Soluções Globais Uniformemente Limitadas para a Equação do Calor Semilinear

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛

■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❙♦❧✉çõ❡s ●❧♦❜❛✐s ❯♥✐❢♦r♠❡♠❡♥t❡ ▲✐♠✐t❛❞❛s ♣❛r❛ ❛

❊q✉❛çã♦ ❞♦ ❈❛❧♦r ❙❡♠✐❧✐♥❡❛r

♣♦r

●✐❧❜❡rt♦ ❞❡ ❆ss✐s P❡r❡✐r❛

❇r❛sí❧✐❛

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ♠❛t❡♠át✐❝❛

❙♦❧✉çõ❡s ●❧♦❜❛✐s ❯♥✐❢♦r♠❡♠❡♥t❡

▲✐♠✐t❛❞❛s ♣❛r❛ ❛ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r

❙❡♠✐❧✐♥❡❛r

♣♦r

●✐❧❜❡rt♦ ❞❡ ❆ss✐s P❡r❡✐r❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡

▼❊❙❚❘❊ ❊▼ ▼❆❚❊▼➪❚■❈❆

❇r❛sí❧✐❛✱ ✶✾ ❞❡ ❛❜r✐❧ ❞❡ ✷✵✶✷

❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛✿

❉r❛✳ ▲✐❧✐❛♥❡ ❞❡ ❆❧♠❡✐❞❛ ▼❛✐❛ ✲ ❯♥❇ ✲ ❖r✐❡♥t❛❞♦r❛

❉r✳ ❖❧í♠♣✐♦ ❍✐r♦s❤✐ ▼✐②❛❣❛❦✐ ✲ ❯❋❏❋ ✲ ❊①❛♠✐♥❛❞♦r

❉r❛✳ ❙✐♠♦♥❡ ▼❛③③✐♥✐ ❇r✉s❝❤✐ ✲ ❯♥❇ ✲ ❊①❛♠✐♥❛❞♦r❛

❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r❛♠❡♥t❡✱ ❛ ❉❡✉s t♦❞♦ ♣♦❞❡r♦s♦✱ q✉❡ s❡ ❤✉♠✐❧❤❛✱ t♦♠❛ ❛ ❢♦r♠❛ ❞❡ s❡r✈♦✱ ❡ ♥♦s ♣❡r❣✉♥t❛✿ ❵❖ q✉❡ q✉❡r❡s q✉❡ ❊✉ t❡ ❢❛ç❛❄✬✳ ❊❧❡ ❜✉s❝❛ ❛t❡♥❞❡r às ♥♦ss❛s ♥❡❝❡ss✐❞❛❞❡s✱ ♠❡s♠♦ s❡♥❞♦ ♠❛✐♦r q✉❡ ♥ós✳

❆♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s✱ ❛ ♠✐♥❤❛ ❡t❡r♥❛ ❣r❛t✐❞ã♦ ♣♦r s✉❛s ♣r❡s❡♥ç❛s ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❡ ♠✐♥❤❛ ✈✐❞❛✱ ❞❛♥❞♦✲♠❡ ❢♦rç❛✱ ❛✉①✐❧✐❛♥❞♦✲♠❡✱ ❝♦♠♣r❡❡♥❞❡♥❞♦✲♠❡ ❡ ♠❡ ❢♦rt❛❧❡❝❡♥❞♦ ♥❛s ❤♦r❛s ❞✐❢í❝❡✐s✳

➚ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ Pr♦❢❛_{✳ ▲✐❧✐❛♥❡✱ ❛❣r❛❞❡ç♦ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ♣❡❧❛ ❛t❡♥çã♦ ❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♣❡❧❛}

s✉❛ s❛❜❡❞♦r✐❛✱ q✉❡ ♠✉✐t♦ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❝♦♥❝❧✉✐r ✉♠❛ ❣r❛♥❞❡ ❡ ✐♠♣♦rt❛♥t❡ ❡t❛♣❛ ❞❡ ♠✐♥❤❛ ✈✐❞❛✳ ❖❜r✐❣❛❞♦ ❛♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥❇ ❝♦♠ ♦s q✉❛✐s ❝♦♥✈✐✈✐✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛♦s ♣r♦❢❡ss♦r❡s ▲✉✐s ❍❡♥r✐q✉❡ ❞❡ ▼✐r❛♥❞❛✱ ❘✐❝❛r❞♦ ❘✉✈✐❛r♦ ❡ ❙✐♠♦♥❡ ▼❛③③✐♥✐ ❇r✉s❝❤✐✱ ♣❡❧❛s s✉❣❡stõ❡s ❡ ❛t❡♥çã♦ ❞✉r❛♥t❡ t♦❞♦ ♦ tr❛❜❛❧❤♦✳

❖❜r✐❣❛❞♦ ❛♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❱✳ ❊♠ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢❡ss♦r P❛✉❧♦ ❚❛❞❡✉✱ ♣❡❧❛ ❛♠✐③❛❞❡✱ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s ❡ ♣❡❧♦ ❡①❡♠♣❧♦ ❞❡ s❡r ❤✉♠❛♥♦✳ ❆♦ ♣r♦❢❡ss♦r ❖❧í♠♣✐♦ ❍✐r♦s❤✐ ▼✐②❛❣❛❦✐ ♣❡❧♦s ❝♦♥s❡❧❤♦s✱ ❛t❡♥çã♦ ❡ ❛♠✐③❛❞❡✳

❆♦s ❛♠✐❣♦s q✉❡ ✜③ ♥❡st❛ ❡t❛♣❛ ❡ q✉❡ ♥✉♥❝❛ ❡sq✉❡❝❡r❡✐✱ ❊❞✐♠✐❧s♦♥ ❞♦s ❙❛♥t♦s ❞❛ ❙✐❧✈❛✱ ❈❧♦❞♦♠✐r ◆❡t♦ ❡ ❆r✐stót❡❧❡s ❏ú♥✐♦r✱ q✉❡ ♠✉✐t❛s ✈❡③❡s ❞❡✐①❛✈❛♠ ❞❡ ❡st✉❞❛r ♣❛r❛ ♠❡ ❛❥✉❞❛r ❝♦♠ ♣r♦❜❧❡♠❛s ❝♦♠♣✉t❛❝✐♦♥❛✐s✳ ❙♦✉ ❡t❡r♥❛♠❡♥t❡ ❣r❛t♦ ❛♦ ♠❡✉ ❛♠✐❣♦ ❆rt✉r ❋❛ss♦♥✐ ♣❡❧♦s q✉❛tr♦s ❛♥♦s ❞❡ ❝♦♥✈✐✈ê♥❝✐❛ ♥❛ ❣r❛❞✉❛çã♦✱ ♣❡❧❛s ❝♦♥✈❡rs❛s✱ ♣♦r s❡r ♠❡✉ ✐r♠ã♦ ❞❡ ❝♦♥s✐❞❡r❛çã♦✱ ❡ ❛♦s ♠❡✉s ❝♦❧❡❣❛s ❞♦ ❛♣❛rt❛♠❡♥t♦ ✶✾✶✶ ♣♦r t✉❞♦ ❞❡ ❜♦♠ q✉❡ ♣❛ss❛♠♦s ❥✉♥t♦s✳

❯♠ ❛❣r❛❞❡❝✐♠❡♥t♦ ❡s♣❡❝✐❛❧ ❣♦st❛r✐❛ ❞❡ ❢❛③❡r ❛ ❞✉❛s ♣❡ss♦❛s✿ ❏♦sé ●❡r❛❧❞♦ ❚❡✐①❡✐r❛✱ q✉❡ ♠❡ ❛❝♦❧❤❡✉ ❝♦♠♦ ✜❧❤♦ ❡♠ ❱✐ç♦s❛✱ ❞❛♥❞♦✲♠❡ ♠♦r❛❞✐❛✱ ❝♦♥s❡❧❤♦s✱ s❛❜❡❞♦r✐❛✱ t♦r♥❛♥❞♦✲s❡ ✉♠ ♠♦❞❡❧♦ ✐❞❡❛❧ ❞❡ ♣❡ss♦❛ ❛ s❡r s❡❣✉✐❞♦✱ ❡ ❛ ♠❡✉ ♣r♦❢❡ss♦r ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ♠é❞✐♦✱ ❈❧❛ú❞✐♦✱ q✉❡ s❡♠♣r❡ ♠❡ ❛❥✉❞♦✉✳

❆❣r❛❞❡ç♦ ❛♦ ❈◆Pq ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❛ ❡st❡ tr❛❜❛❧❤♦✳

❊♥✜♠✱ ❛❣r❛❞❡ç♦ ❛ t♦❞♦s q✉❡ r❡③❛r❛♠ ♣♦r ♠✐♠ ❡ ♣❡ç♦ ❞❡s❝✉❧♣❛s ❛ t♦❞♦s q✉❡ ♥ã♦ ♣✉❞❡ ❝✐t❛r✱ ♣♦✐s sã♦ t❛♥t❛s ❛s ♣❡ss♦❛s ❡s♣❡❝✐❛✐s q✉❡ ♠❡✉s ❛❣r❛❞❡❝✐♠❡♥t♦s s❡r✐❛♠ ❛ ♠❛✐♦r ♣❛rt❡ ❞❡ ♠✐♥❤❛ ❞✐ss❡rt❛çã♦✳✳✳

❘❡s✉♠♦

❆♣r❡s❡♥t❛♠♦s ❛ t❡♦r✐❛ ❞❡ s❡♠✐❣r✉♣♦s ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s✱ ❝♦♠ ♦❜❥❡t✐✈♦ ❞❡ ❡st✉❞❛r ♦ ❝♦♠♣♦rt❛✲ ♠❡♥t♦ ❞❛ ❡q✉❛çã♦ ❞♦ ❝❛❧♦r ❤♦♠♦❣ê♥❡❛✱ ❧✐♥❡❛r ❡ s❡♠✐❧✐♥❡❛r✳ ❈♦♥s✐❞❡r❛♠♦s ♦ ♣r♦❜❧❡♠❛ ♣❛r❛ ❛ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r s❡♠✐❧✐♥❡❛r



 

 

∂u

∂t −∆u+mu = g(u), ❡♠ (0,∞)×Ω, u = 0, ❡♠ [0,_∞)_×∂Ω, u(0, x) = u0(x), ❡♠ Ω,

✭✶✮

❡♠ q✉❡ Ω_⊂RN _{é ❛❜❡rt♦✱ ❧✐♠✐t❛❞♦ ❡ r❡❣✉❧❛r✱N ≥3✱ m >−λ}

1✱ ♦♥❞❡ λ1 é ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ❞❡ −∆

❡♠ H1

0(Ω)❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ (·,·)✳ ❆ ❢✉♥çã♦g∈C1(R,R)é t❛❧ q✉❡

|g(x)_|6C1|x|+C2|x|p, ✭✷✮

❝♦♠1< p < N+ 2 N−2 = 2

∗

−1✱C1❡C2❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s❀gs❛t✐s❢❛③ t❛♠❜é♠

(g(u), u)≥(2 +ε)G(u), ✭✸✮

♣❛r❛ ❛❧❣✉♠ε >0✳

▼♦str❛♠♦s q✉❡ ❡①✐st❡ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❡ q✉❡ ❡❧❛ é ✉♥✐❢♦r♠❡♠❡♥t❡ ❧✐♠✐t❛❞❛ ♥♦ t❡♠♣♦ t_≥δ✱

