UNIVERSIDADE FEDERAL DE SÃO CARLOS

P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

▼✐❣✉❡❧ ❆♥❣❡❧ ❈✉❛②❧❛ ❩❛♣❛t❛

❊st✐♠❛t✐✈❛ ❞❡ ❡♥❡r❣✐❛ ♥♦ ✐♥✜♥✐t♦ ♣❛r❛ ❡q✉❛çõ❡s

❤✐♣❡r❜ó❧✐❝❛s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ♦s❝✐❧❛♥t❡s

❙ã♦ ❈❛r❧♦s ✲ ❙P

❆❣♦st♦ ❞❡ ✷✵✶✷

P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

❊st✐♠❛t✐✈❛ ❞❡ ❡♥❡r❣✐❛ ♥♦ ✐♥✜♥✐t♦ ♣❛r❛ ❡q✉❛çõ❡s

❤✐♣❡r❜ó❧✐❝❛s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ♦s❝✐❧❛♥t❡s

▼✐❣✉❡❧ ❆♥❣❡❧ ❈✉❛②❧❛ ❩❛♣❛t❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢ ❉r✳ ❏♦s❡ ❘✉✐❞✐✈❛❧ ❞♦s ❙❛♥t♦s ❋✐❧❤♦

❇♦❧s✐st❛ ❈♥♣q

Pr♦❝❡ss♦ ✷✵✶✷

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❙❈❛r ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛

❙ã♦ ❈❛r❧♦s ✲ ❙P

❆❣♦st♦ ❞❡ ✷✵✶✷

Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária da UFSCar

C961ee

Cuayla Zapata, Miguel Angel.

Estimativa de energia no infinito para equações hiperbólicas com coeficientes oscilantes / Miguel Angel Cuayla Zapata. -- São Carlos : UFSCar, 2012.

59 f.

Dissertação (Mestrado) -- Universidade Federal de São Carlos, 2012.

1. Equações diferenciais parciais. 2. Equação da onda. 3. Estimativa de energia. 4. Coeficiente oscilante. I. Título.

➚ ❉❡✉s✱ ♣❡❧❛ ✈✐❞❛✱ ♣❛③ ❡ s❛ú❞❡✳

❆♦s ♠❡✉s ♣❛✐s ❖❧❣❛ ❡ ❆❣✉stí♥✱ ♣♦r s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❡♠ ❡ ♠♦t✐✈❛r❡♠ ❞✉r❛♥t❡ ♠❡✉s ❡st✉❞♦s ❡ ❡♠ t♦❞♦s ♦s ♦✉tr♦s ❛s♣❡❝t♦s ❞❛ ♠✐♥❤❛ ✈✐❞❛✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛♦s ♣r♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❙❈❛r✱ q✉❡ ♠❡ ❛❥✉❞❛r❛♠ ❝♦♠ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❞✉r❛♥t❡ ❛ ♦ ♠❡str❛❞♦✳

❚♦❞♦s ❢♦r❛♠ s❡♠♣r❡ ♠✉✐t♦ ❛t❡♥❝✐♦s♦s ❡ ❡①❡♠♣❧♦s ❞❡ ♣r♦❢❡ss♦r❡s ♣❛r❛ ♠✐♠✳ ❊♥tr❡ t♦❞♦s ❡st❡s✱ ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❡♠ ❡s♣❡❝✐❛❧ ♠❡✉ ♦r✐❡♥t❛❞♦r ❘✉✐❞✐✈❛❧✱ ♣♦r ♠❡ ❛✉①✐❧✐❛r ♥❡st❡ tr❛❜❛❧❤♦ ❡ ♣♦r t♦❞❛ ❛❥✉❞❛ ❛ ♠✐♠ ❞❡❞✐❝❛❞❛✳

◆ós ❡st✉❞❛♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❡♥❡r❣✐❛✱ ♣❛r❛ t _{→ ∞}✱ ❞❛s s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ♣❛r❛ ❛❧❣✉♠❛s ❡q✉❛çõ❡s ❡str✐t❛♠❡♥t❡ ❤✐♣❡r❜ó❧✐❝❛s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s q✉❡ ♦s❝✐❧❛♠ r❛♣✐❞❛♠❡♥t❡✳

❲❡ st✉❞② t❤❡ ❜❡❤❛✈✐♦r✱ ❛s t _{→ ∞}✱ ♦❢ t❤❡ ❡♥❡r❣② ❢♦r t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❢♦r s♦♠❡ str✐❝t❧② ❤②♣❡r❜♦❧✐❝ ❧✐♥❡❛r s❡❝♦♥❞ ♦r❞❡r ❡q✉❛t✐♦♥s ✇✐t❤ ❝♦❡✣❝✐❡♥ts ✈❡r② r❛♣✐❞❧② ♦s❝✐❧❧❛t✐♥❣✳

❆❣r❛❞❡❝✐♠❡♥t♦s ✐✐✐

❘❡s✉♠♦ ✐✈

❆❜str❛❝t ✈

■♥tr♦❞✉çã♦ ✈✐✐✐

✶ Pr❡❧✐♠✐♥❛r❡s ✷

✶✳✶ ◆♦t❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷

✶✳✷ ❊s♣❛ç♦s Lp _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷}

✶✳✷✳✶ ❖♣❡r❛çõ❡s ❝♦♠ ❉✐str✐❜✉✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

✶✳✸ ❉✐str✐❜✉✐çõ❡s ❚❡♠♣❡r❛❞❛s ❡ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✶✳✹ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✶✳✹✳✶ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ❍♦♠♦❣ê♥❡♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✷ Pr♦❜❧❡♠❛ ■♥✐❝✐❛❧ ✶✽

✸ Pr♦❜❧❡♠❛ ❣❡r❛❧ ✷✽

✸✳✶ ❙♦❧✉❝ã♦ ❞❛ ❡q✉❛çã♦ ❞❛ ♦♥❞❛ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ♦s❝✐❧❛♥t❡s ❡♠ ✐♥t❡r✈❛❧♦s ❞❡ ❝♦♠♣r❡♠❡♥t♦ ✉♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✸✳✷ ❙♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❛ ♦♥❞❛ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ♦s❝✐❧❛♥t❡s ♥✉♠❛ s❡q✉ê♥✲ ❝✐❛ ❞❡ ✐♥t❡r✈❛❧♦s ❝r❡s❝❡♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✹ ❆♣ê♥❞✐❝❡ ✺✸

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✾

❈♦♥s✐❞❡r❛♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡♠[0,+_∞)_×R✱ ❝♦♠ u=u(t, x)✱

  

∂2

tu−a(t)∂x2u= 0,

u(0, x) = u0(x), ∂tu(0, x) =u1(x),

✭✶✮

❝♦♠ ✈❛❧♦r ✐♥✐❝✐❛✐s u0 ∈ Hs(R), u1 ∈Hs−1(R), s >0✱ ❝♦♠ ❛ ❤✐♣ót❡s❡ ❞❡ ❤✐♣❡r❜♦❧✐❝✐✲

❞❛❞❡

0< λ_≤a(t)_≤Λ, _∀t. ✭✷✮

❖❜s❡r✈❛♠♦s q✉❡ s❡a(t)é ♣♦s✐t✐✈❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠ ❞❡r✐✈❛❞❛ ❧✐♠✐t❛❞❛✱ ❡♥tã♦ t❡♠✲s❡ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❧♦❝❛❧ ♥♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ ❡✱ ♥❡ss❡ ❝❛s♦✱ s❡❣✉❡✲s❡ ❛ ❡st✐♠❛t✐✈❛ ❞❡ ❡♥❡r❣✐❛

Es(u)(t)≤Cs,TEs(u)(0), ∀t∈[0, T] ✭✸✮

❝♦♠

Es(u)(t) :=ku(t)k2Hs +k∂_tu(t)k2_Hs−1. ✭✹✮

❖✉tr❛ ❢♦r♠❛ ❞❡ ✈❡r ♦ ♣r♦❜❧❡♠❛ ✭✶✮ é ❝♦♥s✐❞❡r❛r ❛♦ ✐♥✈és ❞❛ ❝♦♥t✐♥✉✐❞❛❞❡ ▲✐♣s❝❤✐t③ ❞♦ ❝♦❡✜❝✐❡♥t❡ a(t)✱ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ▲♦❣✲▲✐♣✳

❆q✉✐ ▲♦❣✲▲✐♣ é ❞❡✜♥✐❞♦ ♣♦r✿

❉❡✜♥✐çã♦ ✵✳✵✳✶✳ ❙❡❥❛ f : I → R✱ ❝♦♠ I ✉♠ ✐♥t❡r✈❛❧♦✱ ❞✐r❡♠♦s q✉❡ f é ▲♦❣✲▲✐♣ ❝♦♥tí♥✉❛ s❡ ❡❧❛ s❛t✐s❢❛③✿

kfkLL(I):= sup

t,t+τ∈I

0<|τ|<1/2

|f(t+τ)−f(t)|

|τ_||log_|τ_|| <+∞ ✭✺✮

❈♦♠ ❡st❛ r❡❣✉❧❛r✐❞❛❞❡ ❞♦s ❝♦❡✜❝✐❡♥t❡s a(t) ❞❡♠♦♥str❛✲s❡ q✉❡ ✭✶✮ é C∞_✲❜❡♠ ♣♦st♦✱ ✈❡❥❛ ❬✷❪✳

❆❧é♠ ❞✐ss♦ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ▲♦❣✲▲✐♣ ♣♦❞❡ s❡r ♣❡♥s❛❞❛ ❝♦♠♦ ✉♠❛ ❤✐♣ót❡s❡ ♠í♥✐♠❛ ♣❛r❛ ❛ r❡❣✉❧❛r✐❞❛❞❡ ♥♦s ❝♦❡✜❝✐❡♥t❡s ❛✜♠ ❞❡ t❡rC∞_{✲❜❡♠ ♣♦st♦ ♣❛r❛ ✭✶✮✳ P♦r ❡①❡♠✲} ♣❧♦✱ ❬✷❪ ❡①✐❜❡ ✉♠❛ s✐t✉❛çã♦ ❡♠ q✉❡ ❛ ❤✐♣ót❡s❡ ▲♦❣✲▲✐♣ ♥ã♦ ♣♦❞❡ s❡r ❡♥❢r❛q✉❡❝✐❞❛ ♣❛r❛ ❛ ❝❧❛ss❡ ❍ö❧❞❡r ❝♦♠ ❡①♣♦♥❡♥t❡ ♠❡♥♦r q✉❡ ✶✳ ▲á✱ t❛❧ ❡①❡♠♣❧♦ é ❣❡♥❡r❛❧✐③❛❞♦✱ ♣r♦✈❛♥❞♦ q✉❡ ❡♠ ❣❡r❛❧ ❛ ❤✐♣ót❡s❡s ▲♦❣✲▲✐♣ r❡❣✉❧❛r✐❞❛❞❡ ♥ã♦ ♣♦❞❡ s❡ ❡♥❢r❛q✉❡❝✐❞❛ ♥♦ s❡♥t✐❞♦ ❡s♣❡❝✐✜❝❛❞♦✳

