim h↓0 ✈➻ Nξ (0, ξ; t − τ ) ≤ h−1 Const, ✈ỵ✐ ≤ ξ ≤ h ❑➳t ❤đ♣ ❝→❝ ✤→♥❤ ❣✐→ tr➯♥✱ t❛ t❤✉ ✤÷đ❝ ✤→♥❤ ❣✐→ s❛✉ ✈ỵ✐ ♠å✐ t, ≤ t ≤ T∗ ✱ T∗ ∞ ∂t u(0, t) ≥ g(0) (πt) − M C2 sup |∂x u| QT∗ Nξ (0, ξ, t − τ )dξdτ 0 ❤♦➦❝ ∂t u(0, t) ≥ g(0) (πt) − M C2 G(T∗ ) 4T∗ π ✶✼ , ≤ t ≤ T∗ ❙û ❞ö♥❣ ✭✷✳✹✶✮✱ ✭✷✳✹✷✮✱ ✭✷✳✹✸✮ t❛ ❝â ∂t u(0, t) > 1/M [0,T∗ ] ❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû u(x, t) t❤ä❛ ♠➣♥ ✭✷✳✶✸✮ ợ g, F1 tữỡ ự tọ ✭✷✳✶✼✮ ✈➔ φ t❤✉ë❝ S(M, T∗ )✳ ✣➸ ✤ì♥ ❣✐↔♥ t❛ s➩ ❦➼ ❤✐➺✉ SM t❤❛② ❝❤♦ S(M, T∗ )✱ tr♦♥❣ ✤â T∗ ♣❤ö t❤✉ë❝ ✈➔♦ M ✳ ◆➳✉ ≤ t ≤ T∗ , f (t) = u(0, t), t❤➻ tø ✭✷✳✸✷✮ ✈➔ ✭✷✳✸✻✮ t❛ ❝â ≤ f (t) ≤ µT∗ ≤ t ≤ T∗ ✭✷✳✹✺✮ ❍ì♥ ♥ú❛ tø ✭✷✳✹✹✮ t❛ ❝â ≤ t ≤ T∗ f (t) > 1/M, ✭✷✳✹✻✮ ❉♦ ✤â ❤➔♠ ♥❣÷đ❝ f −1 ❤♦➔♥ t♦➔♥ ✤÷đ❝ ①→❝ ✤à♥❤ tr➯♥ [0, T∗ ] ✈➔ ♥➳✉ t❛ ❦➼ ❤✐➺✉ ❤➔♠ ♥❣÷đ❝ ♥➔② ❧➔ ψ t❤➻ t❛ ❝â ψ(f (t)) = t ✈ỵ✐ ♠å✐ ≤ t ≤ T∗ , ✭✷✳✹✼✮ ✈ỵ✐ ≤ u ≤ µT∗ ✭✷✳✹✽✮ ✈➔ ≤ ψ(u) ≤ T∗ ❈❤✐ t✐➳t ❤ì♥ t❛ ❝â ✈ỵ✐ ≤ u ≤ u, ≤ ψ(u) ≤ T∗ tr♦♥❣ ✤â u = f (T∗ ) ≤ µT∗ , ✈➔ t❛ rở ữ s (u) = ợ u ≤ 0, ψ(u) = T∗ ✈ỵ✐ u ≥ u ✶✽ ✭✷✳✹✾✮ ❚ø ✭✷✳✷✵✮ t❛ ❝â < t2 − t1 < M (f (t2 ) − f (t1 )), ✈ỵ✐ ≤ t1 < t2 ≤ T∗ ✳ ❉♦ ✤â ✭✷✳✹✼✮ ❦➨♦ t❤❡♦ < ψ(f (t2 )) − ψ(f (t1 )) < M (f (t2 ) − f (t1 )), ❤♦➦❝ ❜➡♥❣ ❝→❝❤ ✤➦t ui = f (ti ), i = 1, 2, t❛ ❝â < ψ(u2 ) − ψ(u1 ) ≤ M (u2 − u1 ), ✭✷✳✺✵✮ ✈ỵ✐ ≤ u1 < u2 ≤ u ❉♦ ✤â ψ t❤✉ë❝ SM ✳ ❱ỵ✐ ♠ët ❤➔♠ h ∈ SM ❝❤✉➞♥ ❝õ❛ ❤➔♠ h ✤÷đ❝ ❤✐➸✉ ❧➔ ❝❤✉➞♥ s✉♣r❡♠✉♠✳ ❑❤✐ ✤â t❛ ❝â ❜ê ✤➲ s❛✉ ✷✳✷✳✻ ❇ê ✤➲✳ ●✐↔ sỷ r g, F1 tữỡ ự tọ ợ ộ tở SM tỗ t t u = u(x, t) ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✸✮ ✲ ✭✷✳✶✺✮ ✈➔ ♥❣❤✐➺♠ u ①→❝ ✤à♥❤ ♠ët ❤➔♠ ❞✉② ♥❤➜t ψ t❤✉ë❝ SM q✉❛ ✭✷✳✹✼✮✱ ✭✷✳✹✾✮✳ ◆➳✉ t❛ ❜✐➸✉ t❤à ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ φ ✈➔ ψ ❜➡♥❣ ❝→❝❤ ✈✐➳t ψ = f(φ) t❤➻ f ❧➔ ♠ët →♥❤ ①↕ ❝♦ tø SM ✈➔♦ ❝❤➼♥❤ ♥â✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❜✐➳t r➡♥❣ ✈ỵ✐ ❣✐↔ t❤✐➳t ✭✷✳✶✻✮ ✤è✐ ✈ỵ✐ g ✱ ❣✐↔ t❤✐➳t ✭✷✳✶✼✮ ✤è✐ ợ F1 tở SM ổ tỗ t ♥❤➜t ♥❣❤✐➺♠ u = u(x, t) ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✸✮ ✲ ✭✷✳✶✺✮✳ ❚❛ t❤➜② r➡♥❣ ψ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ t❤❡♦ ❝æ♥❣ t❤ù❝ ✭✷✳✹✼✮ ✲ ✭✷✳✹✾✮ t❤✉ë❝ SM ✳ ✣➸ ❝❤ù♥❣ tä f ❧➔ →♥❤ ①↕ ❝♦✱ ❧➜② φ1 , φ2 ❧➔ ❤❛✐ ♣❤➛♥ tû ❝õ❛ SM ✈➔ u1 , u2 t÷ì♥❣ ù♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✶✸✮ ✲ ✭✷✳✶✺✮ tr♦♥❣ DT∗ ✳ ❑❤✐ ✤â t ∞ u1 (0, t)−u2 (0, t) = N (0, ξ; t−τ )[F1 (φ1 (u1 (ξ, τ )))−F1 (φ2 (u2 (ξ, τ )))]dξdτ, 0 ✶✾ ✈➔ t ∞ |f1 (t) − f2 (t)| ≤ C2 N (0, ξ; t − τ )|φ1 (u1 (ξ, τ )) − φ2 (u2 (ξ, τ ))|dξdτ 0 t ∞ ≤ C2 N (0, ξ; t − τ )|φ1 (u1 (ξ, τ )) − φ2 (u1 (ξ, τ ))|dξdτ 0 t ∞ N (0, ξ; t − τ )|φ2 (u1 (ξ, τ )) − φ2 (u2 (ξ, τ ))|dξdτ +C2 0 ❱➻ t ∞ t ∞ N (0, ξ; t − τ )dξdτ = 0 0 t ∞ π(t − τ ) = 0 t ∞ π(t − τ ) ξ2 e− 4(t−τ ) dξdτ 4(t − τ )e−z dzdτ t 2 √ e−z dzdτ = π = 0 dτ = t, t❛ ❝â sup |f1 (t) − f2 (t)| ≤ C2 T∗ sup |φ1 (u) − u2 (u)| + M T∗ C2 sup |f1 (t) − f2 (t)| [0,T∗ ] [0,µT∗ ] ✈➔ sup |f1 − f2 | ≤ [0,T∗ ] [0,T∗ ] C2 T∗ sup |φ1 (u) − φ2 (u)| − M T∗ C2 [0,µT∗ ] ú ỵ r t1 < t2 T∗ ✈➔ f1 (t1 ) = u = f2 (t2 ) t t ỵ r t õ f1 (t2 ) − f1 (t1 ) f1 (t2 ) − f1 (t1 ) f1 (t2 ) − f2 (t2 ) = = = f1 (τ ) > ψ2 (u) − ψ1 (u) ψ2 (f2 (t2 ) − ψ1 ((f1 (t1 )) (t2 ) − (t1 ) M ✷✵ ✈ỵ✐ τ t❤ä❛ ♠➣♥ t1 ≤ τ ≤ t2 ✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ sup |f1 − f2 | > [0,T∗ ] sup |ψ1 (u) − ψ2 (u)|, M [0,µT∗ ] ✈➔ ❞♦ ✤â sup |ψ1 (u) − ψ2 (u)| ≤ [0,U∗ ] M C T∗ sup |φ1 (u) − φ2 (u)| − M C2 T∗ [0,µT∗ ] ❚ø ✭✷✳✹✸✮ t❛ ❝â ||ψ1 − ψ2 || ≤ ||φ1 − φ2 || ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä f ❧➔ ♠ët →♥❤ ①↕ ❝♦✳ ❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❚ø ❝→❝ ❜ê ✤➲ tr➯♥ t❛ ❝â ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ữ s ỗ t ♥❤➜t ♠ët ♣❤➛♥ tû ψ∞ t❤✉ë❝ SM s❛♦ ❝❤♦ ❜➔✐ t♦→♥ tr♦♥❣ ∂t u(x, t) = ∂xx u(x, t) + F1 (ψ∞ (u(x, t))) u(x, 0) = 0, QT∗ , x > 0, −∂x (0, t) = g(t) < t < T∗ , ❝â ♠ët ♥❣❤✐➺♠ ❞✉② ♥❤➜t u∞ (x, t)✱ ✈➔ ❤ì♥ ♥ú❛ ✈ỵ✐ ≤ t ≤ T∗ , ψ∞ (u∞ (0, t)) = t tr♦♥❣ ✤â g, F1 t÷ì♥❣ ù♥❣ t❤ä❛ ♠➣♥ ✭✷✳✶✻✮✱ ✭✷✳✶✼✮✳ ✷✳✸ ❇➔✐ t♦→♥ ■■ ❱ỵ✐ T > 0✱ ❣✐↔ sû r➡♥❣ u(x, t) t❤ä❛ ♠➣♥ ∂t u(x, t) = ∂xx u(x, t) + F2 (u(h, φ(u(x, t)))) u(x, 0) = 0, tr♦♥❣ QT , ✭✷✳✺✶✮ x > 0, −∂x u(0, t) = g(t), < t < T ✷✶ ✭✷✳✺✷✮ ✭✷✳✺✸✮ ❈❤ó♥❣ t❛ t✐➳♣ tö❝ ❣✐↔ t❤✐➳t r➡♥❣ g t❤ä❛ ♠➣♥ ✭✷✳✶✻✮ ✈➔ F2 t❤ä❛ ♠➣♥ ✭✷✳✶✼✮✳ ◆❣♦➔✐ r❛✱ ❣✐↔ t❤✐➳t φ t❤✉ë❝ ❧ỵ♣ ❤➔♠ S(M, T ) ✈ỵ✐ ❝→❝ ❤➡♥❣ sè ❞÷ì♥❣ M, T ✳ ❑❤✐ ✤â ❝→❝ ❇ê ✤➲ ✷✳✷✳✸ ✈➔ ✷✳✷✳✹ →♣ ❞ư♥❣ ✤÷đ❝ ❝❤♦ ♥❣❤✐➺♠ u(x, t) ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✺✶✮ ✲ ✭✷✳✺✸✮✳ ❍ì♥ ♥ú❛ t❛ ❝â ✷✳✸✳✶ ợ g, F2 tữỡ ự tọ ✈➔ ✭✷✳✶✼✮✱ ❝❤♦ h > ✈➔ T > ✤õ ♥❤ä✱ t❛ ❝â ✤→♥❤ ❣✐→ sup |∂t u(h, t)| ≤ (4/eh)(g(0) + KT ) + 2C1 ✭✷✳✺✹✮ [0,T ] tr♦♥❣ ✤â K ✈➔ C1 ❧➔ ❝→❝ ❤➡♥❣ sè tr♦♥❣ ❝→❝ ❝ỉ♥❣ t❤ù❝ ✭✷✳✶✻✮ ✈➔ ✭✷✳✶✼✮ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ h > 0✱ tø ✭✷✳✸✽✮ t❛ ❝â t k(h, τ )g (t − τ )dτ + F2 (u(h, φ(u(h, t)))) ∂t u(h, t) = k(h, t)g(0) + t ∞ Nξξ (h, ξ; t − τ )F2 (u(h, φ(u(ξ, τ ))))dξdτ + 0 ▼➦t ❦❤→❝ t❛ ❝â e−h /2t k(h, t) = ≤ eh (πt)1/2 ♠å✐ t ≥ ❞♦ ✤â sup |∂t u(h, t)| ≤ (2/eh)(g(0) + KT ) + C1 + sup |ω(h, t)|, [0,T ] ✭✷✳✺✺✮ [0,T ] tr♦♥❣ ✤â t ∞ Nξξ (h, ξ; t − τ )F2 (u(h, φ(u(ξ, τ ))))dξdτ ω(h, t) = ✭✷✳✺✻✮ ✣➸ ✤ì♥ ❣✐↔♥ ❤â❛ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ t❛ ❣✐↔ t❤✐➳t F2 , φ ❧➔ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ t❤❛② ✈➻ ❝❤➾ ❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ ✭❈❤ù♥❣ tr trữớ ủ st tữỡ tỹ ợ ❝❤ù♥❣ ♠✐♥❤ ❇ê ✤➲ ✷✳✷✳✺✮✳ ❱ỵ✐ ❣✐↔ t❤✐➳t F2 ✈➔ φ ❧➔ ❝→❝ ❤➔♠ ❦❤↔ ✈✐✱ t❤❡♦ ❝æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ t ξ=∞ Nξ (h, ξ; t − τ )F2 (u, φ(u, (ξ, τ ))) ω(h, t) = dτ ✭✷✳✺✼✮ ξ=0 t ∞ − Nξ (h, ξ; t − τ )F2 (u)∂t u(h, φ(u))φ (u)∂ξ u(ξ, τ )dξdτ 0 ❚❛ ❝â lim Nξ (h, ξ; t − τ ) = 0, ✭✷✳✺✽✮ lim Nξ (h, ξ; t − τ ) = 0, ✭✷✳✺✾✮ ξ→0 ξ→∞ ✈➔ t ∞ Nξ (h, ξ; t − τ )dξdτ ≤ ( 4t 1/2 ) , t > π ✭✷✳✻✵✮ ❱➻ F2 t❤ä❛ ♠➣♥ ✭✷✳✶✼✮ ✈➔ φ ∈ SM ♥➯♥ |F2 | ≤ C2 ✱ |φ | ≤ M ✳ ✣➦t B2 (T ) = max |∂t u(h, t)|, ✭✷✳✻✶✮ B3 (T ) = max |ω(h, t)| ✭✷✳✻✷✮ [0,T ] ✈➔ [0,T ] ❑❤✐ ✤â✱ ❝❤ó♥❣ t❛ ❝â t t ữợ B2 (T ) (2/eh)(g(0) + KT ) + C1 + B3 (T ) ✭✷✳✻✸✮ ❇➙② ❣✐í✱ t❛ ✤→♥❤ ❣✐→ B3 (T )✳ ❚ø ✭✷✳✺✼✮ ✲ ✭✷✳✻✵✮✱ ✭✷✳✸✼✮ ✈➔ ✭✷✳✹✵✮ t❛ ❝â B3 (T ) ≤ M C2 G(T ) ✈➔ 4T π 1/2 (g(0) + KT ) + C1 + B3 (T ) , eh M C2 G(T )(4T /π)1/2 (2/eh)(g(0) + KT ) + C1 B3 (T ) ≤ + M C2 G(T )(4T /π)1/2 ✷✸ ✭✷✳✻✹✮ ✈ỵ✐ G(T ) ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝ ✭✷✳✹✵✮ ❚❛ ❝â t❤➸ ❝❤å♥ ♠ët ❣✐→ trà T0 > s❛♦ ❝❤♦ M C2 G(T0 )(4T0 /π)1/2 = ✭✷✳✻✺✮ ❦❤✐ ✤â ✈ỵ✐ ♠å✐ T ≤ T0 t❛ ❝â B3 (T ) ≤ (2/eh)(g(0) + KT ) + C1 ✭✷✳✻✻✮ ❑❤➥♥❣ ✤à♥❤ tr♦♥❣ ❜ê ✤➲ ✤÷đ❝ ❦➨♦ t❤❡♦ tø ✭✷✳✻✸✮ ✈➔ ✭✷✳✻✻✮✳ ✷✳✸✳✷ ❇ê ✤➲✳ ❈❤♦ g, F2 t÷ì♥❣ ù♥❣ t❤ä❛ ♠➣♥ ✭✷✳✶✻✮ ✈➔ ✭✷✳✶✼✮ ✈➔ φ t❤✉ë❝ S(M, T ) ❦❤✐ õ tỗ t T1 , < T1 < T s❛♦ ❝❤♦ ∂t u(0, t) > 1/M [0,T1 ] ✭✷✳✻✼✮ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â t ∂t u(0, t) k(0, τ )g (t − τ )dτ + F2 (u(h, φ(u(0, t)))) = k(0, t).g(0) + t ∞ Nξξ (0, ξ; t − τ )F2 (u(h, φ(u(ξ, τ ))))dξdτ + 0 ❉♦ ✤â ∂t u(0, t) t ∞ ≥ g(0) − (πt)1/2 |Nξ (0, ξ; t − τ )F2 (u)∂t u(h, φ(u))φ (u)∂ξ u(ξ, τ )|dξdτ, 0 ❚❤❡♦ ❇ê ✤➲ ✷✳✸✳✶✈➔ ✭✷✳✸✼✮✱ ✭✷✳✹✵✮✱ ✭✷✳✻✵✮ t❛ ❝â g(0) ∂t u(0, t) ≥ − M C2 G(T ) (πt)1/2 4T π ✷✹ 1/2 (g(0) + KT ) + 2C1 eh ✭✷✳✻✽✮ ✈ỵ✐ ♠å✐ T ≤ T0 ✈➔ ♠å✐ t t❤ä❛ ♠➣♥ < t < T − T0 ✳ ✣➦t g(0) H(T ) = − M C2 G(T ) (πT )1/2 4T π 1/2 (g(0) + KT ) + 2C1 eh õ tỗ t ởt ❤➡♥❣ sè TM > s❛♦ ❝❤♦ H(TM ) = 1/M ✳ ❉♦ ✤â✱ ♥➳✉ ❝❤ó♥❣ t❛ ❝❤å♥ T1 = T0 , TM M M +1 ✭✷✳✼✵✮ ợ T0 ữủ ổ tự t ❜ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❚r♦♥❣ ♠ư❝ ♥➔②✱✤➸ ❝❤♦ ❣å♥ t❛ ❦➼ ❤✐➺✉ SM t❤❛② ❝❤♦ S(M, T1 ) ✈ỵ✐ T1 ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ ✭✷✳✼✵✮✱ ❣✐↔ sû r➡♥❣ u(x, t) t❤ä❛ ♠➣♥ tø ✭✷✳✺✶✮ ✤➳♥ ✭✷✳✺✸✮ ❝❤♦ g ✱ F2 t❤ä❛ ♠➣♥ ✭✷✳✶✻✮✱ ✭✷✳✶✼✮ ✈➔ φ ∈ SM ✳ ❈❤♦ f (t) = u(0, t), ✭✷✳✼✶✮ ✈➔ ❣✐↔ sû ψ(u) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ tr♦♥❣ ✭✷✳✹✼✮ ✈➔ ✭✷✳✹✾✮✳ ❙❛✉ ✤â t❛ ❝â ψ ∈ SM ✳ ❙✉② r❛ ❜ê ✤➲ s❛✉ ✷✳✸✳✸ ❇ê ✤➲✳ ●✐↔ sû g, F2 ❧➛♥ ❧÷đt t❤ä❛ ♠➣♥ ✭✷✳✶✻✮ ✈➔ ✭✷✳✶✼✮ t÷ì♥❣ ù♥❣✳ ❑❤✐ õ ợ ộ SM tỗ t t ♥❣❤✐➺♠ u = u(x, t) ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✺✶✮ ✲ ✭✷✳✺✸✮ ✈➔ ♥❣❤✐➺♠ ♥➔② ①→❝ ✤à♥❤ ♠ët ❤➔♠ ❞✉② ♥❤➜t ψ ∈ SM q✉❛ ✭✷✳✼✶✮✱ ✭✷✳✹✼✮ ✈➔ ✭✷✳✹✾✮✳ ◆➳✉ t❛ ❜✐➸✉ t❤à ψ = f (φ) t❤➻ f ❧➔ ♠ët →♥❤ ①↕ ❝♦ tø SM ✈➔♦ ❝❤➼♥❤ ♥â✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ♥➔② ❝❤➾ ❦❤→❝ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❇ê ✤➲ ✷✳✷✳✻ ð ♣❤➛♥ ❝❤ù♥❣ ♠✐♥❤ f ❧➔ →♥❤ ①↕ ❝♦✳ ▲➜② φ1 , φ2 ❧➔ ❤❛✐ ♣❤➛♥ tû ❝õ❛ SM ✈➔ u1 (x, t), u2 (x, t) ❧➔ ❝→❝ ♥❣❤✐➺♠ t÷ì♥❣ ù♥❣ ❝õ❛ ✭✷✳✺✶✮ ✲ ✭✷✳✺✸✮ tr♦♥❣ QT1 ợ T1 ữủ t õ t ❝â u1 (x, t) − u2 (x, t) ∞ t N (x, ξ; t − τ )[F2 (u1 (h, φ1 (ξ, τ ))) − F2 (u2 (h, φ2 (ξ, τ )))]dξdτ, = 0 ✈ỵ✐ ♠å✐ (x, t) ∈ QT1 ✈➔ t❛ ❝â t❤➸ ✈✐➳t ✷✺ u1 (x, t) − u2 (x, t) ∞ t N (x, ξ; t − τ )[F2 (u1 (h, φ1 (u1 ))) − F2 (u1 (h, φ1 (u2 )))]dξdτ = 0 ∞ t N (x, ξ; t − τ )[F2 (u1 (h, φ1 (u2 ))) − F2 (u2 (h, φ1 (u2 )))]dξdτ + 0 ∞ t N (x, ξ; t − τ )[F2 (u2 (h, φ1 (u2 ))) − F2 (u2 (h, φ2 (u2 )))]dξdτ + 0 ▼➦t ❦❤→❝✱ t❛ ❝â ∞ t N (x, ξ; t − τ )dξdτ = t ✈ỵ✐ t > 0, ✭✷✳✼✷✮ ❞♦ ✤â tø ✭✷✳✶✼✮✱ ✷✳✸✹ ✈➔ ✭✷✳✻✶✮ t❛ ❝â ✭✷✳✼✸✮ sup |u1 − u2 | ≤ C2 B2 (T1 )M T1 sup |u1 − u2 | QT1 QT1 + C2 T1 sup |u1 − u2 | + C2 B2 (T1 )T1 sup |φ1 − φ2 |, QT1 QT1 ❚ø ✭✷✳✼✸✮ t❛ s✉② r❛ sup |u1 − u2 | ≤ QT1 C2 B2 (T1 )T1 sup |φ1 − φ2 | − C2 T1 (1 + M B2 (T1 )) [0,µT ] ✈➔ sup |f1 (t) − f2 (t)| ≤ [0,T1 ] C2 B2 (T1 )T1 sup |φ1 − φ2 | − C2 T1 (1 + M B2 (T1 )) [0,µT ] ✭✷✳✼✹✮ ❚÷ì♥❣ tü ♥❤÷ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❇ê ✤➲ ✷✳✷✳✻✱ t❛ ❝â sup |f1 − f2 | ≥ (1/M ) sup |ψ1 − ψ2 |, [0,T1 ] [0,µT1 ] ð ✤➙② µT1 = P+ (T1 ) ✷✻ ✭✷✳✼✺✮ ❚ø ✭✷✳✼✹✮✱ ✭✷✳✼✺✮ t❛ ❝â sup |ψ1 − ψ2 | ≤ [0,µT1 ] ✭✷✳✼✻✮ ❈❤å♥ T1 > s❛♦ ❝❤♦ ①↕ ❝♦✳ C2 M B2 (T1 )T1 sup |φ1 − φ2 | − C2 T1 (1 + M B2 (T1 )) [0,µT ] C2 M B2 (T1 )T1 