♣❛r❛ q✉❛❧q✉❡rδ >0✳

P❛❧❛✈r❛s✲❈❤❛✈❡s✿ ❙❡♠✐❣r✉♣♦s ❞❡ ♦♣❡r❛❞♦r❡s❀ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r ❙❡♠✐❧✐♥❡❛r❀ ❖♣❡r❛❞♦r ♠✲❛❝r❡t✐✈♦❀ ❖♣❡r❛❞♦r ▼❛①✐♠❛❧ ▼♦♥ót♦♥♦❀ ❇♦♦tstr❛♣✳

❆❜str❛❝t

❲❡ ♣r❡s❡♥t t❤❡ t❤❡♦r② ♦❢ s❡♠✐❣r♦✉♣s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs ✇✐t❤ t❤❡ ❛✐♠ ♦❢ st✉❞②✐♥❣ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❤♦♠♦❣❡♥❡♦✉s✱ ❧✐♥❡❛r ❛♥❞ s❡♠✐❧✐♥❡❛r ❤❡❛t ❡q✉❛t✐♦♥✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ s❡♠✐❧✐♥❡❛r ❤❡❛t ❡q✉❛t✐♦♥



 

 

∂u

∂t −∆u+mu = g(u), ✐♥ (0,∞)×Ω, u = 0, ✐♥ [0,_∞)_×∂Ω, u(0, x) = u0(x), ✐♥ Ω,

✭✹✮

✇❤❡r❡ Ω_⊂RN _{✐s ❛♥ ♦♣❡♥✱ ❜♦✉♥❞❡❞ r❡❣✉❧❛r ❞♦♠❛✐♥✱N}

≥3✱m >₋λ1✱λ1t❤❡ ✜rst ❡✐❣❡♥✈❛❧✉❡ ♦❢ t❤❡ −∆

✐♥ H1

0(Ω) t❤❡ ❍✐❧❜❡rt s♣❛❝❡ ✇✐t❤ s❝❛❧❛r ♣r♦❞✉❝t(·,·)✳ ❚❤❡ ❢✉♥❝t✐♦♥g∈C1(R,R)✐s s✉❝❤ t❤❛t

|g(x)_|6C1|x|+C2|x|p, ✭✺✮

✇✐t❤ 1< p < N+ 2 N₋2 = 2

∗

−1✱C1 ❛♥❞C2♣♦s✐t✐✈❡ ❝♦♥st❛♥ts ❀g❛❧s♦ s❛t✐s✜❡s

(g(u), u)≥(2 +ε)G(u), ✭✻✮

❢♦r s♦♠❡ ε > 0. ❲❡ s❤♦✇ t❤❛t t❤❡r❡ ❡①✐sts ❛ s♦❧✉t✐♦♥ ❢♦r t❤❡ ♣r♦❜❧❡♠ ❛♥❞ t❤✐s s♦❧✉t✐♦♥ ✐s ✉♥✐❢♦r♠❧②

❜♦✉♥❞❡❞ ❢♦r ❛❧❧ t✐♠❡t_≥δ,❢♦r ❛♥②δ >0.

❑❡②✲❲♦r❞s✿ ❙❡♠✐❣r♦✉♣s ♦❢ ♦♣❡r❛t♦rs❀ ❙❡♠✐❧✐♥❡❛r ❍❡❛t ❊q✉❛t✐♦♥❀ ❖♣❡r❛t♦r ♠✲❛❝❝r❡t✐✈❡❀ ▼❛①✐♠❛❧ ▼♦♥♦t♦♥❡ ❖♣❡r❛t♦r❀ ❇♦♦tstr❛♣✳

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

◆♦t❛çõ❡s ✸

✶ ❙❡♠✐❣r✉♣♦s ❞❡ ❈❧❛ss❡ C0 ✺

✶✳✶ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r ❍♦♠♦❣ê♥❡❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✷ ❈❛s♦ ▲✐♥❡❛r ❞❛ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r ✷✷

✸ ❊①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ❞❛ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r ♥ã♦✲❧✐♥❡❛r ✸✼

✸✳✶ ❆♣❧✐❝❛çã♦ ❞♦s ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✹ ❊st✐♠❛t✐✈❛ ✉♥✐❢♦r♠❡ ♣❛r❛ ❛ s♦❧✉çã♦ ❞❛ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r ♥ã♦✲❧✐♥❡❛r ✹✼

❆ ❘❡s✉❧t❛❞♦s ❆✉①✐❧✐❛r❡s ✻✸

■♥tr♦❞✉çã♦

❙❡❥❛ Ω _⊂ RN ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❡ ❧✐♠✐t❛❞♦ ❝♦♠ ❢r♦♥t❡✐r❛ _∂Ω✱ _{N ≥ 1} ✱ ❝♦♥s✐❞❡r❛♠♦s ♦ s❡❣✉✐♥t❡

♣r♦❜❧❡♠❛✿ ❡♥❝♦♥tr❛r ✉♠❛ ❢✉♥çã♦ u: [0,_∞)_×Ω_→Rt❛❧ q✉❡



  

  

∂u

∂t −∆u = 0, ❡♠(0,∞)×Ω, u = 0 ❡♠(0,∞)×∂Ω, u(0, x) = u0(x) ❡♠Ω,

✭✼✮

♦♥❞❡ ∆ =

N

X

i=1

∂2

∂xi ❞❡♥♦t❛ ♦ ▲❛♣❧❛❝✐❛♥♦ ♥❛s ✈❛r✐á✈❡✐s ❡s♣❛❝✐❛✐s

x✱ t é ❛ ✈❛r✐á✈❡❧ t❡♠♣♦ ❡ u0(x) é ✉♠❛

❢✉♥çã♦ ❞❛❞❛✱ ❝❤❛♠❛❞❛ ❞❛❞♦ ✐♥✐❝✐❛❧ ❞❡ ❈❛✉❝❤② ✈❡r [✹] ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❡♠ (7) é ❝❤❛♠❛❞❛ ❊q✉❛çã♦

❞♦ ❈❛❧♦r ♣♦rq✉❡ ♠♦❞❡❧❛ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ t❡♠♣❡r❛t✉r❛ u♥♦ ❞♦♠í♥✐♦Ω❡ ♥♦ t❡♠♣♦t✳

❖ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❛♣r❡s❡♥t❛r ❡st✐♠❛t✐✈❛s ❣❧♦❜❛✐s ✉✐♥✐❢♦r♠❡s ♥♦ t❡♠♣♦✱ ♦✉ s❡❥❛

||u(t)||L∞

(Ω)≤C(δ)

♣❛r❛ t ≥δ. ❖s r❡s✉❧t❛❞♦s s❡ ❜❛s❡✐❛♠ ❞♦ ❝❧áss✐❝♦ ❡ ✐♠♣♦rt❛♥t❡ ❛rt✐❣♦ ❞❡ ❈❛③❡♥❛✈❡ ❡ ▲✐♦♥s ❡♠ ❈✳P✳❉✳❊

✶✾✽✹ ❞❛❞♦ ❡♠ [✽]✳ ◆❡st❡ ❛rt✐❣♦✱ ♦s ❛✉t♦r❡s ❝♦♥s✐❞❡r❛r❛♠ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛







∂u

∂t −∆u+mu = g(u), ❡♠(0,∞)×Ω,

u = 0 ❡♠(0,∞)×∂Ω, ✭✽✮

s❡g_∈C1_{(R,R)❡ ✈❡r✐✜❝❛}

✭✐✮ |g(x)| ≤A|x|+B|x|p_{✱ ❝♦♠1< p <} N+ 2

N₋2 ♣❛r❛ t♦❞♦x∈R✳

✭✐✐✮ xg(x)≥(2 +ε)G(x)♣❛r❛ t♦❞♦x∈R✱ ♦♥❞❡ ❡_{G(x) =}

Z x 0

g(s)ds✱

♦s ♠❡s♠♦s ❛✜r♠❛r❛♠ q✉❡ ♣r♦❜❧❡♠❛ ♣♦ss✉✐ s♦❧✉çã♦ ✉✐♥❢♦r♠❡♠❡♥t❡ ❧✐♠✐t❛❞❛ ❡♠ C2_{(Ω)♣❛r❛ t♦❞♦ t}

≥δ✱

s❡♥❞♦δ >0 ❛r❜✐trár✐♦✳

◆❡st❡ tr❛❜❛❧❤♦ ♥♦ss♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ s❡rá ♦❜t❡r ✉♠❛ ❡st✐♠❛t✐✈❛ ✉♥✐❢♦r♠❡ ❡♠t ♥❛ ♥♦r♠❛L∞_(Ω)✱ ♦❜❡❞❡❝❡♥❞♦ ❛s ❤✐♣ót❡s❡s ❛♣r❡s❡♥t❛❞❛s ❡♠ [✽]✱ ❢❛t♦ ❡st❡ ♦❜t✐❞♦ s❡♠ ♦ ✉s♦ ❞♦ ❡s♣❛ç♦ ❞❡ ▲♦r❡♥t③✱ ❛♣❡♥❛s

■♥tr♦❞✉çã♦ ✷

◆♦ss♦ tr❛❜❛❧❤♦ ❡stá ♦r❣❛♥✐③❛❞♦ ❝♦♠♦ s❡❣✉❡✳

❯♠ ❜❧♦❝♦ ❞❡ ♥♦t❛çõ❡s q✉❡ s❡rã♦ ✉t❡✐s ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✳

◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❞❡✜♥✐r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ s❡♠✐❣r✉♣♦s ❞❡ ❝❧❛ss❡C0 ❡ ❡①♣❧♦r❛r❡♠♦s ❛s s✉❛s ♣r♦✲

♣r✐❡❞❛❞❡s✳ ❊♠ s❡❣✉✐❞❛✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ s❡♠✐❣r✉♣♦s ❞❡ ❝♦♥tr❛çã♦✳ ❊ ♥♦ ✜♥❛❧ ❞♦ ♠❡s♠♦ ❢❛r❡♠♦s ✉♠❛ ❛♣❧✐❝❛çã♦ ❛♦ ♣r♦❜❧❡♠❛ ❞❛ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r ♥♦ ❝❛s♦ ❤♦♠♦❣ê♥❡♦✳

◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❡st✉❞❛r❡♠♦s ❛ t❡♦r✐❛ ❞❡ s❡♠✐❣r✉♣♦s ❞❡ ❝♦♥tr❛çã♦ ❣❡r❛❞♦ ♣♦r ✉♠ ♦♣❡r❛❞♦r✱ ❝♦♠ ✐♥t✉✐t♦ ❞❡ ♦❜t❡r ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞♦ ❝❛❧♦r ♥♦ ❝❛s♦ ❧✐♥❡❛r ♥ã♦ ❤♦♠♦❣ê♥❡♦✳ ❆❧é♠ ❞✐ss♦ ❛♣r❡s❡♥t❛r❡♠♦s ❡st✐♠❛t✐✈❛s Lp _{❡ ❡st✐♠❛t✐✈❛s ❞❡ ❙❝❤❛✉❞❡r ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ♣❛r❛❜ó❧✐❝♦✳}

◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ ❡st✉❞❛r❡♠♦s ♦♣❡r❛❞♦r❡s ♠♦♥ót♦♥♦s ♠❛①✐♠❛✐s✱ ❜❡♠ ❝♦♠♦ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ ❛ ❡q✉❛çã♦ ❞♦ ❝❛❧♦r ♥♦ ❝❛s♦ ♥ã♦ ❧✐♥❡❛r✳ ❆ s❡❣✉✐r ❞❡s❡♥✈♦❧✈❡r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s s♦❜r❡ ♦♣❡r❛❞♦r❡s ♠♦♥ót♦♥♦s ❡ ♠♦♥ót♦♥♦s ♠❛①✐♠❛✐s ❝♦♠♦ ❡♠[✺]✳

◆♦t❛çõ❡s

• I ✐♥t❡r✈❛❧♦ ❞❛ r❡t❛✳

• 1X é ❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r

1X(x) =

(

1, x_∈X

0, x /_∈X✳

• Lp_{(Ω) =}







f : Ω_→R♠❡♥s✉rá✈❡✐s_;

Z

Ω

|f_|p_{dx <}

∞







, 1_≤p <_∞,

• Wk,p_{(Ω) =}

u∈L1

loc(Ω) ; ♣❛r❛ t♦❞♦ ♠✉❧t✐í♥❞✐❝❡|α| ≤k, Dαu❡①✐st❡ ❡Dαu∈Lp(Ω) ,1≤p≤ ∞.