❆♦ ❝♦♥s✐❞❡r❛r ♦ ❝♦❡✜❝✐❡♥t❡ a(t) ❡♠ ▲♦❣✲▲✐♣ ❛✐♥❞❛ t❡♠✲s❡ ❛ ❡st✐♠❛t✐✈❛ ❞❛ ❡♥❡r❣✐❛✱ ♠❛s ❝♦♠ ♣❡r❞❛ ❞❡ ❞❡r✐✈❛❞❛✱ ✈❡r ❬✺❪✳

❖✉ s❡❥❛✱ ♣❛r❛ ❛ s♦❧✉çã♦ u ❡ a(t) ▲♦❣✲▲✐♣ ❝♦♥tí♥✉♦✱ ❝♦♠ T > 0✱ s _∈ R q✉❛❧q✉❡r ❡ ♣❛r❛ t♦❞♦t ∈[0, T]✱ t❡♠✲s❡✿

Es−βt(u)(t)≤Cs,T∗ Es(u)(0), ✭✻✮

❝♦♠ C∗

s,T ❝♦♥st❛♥t❡ q✉❡ só ❞❡♣❡♥❞❡ ❞❡ s✱ ❞❛ ❞✐♠❡♥sã♦ ♥✱ ❚ ❡Λ ✱ ❡ β é ❞❛❞♦ ♣♦r β = 1

λC ∗

ka_kLL([0,t]), ✭✼✮ s❡♥❞♦C∗ _{✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ q✉❡ ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞❡n❡λ ✭ ♠❡♥♦r ❝♦t❛ s✉♣❡r✐♦r} ❞♦ ❝♦❡✜❝✐❡♥t❡ a(t)✮✳

❆❣♦r❛ s❡ ❛♠♣❧✐❛♠♦s ♦ ❡s♣❛ç♦ ❞❡ ▲♦❣✲▲✐♣ ♣❛r❛ ♦ ❡s♣❛ç♦ Ω✲▲♦❣✲▲✐♣✱ ❛♦ q✉❛❧ ❞❡✜♥❡✲s❡ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✳

❉❡✜♥✐çã♦ ✵✳✵✳✷✳ ❯♠❛ ❢✉♥çã♦ f : I _→ R✱ ♦♥❞❡ I é ✉♠ ✐♥t❡r✈❛❧♦ r❡❛❧✱ é ❞✐t❛ s❡r ❝♦♥tí♥✉❛ Ω✲▲♦❣✲▲✐♣ q✉❛♥❞♦✿

kfkΩLL(I) := sup

t,t+τ∈I

0<|τ|<δ

|f(t+τ)−f(t)|

|τ_||log_|τ_||Ω(_|τ_|) <+∞.

❆q✉✐Ω s❛t✐s❢❛③✳

❉❡✜♥✐çã♦ ✵✳✵✳✸✳ ❙❡❥❛ ❛ ❢✉♥çã♦Ω_∈C1_{((0, δ])✱ ♣❛r❛ ❛❧❣✉♠δ >0✭ ♣♦❞❡♠♦s ❛ss✉♠✐r}

s❡♠♣r❡ q✉❡ δ < 1/2✮ ✱ é ✉♠❛ ❢✉♥çã♦ ♠ó❞✉❧♦✱ s❡ é ✉♠❛ ❢✉♥çã♦ ❝♦♥✈❡①❛✱ ♣♦s✐t✐✈❛✱ ❞❡❝r❡s❝❡♥t❡ t❛❧ q✉❡

limτ→0+Ω(τ) = +∞, 0<−Ω′(τ)≤τ−1, Ω(δ)≥1.

❈♦♠ ❛❥✉❞❛ ❞❡st❛s ❞❡✜♥✐çõ❡s✱ tr❛t❛♠♦s ❞❡ r❡s♣♦♥❞❡r ❡♠ ♣❛rt❡ s❡ ❡①✐st❡ ✉♠❛ ❡st✐♠❛t✐✈❛ ❣❧♦❜❛❧ ❞❛ ❡♥❡r❣✐❛ ♣❛r❛ ❝♦❡✜❝✐❡♥t❡s ♥ã♦ ▲✐♣s❝❤✐t③✳ ❖ ♦❜❥❡t✐✈♦ ❞❛ ❞✐ss❡r✲ t❛çã♦ é ❡st✉❞❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♠♣tót✐❝❛ ❞❛ ❡♥❡r❣✐❛ ♥♦ ❝❛s♦ ❞❡ t❡r✲s❡ ❝♦❡✜✲ ❝✐❡♥t❡s a(t)♥ã♦ ❞✐❢❡r❡♥❝✐á✈❡✐s✳ ❉❡ ❢❛t♦✱ ❞❡♠♦♥str❛r❡♠♦s q✉❡ ❛ ❡①♣❧♦sã♦ ❞❛ ❡♥❡r❣✐❛ ♥♦ ✐♥✜♥✐t♦ ❛❝♦♥t❡❝❡✱ ❡ ✐st♦ ✐♥❞❡♣❡♥❞❡ ❞❛ r❡❣✉❧❛r✐❞❛❞❡ ❞♦s ❝♦❡✜❝✐❡♥t❡s✳ ❖ ❢❛t♦ ✐♠♣♦r✲ t❛♥t❡ ♣❛r❛ s❡ t❡r ✉♠❛ ❡st✐♠❛t✐✈❛ ❛ss✐♥tót✐❝❛ ❞❛ ❡♥❡r❣✐❛ ♥♦ ✐♥✜♥✐t♦ é ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ♦s❝✐❧❛♥t❡ ❞♦s ❝♦❡✜❝✐❡♥t❡s✳

P❛r❛ ✐ss♦ ❝♦♥str✉✐r❡♠♦s ✉♠ ❝♦❡✜❝✐❡♥t❡ a(t) t❛❧ q✉❡ ❛❝♦♥t❡❝❡ ❛ ❡①♣❧♦sã♦ ❞❛ s♦❧✉çã♦ ❞❡ ✭✶✮ ❡♠ ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ❤♦♠♦❣ê♥❡♦✳ ❆ss✐♠ ❝♦♥s✐❞❡r❛r❡♠♦s ❞♦✐s ❝❛✲ s♦s✱ ♥❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ♦s ❝♦❡✜❝✐❡♥t❡s ♦s❝✐❧❛♠ ❡♠ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✐♥t❡r✈❛❧♦s ❞❡ ❝♦♠♣r❡♠❡♥t♦ ✶✱ ♥❛ s❡❣✉♥❞❛ ♣❛rt❡ ♦s ❝♦❡✜❝✐❡♥t❡s ♦s❝✐❧❛♠ ❡♠ ✉♠❛ s❡q✉❡♥❝✐❛ ❞❡ ✐♥✲ t❡r✈❛❧♦s ❝✉❥♦ ❝♦♠♣r❡♠❡♥t♦ ❝r❡s❝❡ r❛♣✐❞❛♠❡♥t❡✱ ❡ ❡♠ ❛♠❜♦s ❝❛s♦s t❡♠✲s❡ ❡①♣❧♦sã♦ ❞❛ ❡♥❡r❣✐❛✳ ❆ r❡❢❡r❡♥❝✐❛ ❜ás✐❝❛ é ♦ ❛rt✐❣♦ ❞❡ ❋✳ ❈♦❧♦♠❜✐♥✐✭✈❡❥❛ ❬✶❪✮✳

▼❛✐s r❡❝❡♥t❡♠❡♥t❡ ❋✳ ❍✐r♦s❛✇❛ ✱ ✈❡r ❬✽❪✱ ♦❜t❡r❡♠ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿ ❚❡♦r❡♠❛ ✵✳✵✳✹✳ P❛r❛ q✉❛❧q✉❡r q <1✱ ❡①✐st❡ ✉♠ a(t)_∈C∞_{([0,∞)) s❛t✐s❢❛③❡♥❞♦}

a(k)(t)_≤Ck(1 +t)−kq, ✭✽✮

♣❛r❛ q✉❛❧q✉❡r k∈N✱ t❛❧ q✉❡ ❛ ❝♦♥s❡r✈❛çã♦ ❣❡♥❡r❛❧✐③❛❞❛ ❞❛ ❡♥❡r❣✐❛✶_{✱ ♥ã♦ ✈❛❧❡✳}

❖❜s❡r✈❡ q✉❡ ❡st❡ r❡s✉❧t❛❞♦ ❣❛r❛♥t❡ q✉❡ ♣❛r❛ ❢✉♥çõ❡s ❣❧♦❜❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③✱ ❛ ❡♥❡r❣✐❛ ♥ã♦ é ❧✐♠✐t❛❞❛ ❡♠ [0,∞)]✳ ❯♠ ❡①❡♠♣❧♦ ❡①♣❧í❝✐t♦ ♣♦❞❡ s❡r ✈✐st♦ ♥♦ ❛rt✐❣♦

❞❡ ▼✳ ❘❡✐ss✐♥❣ ❡ ❏✳ ❙♠✐t❤✱ ✈❡r ❬✶✸❪✳

✶_{●❊❈ q✉❡r ❞✐③❡r q✉❡ t❡♠✲s❡ ❛ s❡❣✉✐♥t❡ ❡st✐♠❛t✐✈❛ C1E(0) ≤ E(t) ≤ C2E(0)✭ ♦♥❞❡ E(t) =}

1 2

a(t)2

k∇u(t,_·)_k2+_k∂tu(t,·)k 2

✮

Pr❡❧✐♠✐♥❛r❡s

◆❡st❡ ❝❛♣ít✉❧♦ ✈❛♠♦s ❛♣r❡s❡♥t❛r ♦s ❢❛t♦s ❜ás✐❝♦s ♥❡❝❡ssár✐♦s ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦s ❝❛♣ít✉❧♦s s✉❜s❡q✉❡♥t❡s✳ ❆♣r❡s❡♥t❛♠♦s ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ ❚r❛♥s❢♦r✲ ♠❛❞❛ ❞❡ ❋♦✉r✐❡r✱ ❞❡✜♥✐♠♦s ♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ❡ ❡①♣❧♦r❛♠♦s ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❆s r❡❢❡rê♥❝✐❛s ❜ás✐❝❛s sã♦ ❬✾❪✱ ❬✶✶❪✳

✶✳✶ ◆♦t❛çõ❡s

✶✳✷ ❊s♣❛ç♦s

L

❋✐①❡♠♦s ✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛ (X,M, µ)✱ ♦✉ s❡❥❛✱ X é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦✲ ✈❛③✐♦✱ M é ✉♠❛σ₋á❧❣❡❜r❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ X ❡µ:_{M →}[0,_∞] ✉♠❛ ♠❡❞✐❞❛✳ ❙❡f é ✉♠❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ s♦❜r❡ ❳ ❡1≤p <∞✱ ❞❡✜♥✐♠♦s

kf_kLp = Z

X| f_|pdµ

1/p

P❛r❛ p=_∞✱ t♦♠❛♠♦s

kfkL∞ = inf{a≥0 :µ({x:|f(x)|> a}) = 0}

❝♦♠ ❛ ❝♦♥✈❡♥çã♦ ❞❡ q✉❡ inf_∅ = _∞✱ _kf_kL∞ é ❝❤❛♠❛❞♦ s✉♣r❡♠♦ ❡ss❡♥❝✐❛❧ ❞❡ f ❡

❡s❝r❡✈❡♠♦s

kf_kL∞ =s✉♣❡ss_x∈X|f(x)|.

