H solutions for a kuramoto-velarde type equation

HIGHLIGHTS

SUMMARY

(2.10) Observe that - 2δ R uε (∂x uε )2 dx=-2δ R uε ∂x uε ∂x uε dx=δ R u2ε ∂x2 uε dx. (2.17) Thanks to the Hölder inequality, u2ε (t, x)=2 x -∞ uε ∂x uε dy ≤ 2 x -∞ |uε ||∂x uε |dy ≤ 2 R |uε ||∂x uε |dx ≤ 2 uε (t, ·) L2 (R) ∂x uε (t, ·) L2 (R) 2 2 2=√ uε (t, ·) L2 (R) A ∂x uε (t, ·) L2 (R) A 2 ≤ √ Xε (t). Thanks to Lemma 2.1, 2 uε (t, ·) H 1 (R)=uε (t, ·) L2 . . .

 



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