• Hk_{(Ω) =W}k,2_(Ω)

• H1

0(Ω) é ♦ ❢❡❝❤♦ ❞❡C0∞(Ω) ♥❛ ♥♦r♠❛ ❞♦ ❡s♣❛ç♦H1(Ω).

• λ1= inf

u∈H1 0(Ω)\{0}

Z

Ω|∇

u_|2

Z

Ω|

u_|2

• L(X, Y) =_{T :X _→Y ; T é ❧✐♥❡❛r ❡ ❧✐♠✐t❛❞❛_}✳

• L(X) ={T :X →X ; T é ❧✐♥❡❛r ❡ ❧✐♠✐t❛❞❛}✳ ❉♦t❛❞♦ ❝♦♠ ❛ s❡❣✉✐♥t❡ ♥♦r♠❛

||T_||L(X)= sup

x∈X:||x||X=1||

T(x)_||X= sup x∈X\{0}

||T(x)||X

||x_||X

• |Ω|❞❡♥♦t❛ ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❞♦ ❝♦♥❥✉♥t♦Ω✳

• ∇é ♦ ♦♣❡r❛❞♦r ❣r❛❞✐❡♥t❡✳

• ∆é ♦ ♦♣❡r❛❞♦r ❧❛♣❧❛❝✐❛♥♦✳

• k · k❞❡♥♦t❛ ❛ ♥♦r♠❛ ❞♦ ❡s♣❛ç♦H1 0✳

• k · kX ❞❡♥♦t❛ ❛ ♥♦r♠❛ ❞♦ ❡s♣❛ç♦X✳

• X⊥ ₌

◆♦t❛çõ❡s ✹

• C(I, X) =_{u:I_→X; u é ❝♦♥tí♥✉❛_} ❝♦♠ ❛ ♥♦r♠❛ ❞♦ s✉♣r❡♠♦✱ ❞❛❞❛ ♣♦r

||u_||∞= sup

I ||

u(t)_||X

• C(X, X) =_{u:X_→X ; ué ❝♦♥tí♥✉❛_}

• C1_{(X, X) ={u:X →X ; u}′_{∈C(X, X)}}

• B[I, X] =_{u:I_→X ; u é ❧✐♠✐t❛❞❛_}❝♦♠ ❛ ♥♦r♠❛ ❞♦ s✉♣r❡♠♦✱ ❞❛❞❛ ♣♦r

||u||∞= sup

I ||

u(t)||X

• 0< α <1✱Cα_{(I, X) =}

(

u:I_→X; [u]Cα_(I,X)= sup

s,t∈I\{s6=t}

||u(t)₋u(s)_||X

|t−s|α <∞

)

✱ é ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡sα₋❍ö❧❞❡r ❝♦♥tí♥✉❛s ❝♦♠ ❛ ♥♦r♠❛ ❞❛❞❛ ♣♦r

||u_||Cα_(I,X)=||u||_∞+ [u]_Cα_(I,X).

• A:D(A)⊂X →X♦♣❡❛r❛❞♦r s❡t♦r✐❛❧✱0< α <1❡DA(α,∞) =

x∈X; [x]α= sup

0<t≤1||

t1−αAT(t)x||X<∞

❝♦♠ ❛ ♥♦r♠❛

||x_||DA(α,∞)=||x||X+ [x]α ❡

DA(1 +α,∞) ={x∈D(A); Ax∈DA(α,∞)}

• Cb(Rn) ={u∈C(Rn) ; u é ❧✐♠✐t❛❞❛}

• C0(Ω) ={u∈C(Ω) ; u= 0 ❡♠ ∂Ω}

• 0< α <1✱Cθ2,θ(I,Rn) =

n

u:I→Rn_{; u(·, x)∈C}θ

2(I), ∀x∈Rn ❡ u(t,·)∈Cθ(Rn) ∀t∈I

o

• u+_{= max}

{u(x),0_}

• u− _{= min}

{u(x),0_}

• A⊂⊂B s✐❣♥✐✜❝❛ q✉❡A❡stá ❝♦♠♣❛❝t❛♠❡♥t❡ ❝♦♥t✐❞♦ ❡♠B✱ ✐st♦ éA é ❝♦♠♣❛❝t♦ ❡A⊂B✳

• ut=

∂u ∂t =

du dt

• X∗ _{♦ ❞✉❛❧ t♦♣♦❧ó❣✐❝♦ ❞♦ ❡s♣❛ç♦X✳}

• Lp_{(I, X)❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❞❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s u:I→X t❛❧ q✉❡}

Z

I||

u(s)_||p_Xds <_∞s❡

1≤p <∞✱ ♦✉sup

I ||

u(t)||X<∞s❡p=∞✳ Lp(I, X)é ❡q✉✐♣❛❞♦ ❝♦♠ ❛ ♥♦r♠❛

||u||Lp =



  

  

Z

I||

u(s)_||p_Xds

1p

, s❡p <_∞;

sup

I ||

❈❛♣ít✉❧♦

1

❙❡♠✐❣r✉♣♦s ❞❡ ❈❧❛ss❡

C

₀

◆❡st❡ ❝❛♣ít✉❧♦ ❛❜♦r❞❛r❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❛ t❡♦r✐❛ ❞❡ s❡♠✐❣r✉♣♦s✱ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❛ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞♦ ❝❛❧♦r ♥♦ ❝❛s♦ ❤♦♠♦❣ê♥❡♦✳

■♥✐❝✐❛❧♠❡♥t❡ ❞❡✜♥✐r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ s❡♠✐❣r✉♣♦s ❞❡ ❝❧❛ss❡C0 ❡ ❡①♣❧♦r❛r❡♠♦s ❛s s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❊♠

s❡❣✉✐❞❛✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦s r❡s✉❧t❛❞♦s ❞❡ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ s❡♠✐❣r✉♣♦s ❞❡ ❝♦♥tr❛çã♦✱ ❚❡♦r❡♠❛ ❞❡ ❍✐❧❧❡✲ ❨♦s✐❞❛ ❞❡ ▲✉♠❡r✲P❤✐❧❧✐♣s✳ ❋✐♥❛❧♠❡♥t❡ ❢❛r❡♠♦s ✉♠❛ ❛♣❧✐❝❛çã♦ ❛♦ ♣r♦❜❧❡♠❛ ❞❛ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r ♥♦ ❝❛s♦ ❤♦♠♦❣ê♥❡♦✳

❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❝♦♠ ♥♦r♠❛_{|| · ||}❡_L(X)❛ á❧❣❡❜r❛ ❞♦s ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s

❧✐♠✐t❛❞♦s ❞❡ X✳ ❉✐③✲s❡ q✉❡ ✉♠❛ ❛♣❧✐❝❛çã♦ T : R+ _{→ L(X)} é ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s

❧✐♠✐t❛❞♦s ❞❡ X s❡✿

✐✮ T(0) =I✱ ♦♥❞❡ I é ♦ ♦♣❡r❛❞♦r ✐❞❡♥t✐❞❛❞❡ ❞❡ _L(X)✳

✐✐✮ T(t+s) =T(t)T(s), _∀ t, s_∈R+✳

❖❜s❡r✈❛çã♦ ✶✳✶✳ ❖ ❡s♣❛ç♦ L(X) é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❝♦♠ ❛ ♥♦r♠❛ _{|| · ||}L(X)✳ P♦r s✐♠♣❧✐❝✐❞❛❞❡✱

❞❡♥♦t❛r❡♠♦s ❛ s✉❛ ♥♦r♠❛ t❛♠❜é♠ ♣♦r|| · ||✳

❉❡✜♥✐çã♦ ✶✳✷✳ ❉✐③❡♠♦s q✉❡ ♦ s❡♠✐❣r✉♣♦ T é ❞❡ ❝❧❛ss❡ C0 s❡✿

lim

t→0+k(T(t)−I)xk= 0, ∀ x∈X. ✭✶✳✶✮

❉❡✜♥✐çã♦ ✶✳✸✳ ❉✐③✲s❡ q✉❡ ♦ s❡♠✐❣r✉♣♦T é ❞❡ ❝♦♥tr❛çã♦ q✉❛♥❞♦_kT(t)_k61, _∀t_∈R+_.

❉❡✜♥✐çã♦ ✶✳✹✳ ❯♠ s❡♠✐❣r✉♣♦ ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❧✐♠✐t❛❞♦ T(t)é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥tí♥✉♦ s❡

lim

t→0+kT(t)−Ik= 0. ✭✶✳✷✮

❖❜s❡r✈❛çã♦ ✶✳✷✳ ❖❜s❡r✈❡ q✉❡ ♥❛ ❉❡✜♥✐çã♦ ✶✳✹✱ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ✐♥❞❡♣❡♥❞❡ ❞♦ ♣♦♥t♦x_∈X✳

❉❡✜♥✐çã♦ ✶✳✺✳ ❖ ♦♣❡r❛❞♦r A:D(A)_→X ❞❡✜♥✐❞♦ ♣♦r

D(A) =

x_∈X _| lim

h→0+

T(h)₋I

h x existe

Ax= lim

h→0+

T(h)₋I

✻

é ❞✐t♦ ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞♦ s❡♠✐❣r✉♣♦ T✳

Pr♦♣♦s✐çã♦ ✶✳✶✳ ❖ ❝♦♥❥✉♥t♦D(A)é ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ X ❡Aé ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r✳

❉❡♠♦♥str❛çã♦✳ ❙❡❣✉❡ ❞✐r❡t♦ ❞❛ ❞❡✜♥✐çã♦ ✶✳✸✳

❚❡♦r❡♠❛ ✶✳✶✳ ❙❡ A_{∈ L}(X)❡♥tã♦Aé ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥tí♥✉♦✳

❉❡♠♦♥str❛çã♦✳ ❚♦♠❡♠♦s

T(t) =etA_:=

∞

X

n=0

(tA)n

n! . ✭✶✳✹✮

P❛r❛ ❝❛❞❛t>0✱ t❡♠♦s q✉❡

∞

X

n=0

(tA)n

n! ❝♦♥✈❡r❣❡ ♥❛ t♦♣♦❧♦❣✐❛ ✉♥✐❢♦r♠❡ ❞❡L(X)✱ ♣♦✐sAé ♦♣❡r❛❞♦r ❧✐♠✐t❛❞♦

❡ ♣❡❧♦ t❡st❡ ❞❡ ❝♦♠♣❛r❛çã♦ ❞❡ ❲❡✐❡rstr❛ss✳ ❆ss✐♠✱ T é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ R+ ❡♠_L(X)✱ s❛t✐s❢❛③❡♥❞♦

T(0) =I ❡T(t+s) =T(t)T(s).