❉❡✜♥✐♠♦s

Lp(X,_M, µ) = _{f :X _→C; ❢ é ♠❡♥s✉rá✈❡❧ ❡ _kf_kLp <∞}.

❉✐③❡♠♦s q✉❡ ❞✉❛s ❢✉♥çõ❡s ❞❡✜♥❡♠ ♦ ♠❡s♠♦ ❡❧❡♠❡♥t♦ ❞❡ Lp _{q✉❛♥❞♦ ❡❧❛s}

sã♦ ✐❣✉❛✐s q✉❛s❡ t♦❞❛ ❛ ♣❛rt❡✳ ▼❡❞✐❛♥t❡ ❡st❛ ✐❞❡♥t✐✜❝❛çã♦✱ t❡♠♦s q✉❡ ♦ ❡s♣❛ç♦ Lp_{(X,M, µ)✱ ♠✉♥✐❞♦ ❞❛ ♥♦r♠❛ k · k}

Lp✱ é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ✭♣❛r❛ ❛ ♣r♦✈❛✱ ✈❡❥❛

❬✶✹❪✱ ♣♦r ❡①❡♠♣❧♦✮✳

❘❡❝♦r❞❡♠♦s q✉❡ ❞♦✐s ♥ú♠❡r♦s r❡❛✐s p ❡ p′ _{s❛t✐s❢❛③❡♥❞♦} 1 p +

1

p′ = 1✱ ❝♦♠ p, p′ _{>1 sã♦ ❞✐t♦s ❡①♣♦❡♥t❡s ❝♦♥❥✉❣❛❞♦s✳ ❆❧é♠ ❞✐ss♦✱ ✶ ❡ ∞sã♦ ❝♦♥❥✉❣❛❞♦s✳} Pr♦♣♦s✐çã♦ ✶✳✷✳✶ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✮✳ ❙❡❥❛♠ 1 _≤ p, p′ _{≤ ∞ ❡①♣♦❡♥t❡s} ❝♦♥❥✉❣❛❞♦s✳ ❙❡ ❢ ❡ ❣ sã♦ ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s s♦❜r❡ X✱

kf g_kL1 ≤ kfk_Lpkgk

Lp′ ✭✶✳✶✮

P❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ✈❡❥❛ ❬✼❪✱ ♣á❣✳ ✶✼✹✳

❙❡p=p′ _{= 2✱ ✭✶✳✶✮ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛rt③✳} Pr♦♣♦s✐çã♦ ✶✳✷✳✷ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❤❡❜②s❤❡✈✮✳ ❙❡ f _∈ Lp _{(1 ≤ p < ∞)✱ ❡♥tã♦}

♣❛r❛ t♦❞♦ λ >0✱

µ(_{x:_|f(x)_|> λ_})_≤

kf_kLp

λ

p

❆ ♣r♦✈❛ ❞❡st❡s ❞♦✐s r❡s✉❧t❛❞♦s ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✼❪✱ ♥❛s ♣á❣✐♥❛s ✶✼✽ ❡ ✶✽✺✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥✉♥❝✐❛♠♦s ❛ s❡❣✉✐r ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ♥♦r♠❛ ❡♠ Lp

❛tr❛✈és ❞❡ ✐♥t❡❣r❛✐s s♦❜r❡ [0,_∞)✿

❚❡♦r❡♠❛ ✶✳✷✳✸✳ ❙❡p_∈[1,_∞)✱ ❡♥tã♦ ♣❛r❛ t♦❞❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ ❢ s♦❜r❡(X,_M, µ)✱

kfkpLp =p

Z ∞

0

λp−1µ({x:|f(x)|> λ})dλ

P❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ✈❡r ❬✼❪✱ ♣á❣✳ ✶✾✶✳

❙❡rá út✐❧✱ ❡♠ ❛❧❣✉♠❛s ❡st✐♠❛t✐✈❛s✱ ❛ ♣r♦♣♦s✐çã♦ ❛❜❛✐①♦ ❡ s❡✉ r❡s♣❡❝t✐✈♦ ❝♦r♦❧ár✐♦✱ q✉❡ ❣❡♥❡r❛❧✐③❛ ❛ ✐❞é✐❛ ❞❛ ✐♥t❡❣r❛çã♦ ❝♦♠ ❝♦♦r❞❡♥❛❞❛s ♣♦❧❛r❡s ♣❛r❛Rd_{✳ ◆♦✈❛♠❡♥t❡✱}

❛ ♣r♦✈❛ ♣♦❞❡ s❡r ✈✐st❛ ❡♠ ❬✼❪✱ ♥❛ ♣á❣✐♥❛ ✼✺✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✹✳ ❙❡❥❛ ❢ é ✉♠❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ s♦❜r❡ Rd_{✱ ♥ã♦✲♥❡❣❛t✐✈❛ ♦✉ ✐♥✲}

t❡❣rá✈❡❧ t❛❧ q✉❡ f(x) =g(_|x_|)✱ ♣❛r❛ ❛❧❣✉♠❛ ❢✉♥çã♦ g ❡♠ (0,_∞)✳ ❊♥tã♦

Z

Rd

f(x)dx=σ(Sd−1)

Z ∞

0

g(r)rd−1dr,

♦♥❞❡ σ(Sd−1_{) ❡①♣r❡ss❛ ❛ ♠❡❞✐❞❛ ❞❛ ár❡❛ ❞❡ S}d−1_✳

❈♦r♦❧ár✐♦ ✶✳✷✳✺✳ ❙❡❥❛ s_∈R✳ ❙❡ s > d/2✱ ❡♥tã♦

Z

Rd

dξ

(1 +_|ξ_|2₎s <∞.

❖ s❡❣✉✐♥t❡ ❚❡♦r❡♠❛ ❞❡ ❘❛❞❡♠❛❝❤❡r s❡rá ✉t✐❧✐③❛❞♦ ❡♠ ✭✹✳✶✮✳

❚❡♦r❡♠❛ ✶✳✷✳✻✳ ✭❚❡♦r❡♠❛ ❞❡ ❘❛❞❡♠❛❝❤❡r✮ ❙❡❥❛ f : Rn _{→ R}m _{✉♠❛ ❢✉♥çã♦ ❧♦❝❛❧✲}

♠❡♥t❡ ▲✐♣s❝❤✐t③✳ ❊♥tã♦ f é ❞✐❢❡r❡♥❝✐á✈❡❧ q✳t✳♣✳

P❛r❛ ✈❡r ❛ ❞❡♠♦♥str❛çã♦ ✈❡r ❬✻❪✳

❙❡❥❛ Ω ⊂ Rd _{❛❜❡rt♦ ❡ φ : Ω → C ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳ ❉❡✜♥✐♠♦s ♦ s✉✲}

♣♦rt❡ ❞❡ φ✱ ♦ q✉❛❧ ❞❡♥♦t❛r❡♠♦s ♣♦r S(φ)✱ ❝♦♠♦ s❡♥❞♦ ♦ ❢❡❝❤♦ ❡♠ Ω ❞♦ ❝♦♥❥✉♥t♦

{x∈Ω; φ(x)6= 0}✳

❉❡✜♥✐çã♦ ✶✳✷✳✼✳ ❙❡❥❛ Ω ✉♠ ❛❜❡rt♦ ❞❡ Rn_{✳ ❖ ❝♦♥❥✉♥t♦}

C_c∞(Ω) ={φ: Ω→C; u∈C∞ ❡ S(φ) é ❝♦♠♣❛❝t♦}

é ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s t❡st❡s✳

P❛r❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❢✉♥çõ❡s t❡st❡s ♥ã♦✲♥✉❧❛s✱ s❡rá út✐❧ ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✱ ❝✉❥❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ✈✐st❛ ♥❛ ♣á❣✐♥❛ ✼ ❞❡ ❬✾❪✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✽✳ ❙❡❥❛K ✉♠ s✉❜❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ ❞❡ ✉♠ ❛❜❡rt♦ Ω_⊂Rd_{✳ ❊①✐st❡} ψ ∈C∞

c (Ω) t❛❧ q✉❡ 0≤ψ ≤1 ❡ ψ = 1 ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ K✳

❚❡♠♦s q✉❡ C∞

c (Ω) é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡♥s♦ ❡♠ Lp(Ω)✱ ❝♦♠ 1 ≤ p < ∞✱

♣♦r ❬✼❪✳ ❈♦♥❢♦r♠❡ ❬✶✷❪✱ é ♣♦ssí✈❡❧ ❡q✉✐♣á✲❧♦ ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ t♦♣♦❧ó❣✐❝♦✱ ♥ã♦✲♠❡tr✐③á✈❡❧✱ ❞❡ ♠♦❞♦ q✉❡C∞

c (Ω)t♦r♥❡✲s❡ ✉♠ ❡s♣❛ç♦ ❝♦♠♣❧❡t♦✳ ❈♦♠

❡st❛ ❡str✉t✉r❛ t♦♣♦❧ó❣✐❝❛✱ t❡r❡♠♦s q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛(φj)j∈N ❞❡ ❢✉♥çõ❡s t❡st❡ ❝♦♥✲ ✈❡r❣❡ ❛ ③❡r♦ ❡♠ C∞

c (Ω) s❡ ❡①✐st❡ ✉♠ ❝♦♠♣❛❝t♦ K ⊂ K t❛❧ q✉❡ S(φj) ⊂Ω, ∀j ∈ N

❡✱ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ m✱ ❛s ❞❡r✐✈❛❞❛s ❞❡ ♦r❞❡♠ m ❝♦♥✈❡r❣❡ ✉♥✐❢♦r♠❡♠❡♥t❡ ❛ ③❡r♦ q✉❛♥❞♦j → ∞✳

❉❡✜♥✐çã♦ ✶✳✷✳✾✳ ❙❡❥❛Ω_⊆Rd_{❛❜❡rt♦✳ ❯♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❡ ❝♦♥tí♥✉♦u:C}∞ c (Ω) →

C é ❞✐t♦ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡♠ Ω✳ ❖ ❡s♣❛ç♦ ❞❛s ❞✐str✐❜✉✐çõ❡s ❡♠ Ω s❡ ❞❡♥♦t❛ ❝♦♠

D′_(Ω)✳

❉❡♥♦t❛♠♦s ♦ ✈❛❧♦r ❞❛ ❞✐str✐❜✉✐çã♦u ♥❛ ❢✉♥çã♦ t❡st❡ φ ♣♦r _hu, φ_i✳

❊①❡♠♣❧♦ ✶✳✷✳✶✵✳ ❙❡❥❛f ∈L1_❧♦❝(Ω) ❡ ❞❡✜♥❛♠♦s ♦ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r

hTf, φi=

Z

Ω

f(x)φ(x)dx, φ_∈C_c∞(Ω) ✭✶✳✷✮

❙❡ (φj)j∈N é ✉♠❛ s❡qüê♥❝✐❛ ❝♦♥✈❡r❣✐♥❞♦ ❛ ③❡r♦ ❡♠ Cc∞(Ω)✱ s❡❥❛ K ⊂ Ω ❝♦♠♣❛❝t♦

t❛❧ q✉❡ S(φj)⊂K✳ ❊♥tã♦

|hTf, φji| ≤

Z

K|

f(x)_||φj(x)|dx

≤ sup

x∈K| φj(x)|

Z

K|

f(x)_|dxj_−→→∞ 0

P♦rt❛♥t♦✱ ❛ ❡①♣r❡ssã♦ ✭✶✳✷✮ ❞❡✜♥❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡♠ Ω ✳ ❚❛♠❜é♠✱ s❡ f, g _∈ L1