1−C2 T1 (1+M B2 (T1 )) < 1✱ t❤➻ tø ✭✷✳✼✻✮ t❛ s✉② r❛ f ❧➔ →♥❤ ❚ø ❝→❝ ❜ê ✤➲ tr t õ ỵ s ỗ t↕✐ ♠ët ♣❤➛♥ tû ❞✉② ♥❤➜t ψ∞ tr♦♥❣ SM s❛♦ ❝❤♦ ❜➔✐ t♦→♥ ∂t u(x, t) = ∂xx u(x, t) + F2 (u(h, ψ∞ (u(x, t)))) u(x, 0) = 0, tr♦♥❣ QT1 , x > 0, −∂x u(0, t) = g(t), < t < T1 , ❝â ❞✉② ♥❤➜t ♥❣❤✐➺♠ u∞ (x, t) t❤ä❛ ♠➣♥ ✈ỵ✐ ≤ t ≤ T1 , ψ∞ (u∞ (0, t)) = t ð ✤➙② F2 , g t÷ì♥❣ ù♥❣ t❤ä❛ ♠➣♥ ✭✷✳✶✻✮✱ ✭✷✳✶✼✮✳ ỗ t ú t q t ởt ỗ ữ ữủ t S(U ) tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ tr✉②➲♥ ♥❤✐➺t tø ❞ú ❦✐➺♥ ✤÷đ❝ ✤♦ ✤↕❝ tr➯♥ ❜✐➯♥✳ ❈❤ó♥❣ t❛ ①❡♠ ①➨t ❜➔✐ t♦→♥ ✭✷✳✽✮✱ ✭✷✳✷✮✱ ✭✷✳✸✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✭✷✳✹✮✳ ❚❛ t✐➳♣ tö❝ sû ❞ö♥❣ ✣à♥❤ ♥❣❤➽❛ ✷✳✻ ❝❤♦ ❤➔♠ φ ✈➔ ❣✐↔ t❤✐➳t r➡♥❣ ❤➔♠ f tr♦♥❣ ✭✷✳✹✮ t❤ä❛ ♠➣♥ f ∈ C1 [0, ∞) ✈ỵ✐ f (0) = ✭✷✳✼✼✮ ≤ w < ∞, ✭✷✳✼✽✮ ◆❣♦➔✐ r❛ f ✈➔ f t❤ä❛ ♠➣♥ ✭✷✳✶✼✮✳ ❱ỵ✐ h > 0✱ t❛ ✤à♥❤ ♥❣❤➽❛ F1 (w) = f (w) − h−2 f (w), ✷✼ t❤➻ F1 t❤ä❛ ♠➣♥ ✭✷✳✶✼✮ ♥➳✉ f t❤ä❛ ♠➣♥ ✭✷✳✼✼✮✳ ❚❤➟t ✈➟②✱ ✈➻ f ✈➔ f t❤ä❛ ♠➣♥ ✭✷✳✶✼✮ tỗ t số p, q > s❛♦ ❝❤♦ |f (x)| p, |f (x)| |f (x) − f (y)| p, ∀x q|x − y|, |f (x) − f (y)| q|x − y|, ∀x |F1 (x)| = |f (x) − h−2 f (x)| |f (x)| + h−2 |f (x)| (p + h−2 p) = C1 ∀x |F1 (x) − F1 (y)| = |f (x) − f (y) − h−2 |f (x) − f (y)| |f (x) − f (y)| + h−2 q|x − y| C2 |x − y| C1 ❱ỵ✐ h, T > 0✱ t❛ ✤à♥❤ ♥❣❤➽❛ F2 (w) = ♥➳✉ w < 0, = h−2 w ♥➳✉ ≤ w − T, = h−2 T ♥➳✉ w > T ✭✷✳✼✾✮ ❑❤✐ ✤â F2 ❝ơ♥❣ t❤ä❛ ♠➣♥ ✭✷✳✶✼✮✳ ❇➙② ❣✐í✱ ❝❤ó♥❣ t❛ ❝â t❤➸ t ữợ t u(x, t) ∂xx u(x, t) = F1 (φ(u(x, t))) + 2F2 (u(h, φ(u(x, t)))) − F2 (u(2h, φ(u(x, t))) tr♦♥❣ Qt ứ ỵ trữợ t õ ỵ s sỷ g t❤ä❛ ♠➣♥ ✭✷✳✶✻✮✱ f t❤ä❛ ♠➣♥ ✭✷✳✼✼✮✱ h > 0, T > ✤➣ ❝❤♦✱ F1 ✈➔ F2 t÷ì♥❣ ù♥❣ ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ ✭✷✳✼✽✮ ✈➔ ✭✷✳✼✾✮✳ ỗ t T > ởt t φ∗ tr♦♥❣ SM s❛♦ ❝❤♦ ❜➔✐ t♦→♥ ✭✷✳✽✶✮ ∂t u(x, t) = ∂xx u(x, t) + F1 (φ∗ (u(x, t))) +2F2 (u(h, φ∗ (u(x, t)))) − F2 (u(2h, φ∗ (u(x, t)))), tr♦♥❣ QT ∗ , u(x, 0) = 0, x > 0, ✭✷✳✽✷✮ −∂x u(0, t) = g(t), < t < T∗ , ✭✷✳✽✸✮ ✷✽ ❝â ❞✉② ♥❤➜t ♥❣❤✐➺♠ u∗ (x, t) ✈➔ ♥❣♦➔✐ r❛✱ ✈ỵ✐ < t < T∗ φ∗ (u∗ (x, t)) = t ✭✷✳✽✹✮ t ứ ỵ ổ tự t t r õ t ỗ t ❝❤÷❛ ✤÷đ❝ ❜✐➳t S(u) ①✉➜t ❤✐➺♥ tr♦♥❣ ❜✐➸✉ t❤ù❝ ✭✷✳✶✮ ❜ð✐ ✤↕✐ ❧÷đ♥❣ S∗ (u) = f (φ∗ (u)) − h−2 [f (φ∗ (u)) − 2u∗ (h, φ∗ (u)) + u∗ (2h, φ∗ (u))] ✭✷✳✽✺✮ ✷✾ ❑➌❚ ▲❯❾◆ ❑➳t q✉↔ ✤↕t ✤÷đ❝ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ ✶✳ ❚r➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✤➦t ❦❤ỉ♥❣ ❝❤➾♥❤ ✈➔ ❝→❝ ✈➼ ❞ư ♠✐♥❤ ❤å❛✳ ✷✳ ❚r➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ♣❤÷ì♥❣ ♣❤→♣ ❝❤➾♥❤ ❤â❛ ✈➔ ❝→❝ ✈➼ ❞ư ♠✐♥❤ ❤å❛✳ ✸✳ ●✐ỵ✐ t❤✐➺✉ ❜➔✐ t ỗ ữỡ tr tr t ❚r➻♥❤ ❜➔② ❇➔✐ t♦→♥ ■✱ ❇➔✐ t♦→♥ ■■ ✈➔ ❇➔✐ t ỗ t tr ỡ s t ❬✸❪ ✳ ✸✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ P❤↕♠ ❑ý ❆♥❤ ✭✷✵✵✼✮✱ ❇➔✐ t♦→♥ ✤➦t ❦❤æ♥❣ ❝❤➾♥❤✱ ✣❍◗● ❍➔ ◆ë✐✳ ❬✷❪ ❏✳ ❇❛✉♠❡✐st❡r ✭✶✾✽✼✮✱ ❙t❛❜❧❡ s♦❧✉t✐♦♥ ♦❢ ■♥✈❡rs❡ ♣r♦❜❧❡♠s✱ ❋r✐❡❞r✳❱✐❡✇❡❣ & ❙♦❤♥✱ ❇r❛✉♥s❝❤✇❡✐❣✳ ❬✸❪ ❏✳ ❘✳ ❈❛♥♥♦♥ ✭✶✾✽✵✮✱ ✧❆♥ ■♥✈❡rs❡ Pr♦❜❧❡♠ ❢♦r ❛♥ ❯♥❦♥♦✇♥ ❙♦✉r❝❡ ✐♥ ❛ ❍❡❛t ❊q✉❛t✐♦♥✧✱ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✼✺✱ ✹✻✺✕✹✽✺✳ ❬✹❪ ▲✳❈✳ ❊✈❛♥s ✭✶✾✾✽✮✱ P❛rt✐❛❧ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ❆♠❡r✐❝❛♥ ▼❛t❤✳ ❙♦✲ ❝✐❡t②✳ ❬✺❪ ❆✳ ❑✐rs❝❤ ✭✶✾✾✻✮✱ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ ▼❛t❤❡♠❛t✐❝❛❧ ❚❤❡♦r② ♦❢ ■♥✈❡rs❡ Pr♦❜❧❡♠s✱ ❙♣r✐♥❣❡r✳ ✸✶