❊st✐♠❛♥❞♦ ❛ sér✐❡ ❞❡ ♣♦tê♥❝✐❛✱ t❡♠♦s✿

kT(t)−Ik =

∞ X n=0

(tA)n

n! −I

= ∞ X n=1

(tA)n

n! = tA ∞ X n=0

(tA)n

n!

6 tkAket||A||

❡

T(t)₋I t −A

= A ∞ X n=0

(tA)n

n! −A

6 _kA_kkT(t)₋I_k.

❚♦♠❛♥❞♦ ♦ ❧✐♠✐t❡t _→0+_{✱ ♦❜t❡♠♦s(1.3) ❡(1.2)✳ ■ss♦ ✐♠♣❧✐❝❛ q✉❡ T(t)é ✉♠ s❡♠✐❣r✉♣♦ ✉♥✐❢♦r♠❡♠❡♥t❡}

❝♦♥tí♥✉♦ ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❧✐♠✐t❛❞♦s ❡ q✉❡ Aé ♦ s❡✉ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧✳

Pr♦♣♦s✐çã♦ ✶✳✷✳ ❙❡ T(t)é ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝❧❛ss❡ C0✱ ❡♥tã♦ ❡①✐st❡♠ M >1 ❡w >0 t❛✐s q✉❡

kT(t)_k6M ewt_{, t >0.}

❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ♠♦str❛r ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡ ❡①✐st❡ α >0 ❞❡ ♠♦❞♦ q✉❡ _k T(t)_k é ❧✐♠✐t❛❞♦ ♣❛r❛

t♦❞♦ t∈[0, α]✳ ❙✉♣♦♥❞♦ q✉❡ ♦ r❡s✉❧t❛❞♦ ♥ã♦ ❛❝♦♥t❡ç❛✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛(tn), tn→0+ t❛❧ q✉❡

k T(tn)k>n✱ ∀n∈N✳ P❡❧♦ ❚❡♦r❡♠❛ ❞❛ ▲✐♠✐t❛çã♦ ❯♥✐❢♦r♠❡✱ kT(tn)xk é ✐❧✐♠✐t❛❞❛✱ ♣❛r❛ ♣❡❧♦ ♠❡♥♦s

✉♠x_∈X✱ ♦ q✉❡ ❝♦♥✜❣✉r❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ♣❡❧♦ ❢❛t♦ ❞❡ lim

tn→0+kT(t)x−xk= 0, ∀x∈X✳

❆ss✐♠✱ k T(t) k6 M ♣❛r❛ 0 6 t 6 α✳ ❈♦♠♦ k T(0) k=k I k= 1✱ t❡♠✲s❡ q✉❡ M > 1✳ ❙❡❥❛

✼

s❡♠✐❣r✉♣♦ t❡♠♦s q✉❡

kT(t)_k=_kT(nα+δ)_k=_kT(δ)T(α)n

k6Mn+1

6M_·Mαt =M ewt.

❈♦r♦❧ár✐♦ ✶✳✶✳ ❙❡ T(t)é ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝❧❛ss❡C0✱ ❡♥tã♦t7→T(t)x✱ ♣❛r❛ t♦❞♦x∈X✱ é ✉♠❛ ❢✉♥çã♦

❝♦♥tí♥✉❛ ❞❡R+ ❡♠ _X✳

❉❡♠♦♥str❛çã♦✳ ❉❡✈❡♠♦s ♠♦str❛r q✉❡ lim

s→tT(s)x =T(t)x,∀x∈ X✳ ❈♦♠ ❡❢❡✐t♦✱ s❡❥❛♠ t > 0 ❡ h > 0✳

❆ss✐♠✱ t♦♠❛♥❞♦h→0✱ t❡♠♦s q✉❡

kT(t+h)x₋T(t)x_k = _kT(t)(T(h)₋I)x_k

6 _kT(t)kk(T(h)−I)xk

6 M ewt kT(h)x−xk→0.

P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛t>h>0 ❡h→0✱ t❡♠♦s q✉❡

kT(t−h)x−T(t)xk 6 kT(t−h)xkkx−T(h)xk

6 M ewtkx−T(h)xk→0.

▲♦❣♦✱t7→T(t)xé ❝♦♥tí♥✉❛✳

Pr♦♣♦s✐çã♦ ✶✳✸✳ ❙❡❥❛T(t) ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝❧❛ss❡C0 ❡ As❡✉ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧✱ ❡♥tã♦✿

❛✮ lim

h→0

1

h

Z t+h

t

T(s)xds=T(t)x✱ ♣❛r❛ x_∈X,

❜✮ Z t

0

T(s)xds_∈D(A)❡A

Z t

0

T(s)xds

=T(t)x₋x✱ ♣❛r❛ x_∈X,

❝✮ T(t)x_∈D(A)❡

d

dtT(t)x=AT(t)x=T(t)Ax, para x∈D(A). ✭✶✳✺✮

❞✮ T(t)x₋T(s)x=

Z t

s

AT(r)xdr, para x_∈D(A),

❉❡♠♦♥str❛çã♦✳ ❛✮ ❙❡❣✉❡ ❞❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❛♣❧✐❝❛çã♦ t _7→ T(t)x✱ ❞♦♥❞❡ q✉❡ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ s❡❣✉❡ ❞♦

✽

❜✮ ❉❛❞♦sx_∈X ❡h >0✱ ❡♥tã♦

T(h)₋I h

Z t

0

T(s)xds

= 1

h

Z t 0

(T(s+h)x₋T(s)x)ds

= 1

h

Z t 0

T(s+h)xds₋1 h

Z t 0

T(s)xds

= 1

h

Z t+h

h

T(r)xdr−_h1

Z t 0

T(s)xds

= 1

h

Z t+h 0

T(r)xdr₋ 1 h

Z t 0

T(r)xdr₋1 h

Z h 0

T(r)xdr

= 1

h

Z t+h

t

T(r)xdr−_h1

Z h 0

T(r)xdr ✭✶✳✻✮

t♦♠❛♥❞♦ h_→0+ _{❡♠ (1.6)✱ ❡ ✉s❛♥❞♦ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❛♣❧✐❝❛çã♦ t}

7→T(t)x✱ t❡♠♦s ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ A

❡♠ (1.3)q✉❡

A

Z t

0

T(s)xds

= lim

h→0+

T(h)₋I h

Z t 0

T(s)xds=T(t)x₋x.

❝✮ ❉❛❞♦sx_∈D(A)❡h >0✱ ❡♥tã♦ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✭✐✐✮ ❞❛ ❉❡✜♥✐çã♦ ✶✳✶✱ ❞❛ ❧✐♥❡❛r✐❞❛❞❡ ❞❡T(t)❡ ❞❛

❞❡✜♥✐çã♦ ❞❡ A✱ ♦❜t❡♠♦s✿

T(h)₋I

h T(t)x=T(t)

_T(h)

−I h

x_→T(t)Ax,q✉❛♥❞♦h_→0+_.

❆ss✐♠✱

T(t)x_∈D(A), AT(t)x=T(t)Ax e d

+

dtT(t)x=AT(t)x=T(t)Ax,

❛❧é♠ ❞✐ss♦

d−

dtT(t)x= limh→0

_T(t)x

−T(t₋h)

h

= lim

h→0+T(t−h)

_T(h)x

−x h

=T(t)Ax.

◆♦t❡ q✉❡ ♥❡ss❛ ❞❡♠♦♥str❛çã♦✱ ✉s❛♠♦s ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡t 7→T(t−h)x❡ ❛ ✐❞❡♥t✐❞❛❞❡ (1.3)✳ ❆❧é♠

❞✐ss♦✱ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡♠(1.5)é ✈á❧✐❞❛ s❡x_∈D(A)✳ ❊♥tr❡t❛♥t♦✱ ♥ã♦ é ♣♦ssí✈❡❧ s❡ x /_∈D(A)✳

❞✮ ❙❡x∈D(A)✱ s❛❜❡♠♦s ♣❡❧♦ ✐t❡♠ ❛♥t❡r✐♦r q✉❡

d

dtT(t)x=AT(t)x=T(t)Ax. ✭✶✳✼✮

❆ss✐♠✱ ✐♥t❡❣r❛♥❞♦(1.7) ❞❡s❛t♦❜t❡♠♦s✿

T(t)x₋T(s)x=

Z t

s

d

dtT(r)xdr=

Z t

s

AT(r)xdr. ✭✶✳✽✮

❈♦r♦❧ár✐♦ ✶✳✷✳ ❙❡Aé ✉♠ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦T(t)❞❡ ❝❧❛ss❡C0✱ ❡♥tã♦D(A)é ❞❡♥s♦

❡♠ X ❡A é ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❢❡❝❤❛❞♦✳

❉❡♠♦♥str❛çã♦✳ P❛r❛ t♦❞♦ x_∈X✱ t♦♠❡♠♦s xt =

1

t

Z t 0

T(s)xds✳ P❡❧♦ ✐t❡♠(b) ❞❛ Pr♦♣♦s✐çã♦ ✶✳✸✱ xt ∈

✾

❢♦r♠❛✱D(A) =X✳

❆ ❧✐♥❡❛r✐❞❛❞❡ ❞❡ A s❡❣✉❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧✳ P❛r❛ ♣r♦✈❛r q✉❡ A é ❢❡❝❤❛❞♦✱ s❡❥❛ xn ∈D(A)✱ t❛❧ q✉❡ xn →x❡Axn →y✱ q✉❛♥❞♦n→ +∞✳ P❡❧♦ ✐t❡♠ (d)❞❛ ♣r♦♣♦s✐çã♦ ✶✳✸✱ ✐♥❢❡r✐♠♦s

q✉❡

T(t)xn−xn=

Z t 0

T(s)Axnds. ✭✶✳✾✮

❖ ✐♥t❡❣r❛♥❞♦ ❡♠(1.9) ❝♦♥✈❡r❣❡ ♣❛r❛T(s)y ✉♥✐❢♦r♠❡♠❡♥t❡✱ ♥♦ ✐♥t❡r✈❛❧♦ ❧✐♠✐t❛❞♦[0, t]✱ ♣♦✐s

kT(s)Axn−T(s)yk6kT(s)k kAxn−yk.