❧♦❝(Ω) ❡ hTf, φi =hTg, φi✱ ♣❛r❛ t♦❞❛ φ ∈ Cc∞(Ω)✱ ❡♥tã♦ f = g q✳t✳♣ ✭✈❡r ♣r♦✈❛ ♥❛

♣á❣✳ 11❞❡ ❬✾❪✮✳

❉❡st❡ ♠♦❞♦✱ ❛ ❛♣❧✐❝❛çã♦ ✐♥❥❡t✐✈❛f _7→Tf ♥♦s ♣❡r♠✐t❡ ❝♦♥s✐❞❡r❛r ✈ár✐♦s ❡s♣❛ç♦s

❞❡ ❢✉♥çõ❡s ❝♦♠♦ s✉❜❡s♣❛ç♦s ❞❡ D′_{(Ω)✳ ➱ ❝♦♠✉♠ ❡s❝r❡✈❡r s✐♠♣❧❡s♠❡♥t❡ hf, φi ❛♦} ✐♥✈és ❞❡ hTf, φi✳

❉✐③❡♠♦s q✉❡ ✉♠❛ ❞✐str✐❜✉✐çã♦u_{∈ D}′_{(Ω) é ✐❣✉❛❧ ❛ ③❡r♦ ♥✉♠ ❛❜❡rt♦U ⊂Ωs❡}

hu, φi= 0✱ ♣❛r❛ t♦❞❛φ ∈C∞

c (U)✳ ❉❡✜♥✐♠♦s ❡♥tã♦ ♦ s✉♣♦rt❡ ❞❡ u✱ ❡ ❞❡♥♦t❡♠♦s ♣♦r S(u)✱ ❝♦♠♦ ❛ ✐♥t❡rs❡çã♦ ❞❡ t♦❞♦s ♦s ❢❡❝❤❛❞♦s ❞❡Ω❢♦r❛ ❞♦s q✉❛✐sué ♥✉❧❛✳ ❉❡♥♦t❛♠♦s ❝♦♠ E′_{(Ω) ♦ s✉❜❡s♣❛ç♦ ❞❡ D}′_{(Ω) ❞❛s ❞✐str✐❜✉✐çõ❡s ❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✳}

❚❡♦r❡♠❛ ✶✳✷✳✶✶✳ ❙❡❥❛ u_{∈ D}′_{(Ω)✳ ❊♥tã♦ S(u) é ❝♦♠♣❛❝t♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡} ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❡ ❝♦♥t✐♥✉♦ v ❡♠ C∞_{(Ω) ❝✉❥❛ r❡str✐çã♦ ❛ C}∞

c (Ω) é ✐❣✉❛❧ ❛ u✳

❆ ♣r♦✈❛ ❞❡st❡ ❚❡♦r❡♠❛ s❡ ❡♥❝♦♥tr❛ ♥❛ ♣á❣✐♥❛ ✹✶ ❞❡ ❬✾❪✳ ❆q✉✐✱ ❛ ♥♦çã♦ ❞❡ s❡q✉❡♥❝✐❛❧✲ ♠❡♥t❡ ❝♦♥tí♥✉❛ é ❛ s❡❣✉✐♥t❡✿ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s C∞_{(Ω) ❝♦♥✈❡r❣❡ ♣❛r❛ ③❡r♦} s❡✱ ♣❛r❛ t♦❞♦ ❝♦♠♣❛❝t♦ K ❡ t♦❞♦ ✐♥t❡✐r♦ m✱ ❛s ❞❡r✐✈❛❞❛s ❞❡ ♦r❞❡♠ m ❝♦♥✈❡r❣❡♠ ✉♥✐❢♦r♠❡♠❡♥t❡ ❛ ③❡r♦ ❡♠ K q✉❛♥❞♦ j _{→ ∞}✳

❉❡✜♥✐çã♦ ✶✳✷✳✶✷✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ uj ∈ D′(Ω), j ∈ N✱ ❝♦♥✈❡r❣❡ ❛ u_{∈ D}′_{(Ω) s❡ hu}

j, φi ❝♦♥✈❡r❣❡ ❛ hu, φi✱ ♣❛r❛ t♦❞❛ φ ∈Cc∞(Ω)✳

❙✉♣♦♥❤❛♠♦s q✉❡un✱ n= 1,2, ... é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❞✐str✐❜✉✐çõ❡s ❡♠D′(Ω) t❛❧ q✉❡ un(φ) é ❝♦♥✈❡r❣❡♥t❡ ♣❛r❛ ❝❛❞❛ φ ∈Cc∞(Ω)✳ ❙❡ ❞❡✜♥✐r♠♦s u(φ) = lim

n→∞un(φ)✱ t❡♠♦s q✉❡u é ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r✳ ▼❛✐s ❛✐♥❞❛✱ ur❡s✉❧t❛ ❝♦♥tí♥✉♦ ❡♠ C∞

c (Ω)✳

❚❡♦r❡♠❛ ✶✳✷✳✶✸✳ ❙❡❥❛(un)n∈N ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❞✐str✐❜✉✐çõ❡s ❡♠Ω✱ ❡ s✉♣♦♥❤❛♠♦s q✉❡✱ ♣❛r❛ t♦❞❛ φ _∈ C∞

c (Ω)✱ hun, φi é ✉♠❛ s❡q✉ê♥❝✐❛ ♥✉♠ér✐❝❛ ❞❡ ❈❛✉❝❤②✳ ❊♥tã♦

(un)n∈N é ❝♦♥✈❡r❣❡♥t❡ ❡♠ D′(Ω)✳

❱❡r ❬✾❪✱ ♣á❣✐♥❛ ✺✻✱ ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦✳

✶✳✷✳✶ ❖♣❡r❛çõ❡s ❝♦♠ ❉✐str✐❜✉✐çõ❡s

❙❡❥❛ u _∈ C∞

c (Ω)✳ ❈♦♠♦ u é ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧✱ ❛ ❡①♣r❡ssã♦ ✭✶✳✷✮ ♥♦s

♣❡r♠✐t❡ ❝♦♥s✐❞❡rá✲❧❛ ❝♦♠♦ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡♠Ω✳ P♦r ✐♥t❡❣r❛çã♦ ♣♦r ♣❛rt❡s✱ t❡♠♦s

Z

∂u ∂xj

(x)φ(x)dx=−

Z

u(x)∂φ ∂xj

(x)dx,

♣❛r❛ t♦❞❛ ❢✉♥çã♦ t❡st❡ φ✳ ❉❡st❛ ♠❛♥❡✐r❛✱ ♣♦r ❞✉❛❧✐❞❛❞❡✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❞✐s✲ tr✐❜✉✐çã♦

h∂xju, φi=−hu, ∂xjφi,

♣❛r❛ t♦❞❛u_{∈ D}′_{(Ω)✳ P♦r ✐♥❞✉çã♦ ❡♠ |α|✱}

h∂αu, φi= (−1)|α|hu, ∂αφi

P❡❧♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦✱ ❞❡✜♥✐♠♦s ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ✉♠❛ ❢✉♥çã♦f ∈C∞_(Ω) ❝♦♠♦ s❡♥❞♦ ❛ ❞✐str✐❜✉✐çã♦

hf u, φi=hu, f φi

❆❣♦r❛✱ s❡❥❛♠ ❢✱ ❣ ❞✉❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❡♠ Rd _{t❛❧ q✉❡ ✉♠❛ ❞❡❧❛s t❡♥❤❛}

s✉♣♦rt❡ ❝♦♠♣❛❝t♦✳ ❊♥tã♦ ❛ ❝♦♥✈♦❧✉çã♦ ❞❡ ❢ ❡ ❣ s❡ ❞❡✜♥❡ ❝♦♠♦

f_∗g(x) =

Z

f(x₋y)g(y)dy =

Z

f(y)g(x₋y)dy

❆ ✜♠ ❞❡ ❡st❡♥❞❡r ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ ♣❛r❛ ♦ ❝♦♥t❡①t♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✱ ❝♦♥✲ s✐❞❡r❡♠♦s ❛

❉❡✜♥✐çã♦ ✶✳✷✳✶✹✳ ❙❡❥❛ u ∈ D′_{(Ω) ❡ φ ∈ C}∞

c (Ω) (♦✉ u ∈ E′(Ω) ❡ φ ∈ C∞(Ω))✳

❉❡✜♥✐♠♦s ❛ ❝♦♥✈♦❧✉çã♦ ❞❡ u ❡ φ✱ ❞❡♥♦t❛❞❛ ♣♦r u_∗φ✱ ❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r

u∗φ(x) = hu,φ˘xi,

♦♥❞❡ _φ˘_{x(y) = φ(x−y)✳}

❱❛❧❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✭✐✮ u_∗φ_∈C∞_{(Ω) ❡ s✉❛s ❞❡r✐✈❛❞❛s sã♦ ❞❛❞❛s ♣♦r}

∂α(u∗φ) =∂αu∗φ =u∗∂αφ

✭✐✐✮ S(u_∗φ)_⊂S(u) +S(φ) ❚❡♦r❡♠❛ ✶✳✷✳✶✺✳ C∞

c (Ω) é ❞❡♥s♦ ❡♠ D′(Ω)✳

P❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ✈❡r ❬✾❪✱ ♣á❣✳ ✻✹

✶✳✸ ❉✐str✐❜✉✐çõ❡s ❚❡♠♣❡r❛❞❛s ❡ ❚r❛♥s❢♦r♠❛❞❛ ❞❡

❋♦✉r✐❡r

❙❡f _∈L1_(Rd_{)✱ ❞❡✜♥✐♠♦s ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❡ ❢ ♣♦r}

Ff(ξ) = fb(ξ) =

Z

e−ix.ξf(x) dx, ξ _∈Rd

♦♥❞❡ i é ❛ ✉♥✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛ ❡ x.ξ é ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❝❛♥ô♥✐❝♦✳

