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Capitolul 1 Ecuatii diferentiale de ordinul ˆıntâi rezolvabile prin ...

Capitolul 1 Ecuatii diferentiale de ordinul ˆıntâi rezolvabile prin ...

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260 CAPITOLUL 7<br />

¸si faptul cǎ funct¸ia ∂u<br />

este continuǎ pe [0, +∞) × Ω. Din continuitatea<br />

∂t<br />

funct¸iei ∂u<br />

rezultǎ cǎ aceasta este uniform continuǎ pe o mult¸ime <strong>de</strong> forma<br />

∂t<br />

[t0 − η, t0 + η] × Ω, (y > 0) ¸si <strong>prin</strong> urmare:<br />

(∀)ε > 0, (∃)δ(ε) a.î.(∀)(t ′ , X ′ ), (t”, X”) ∈ [t0 − η, t0 + η] × Ω<br />

dacǎ |t ′ − t”| < δ ¸si |X ′ − X”| < δ atunci<br />

<br />

<br />

<br />

∂u<br />

∂t (t′ , X ′ ) − ∂u<br />

<br />

<br />

(t”, X”) <br />

∂t <<br />

ε<br />

|Ω| .<br />

Rezultǎ <strong>de</strong> aici cǎ, dacǎ |t − t0| < δ(ε) avem: U ′ (t ′ ) − U ′ (t0) < ε.<br />

ii) Sǎ arǎtǎm cǎ U ca funct¸ie cu valori în spat¸iul:<br />

H 1 <br />

0 = u ∈ L 2 <br />

<br />

(Ω) <br />

∂u<br />

(∃)<br />

∂xi<br />

∈ L 2 (Ω) ¸si u|∂Ω = 0<br />

este continuǎ. Faptul cǎ, pentru orice t ∈ [0, +∞) funct¸ia U(t) apart¸ine<br />

spat¸iului H 1 0 rezultǎ din proprietǎt¸ile solut¸iei u(t, X) = U(t)(X). Trebuie<br />

doar sǎ evaluǎm norma U(t) − U(t0) H 1 0 ¸si sǎ arǎtǎm cǎ aceasta tin<strong>de</strong> la<br />

zero dacǎ t → t0.<br />

Avem:<br />

U(t) − U(t0) 2<br />

H 1 0<br />

= U(t) − U(t0) 2<br />

L 2 (Ω) +<br />

<br />

=<br />

+<br />

Ω<br />

n<br />

<br />

i=1<br />

<br />

=<br />

+<br />

Ω<br />

n<br />

<br />

<br />

<br />

∂U(t)<br />

∂xi<br />

i=1<br />

|u(t, X) − u(t0, X)| 2 dX+<br />

Ω<br />

<br />

<br />

<br />

∂u<br />

(t, X) −<br />

∂xi<br />

∂u<br />

<br />

<br />

(t0, X) <br />

∂xi<br />

<br />

<br />

<br />

<br />

∂u <br />

(t(X), X) <br />

∂t <br />

n<br />

<br />

i=1<br />

Ω<br />

<br />

<br />

<br />

∂<br />

<br />

2u ∂t∂xi<br />

(t ∗ i<br />

2<br />

2<br />

· |t − t0| 2 dX+<br />

<br />

− ∂U(t0)<br />

∂xi<br />

dX =<br />

2<br />

<br />

(X), X) <br />

· |t − t0| 2 dX<br />

<br />

<br />

<br />

<br />

2<br />

L 2 (Ω)<br />

=

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