❆ss✐♠✱ ♣♦r ✭✶✳✽✮ t❡♠♦s✿

T(t)x−x=

Z t 0

T(s)yds. ✭✶✳✶✵✮

❆❣♦r❛✱ ❞✐✈✐❞✐♥❞♦ ❛ ❡①♣r❡ssã♦ ❞❛❞❛ ❡♠(1.10)♣♦rt >0✱ ♦❜t❡♠♦s q✉❡

T(t)x₋x

t =

1

t

Z t 0

T(s)yds.

❉❡ss❛ ❢♦r♠❛✱ ❢❛③❡♥❞♦ t_→0+ _{❡ ✉s❛♥❞♦ ♦ ✐t❡♠(a)❞❛ Pr♦♣♦s✐çã♦ ✶✳✸✱ t❡♠♦s q✉❡x}

∈D(A)❡Ax=y✳

Pr♦♣♦s✐çã♦ ✶✳✹✳ ❙❡❥❛♠ T(t) ❡ S(t) s❡♠✐❣r✉♣♦s ❞❡ ❝❧❛ss❡ C0 ❝♦♠ ♦ ♠❡s♠♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ A✱

❡♥tã♦ T(t) =S(t)✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡ A s❡❥❛ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞♦s s❡♠✐❣r✉♣♦s T1 ❡T2 ❞❡ ❝❧❛ss❡ C0✳ ❙❡

06s6t <+∞✱ ♣❛r❛ ❝❛❞❛x∈D(A)✱ ❡♥tã♦ ❛ ❢✉♥çã♦φ(s) =T1(t−s)T2(s)xé ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ✐♥t❡r✈❛❧♦

06s <+_∞✱ ❡

φ′(s) = ₋AT1(t−s)T2(s)x+T1(t−s)AT2(s)x

= ₋AT1(t−s)T2(s)x+AT1(t−s)T2(s)x

= 0.

▲♦❣♦✱φ(s)é ❝♦♥st❛♥t❡ ♣❛r❛06s6t✳ ❚❡♠♦s ❡♥tã♦

T1(t)x=φ(0) =φ(t) =T2(t)x,∀x∈D(A).

P♦r ❞❡♥s✐❞❛❞❡✱ ❝♦♥❝❧✉í♠♦s q✉❡

T1(t)x=T2(t)x,∀x∈X.

❉❡✜♥✐çã♦ ✶✳✻✳ ❙❡❥❛ A ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❡♠ X✱ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❧✐♠✐t❛❞♦✳ ❖ ❝♦♥❥✉♥t♦ ❞❛❞♦ ♣♦r ρ(A) =_{λ_∈C_|λI−Aé ✐♥✈❡rtí✈❡❧_} é ❝❤❛♠❛❞♦ ♦ r❡s♦❧✈❡♥t❡ ❞❡ _A✳ ❉✐③❡r q✉❡ _λ∈ρ(A) é ❡q✉✐✈❛❧❡♥t❡ ❛

(λI₋A)−1 _{s❡r ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❡♠ X ❡ ❧✐♠✐t❛❞♦✳ ❉❡♥♦t❛♠♦s ♣♦rR(λ:A) = (λI}

−A)−1_{, λ}

∈ρ(A)✳

❚❡♦r❡♠❛ ✶✳✷✳ ✭❍✐❧❧❡✲❨♦s✐❞❛✮ ❯♠ ♦♣❡r❛❞♦r ❧✐♥❡❛rAé ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝❧❛ss❡ C0 ❞❡ ❝♦♥tr❛çã♦ T(t), t>0 s❡✱ ❡ s♦♠❡♥t❡✱ s❡✿

✶✵

✐✐✮ ❖ ❝♦♥❥✉♥t♦ r❡s♦❧✈❡♥t❡ρ(A)❞❡A ❝♦♥té♠R+ _{❡ ♣❛r❛ ❝❛❞❛ λ >0✱ t❡♠✲s❡ kR(λ:A)k6} 1

λ✳

❉❡♠♦♥str❛çã♦✳ ✭❈♦♥❞✐çã♦ ♥❡❝❡ssár✐❛✮ ❙❡Aé ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦T(t)❞❡ ❝❧❛ss❡C0❞❡

❝♦♥tr❛çã♦✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✷Aé ❢❡❝❤❛❞♦ ❡D(A) =X✳ ❆❣♦r❛✱ ♣❛r❛ ❝❛❞❛ λ >0 ❡x_∈X✱ s❡❥❛

R(λ)x=

Z ∞

0

e−λt_{T(t)xdt. ✭✶✳✶✶✮}

❯♠❛ ✈❡③ q✉❡t_7→T(t)xé ❝♦♥tí♥✉❛ ❡ ✉♥✐❢♦r♠❡♠❡♥t❡ ❧✐♠✐t❛❞❛✱ ❡♥tã♦ ❛ ✐♥t❡❣r❛❧ ❡①✐st❡ ❝♦♠♦ ✉♠❛ ✐♥t❡❣r❛❧

❞❡ ❘✐❡♠❛♥♥✱ ❞❡✜♥✐♥❞♦ ❛ss✐♠ ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❧✐♠✐t❛❞♦✳ ▲♦❣♦✱

kR(λ)xk6

Z ∞

0

e−λtkT(t)xk6

Z ∞

0

e−λtkxkdt= 1

λ kxk.

❆❧é♠ ❞✐ss♦✱ ♣❛r❛h >0✱ t❡♠♦s q✉❡

T(h)₋I

h R(λ)x =

1

h

Z +∞

0

e−λt_(T(t+h)x

−T(t)x)dt

= 1

h

Z +∞

0

e−λtT(t+h)xdt−1_h

Z +∞

0

e−λtT(t)xdt

= 1

h

Z +∞

h

e−λ(t−h)_T(t)xdt

−_h1

Z +∞

0

e−λt_T(t)xdt

= e

λh

h

Z +∞

h

e−λt_T(t)xdt

−_h1

Z +∞

0

e−λt_T(t)xdt

= e

λh

h

Z +∞

0

e−λtT(t)xdt−e

λh

h

Z h 0

e−λtT(t)xdt−_h1

Z +∞

0

T(t)xdt

= e

λh

−1

h

Z +∞

0

e−λt_T(t)xdt

−e

λh

h

Z h 0

e−λt_T(t)xdt

= e

λh₋₁

h R(λ)x− eλh

h

Z h 0

e−λtT(t)xdt. ✭✶✳✶✷✮

◗✉❛♥❞♦ h_→ 0+_{✱ ❛ ❡q✉❛çã♦ ✭✶✳✶✷✮ ❝♦♥✈❡r❣❡ ♣❛r❛ λR(λ)x}

−x✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ A✱ s❡❣✉❡ q✉❡ ♣❛r❛

t♦❞♦ x∈X ❡λ >0, R(λ)x∈D(A)❡AR(λ) =λR(λ)−I✱ ♦✉ s❡❥❛✱

(λI₋A)R(λ) =I. ✭✶✳✶✸✮

P❛r❛x_∈D(A)✱ ✉s❛♥❞♦ ♦ ✐t❡♠(c)❞❛ Pr♦♣♦s✐çã♦ ✶✳✸ ❡ ♦ ❢❡❝❤❛♠❡♥t♦ ❞❡ A✱ t❡♠♦s

R(λ)Ax =

Z +∞

0

e−λt_T(t)Axdt

=

Z +∞

0

e−λtAT(t)xdt

= A

Z +∞

0

e−λt_T(t)xdt

= AR(λ)x. ✭✶✳✶✹✮

❉❛s ❡q✉❛çõ❡s ✭✶✳✶✸✮ ❡ ✭✶✳✶✹✮✱ s❡❣✉❡ q✉❡

✶✶

❊♥tã♦✱ ♣❛r❛ t♦❞♦λ >0s❛t✐s❢❛③❡♥❞♦(i)❡(ii)✱ t❡♠♦s q✉❡ ❡①✐st❡ ♦ ✐♥✈❡rs♦ ❞❡λI₋A❞❛❞♦ ♣♦rR(λ)✳

P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ ❞❡♠♦♥str❛r q✉❡ ❛s ❝♦♥❞✐çõ❡s (i)❡ (ii) sã♦ s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡A s❡❥❛ ❣❡r❛❞♦r

✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝♦♥tr❛çã♦ ❞❡ ❝❧❛ss❡ C0✱ ♣r♦✈❛r❡♠♦s ♦s s❡❣✉✐♥t❡s ❧❡♠❛s✳

▲❡♠❛ ✶✳✶✳ ❙❡❥❛♠A s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s(i)❡(ii)❞♦ ❚❡♦r❡♠❛ ✶✳✷ ❡R(λ:A) = (λI−A)−1✱ ❡♥tã♦

lim

λ→+∞λR(λ:A)x=x,∀x∈X. ✭✶✳✶✺✮

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛♠♦s ♣r✐♠❡✐r♦ q✉❡ x∈D(A)✳ ❊♥tã♦

kλR(λ:A)x₋x_k=_kAR(λ:A)x_k=_kR(λ:A)Ax_k6 1

λ kAxk→0

s❡λ_→+_∞✳ ▼❛sD(A)é ❞❡♥s♦ ❡♠X ❡_kλR(λ:A)_k61✳ P♦rt❛♥t♦✱

λR(λ:A)x_→x,

q✉❛♥❞♦λ_→+_∞✱ ♣❛r❛ t♦❞♦x_∈X✳

❉❡✜♥✐♠♦s ❛❣♦r❛✱ ♣❛r❛ t♦❞♦λ >0✱ ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❨♦s✐❞❛ ❞❡A✱ q✉❡ é ❞❛❞❛ ♣♦r

Aλ:=λAR(λ:A) =λ2R(λ:A)−λI ✭✶✳✶✻✮

♦♥❞❡Aλ é ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❞❡A♥♦ s❡❣✉✐♥t❡ s❡♥t✐❞♦✿

▲❡♠❛ ✶✳✷✳ ❙❡❥❛As❛t✐s❢❛③❡♥❞♦(i)❡(ii)❞♦ ❚❡♦r❡♠❛ ✶✳✷✳ ❙❡Aλ é ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❨♦s✐❞❛ ❞❡A✱ ❡♥tã♦

lim

λ→+∞Aλx=Ax✱ ♣❛r❛x∈D(A)✳

❉❡♠♦♥str❛çã♦✳ P❛r❛ x_∈D(A)✱ t❡♠♦s ♣❡❧♦ ▲❡♠❛ ✶✳✷ ❡ ❛ ❞❡✜♥✐çã♦ ❞❡Aλ✱ q✉❡

lim

λ→+∞Aλx= limλ→+∞λR(λ:A)Ax=Ax.