❙❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ q✉❡ ❛ ❛♣❧✐❝❛çã♦ f 7→ fb❞❡✜♥❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❞❡ L1_(Rd_{) ❡♠ L}∞_(Rd_{)s❛t✐s❢❛③❡♥❞♦}

kfb_kL∞ ≤ kfk_L1. ✭✶✳✸✮

▼❛✐s ❛✐♥❞❛✱ ❡stá é ✉♠❛ ❛♣❧✐❝❛çã♦ q✉❡ ❧❡✈❛L1 _{♥♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s q✉❡}

s❡ ❛♥✉❧❛♠ ♥♦ ✐♥✜♥✐t♦✳

▲❡♠❛ ✶✳✸✳✶ ✭❘✐❡♠❛♥♥✲▲❡❜❡s❣✉❡✮✳ ❙❡❥❛ f _∈ L1_(Rd_{)✳ ❊♥tã♦ f}b_{é ✉♠❛ ❢✉♥çã♦ ❝♦♥✲}

tí♥✉❛ s❛t✐s❢❛③❡♥❞♦ fb(ξ)_→0 q✉❛♥❞♦ _|ξ_{| → ∞}✳

❱❡r ❬✾❪✱ ♣á❣✳ ✼✺✱ ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦✳

❙❡φ é ✉♠❛ ❢✉♥çã♦ t❡st❡✱ ♣r♦✈❛✲s❡ q✉❡ s✉❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r é ❛♥❛❧ít✐❝❛ ❝♦♠♣❧❡①❛ ❡♠ Cd_{✳ ❆ss✐♠✱ φ}b _{♥ã♦ t❡rá s✉♣♦rt❡ ❝♦♠♣❛❝t♦✱ ❛ ♠❡♥♦s q✉❡ φ s❡❥❛ ♥✉❧❛✱}

✉♠❛ ✈❡③ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ③❡r♦s ❞❡ ✉♠❛ ❢✉♥çã♦ ❛♥❛❧ít✐❝❛ ❝♦♠♣❧❡①❛ ♥ã♦✲♥✉❧❛ ❡♠

Cd _{t❡♠ ✐♥t❡r✐♦r ✈❛③✐♦✳}

❈♦♥s✐❞❡r❡♠♦s ❡♥tã♦ ✉♠ ❡s♣❛ç♦ q✉❡ ❝♦♥té♠ ❛s ❢✉♥çõ❡s ❞❡ s✉♣♦rt❡ ❝♦♠♣❛❝t♦ ❡ q✉❡ s❡❥❛ ✐♥✈❛r✐❛♥t❡ ♣❡❧❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✳

❉❡✜♥✐çã♦ ✶✳✸✳✷ ✭❊s♣❛ç♦ ❞❡ ❙❝❤✇❛rt③✮✳ ❉❡♥♦t❛♠♦s ♣♦r S ♦ s✉❜❡s♣❛ç♦ ❞❡ C∞_(Rd₎

❞❛s ❢✉♥çõ❡s φ t❛✐s q✉❡

kφkN,α = sup x∈Rd

(1 +|x|)N|∂αφ|<∞ ✭✶✳✹✮

♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ ♥ã♦✲♥❡❣❛t✐✈♦ N ❡ ♣❛r❛ t♦❞♦ α_∈Nd_✳

❚❛♥t♦ ❛s ❢✉♥çõ❡s ❞❡ S q✉❛♥t♦ ❛s s✉❛s ❞❡r✐✈❛❞❛s ❞❡❝r❡s❝❡♠ ♥♦ ✐♥✜♥✐t♦ ♠❛✐s

r❛♣✐❞❛♠❡♥t❡ ❞♦ q✉❡ q✉❛❧q✉❡r ♣♦tê♥❝✐❛ ♥❡❣❛t✐✈❛ ❞❡ |x|✳ ▼✉♥✐r❡♠♦s ♦ ❡s♣❛ç♦ ❞❡

❙❝❤✇❛rt③S ❝♦♠ ❛ t♦♣♦❧♦❣✐❛ ❞❛❞❛ ♣❡❧❛ ❢❛♠í❧✐❛ ❡♥✉♠❡rá✈❡❧ ❞❡ s❡♠✐✲♥♦r♠❛s ❡♠ ✭✶✳✹✮✳

❊①❡♠♣❧♦ ✶✳✸✳✸✳ ❖ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s✲t❡st❡s ❡stá ❝♦♥t✐❞♦ ❞❡♥s❛♠❡♥t❡ ❡♠ S✱ ♠❛s

♣❛r❛ x _∈ Rd_{✱ φ(x) = e}−|x|2

♣❡rt❡♥❝❡ ❛ S✱ ♣♦ré♠ ♥ã♦ ♣♦ss✉✐ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✳

❆ss✐♠✱ C∞ c S✳

❚❡♥❞♦ ❡♠ ✈✐st❛ ❛ Pr♦♣♦s✐çã♦ ✶✳✷✳✹✱ s❡❣✉❡ q✉❡ s❡ φ∈ S✱

k∂αφ_kLp = Z

|∂αφ(x)_|p(1 +|x|)

n+1

(1 +_|x_|)n+1dx 1/p

≤ C sup

x∈Rd

(1 +_|x_|)n+1p |∂α(x)φ| ≤Ckφk

N,α, ✭✶✳✺✮

♦♥❞❡ ◆ é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ♠❛✐♦r q✉❡ n+1

p ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ S ֒→L p_✳

❚❡♦r❡♠❛ ✶✳✸✳✹✳ ❆ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r F : S → S é ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❡

❝♦♥tí♥✉♦✱ ❝♦♥t✐♥✉❛♠❡♥t❡ ✐♥✈❡rsí✈❡❧✱ ❝✉❥❛ tr❛♥s❢♦r♠❛❞❛ ✐♥✈❡rs❛ é ❞❛❞❛ ♣♦r

F−1_{φ(x) = ˇφ(x) =} 1

(2π)n

Z

eix.ξφ(ξ)dξ, φ∈ S

❆ ♣r♦✈❛ ❞❡st❡ r❡s✉❧t❛❞♦ ❡♥❝♦♥tr❛✲s❡ ♥❛ ♣á❣✐♥❛ ✼✼ ❞❡ ❬✾❪✳ ❙❡ φ✱ ψ ∈ S✱ ✉t✐❧✐③❛♥❞♦ ♦

t❡♦r❡♠❛ ❛♥t❡r✐♦r ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❝♦♥✈♦❧✉çã♦✱ ♣r♦✈❛✲s❡ ❛s s❡❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s✿

✭✐✮ ∂dα_{φ(ξ) = (iξ)}α_φb_(ξ)

✭✐✐✮ F(xα_{φ(x))(ξ) = i}|α|_∂α_φb_(ξ)

✭✐✐✐✮ Z φψ dxb =

Z

φψ dxb

✭✐✈✮ φ[_∗ψ =φbψb

✭✈✮ _φψc _{= (2π)}−d_φb_∗ψb

✭✈✐✮ bbu= (2π)−d_{u˘✱ ❝♦♠ u˘(ξ) = u(−ξ)✳}

❚❛♠❜é♠✱ ♣❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✱ ❣❛r❛♥t✐❞❛ ♣❡❧♦ ❚❡♦✲ r❡♠❛ ✶✳✸✳✹✱ ♦✉tr❛s ❢❛♠í❧✐❛s ❞❡ s❡♠✐✲♥♦r♠❛s q✉❡ ♣♦❞❡♠ s❡r ✉s❛❞❛s ♣❛r❛ ❞❡✜♥✐r ❛ t♦♣♦❧♦❣✐❛ ❡♠ S sã♦ ❞❛❞❛s ♣♦r

kf_kk,S = sup |α|≤k x∈Rd

(1 +_|ξ_|)k _|∂αf(x)_|, k _∈N

kfbkk = sup |α|≤k ξ∈Rd

(1 +|ξ|)k |∂αfb(ξ)|, k ∈N

❉❡✜♥✐çã♦ ✶✳✸✳✺✳ ❯♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❡ ❝♦♥t✐♥✉♦ ❡♠ S é ❞✐t♦ ✉♠❛ ❞✐str✐❜✉✐çã♦

t❡♠♣❡r❛❞❛✳ ❖ ❡s♣❛ç♦ ❞❛s ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛❞❛s s❡ ❞❡♥♦t❛ ❝♦♠ S′_✳

❊①❡♠♣❧♦ ✶✳✸✳✻✳ ❯♠❛ ✈❡③ q✉❡ t♦❞❛ ❞✐str✐❜✉✐çã♦ ❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦ s❡ ❡st❡♥❞❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ❛C∞_(Rd_{)✱ ✈❛❧❡ ❛ ✐♥❝❧✉sã♦ E}′ _{⊂ S}′_{✳ P♦r r❡str✐çã♦ ❛ C}∞

c (Rd)✱ ❡ ❝♦♠♦

❡st❡ é ❞❡♥s♦ ❡♠ S✱ t❡♠♦s q✉❡ S′ _{⊂ D}′_{✳ ❙❡ f ∈L}p_{✱ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝á✲❧❛ ❝♦♠♦ ✉♠❛}

❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛ ❞❡✜♥✐♥❞♦✱ ♣❛r❛ ❝❛❞❛ φ_{∈ S}✱

hTf, φi=

Z

f(x)φ(x)dx.

❆ ❧✐♥❡❛r✐❞❛❞❡ é ✐♠❡❞✐❛t❛ ❡ ❛ ✐♥t❡❣r❛❧ ❛❝✐♠❛ é ✜♥✐t❛✱ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✳ P❛r❛ ✈❡r✐✜❝❛r ❛ ❝♦♥t✐♥✉✐❞❛❞❡✱ s❡φ _{∈ S}✱ s❡❣✉❡ ❞❡ ✭✶✳✺✮ q✉❡

|hTf, φi| ≤ kfkLpkφk_Lp′

≤ C_kf_kLpkφk_N,1

❉❛ ❡st✐♠❛t✐✈❛ ❛❝✐♠❛✱ ❝♦♥❝❧✉í♠♦s t❛♠❜é♠ q✉❡ Lp _{֒→ S}′_✳

❚❡♦r❡♠❛ ✶✳✸✳✼ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣✮✳ ❙❡ f _∈Lp _{❡ g ∈L}q_{✱ ❡♥tã♦ ❛ ❝♦♥✈♦❧✉çã♦} f∗g ∈Lr_{✱ ❝♦♠} 1

r + 1 = 1 p+

1

q✱ ❡ ✈❛❧❡

kf ∗gkLr ≤ kfk_Lpkgk_Lq

❱❡r ❬✼❪✱ ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦✳

❉❡✜♥✐çã♦ ✶✳✸✳✽✳ ❙❡ u∈ S′_{✱ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ub ❞❡ u s❡ ❞❡✜♥❡ ♣♦r}

hbu, φ_i=_hu,φb_i

P❡❧♦ ❚❡♦r❡♠❛ ✶✳✸✳✹✱ bu ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡ ❞❡t❡r♠✐♥❛ ✉♠❛ ♥♦✈❛ ❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛✳ ▼❛✐s ❛✐♥❞❛✱F r❡s✉❧t❛ ❝♦♥tí♥✉❛ ❡ ✐♥✈❡rsí✈❡❧ ❡♠S′_✳

Pr♦♣♦s✐çã♦ ✶✳✸✳✾✳ ❙❡❥❛ f ∈ S′_(Rd_)✳

✭✐✮ ❙❡ f _∈ L1_(Rd_{)✱ ❛ tr❛♥s❢♦r♠❛❞❛ f}b _{❞❡ f ❝♦♠♦ ❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛ ❡ ❝♦♠♦}

