Download Communications In Mathematical Physics - Volume 286 by M. Aizenman (Chief Editor) PDF

By M. Aizenman (Chief Editor)

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In the present paper, for the sake of simplicity, we work for the minimal set of indices so that we are able to prove Theorems 1 and 2. Indeed, let us define p = {(k, ) | = 0, 1 ≤ k ≤ p, or = 1, k = 1}. 3, we solve the systems ( following [21]. 4 (Resolution of ( exists (ak, , Ak, , Bk, ) of the form k, ) for (k, ) ∈ 55 p ). 2. 4, we see that by restricting the sum defining v(t, x) to the set of indices p , all the functions Ak, belong to Y and the functions Bk, are bounded with derivatives in Y.

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12), and using the equations of R and Rc , we obtain the following decomposition (see also [21], Proof of Prop. 20) where I = ∂t R + ∂x (∂x2 R − R + f (R)), II = ∂x ( f (R + Rc ) − f (R) − f (Rc )), III = ∂t W − ∂x (LW ), where LW = −∂x2 w + w − f (R)w, IV = ∂x ( f (R + Rc + W ) − f (R + Rc ) − f (R)W ). Now, we follow exactly the same steps as in Sect. 1 and using Taylor expansions. 2) for k0 ≤ p, we have the following Taylor expansion of f and F: f (s) = s p + f 1 (s) = s p + p+1≤k1 ≤k0 1 p+1 s + p+1 F(s) = p+2≤k1 ≤k0 1 k1 (k1 ) s f 1 (0) + s k0 +1 O(1), k1 !

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