▲❡♠❛ ✶✳✸✳ ❙❡❥❛As❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s(i✮ ❡(ii)❞♦ ❚❡♦r❡♠❛ ✶✳✷✳ ❙❡Aλé ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❨♦s✐❞❛

❞❡ A✱ ❡♥tã♦ Aλ é ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥tí♥✉♦ ❞❡ ❝♦♥tr❛çã♦ etAλ✳

❆❧é♠ ❞✐ss♦✱ ♣❛r❛ t♦❞♦ x_∈X, λ, u >0✱ t❡♠♦s

etAλx−etAux

6tkAλx−Auxk. ✭✶✳✶✼✮

❉❡♠♦♥str❛çã♦✳ P♦r ✭✶✳✶✻✮✱ t❡♠♦s q✉❡

Aλ=λAR(λ:A) =λ2R(λ:A)−λI,

❛❧é♠ ❞✐ss♦ Aλ é ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❡ ❧✐♠✐t❛❞♦✳

❆ss✐♠✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✶✱Aλé ✉♠ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦etAλ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥tí♥✉♦

✶✷

❉❡st❡ ♠♦❞♦✱

etAλ

=

e

t(λ2R(λ:A)−λI)

= e−tλ e

tλ2R(λ:A)

6 e−tλ_etλ2kR(λ:A)k 6 e−λt_eλ2t1λ

= 1

❡ ♣♦rt❛♥t♦ etAλ é ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝♦♥tr❛çã♦✳ ➱ ❝❧❛r♦ ❞❛s ❞❡✜♥✐çõ❡s ❞❡etAλ_{, e}tAu_{, A}

λ ❡Auq✉❡ ♦s ♠❡s♠♦s ❝♦♠✉t❛♠ ❡♥tr❡ s✐✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱

etAλx−etAux = Z 1 0 d ds

etsAλ_et(1−s)Au_xds

= Z 1 0

tAλetsAλet(1−s)Aux+etsAλ(−t)Auet(1−s)Aux

ds = Z 1 0

tetsAλ_et(1−s)Au_(A

λx−Aux)

ds 6 Z 1 0 t e

tsAλ_et(1−s)Au_(A

λx−Aux)

ds 6 Z 1 0

tkAλx−Auxkds

= t_kAλx−Auxk

❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✶✼✮ ❡stá ♣r♦✈❛❞❛✳

❆❣♦r❛ ❡①✐❜✐r❡♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❝♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ ❞♦ ❚❡♦r❡♠❛ ✶✳✷✳

❉❡♠♦♥str❛çã♦✳ ✭❈♦♥❞✐çã♦ ❙✉✜❝✐❡♥t❡✮✳ ❙❡❥❛ x∈D(A)✳ ❊♥tã♦

etAλx−etAux

6 tkAλx−Auxk

6 t_kAλx−Axk+tkAx−Auxk. ✭✶✳✶✽✮

❉❛ ❡q✉❛çã♦ (1.18) ❡ ❞♦ ▲❡♠❛ ✶✳✷ s❡❣✉❡ q✉❡ ♣❛r❛ x_∈ D(A), etAλ_x ❝♦♥✈❡r❣❡ q✉❛♥❞♦ _{λ → +∞} ❡ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ é ✉♥✐❢♦r♠❡ ❡♠ ✐♥t❡r✈❛❧♦s ❧✐♠✐t❛❞♦s✳ ❱✐st♦ q✉❡ D(A) =X ❡etAλ

61✱ s❡❣✉❡ q✉❡

lim

λ→+∞e

tAλ_{x=T(t)x,∀x∈X.}

❖ ❧✐♠✐t❡ ❛❝✐♠❛ é s❡♠♣r❡ ✉♥✐❢♦r♠❡ ❡♠ ✐♥t❡r✈❛❧♦s ❧✐♠✐t❛❞♦s✳ ❆ss✐♠✱ T(t)s❛t✐s❢❛③ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡

s❡♠✐❣r✉♣♦✱ ✉♠❛ ✈❡③ q✉❡ T(0) =I ❡_k T(t)_k61✳ ❆❧é♠ ❞✐ss♦✱ t _7→T(t)xé ❝♦♥tí♥✉❛ ♣❛r❛ t>0✱ ♣♦✐s é

❧✐♠✐t❡ ✉♥✐❢♦r♠❡ ❞❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s t_7→etAλ_x✳ ❆ss✐♠✱_T(t)é s❡♠✐❣r✉♣♦ ❞❡ ❝♦♥tr❛çã♦ ❞❡ ❝❧❛ss❡_C

0 ❡♠

X✳

✶✸

❯s❛♥❞♦ q✉❡ T(t)x= lim

λ→+∞e

tAλ_x❡ ❛ Pr♦♣♦s✐çã♦ ✶✳✸✱ t❡♠♦s q✉❡

T(t)x₋x = lim

λ→+∞ e

tAλ_x

−x

= lim

λ→+∞

Z t 0

esAλ_A

λxds

=

Z t 0

T(s)Axds. ✭✶✳✶✾✮

❆ ❡q✉❛çã♦(1.19)s❡❣✉❡ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ✉♥✐❢♦r♠❡ ❞❡etAλ_A

λx♣❛r❛T(t)Ax❡♠ ✐♥t❡r✈❛❧♦s ❧✐♠✐t❛❞♦s✳

❙❡❥❛♠B ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ T(t)❡x_∈D(A)✱ ❡♥tã♦

T(x)₋x

t =

1

t

Z t 0

T(s)Axds.

❋❛③❡♥❞♦t→0+_{✱ t❡♠♦s Bx=Ax✱ ♦✉ s❡❥❛✱D(A)⊂D(B)✳ ❈♦♠♦B é ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡T(t)✱}

s❡❣✉❡ ❞❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ q✉❡ 1 _∈ ρ(B)✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ss✉♠✐♠♦s ❛ ❤✐♣ót❡s❡ ✭✐✐✮ ❞♦ ❚❡♦r❡♠❛ ✶✳✷

❛ss❡❣✉r❛♥❞♦ q✉❡1_∈ρ(A)✳ ❯♠❛ ✈❡③ q✉❡D(A)_⊂D(B), t❡♠♦s

(I−B)D(A) = (I−A)D(A) =X,

♦ q✉❡ ✐♠♣❧✐❝❛D(B) = (I−B)−1_{X =D(A)✱ ❡ ♣♦rt❛♥t♦ A=B✳}

❯♠ ♦✉tr❛ ♠❛♥❡✐r❛ ❞❡ ❝❛r❛❝t❡r✐③❛r ♦s ❣❡r❛❞♦r❡s ✐♥✜♥✐t❡s✐♠❛✐s ❞♦s s❡♠✐❣r✉♣♦s ❞❡ ❝♦♥tr❛çõ❡s ❧✐♥❡❛r❡s ❞❡ ❝❧❛ss❡C0✱ ♣♦❞❡ s❡r ✈✐st♦ ♥♦ ❚❡♦r❡♠❛ ❞❡ ▲✉♠❡r ❡ P❤✐❧❧✐♣s✱ ❝♦♥t✉❞♦✱ ❛♥t❡s ❞❡ ❡♥✉♥❝✐á✲❧♦ ❢❛r❡♠♦s ❛❧❣✉♠❛s

❝♦♥s✐❞❡r❛çõ❡s✳

❙❡❥❛X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱X∗_{♦ ❞✉❛❧ ❞❡X ❡h·,·i❛ ❞✉❛❧✐❞❛❞❡ ❡♥tr❡X ❡X}∗_{✳ ❚♦♠❡♠♦s ♣❛r❛ ❝❛❞❛}

x_∈X✱

J(x) =

x∗∈X∗| hx, x∗i=kxk2₌

kx∗k2 _.

P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❍❛❤♥✲❇❛♥❛❝❤✱ J(x) ₆= _∅,_∀x _∈ X✳ ❯♠❛ ❛♣❧✐❝❛çã♦ ❞✉❛❧✐❞❛❞❡ é ✉♠❛ ❛♣❧✐❝❛çã♦ j :X −→X∗ _{t❛❧ q✉❡ j(x)∈J(x),∀x∈X✳ ◆♦t❡ q✉❡kj(x)k=kxk✳}

❉❡✜♥✐çã♦ ✶✳✼✳ ✐✮ ❉✐③✲s❡ q✉❡ ♦ ♦♣❡r❛❞♦r ❧✐♥❡❛r A : D(A) _⊂ X _−→ X é ❞✐ss✐♣❛t✐✈♦ s❡✱ ♣❛r❛ ❛❧❣✉♠❛

❛♣❧✐❝❛çã♦ ❞✉❛❧✐❞❛❞❡ j✱

RehAx, j(x)i ≤0,∀x∈D(A).

✐✐✮ ❉✐③✲s❡ q✉❡ Aé ♠✲❞✐ss✐♣❛t✐✈♦ s❡ ❢♦r ❞✐ss✐♣❛t✐✈♦ ❡ Im(λI₋A) =X ♣❛r❛ ❛❧❣✉♠λ >0✳

❚❡♦r❡♠❛ ✶✳✸✳ ✭▲✉♠❡r✲P❤✐❧❧✐♣s✮✳ Aé ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝♦♥tr❛çã♦ s❡✱ ❡ s♦♠❡♥t❡

s❡✱A é ♠✲❞✐ss✐♣❛t✐✈♦ ❡ ❞❡♥s❛♠❡♥t❡ ❞❡✜♥✐❞♦✳

❉❡♠♦♥str❛çã♦✳ ❙❡ A é ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝♦♥tr❛çã♦✱ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ❍✐❧❧❡ ✲

❨♦s✐❞❛✱ t❡♠♦s q✉❡ Aé ❞❡♥s❛♠❡♥t❡ ❞❡✜♥✐❞♦✱ ❢❡❝❤❛❞♦ ❡ q✉❡(0,+_∞)_⊂ρ(A)✱ ❞♦♥❞❡ ❞❛❞♦sλ >0❡f _∈X✱

❡①✐st❡u_∈D(A)t❛❧ q✉❡λu₋Au=f✱ ✐st♦ é✱Im(λI₋A) =X✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ ❝❛❞❛ ❛♣❧✐❝❛çã♦ ❞✉❛❧✐❞❛❞❡ j✱ t❡♠✲s❡

✶✹

✈✐st♦ q✉❡✱ ♣♦r ❤✐♣ót❡s❡✱||T(t)x_{|| ≤ ||}x_||✱ ♣❛r❛ t♦❞♦x_∈D(A)✳

P♦rt❛♥t♦✱

Re_hT(t)x₋x, j(x)_i = Re_hT(t)x, j(x)_{i − h}x, j(x)_i = Re_hT(t)x, j(x)_{i − ||}x_||2

≤0, ✭✶✳✷✵✮

❞♦♥❞❡✱ ❞✐✈✐❞✐♥❞♦ (1.20)♣♦rt >0 ❡ ♣❛ss❛♥❞♦ ❛♦ ❧✐♠✐t❡ q✉❛♥❞♦t_→0+ _{t❡♠✲s❡}

RehAx, j(x)i ≤0

♣❛r❛ t♦❞♦x∈D(A)❡✱ ❛ss✐♠✱Aé ❞✐ss✐♣❛t✐✈♦ ❡ ♣♦rt❛♥t♦Aé ♠✲❞✐ss✐♣❛t✐✈♦✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡A é ♠✲❞✐ss✐♣❛t✐✈♦ ❡ ❞❡♥s❛♠❡♥t❡ ❞❡✜♥✐❞♦✱ ❡♥tã♦D(A) =X✳ ❆ss✐♠ ❞❡✈❡♠♦s ♠♦str❛r

q✉❡Aé ❢❡❝❤❛❞♦✱(0,+_∞)_⊂ρ(A)❡_||R(λ:A)_{|| ≤} 1

λ ❡ ✉s❛r ♦ ❚❡♦r❡♠❛ ❞❡ ❍✐❧❧❡✲❨♦s✐❞❛✳ P❛r❛ ✐st♦ ❢❛r❡♠♦s

❛❧❣✉♠❛s ❛✜r♠❛çõ❡s✳

❆✜r♠❛çã♦ ✶✳✶✳ ❙❡ Aé ❞✐ss✐♣❛t✐✈♦✱ ❡♥tã♦

||(λI₋A)x_{|| ≥}λ_||x_||

♣❛r❛ t♦❞♦ λ >0❡ ♣❛r❛ t♦❞♦ x∈D(A)✳

❈♦♠ ❡❢❡✐t♦✱ s❡λ >0✱Aé ❞✐ss✐♣❛t✐✈♦ ❡x_∈D(A)✱ ❡ ❡♥tã♦ ❞❡

h(λI₋A)x, j(x)_i=λ_hx, j(x)_{i − h}Ax, j(x)_i=λ_||x_||2_{− h}Ax, j(x)_i

s❡❣✉❡ q✉❡

λ_||x_||2

≤ Re_h(λI₋A)x, j(x)_i

≤ | h(λI₋A)x, j(x)_{i |}

≤ ||(λI₋A)x_{|| ||}j(x)_|| = _||(λI₋A)x_{|| ||}x_||.