❢✉♥çã♦ ✐♥t❡❣rá✈❡❧ ❝♦✐♥❝✐❞❡♠✳

✭✐✐✮ ❙❡ f ∈ E′_(Rd_{)✱ f}b_{é ✉♠❛ ❢✉♥çã♦ ❞❡ ❝❧❛ss❡ C}∞ _{❞❛❞❛ ♣♦r}

b

f(ξ) = _hfx, e−ixξi ✭✶✳✻✮

✭✐✐✐✮ ❙❡ f ∈L2_(Rd_{) ❡♥tã♦ f}b_∈L2_(Rd_{)✱ ❡ ✈❛❧❡}

kf_k2_L2 = (2π)−dkfbk2_L2 (■❞❡♥t✐❞❛❞❡ ❞❡ ❋♦✉r✐❡r✲P❧❛♥❝❤❡r❡❧)

❱❡r ❬✾❪✱ ♣á❣✳ ✽✶✱ ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦✳

❖❜s❡r✈❛çã♦ ✶✳✸✳✶✵✳ ❈♦♥s✐❞❡r❡♠♦s ✉♠❛ ❞✐str✐❜✉✐çã♦ u ∈ S′ _{t❛❧ q✉❡ bu ∈ L}1_(Rd_)✳

P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✸✳✾✱ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❡ubé ❛ ❢✉♥çã♦ ❞❛❞❛ ♣♦r bb

u(ξ) =

Z

e−ix.ξ_bu(x)dx

❈♦♠♦ bbu= (2π)d_{u˘✱ ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ♥♦s ❞á q✉❡}

u(ξ) = 1 (2π)d

Z

eix.ξ_ub(x)dx

❆ss✐♠✱ s❡bu_∈L1_{✱ ♣♦❞❡♠♦s ✧r❡❝✉♣❡r❛r✧u♣❡❧❛ ❢ór♠✉❧❛ ❞❡ ✐♥✈❡rsã♦ ❞❛❞❛ ♣❡❧♦}

❚❡♦r❡♠❛ ✶✳✸✳✹✳ P♦r ❛r❣✉♠❡♥t♦ s❡♠❡❧❤❛♥t❡ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ❞❡ ❘✐❡♠❛♥♥✲ ▲❡❜❡s❣✉❡✱ ♦❜t❡♠♦s q✉❡ u _∈ C0

0(Rd)✱ ♦✉ s❡❥❛✱ u é ❝♦♥tí♥✉❛ ❡ ✈❛✐ ❛ ③❡r♦ q✉❛♥❞♦

|ξ| → ∞✳

❚❛♠❜é♠✱ ♣❡❧❛ ♠❡s♠❛ ❡①♣r❡ssã♦✱

|u(ξ)_{| ≤}(2π)−d_kbu_kL1,

♣❛r❛ q✉❛s❡ t♦❞♦ ξ_∈Rd_{✱ ❞♦♥❞❡}

ku_kL∞ ≤(2π)−dkubk_L1. ✭✶✳✼✮

✶✳✹ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈

◆❡st❡ t❡①t♦✱ ✈❛♠♦s ♥♦s r❡str✐♥❣✐r ❛♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ♠♦❞❡❧❛❞♦s ❡♠ L2_✳

❊st❡s ❡s♣❛ç♦s ❞❡s❡♠♣❡♥❤❛♠ ✉♠ ♣❛♣❡❧ ❝r✉❝✐❛❧ ♥♦ ❡st✉❞♦ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s✱ ❧✐♥❡❛r❡s ♦✉ ♥ã♦✳ ❖ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ s❡rá ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✳

❉❡✜♥✐çã♦ ✶✳✹✳✶✳ ❙❡❥❛ s ✉♠ ♥ú♠❡r♦ r❡❛❧✳ ❯♠❛ ❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛ u ♣❡rt❡♥❝❡ ❛♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ ❞❡ í♥❞✐❝❡ s✱ ❞❡♥♦t❛❞♦ ♣♦r Hs_(Rd_{) s❡✱ ❡ s♦♠❡♥t❡ s❡✱}

b

u∈L2_❧♦❝(Rd_{) ❡ ub(ξ)∈L}2_(Rd_{,(1 +|ξ|}2₎s_dξ)

❊s❝r❡✈❡♠♦s

kuk2

Hs = Z

Rd

(1 +|ξ|2₎s_|ub(ξ)|2_dξ

Pr♦♣♦s✐çã♦ ✶✳✹✳✷✳ P❛r❛ t♦❞♦ ❡ q✉❛❧q✉❡rs∈R✱ ♦ ❡s♣❛ç♦ Hs _{❡q✉✐♣❛❞♦ ❝♦♠ ❛ ♥♦r♠❛}

k · kHs é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✳

❉❡♠♦♥str❛çã♦✿ ➱ ✐♠❡❞✐❛t♦ q✉❡ ❛ ♥♦r♠❛ k · kHs ♣r♦✈é♠ ❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦

hu, v_iHs = Z

Rd

(1 +_|ξ_|2)s_bu(ξ)_bv(ξ)dξ

Pr♦✈❡♠♦s ❡♥tã♦ q✉❡ ❡st❡ ❡s♣❛ç♦ é ❝♦♠♣❧❡t♦✳ ❙❡❥❛(un)n∈N ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠Hs_{✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❛ ♥♦r♠❛✱ (ub}

n)n∈Né ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠L2(Rd,(1+

|ξ_|2₎s_dξ)✳

❈♦♠♦ ❡st❡ é ❝♦♠♣❧❡t♦✱ ❡①✐st❡ u˜_∈L2_(Rd_{,(1 +|ξ|}2₎s_{dξ)t❛❧ q✉❡}

lim

n→∞kubn−u˜kL2(Rd,(1+|ξ|2)sdξ) = 0. ✭✶✳✽✮

❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ s❡q✉ê♥❝✐❛ (u_bn) ❝♦♥✈❡r❣❡ ❛ u˜ ❡♠ S′✳ ❚♦♠❡♠♦s u = F−1u˜✳

❈♦♠♦ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ S′_{✱ s❡❣✉❡ q✉❡ u ∈ S}′_{✳ P♦r}

✜♠✱ un→u ❡♠ Hs ❞❡✈✐❞♦ ❛ ✭✶✳✽✮✳

◆♦t❡♠♦s q✉❡ ♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ❢♦r♠❛♠ ✉♠❛ ❢❛♠í❧✐❛ ❞❡❝r❡s❝❡♥t❡ ❞❡ ❡s✲ ♣❛ç♦s✱ ❝♦♠ r❡s♣❡✐t♦ ❛♦ í♥❞✐❝❡ s✳ ❉❡ ❢❛t♦✱ s _≥ s′ _{✐♠♣❧✐❝❛ (1 +|ξ|}2₎s′

≤ (1 +_|ξ_|2₎s_✳

P♦rt❛♥t♦✱ s❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛ f é t❛❧ q✉❡ fb_∈L2

❧♦❝(Rd)✱ s❡❣✉❡ q✉❡

kf_k2_Hs′ = Z

Rd

(1 +_|ξ_|2)s′_|u_b(ξ)_|2dξ

≤

Z

Rd

(1 +|ξ|2₎s_|bu(ξ)|2_{dξ =kfk}2

Hs.

❆ss✐♠✱ Hs_(Rd_)⊆Hs′

(Rd_{) ❡ t❛❧ ✐♥❝❧✉sã♦ é ❝♦♥tí♥✉❛✳}

❖s t❡♦r❡♠❛s q✉❡ s❡❣✉❡♠ tê♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❝❛r❛❝t❡r✐③❛r ♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ♣❛r❛ ❞❡t❡r♠✐♥❛❞♦s ✈❛❧♦r❡s ❞❡ s s❡♠ ♦ ✉s♦ ❡①♣❧í❝✐t♦ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✳ ❚❡♦r❡♠❛ ✶✳✹✳✸✳ ❙❡❥❛ s ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♥ã♦✲♥❡❣❛t✐✈♦✳ ❖ ❡s♣❛ç♦ Hs_(Rd_{) é ♦}

❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s u ♣❡rt❡♥❝❡♥t❡s ❛ L2 _{t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ α ❡♠ N}d_{✱ ❝♦♠ |α| ≤ s}

t❡♠♦s ∂α_u∈L2_✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ s ∈ N✳ P❡❧♦ ❜✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥✱ t❡♠♦s (1 + |ξ|2₎s ₌ s

X

i=0

s i

|ξ_|2i✳ ❆❣♦r❛✱ ✜①❛❞♦ 0_≤i_≤s ❡ u_∈L2_✱

|ξ|2i_|bu(ξ)|2 _{= (ξ}2

1 +...+ξ2d)i|ub(ξ)|2

= X |α|=i

cα|ξαbu(ξ)|2

= X |α|=i

cα|∂dαu|2,

♣♦✐s ∂dα_{u(ξ) = (iξ)}α_{bu(ξ)✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✸✳✹✳ ❆ss✐♠✱}

(1 +_|ξ_|2)s_|u_b(ξ)_|2 =

s

X

i=0 X

|α|=i cα

s i

|∂dα_u|2

= X |α|≤s

˜

cα|∂dαu|2

■♥t❡❣r❛♥❞♦ ❡♠ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❡ ✉t✐❧✐③❛♥❞♦ ❛ ■❞❡♥t✐❞❛❞❡ ❞❡ ❋♦✉r✐❡r✲P❧❛♥❝❤❡r❡❧✱ s❡❣✉❡ q✉❡

Z

(1 +_|ξ_|2)s_|u_b(ξ)_|2dξ = Z X |α|≤s

˜

cα|∂dαu|2dξ

= X |α|≤s

(2π)d˜cαk∂αuk2L2

❚♦♠❛♥❞♦C1 = min

(2π)d_˜c

α; |α| ≤s ❡C2 = max

(2π)d_˜c

α; |α| ≤s ✱ ♦❜t❡✲

♠♦s

C1 X

|α|≤s

k∂αu_k2_L2 ≤

Z

(1 +_|ξ_|2)s_|u_b(ξ)_|2dξ_≤C2 X

|α|≤s

k∂αu_k2_L2 ✭✶✳✾✮

❈♦r♦❧ár✐♦ ✶✳✹✳✹✳ S é ❝♦♥t✐♥✉❛♠❡♥t❡ ✐♥❝❧✉í❞♦ ❡♠ Hs _{✱ ∀s∈R}

❉❡♠♦♥str❛çã♦✿ ❚❡♥❞♦ ❡♠ ✈✐st❛ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛❞❛ ♣❡❧♦ ❚❡♦r❡♠❛ ❛♥t❡r✐♦r✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✺✮✱ s❡ s_∈N✱

ku_k2

Hs ≤C X

|α|≤s

k∂α_uk2

L2 ≤C

X

|α|≤s

kφ_k2

N,α,

♣❛r❛ t♦❞❛ ❢✉♥çã♦u∈ S✳ ❆ss✐♠✱ S ֒→Hs_{✱ s❡ s é ♥❛t✉r❛❧✳}

P♦r ✜♠✱ s❡ s ∈ R q✉❛❧q✉❡r✱ ❞❡♥♦t❛♥❞♦ ♣♦r ⌈s⌉ ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ♠❛✐♦r ♦✉

✐❣✉❛❧ q✉❡s✱ s❡❣✉❡ q✉❡ _S ֒_→H⌈s⌉ _֒→Hs_{✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❡♥❝❛✐①❡✳}

