Gevrey regularity in time for generalized KdV type equations

Abstract

Given s ≥ 1 we present initial data that belong to the Gevrey space G s for which the solution to the Cauchy problem for the generalized mk-KdV equation does not belongs to G s in the time variable. Also, for the KdV, in the periodic case, we show that the solution to the Cauchy problem withanalytic initial data (Gevrey class G 1) belongs to G 3 in time.

Formal solutions of Burgers type equations

“…Observe that the homogeneity degree of every summand on the right-hand side of ( 18) is equal to 2k + 1 r−1 . A similar lemma for the case of generalized KdV equation was obtained by H. Hannah, A. Himonas and G. Petronilho [9], Lemma 2.2, see also [12], Lemma 2. So we omit its proof.…”

Formal solutions of semilinear heat equations

“…A similar lemma for the case of generalized KdV equation was obtained by H. Hannah, A. Himonas and G. Petronilho, see [7,Lemma 2.2]. However, for the sake of completeness we give its proof.…”

Analytic well-posedness of periodic gKdV

“…. the following modification of an example given in [12] θ(x) = i 1/2 ∞ n=1 e −2δn e inx provides initial data θ ∈ G δ,s (T). Now, following [13] one can prove that the solution to the Cauchy problem (1.1) with initial data θ is not G σ in t, for t near zero and for 1 σ < 3.…”