▲♦❣♦✱||(λI₋A)x_{|| ≥}λ_||x_||✳ ❈♦♠♦λ >0❡x_∈D(A)sã♦ ❛r❜✐trár✐♦s✱ s❡❣✉❡ ❛ ❛✜r♠❛çã♦✳

Pr♦✈❛r❡♠♦s q✉❡Aé ❢❡❝❤❛❞♦ ❡ q✉❡(0,+∞)∩ρ(A)6=∅✳

❙❡❥❛ (xn)⊂D(A)t❛❧ q✉❡xn →x❡Axn →y✱ q✉❛♥❞♦n→+∞✳ ❈♦♠♦A é ♠✲❞✐ss✐♣❛t✐✈♦✱ ❡①✐st❡λ >0

t❛❧ q✉❡ ❛ ✐♠❛❣❡♠ ❞♦ ♦♣❡r❛❞♦rλI₋AéX✳

❆ss✐♠✱ ♣❡❧❛ ❛✜r♠❛çã♦ ✶✳✶✱ t❡♠♦s q✉❡

||x_||=_||(λI₋A)(λI₋A)−1_x

|| ≥λ_||(λI₋A)−1_x

||

♦✉ s❡❥❛✱

✶✺

▲♦❣♦✱

λ_∈ρ(A)

❡

xn= (λI−A)−1(λI−A)(xn) = (λI−A)−1(λxn−Axn),

❛❣♦r❛ t♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ s❡ n_→+_∞✱ ❝♦♠♦(λI₋A)−1_{é ❝♦♥tí♥✉❛✱ t❡♠♦s}

x= (λI₋A)−1(λx₋y).

❆ss✐♠✱λx₋Ax=λx₋y✱ ♦✉ s❡❥❛ x_∈D(A)❡Ax=y✳ P♦rt❛♥t♦✱ s❡❣✉❡ q✉❡A é ❢❡❝❤❛❞♦✳

❋✐♥❛❧♠❡♥t❡✱ ♣r♦✈❛r❡♠♦s q✉❡(0,+∞)⊂ρ(A)✳ ❙❛❜❡♠♦s q✉❡B= (0,+∞)∩ρ(A)é ♥ã♦ ✈❛③✐♦✳ ❆❧é♠ ❞✐ss♦✱

B é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❡♠ (0,+_∞) ❥á q✉❡ ρ(A) é ❛❜❡rt♦ ❡♠ (0,+_∞)✱ ✭✈❡r ❑r❡②s③✐❣ ✲ ❬✶✾❪ ❚❡♦r❡♠❛

✼✳✸✲✷✮✳ ▼♦str❛r❡♠♦s q✉❡ B é t❛♠❜é♠ ❢❡❝❤❛❞♦ ❡♠(0,+_∞)✳ ❙❡❥❛(λn)⊂B ❝♦♠ λn →λ✱ λ∈(0,+∞)✳

P❛r❛ns✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ t❡♠♦s q✉❡_|λn−λ|<

λ

4✱ ❡ ❡♥tã♦✱

||(λn−λ)(λn−A)−1||=|λn−λ|||(λn−A)−1|| ≤ |

λn−λ|

λn ≤

1 3.

❆ss✐♠✱I+ (λn−λ)(λn−A)−1 é ✉♠ ✐s♦♠♦r✜s♠♦ ❡♠X✱ ♣♦✐s é ❛ s♦♠❛ ❞❡ ❞♦✐s ✐s♦♠♦r✜s♠♦s✳ ❊♥tã♦

λI₋A=

I+ (λn−λ)(λn−A)−1 (λn−A)

❧❡✈❛ D(A)s♦❜r❡ X ❡ ❧♦❣♦ λ_∈ ρ(A)✱ ✈❡r✐✜❝❛♥❞♦ ❛ss✐♠ t♦❞❛s ❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛ ❞❡ ❍✐❧❧❡✲❨♦s✐❞❛✳

P♦rt❛♥t♦Aé ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝♦♥tr❛çã♦✳

❖❜s❡r✈❛çã♦ ✶✳✸✳ ❉❛ Pr♦♣♦s✐çã♦ ✶✳✸ ✐t❡♠ c)✱ t❡♠♦s q✉❡ s❡ x _∈ D(A) ❡♥tã♦ T(t)x _∈ D(A),_∀ t _≥ 0.

P♦rt❛♥t♦✱ T(t)D(A)⊂D(A),∀t≥0✳ ❊ss❛ ♣r♦♣r✐❡❞❛❞❡ ♥ã♦ é✱ ❡♠ ❣❡r❛❧✱ ✈á❧✐❞❛ ♣❛r❛ t♦❞♦x∈X✱ ♣♦rq✉❡

❞❡ T(t)x _∈ D(A),_∀ t _≥ 0✱ t❡♠♦s X = IX = T(0)X _⊂ D(A)✱ ✐st♦ é✱ D(A) = X ❡✱ ❛ss✐♠✱ ❞❡ ❛❝♦r❞♦

❝♦♠ ♦ ❚❡♦r❡♠❛ ❞♦ ●rá✜❝♦ ❋❡❝❤❛❞♦✱ A é ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❧✐♠✐t❛❞♦✱ r❡❝❛✐♥❞♦✲s❡ ♥♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r

❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ✉♥✐❢♦r♠❡✱ ❥á ❡st✉❞❛❞♦ ♥♦ ❚❡♦r❡♠❛ ✶✳✶✳ ■st♦ ♥ã♦ ❛❝♦♥t❡❝❡✱ ❝♦♥t✉❞♦✱ s❡ T(t)X _⊂D(A)

❛♣❡♥❛s ♣❛r❛ t >0 ❡✱ ❞❡ ✉♠ ♠♦❞♦ ♠❛✐s ❣❡r❛❧✱ ❛♣❡♥❛s ♣❛r❛ t > t0≥0✳ ➱ ❡ss❡ ❝❛s♦ ♣❛rt✐❝✉❧❛r q✉❡ ✈❛♠♦s

❝♦♥s✐❞❡r❛r ❛❣♦r❛✳

❉❡✜♥✐çã♦ ✶✳✽✳ ❙❡❥❛ s❡♠✐❣r✉♣♦ T(t) ❞❡ ❝❧❛ss❡ C0 ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ X✳ ❖ s❡♠✐❣r✉♣♦ T(t) é

❞✐t♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ♣❛r❛ t > t0✱ s❡ ♣❛r❛ t♦❞♦ x∈ X✱ t →T(t)x é ❞✐❢❡r❡♥❝✐á✈❡❧ ♣❛r❛ t > t0✳ T(t) é ❞✐t♦

❞✐❢❡r❡♥❝✐á✈❡❧ s❡ é ❞✐❢r❡♥❝✐á✈❡❧ ♣❛r❛ t >0✳

Pr♦♣♦s✐çã♦ ✶✳✺✳ ❙❡ A é ✉♠ ♦♣❡r❛❞♦r ♠✲❞✐ss✐♣❛❞♦ ❡ ❛✉t♦✲❛❞❥✉♥t♦ ❞❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rtH ✱ ❡♥tã♦ ♦

s❡♠✐❣r✉♣♦ T(t)❞❡ ❝❧❛ss❡ C0✱ ❣❡r❛❞♦ ♣♦r A✱ é ❞✐❢❡r❡♥❝✐á✈❡❧✳

❆♥t❡s ❞❡ ✐♥✐❝✐❛r♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✺✱ ♣r♦✈❛r❡♠♦s ❛ s❡❣✉✐♥t❡ ❛✜r♠❛çã♦✱ q✉❡ s❡rá ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛ ♣r♦✈❛ ❞❛ ♠❡s♠❛✳

❆✜r♠❛çã♦ ✶✳✷✳ ❙❡❥❛ A ✉♠ ♦♣❡r❛❞♦r ❞✐ss✐♣❛t✐✈♦ ❡ ❛✉t♦✲❛❞❥✉♥t♦ ❞❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H✱ ❡ u ∈

C2_([0,+

∞), H)✉♠❛ ❢✉♥çã♦ q✉❡ s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s

du

dt =Au ❡ d2_u

dt2 =A

✶✻ ❊♥tã♦✱ du dt(t)

< 1

t ku(0)k. ✭✶✳✷✷✮

❉❡ ❢❛t♦✱ ❝♦♠♦du

dt, u

= (Au, u)❡Re

_du

dt, u

=1 2

d dt kuk

2_{✱ t❡♠♦s q✉❡}

1 2

d

dt kuk =Re(Au, u). ✭✶✳✷✸✮

■♥t❡❣r❛♥❞♦(1.23)❞❡s❛t✱ ❝♦♠06s6t✱ t❡♠✲s❡ q✉❡

1

2 ku(t)k

2

−1₂ ku(s)_k2 =

Z t

s

Re(Au(r), u(r))dr60, ✭✶✳✷✹✮

♦♥❞❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ (1.24) s❡❣✉❡ ❞❡ A s❡r ❞✐ss✐♣❛t✐✈♦✳ ❊♥tã♦ ❞❡❞✉③✐♠♦s ❛ ✈❛❧✐❞❛❞❡ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡

ku(t)k6_ku(s)k✳ ❆ss✐♠✱kuk é ❞❡❝r❡s❝❡♥t❡✳

❆❣♦r❛✱ r❡♣❡t✐♥❞♦ ♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦ ❛❝✐♠❛✱ ♣r♦✈❛♠♦s q✉❡

du dt

é ❞❡❝r❡s❝❡♥t❡✳ ❆❧é♠ ❞✐ss♦✱ ❞❛❞♦ τ

t❡♠♦s q✉❡ Z τ 0 Au,du dt tdt = Z τ 0 du dt 2 tdt > Z τ 0 du dt(τ)

2 tdt = τ 2 2 du dt(τ)