❖ r❡s✉❧t❛❞♦ ❛ s❡❣✉✐r ❞❡s❝r❡✈❡ ♦ ❞✉❛❧ t♦♣♦❧ó❣✐❝♦ ❞♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈✱ q✉❡ ♥♦s ♣❡r♠✐t✐rá ✐❞❡♥t✐✜❝❛r✱ ❛ ♠❡♥♦s ❞❡ ✉♠ ✐s♦♠♦r✜s♠♦✱ ♦ ❡s♣❛ç♦H−s _{❝♦♠♦ ♦ ❞✉❛❧ ❞❡} Hs_✳

❚❡♦r❡♠❛ ✶✳✹✳✺ ✭❖ ❉✉❛❧ ❞❡ Hs_{✮✳ ❆ ❢♦r♠❛ ❜✐❧✐♥❡❛r ❇ ❞❡✜♥✐❞❛ ♣♦r}

    

B :S × S −→ C

(u, ϕ) 7→ B(u, ϕ) =

Z

Rd

u(x)ϕ(x)dx

♣♦❞❡ s❡r ❡st❡♥❞✐❞❛ ♣❛r❛ ✉♠❛ ❢♦r♠❛ ❜✐❧✐♥❡❛r ❝♦♥tí♥✉❛ ❞❡ Hs_×H−s _{♣❛r❛ C✳ ❆❧é♠}

❞✐ss♦✱ ❛ ❛♣❧✐❝❛çã♦ δB ❞❡✜♥✐❞❛ ♣♦r

  

δB :H−s −→ (Hs)∗

u _7→ δB(u) :ϕ 7→B(u, ϕ)

é ❧✐♥❡❛r ❡ ✉♠ ✐s♦♠♦r✜s♠♦ ✐s♦♠étr✐❝♦ ✭❛ ♠❡♥♦s ❞❡ ✉♠❛ ❝♦♥st❛♥t❡✮✳

❉❡♠♦♥str❛çã♦✿ P❛r❛u, ϕ _{∈ S}✱ t❡♠♦s

B(u, ϕ) =

Z

u(x)ϕ(x)dx

=

Z

u(x)_F(_F−1ϕ)(x)dx

= (2π)−d

Z b

u(ξ)(_F−1ϕ)(ξ)dξ

= (2π)−d

Z b

u(ξ)ϕ_b(₋ξ)dξ.

▼✉❧t✐♣❧✐❝❛♥❞♦ ❡ ❞✐✈✐❞✐♥❞♦ ♣♦r(1 +|ξ|2₎s/2 _{❡ t♦♠❛♥❞♦ ♦ ♠ó❞✉❧♦✱ s❡❣✉❡ q✉❡}

B(u, ϕ) _≤ (2π)−d

Z

|bu(ξ)_||ϕ_b(₋ξ)_|dξ

= (2π)−d

Z

(1 +_|ξ_|2)s/2_|bu(ξ)_|(1 +_|ξ_|2)−s/2_|ϕ_b(₋ξ)_|dξ

≤ (2π)−d

Z

(1 +|ξ|2₎s_|ub(ξ)|2_dξ

1/2Z

(1 +|ξ|2₎−s_|ϕb(−ξ)|2_dξ 1/2

= (2π)−d_kuk

Hskϕk_H−s

P♦rt❛♥t♦✱ ❛ ❛♣❧✐❝❛çã♦ ❇ é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥tí♥✉❛ ❡♠Hs_×H−s_{✳ ▼❛sS ×S é ❞❡♥s♦}

♥❡st❡ ❡s♣❛ç♦✱ ♣❡❧♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✳ ▲♦❣♦✱ ❡st❛ ❢♦r♠❛ ❜✐❧✐♥❡❛r s❡ ❡st❡♥❞❡ ❛ ✉♠❛ ú♥✐❝❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❞❡Hs_×H−s _{❡♠ C✱ t❛♠❜é♠ ❞❡♥♦t❛❞❛ ♣♦r B✳}

❆❧é♠ ❞✐ss♦✱ ♣❛r❛ ❝❛❞❛u_∈H−s _{✜①❛✱δ}

B(u)é ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❡ ❝♦♥tí♥✉♦ ❡♠ Hs✳

❘❡st❛ ♠♦str❛r♠♦s q✉❡ ❛ ❛♣❧✐❝❛çã♦δB é ✉♠ ✐s♦♠♦r✜s♠♦✳

❆ ❧✐♥❡❛r✐❞❛❞❡ é ✐♠❡❞✐❛t❛✳ ◗✉❛♥t♦ ❛ ✐♥❥❡t✐✈✐❞❛❞❡✱ B(u, ϕ) = 0 é ❡q✉✐✈❛❧❡♥t❡ ❛ ❞✐③❡r

q✉❡ _Z

u(x)ϕ(x)dx= 0,

♣❛r❛ t♦❞❛ ❢✉♥çã♦ϕ _{∈ S}✱ ♦✉ s❡❥❛✱ u= 0 ❡♥q✉❛♥t♦ ❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛✳

P❛r❛ ♣r♦✈❛r ❛ s♦❜r❡❥❡t✐✈✐❞❛❞❡✱ s❡Ψ :Hs_(Rd_{)→Cé ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❞✉❛❧ t♦♣♦❧ó❣✐❝♦}

❞❡Hs_(Rd_{)✱ ❝♦♠♦ ❡st❡ é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ✭Pr♦♣♦s✐çã♦ ✶✳✹✳✷✮✱ ♦ ▲❡♠❛ ❞❛ ❘❡♣r❡✲}

s❡♥t❛çã♦ ❞❡ ❘✐❡s③ ♥♦s ❞✐③ q✉❡ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ h_∈Hs_(Rd_{) t❛❧ q✉❡}

Ψ(f) = hϕ, hiHs = Z

Rd

(1 +|ξ|2₎s_{ϕb(ξ)bh(ξ)dξ,}

♣❛r❛ t♦❞❛ϕ _∈Hs_✳

❙❡❥❛u_{∈ S}′ _{t❛❧ q✉❡ bu(ξ) = (2π)}d_{(1 +|ξ|}2₎sb_h(−ξ)✳

❖❜s❡r✈❡♠♦s q✉❡u_∈H−s_(Rd_{)✱ ♣♦✐s}

Z

(1 +|ξ|2₎−s_|ub(ξ)|2_{dξ = (2π)}d

Z

(1 +|ξ|2₎−s_{(1 +|ξ|}2₎2s_|b_h(−ξ)|2_dξ

= (2π)d

Z

(1 +_|ξ_|2₎s_|b_h(ξ)|2_{dξ= (2π)}d_khk

Hs <∞

❆ss✐♠✱ ♣❛r❛ ϕ ∈Hs_✱

δB(u)(ϕ) = B(u, ϕ) = (2π)−d

Z b

u(ξ)ϕ_b(₋ξ)dξ

=

Z

(1 +_|ξ_|2)sbh(₋ξ)ϕ_b(₋ξ) dξ

=

Z

(1 +_|ξ_|2)sbh(ξ)ϕ_b(ξ) dξ= Ψ(ϕ)

P♦rt❛♥t♦✱ δB(u) = Ψ✱ ❝♦♠♦ q✉❡rí❛♠♦s✳

✶✳✹✳✶ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ❍♦♠♦❣ê♥❡♦

❉❡✜♥✐çã♦ ✶✳✹✳✻✳ ❙❡❥❛ s ✉♠ ♥ú♠❡r♦ r❡❛❧✳ ❖ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ ❤♦♠♦❣ê♥❡♦ H˙s _{é ♦}

❝♦♥❥✉♥t♦ ❞❛s ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛❞❛s t❛✐s q✉❡ ub_∈L1

❧♦❝(Rd) ❡

kuk2 ˙

Hs = Z

Rd|

ξ|2s_|ub(ξ)|2 _{dξ <∞ ✭✶✳✶✵✮}

❆ ♥♦r♠❛k · k_H˙s t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❡s❝❛❧❛

ku(λ_·)_kH˙s =λ− d

2+skuk_˙

Hs, ✭✶✳✶✶✮

❝♦♠♦ ♣♦❞❡ s❡r ✈❡r✐✜❝❛❞❛ ❛tr❛✈és ❞❡ ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ♥❛ ❡①♣r❡ssã♦ ✭✶✳✶✵✮✳

❊♥q✉❛♥t♦ q✉❡ ♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ♥ã♦✲❤♦♠♦❣ê♥❡♦sHs_{❢♦r♠❛♠ ✉♠❛ ❢❛♠í❧✐❛}

❞❡❝r❡s❝❡♥t❡ ❞❡ ❡s♣❛ç♦s ✭❝♦♠ r❡s♣❡✐t♦ ❛♦ í♥❞✐❝❡ s✮✱ ♦s ❡s♣❛ç♦s ❤♦♠♦❣ê♥❡♦s ♥ã♦ sã♦ ❝♦♠♣❛rá✈❡✐s ❡♥tr❡ s✐✳ P♦ré♠✱ é ✐♠❡❞✐❛t♦ q✉❡ s❡s é ♣♦s✐t✐✈♦✱ Hs _{❡stá ❝♦♥t✐❞♦ ❡♠H˙}s

❡ s❡ s é ♥❡❣❛t✐✈♦✱ Hs _{❝♦♥té♠ H}˙s_✳

Pr♦♣♦s✐çã♦ ✶✳✹✳✼✳ ❙❡ s < d/2✱ ❡♥tã♦ H˙s _{é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳}

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛(un)n∈N✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠H˙s✳ ❊♥tã♦ ❛ s❡q✉ê♥❝✐❛ (bun)n∈Né ❞❡ ❈❛✉❝❤② ❡♠L2(Rd\ {0},|ξ|2sdξ)✱ q✉❡ s❛❜❡♠♦s s❡r ❝♦♠♣❧❡t♦✳ ❙❡❥❛f ❡st❡ ❧✐♠✐t❡✱ ✐st♦ é✱

lim

n→∞kubn−fkL2(Rd\{0},|ξ|2sdξ) = 0.

▼♦str❡♠♦s q✉❡ f _∈L1

❧♦❝(Rd)✳

❚❡♠♦s q✉❡ f _∈L1

❧♦❝(Rd\ {0},|ξ|2sdξ)❀ ❛ss✐♠✱ s❡ K ⊂Rd\ {0}é ❝♦♠♣❛❝t♦✱

Z

K|

f(ξ)|dξ =

Z

K|

ξ|−2s_|ξ|2s_{|f(ξ)| dξ}

≤ C

Z

K|

ξ_|2s_|f(ξ)_|dξ < _∞,

♣♦✐s ❛ ❢✉♥çã♦ρ(ξ) =|ξ|2s _{é ❝♦♥tí♥✉❛ ❡♠ K✳}

❘❡st❛ ❡st✐♠❛r♠♦s ❛ ✐♥t❡❣r❛❧ ❞❡ f ❡♠ ❝♦♠♣❛❝t♦s q✉❡ ❝♦♥tê♠ ❛ ♦r✐❣❡♠✳ P❛r❛ ✐st♦✱ é s✉✜❝✐❡♥t❡ ♦❜s❡r✈❛r♠♦s q✉❡ ❛ ✐♥t❡❣r❛❧ ❞❡ f s♦❜r❡ ❛ ❜♦❧❛ ✉♥✐tár✐❛ B(0,1) é ✜♥✐t❛✳ ◆❡st❡ ❝❛s♦✱

Z

B(0,1)|

f(ξ)_| dξ =

Z

B(0,1)|

ξ_|−s_|ξ_|s_|f(ξ)_| dξ

≤

Z

B(0,1)|

ξ|2s_|f(ξ)|2 _dξ

1/2Z

B(0,1)|

ξ|−2s _dξ

1/2

<∞,

✈✐st♦ q✉❡ ρ é ✐♥t❡❣rá✈❡❧ s♦❜r❡ ♦ ❝♦♥❥✉♥t♦B(0,1)✱ ♣♦✐ss < d/2✳

P♦rt❛♥t♦✱ f _∈L1

❧♦❝(Rd)✳

❋✐♥❛❧♠❡♥t❡✱ t♦♠❛♥❞♦ u=_F−1_{f✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳}

Pr♦❜❧❡♠❛ ■♥✐❝✐❛❧

◆❡st❡ ❝❛♣ít✉❧♦✱ ❡st✉❞❛r❡♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✉♠❛ ❞❛❞❛ s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❤✐♣❡r❜ó❧✐❝❛s ❡str✐t❛✱ ♥♦ ❝❛s♦ ❡♠ q✉❡ a(t) é ❞♦ t✐♣♦ ▲✐♣s❝❤✐t③✱ ❞❛♥❞♦ ❝♦♠♦ r❡s✉❧t❛❞♦ ❛ ❡st✐♠❛t✐✈❛ ❞❛ ❡♥❡r❣✐❛✳

❆ss✐♠✱ s❡❥❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡♠ [0,+_∞)_×R✱ ❝♦♠ u=u(t, x)✱

  

∂2

tu−a(t)∂x2u= 0,

u(0, x) = u0(x), ∂tu(0, x) =u1(x),

✭✷✳✶✮

❝♦♠ ✈❛❧♦r ✐♥✐❝✐❛✐s u0 ∈ Hs(R), u1 ∈Hs−1(R), s >0✱ ❝♦♠ ❛ ❤✐♣ót❡s❡ ❞❡ ❤✐♣❡r❜♦❧✐❝✐✲

❞❛❞❡

0< λ≤a(t)≤Λ, ∀t ✭✷✳✷✮

♦♥❞❡ λ ❡ Λ sã♦ ❝♦♥t❛♥t❡s ♣♦s✐t✐✈❛s✳

❙❡ ♦ ❝♦❡✜❝✐❡♥t❡ a(t) ❢♦r ✉♠❛ ❢✉♥çã♦ ❧♦❝❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③ ❡♠ [0,+∞)✱ ❡♥✲

tã♦ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ✭✷✳✶✮ é C∞ _{❜❡♠ ♣♦st♦✭ ✈❡❥❛✲s❡ ❬✶✵❪ ❡❬✶✷❪✮✳ ◆❡st❡ ❝❛s♦✱} t❡r❡♠♦s u(t,·) ❜❡♠ ❞❡✜♥✐❞♦ ❡♠ Hs_{(R) ♣❛r❛ q✉❛❧q✉❡r ✈❛❧♦r ✐♥✐❝✐❛❧ u}

0 ∈ Hs(R) ❡

u1 ∈Hs−1(R)❀ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❛ ú♥✐❝❛ s♦❧✉çã♦ u(t,·)❞❛ ❡q✉❛çã♦ ♣❡rt❡♥❝❡ ❛

C([0,+∞);Hs(R))∩C1([0,+∞);Hs−1(R)).

❉❡ ❢❛t♦✱ s❡ ❞❡♥♦t❛♠♦s

Es(u)(t) :=ku(t)k2Hs +k∂_tu(t)k2_Hs−1, ✭✷✳✸✮

t❡♠♦s ❛ s❡❣✉✐♥t❡ ❛✜r♠❛çã♦✳

❆✜r♠❛çã♦ ✷✳✶✳ ❆ s♦❧✉çã♦ u ❞❡ ✭✷✳✶✮ é t❛❧ q✉❡ ♣❛r❛ q✉❛❧q✉❡r T > 0✱ s _∈ R ❡ t∈[0, T]✱ s❛t✐s❢❛③ ❛ s❡❣✉✐♥t❡ ❡st✐♠❛t✐✈❛

Es(u)(t)≤Cs,TEs(u)(0), ✭✷✳✹✮

❝♦♠ Cs,T >0.

❉❡♠♦♥str❛çã♦✿ ❉❛ ❡q✉❛çã♦

utt−a(t)uxx = 0,

♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ s✐st❡♠❛ s❡❣✉✐♥t❡✿

∂t ut ux =

a(t)uxx utx

=



 0 a

1 0

 

| {z }

A ∂x ut ux ; ✭✷✳✺✮

❞❡♥♦t❛♥❞♦ ❯ =

  ut

ux



 _❡ A =



 0 a

1 0



 _{❡ ❛♣❧✐❝❛♥❞♦ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r}

t❡♠♦s

∂t❯b =iA·ξ❯b, ✭✷✳✻✮

❧♦❣♦ ❞✐❛❣♦♥❛❧✐③❛♥❞♦ ❛ ♠❛tr✐③A_·ξ t❡♠♦s q✉❡✱

N A_·ξ =D(ξ)N,

❝♦♠ N =     1 2pa(t)

1 2

− 1

2pa(t) 1 2

  

, N−1 =

 

p

a(t) ₋pa(t) 1 1

 ,

❡ D=

 

p

a(t)ξ 0

0 ₋pa(t)ξ

 .

❉❡♣♦✐s✱ ❞❡r✐✈❛♥❞♦ NUb✱ t❡♠♦s d

dt(N❯) =b i(N A·ξ)(❯) +b N ′_❯b = iD(N❯) +b N′N−1(N❯)b ,

❢❛③❡♥❞♦ v :=NUb ✱ t❡♠✲s❡ d

dtv = iD v+N

′_N−1_v,

d

dtv = iD v+Bv,

❝♦♠ B =N′_N−1 _{q✉❡ é ❧✐♠✐t❛❞♦ ♣❛r❛ ❝❛❞❛ t✱ ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r ✉s❛♥❞♦ ❛ ♥♦r♠❛ ❞❡}

♦♣❡r❛❞♦r✱✶_{✱ ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ✈❡❥❛ ❆✜r♠❛çã♦ ✹✳✶ ♥♦ ❆♣ê♥❞✐❝❡✳}

P♦r ♦✉tr♦ ❧❛❞♦ ✉s❛r❡♠♦s ♦ s❡❣✉✐♥t❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦

(u, v) =

Z

u(t, ξ)·v(t, ξ)dξ, ✭✷✳✼✮

❡ ❞❡✜♥✐r❡♠♦s ❛ ♥♦r♠❛

kuk2L2 =

Z

|u(t, ξ)|2dξ. ✭✷✳✽✮

❙❡❥❛ ❛ ♥♦r♠❛ ♣❛r❛ v1(t,·), v2(t,·)∈L2



 v1(t,·)

v2(t,·)   1

=_kv1(t,·)k_L2 +kv2(t,·)k_L2. ✭✷✳✾✮

▲♦❣♦✱ ✉s❛♥❞♦ ✭✷✳✼✮✱ t❡♠♦s (v, Bv) = (v, Bv) = (v, Bv) = (Bv, v) ❡ (v,−iDv) = (v,₋iDv) = (₋iDv, v)✱ ❞❡♣♦✐s ❞❡r✐✈❛♥❞♦ t❡r❡♠♦s

d

dtkvkL2 = ( dv

dt, v) + (v, dv

dt)

= ((iD+B)v, v) + (v,(₋iD+B)v)

= (Bv, v) + (v, Bv)

= 2Re_{Bv_·v_{} ≤}2_kBv_kL2kvk_L2 , ❝♦♠ 2kBk ≤eγ,

≤ eγ_kv_k2₁,

❡♥tã♦ kv(t,_·)_kL2 ≤ eγtkv(0,·)k_L2, γ =eγ/2. ✶_kBk=√_{λmax,♦♥❞❡λmax ♦ ♠❛✐♦r ❛✉t♦✈❛❧♦r ❞❛ ♠❛tr✐③B}∗

B❀B∗=❛❞❥✉♥t♦ ❞❡ B✳

♦♥❞❡ eγ ❡γ sã♦ ❝♦♥st❛♥t❡s✳

❊♥tã♦ ❞❛❞❛ ❛ ♥♦r♠❛ ✭✷✳✾✮✱ t❡♠♦skv(t,_·)_k1 =_k(v1(t,·), v2(t,·))k₁ =kv1(t,·)k_L2+

kv2(t,·)kL2 ❡ ❢❛③❡♥❞♦v(t, ξ) =N❯(b t, ξ)✱ s❡❣✉❡ ❛ s❡❣✉✐♥t❡ ❛✜r♠❛çã♦✳

❆✜r♠❛çã♦ ✷✳✷✳

kv(t,·)k1 =

N❯(b t,·)

1 ≡ k❯(t,·)k1.

❉❡♠♦♥str❛çã♦✿ ❇❛st❛ ♣r♦✈❛r q✉❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡s C1, C2 >0t❛✐s q✉❡

C1  v1(t,·)

v2(t,·)   1 ≤ N  v1(t,·)

v2(t,·)   1

≤C2  v1(t,·)

v2(t,·)   1

, _∀t _≤T ✭✷✳✶✵✮

♣♦✐s s❡♥❞♦k·kL2 = (2π)−dkb·k_L2✱ t❡♠♦s q✉❡

 v1(t,·)

v2(t,·)   1

=_kv1(t,·)kL2 +kv2(t,·)kL2 = (2π)−dv\1(t,·)

L2 +

v\2(t,·)

L2

= (2π)−d

 v\1(t,·)

\

v2(t,·)   1 ,

❞♦♥❞❡ s❡❣✉❡ ❛ ❆✜r♠❛çã♦ ✷✳✷✳

❋❛❧t❛ ❡♥tã♦✱ ♣r♦✈❛r ✭✷✳✶✵✮✳ P❛r❛ ✐ss♦ ❢❛③❡♠♦s ♦ s❡❣✉✐♥t❡ ♣r♦❞✉t♦

Nt·N =

  

1

2pa(t) − 1 2pa(t) 1 2 1 2   ·     1 2pa(t)

1 2

− 1

2pa(t) 1 2    =    1 2a(t) 0

0 1 2

  ,

✉s❛♥❞♦ ❛ ♥♦r♠❛ ❞♦ ♦♣❡r❛❞♦r✱ t❡♠✲s❡ q✉❡ kN_k = qmax_{2a1_(t),1_2} ❡ _kN−1_k−1 ₌ q

min_{ 1 2a(t),

1

2}✭ ♦♥❞❡kNk3 = supkxk=1kN xk✮ ✱ ❡♥tã♦ t❡♠♦s ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡ ✭✷✳✶✵✮

kN vk1 ≤ kNk kvk1.

P❛r❛ ♣r♦✈❛r ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❡ ✭✷✳✶✵✮ ✉s❛♠♦s q✉❡ det (N)₆= 0✳

❆ss✐♠✱

kv_k1 =N−1N v_{1 ≤}N−1_kN v_k1,