2 . ✭✶✳✷✺✮

❊♥tr❡t❛♥t♦✱ ❝♦♠♦Aé ❛✉t♦✲❛❞❥✉♥t♦ ❡

Au,du dt

é ✉♠ ♥ú♠❡r♦ r❡❛❧✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s ❤✐♣ót❡s❡s t❡♠♦s q✉❡

d

dt(Au, u) =

Adu dt, u

+ Au,du dt = 2 Au,du dt . ✭✶✳✷✻✮

❯s❛♥❞♦ ✐♥t❡❣r❛çã♦ ♣♦r ♣❛rt❡s ❡(1.26)✱ t❡♠♦s q✉❡

Z τ 0 Au,du dt

tdt = 1 2

Z τ 0

d

dt(Au, u)tdt

= 1

2(Au(τ), u(τ))τ− 1 2

Z τ 0

(Au, u)dt. ✭✶✳✷✼✮

◆♦✈❛♠❡♥t❡✱ ❝♦♠♦Aé ❛✉t♦✲❛❞❥✉♥t♦✱ ❞❛ ✐❣✉❛❧❞❛❞❡

du dt, u

= (Au, u)❡♠(1.21)✱ ❞❡❞✉③✐♠♦s q✉❡

1 2

d dt kuk

2_{= (Au, u),}

❡ ❛ss✐♠

1

2 ku(τ)k

2

−1₂ ku(0)_k2₌Z τ 0

✶✼

P♦rt❛♥t♦✱ ❞❡(1.25)✱(1.27)❡(1.28)t❡♠♦s

τ2 2 du dt(τ)

2 6 Z τ 0 Au,du dt tdt = 1

2(Au(τ), u(τ))τ− 1

4 ku(τ)k

2₊1

4 ku(0)k

2

6 1

2 kAu(τ)kτ ku(τ)k − 1

4 ku(τ)k

2₊1

4 ku(0)k

2 6 1 2      du dt(τ)

2

τ2₊

ku(τ)_k2

2     

−1₄ ku(τ)_k2₊1

4 ku(0)k

2 6 1 4 du dt(τ)

2

τ2+1

4 ku(0)k

2_.

▼❛s ✐ss♦ ✐♠♣❧✐❝❛ q✉❡

τ2 4 du dt(τ)

2 6 1

4 ku(0)k

2_,

❡ ♣♦rt❛♥t♦

k du_dt(τ) k6 1

τ ku(0)k.

❆ ❛✜r♠❛çã♦ ❡stá ♣r♦✈❛❞❛✳

❆ s❡❣✉✐r ❞❛r❡♠♦s ❛ ♣r♦✈❛ ❞❛ Pr♦♣♦s✐çã♦1.5✳

❉❡♠♦♥str❛çã♦✳ ❉❛❞♦ x∈H✱ ❝♦♠♦D(A2_{)é ❞❡♥s♦ ❡♠H ✈❡r ❬✷✶❪✱ ❡①✐st❡(x}

n)⊂D(A2)t❛❧ q✉❡xn →x✳

❊♥tr❡t❛♥t♦✱

kT(t)xn−T(t)xk6kT(t)kkxn−xk,

♣♦✐s ♦ ♦♣❡r❛❞♦r é ❧✐♥❡❛r ❡ ❧✐♠✐t❛❞♦✳ ❆❧é♠ ❞✐ss♦✱ ♣❡❧❛ ❛✜r♠❛çã♦ ✶✳✷✱ t❡♠♦s q✉❡

kAT(t)xn−AT(t)xmk6

1

t kxn−xmk.

▲♦❣♦✱ q✉❛♥❞♦n_→+_∞, T(t)xn❝♦♥✈❡r❣❡ ♣❛r❛T(t)x❡AT(t)xn❝♦♥✈❡r❣❡ ❡♠ t♦❞♦ ✐♥t❡r✈❛❧♦[δ,+∞), δ >0✳

❈♦♠♦ Aé ✉♠ ♦♣❡r❛❞♦r ❢❡❝❤❛❞♦✱T(t)x∈D(A),∀ t>δ >0❡✱ ♣♦rt❛♥t♦✱ ♣❛r❛ t♦❞♦ t >0✳ ▲♦❣♦T é ✉♠

s❡♠✐❣r✉♣♦ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦ ❞❛ ♣r♦♣♦s✐çã♦✳

❚❡♦r❡♠❛ ✶✳✹✳ ❙❡❥❛♠ A ✉♠ ♦♣❡r❛❞♦r ♠✲❞✐ss✐♣❛❞♦ ❡ ❛✉t♦✲❛❞❥✉♥t♦ ❞❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H ❡ T(t) ♦

s❡♠✐❣r✉♣♦ ❞❡ ❝❧❛ss❡ C0 ❣❡r❛❞♦ ♣♦rA✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦x∈H ❡ ✐♥t❡✐r♦s ♥ã♦ ♥❡❣❛t✐✈♦sn❡k✱ ❝♦♥❝❧✉í♠♦s

q✉❡

T(t)x_∈Cn_((0,+

∞);D(Ak_)).

P❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛✱ ♣r❡❝✐s❛r❡♠♦s ♣r♦✈❛r ❛❧❣✉♥s ❧❡♠❛s✳

▲❡♠❛ ✶✳✹✳ ❙❡❥❛ T(t)✉♠ s❡♠✐❣r✉♣♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ♣❛r❛t > t0❡T(n)(t)♦ ♦♣❡r❛❞♦r ❞❡✜♥✐❞♦ ♣♦r T(n)(t) =

An_{T(t), A}0_{=I, n= 0,1,2, . . .✱ ❡♥tã♦}

✐✮ ♦ ♦♣❡r❛❞♦r T(n)_{(t) t❡♠ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ ♣❛r❛ t >(n+ 1)t}

0 ❡ t♦❞♦ s t❛❧ q✉❡t−t0 > s > nt0✱

t❡♠✲s❡ T(n)_(t)x=T(t

−s)T(n)_{(s)x✱ ♣❛r❛ t♦❞♦x}

✶✽

✐✐✮ T(n)_{(t)é ❧✐♠✐t❛❞♦ ♣❛r❛ t♦❞♦t > nt}

0, n= 0,1,2, . . .✳

❉❡♠♦♥str❛çã♦✳ ✭✐✮ ❉❡t > t0 ❡t−t0> s >0✱ t❡♠♦st−s > t0✳ ❈♦♠♦✱t0>0, t−s >0✱ t❡♠♦s ❡♥tã♦

T(0)_{(t)x=T(t)x=T(t}

−s)T(s)x=T(t₋s)T(0)_x,

∀x_∈X.

P♦rt❛♥t♦✱ ❛ ❛✜r♠❛çã♦ é ✈á❧✐❞❛ ♣❛r❛ n = 0✳ ❙✉♣♦♥❤❛♠♦s ✈á❧✐❞❛ ♣❛r❛ n ❡ s❡❥❛♠ t > (n+ 2)t0 ❡ t−

t0 > s > (n+ 1)t0✳ ❖❜s❡r✈❡♠♦s q✉❡ s❡ r > (n+ 1)t0✱ ❡♥tã♦ ❞❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ r❡s✉❧t❛ q✉❡

T(n)_(r)x

∈D(A),_∀x_∈X✳ ▲♦❣♦✱ ❝♦♠♦s >(n+ 1)t0✱ t❡♠♦s q✉❡T(n)(s)x∈D(A),∀x∈X✳

❆ss✐♠✱ ♣❡❧♦ ✐t❡♠(c)❞❛ Pr♦♣♦s✐çã♦ ✶✳✸✱ ❞❡❞✉③✐♠♦s q✉❡

T(t₋s)T(n)(s)x_∈D(A)

❡

AT(t₋s)T(n)_(s)x=T(t

−s)AT(n)_(s)x,

∀x_∈X.

❊♥tr❡t❛♥t♦✱ s❛❜❡♠♦s q✉❡

t >(n+ 2)t0>(n+ 1)t0 ❡ t−t0> s >(n+ 1)t0> nt0.

P❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ s❡❣✉❡✲s❡ ❡♥tã♦ q✉❡

T(n)_(t)x=T(t

−s)T(n)_(s)x,

∀x_∈X,

❞♦♥❞❡

T(n+1)_{(t)x = AT}(n)_(t)x=AT(t

−s)T(n)_(s)x=T(t

−s)AT(n)_(s)x

= T(t₋s)T(n+1)_(s)x,

∀x_∈X.

❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ t❡♠♦s ♦ r❡s✉❧t❛❞♦ ♣❛r❛n+ 1✳ P♦rt❛♥t♦✱ ❛ ❛✜r♠❛çã♦ é ✈á❧✐❞❛ ♣❛r❛n= 0,1,2, . . .✳ ✭✐✐✮ P❛r❛ n = 0✱ t❡♠♦s (ii) tr✐✈✐❛❧♠❡♥t❡✳ ❙✉♣♦♥❤❛♠♦s (ii) ✈á❧✐❞❛ ♣❛r❛ n ❡ s❡❥❛ t > (n+ 1)t0✳

❉✐ss♦✱ t❡♠♦s t > nt0 ❡✱ ♣❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ s❡❣✉❡ q✉❡ T(n)(t) é ✉♠ ♦♣❡r❛❞♦r ❧✐♠✐t❛❞♦ ❞❡ X ❡✱

♣♦rt❛♥t♦✱ ❢❡❝❤❛❞♦✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ Aé ❢❡❝❤❛❞♦✱ ♣♦✐s é ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝❧❛ss❡ C0✳ ▲♦❣♦ T(n+1)(t) é ❢❡❝❤❛❞♦✱ ♣♦✐s T(n+1)(t) = AT(n)(t)✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ t > (n+ 1)t0 t❡♠♦s q✉❡

T(n)_{(t)x∈ D(A), ∀x ∈X✱ ❡ ❡♥tã♦ T}n+1_{(t) é ❞❡✜♥✐❞♦ ❡♠ t♦❞♦ ♦ ❡s♣❛ç♦ X✳ P❡❧♦ ❚❡♦r❡♠❛ ❞♦ ●rá✜❝♦}

❋❡❝❤❛❞♦✱ s❡❣✉❡ q✉❡T(n+1)_{(t)é ✉♠ ♦♣❡r❛❞♦r ❧✐♠✐t❛❞♦✳}

▲❡♠❛ ✶✳✺✳ ❙❡❥❛ T(t) ❞✐❢❡r❡♥❝✐á✈❡❧ ♣❛r❛ t > t0✱ ❡♥tã♦ ♣❛r❛ t♦❞♦ t > nt0 ❛ ❢✉♥çã♦ T(t)x é n ✈❡③❡s

❝♦♥t✐♥✉❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡

dn

dtT(t)x=T

(n)_(t)x=An_{T(t)x, n= 1, . . . , t > nt}

0.

❉❡♠♦♥str❛çã♦✳ P♦r ❤✐♣ót❡s❡✱ s❡ t > t0❡♥tã♦ T(t)x∈D(A),∀x∈X✳ P♦rt❛♥t♦✱ s❡t > t0✱ ❡①✐st❡ ♦ ❧✐♠✐t❡

❞❡

AhT(t)x=

T(t+h)x−T(t)x