Well-posedness of the Cauchy problem for p-evolution equations

Abstract

We consider p-evolution equations in (t,x) with real characteristics. We give sufficient conditions for the well-posedness of the Cauchy problem in Sobolev spaces, in terms of decay estimates of the coefficients as the space variable x goes to infinity

Weighted energy estimates for p-evolution equations in SG classes

“…In this way, we are able to prove that there exists σ > 0 such that for all s 1 , s 2 ∈ R, f ∈ C([0, T ]; H s 1 ,s 2 (R)) and g ∈ H s 1 ,s 2 (R), there is a unique solution u ∈ C([0, T ]; H s 1 −σ ,s 2 (R)) which satisfies the following energy estimate: (1.18) for some C = C(s 1 , s 2 ) > 0. We do not prove here this alternative result, the proof being a repetition of the one of Theorem 1.1 in [6] in our functional setting. REMARK 1.5.…”

“…On the other hand, the assumptions on the coefficients given in our paper are stronger than the ones used in [5,6]. Hence, one can modify the approach and define the functions λ p−k in the following by (2.14) for 1 ≤ k ≤ p − 1 (hence also for k = 1) as in [6] and repeat readily the argument of the proof using the estimates of Lemma 2.1 in [6] instead of Lemma 2.5 in the case k = 1. In this way, we are able to prove that there exists σ > 0 such that for all s 1 , s 2 ∈ R, f ∈ C([0, T ]; H s 1 ,s 2 (R)) and g ∈ H s 1 ,s 2 (R), there is a unique solution u ∈ C([0, T ]; H s 1 −σ ,s 2 (R)) which satisfies the following energy estimate: (1.18) for some C = C(s 1 , s 2 ) > 0.…”

“…Recently, Ascanelli et al [6] extended the results of [14] and [22] to the case p ≥ 4, giving sufficient conditions for H ∞ well posedness of the Cauchy problem for the operator (1.2); results in [6] have then been generalized to pseudo-differential systems in [5] and to higher order equations in [4]; semi-linear three-evolution equations have been then studied in [7]. Recently, in [8], a necessary condition of decay at infinity for the coefficients of (1.2) with arbitrary p ≥ 2 has been given.…”

“…REMARK 1.4. The proof of Theorem 1.2 is in part inspired by [6], but it takes advantage of the fact that, in the new framework we are considering, we can admit initial data with polynomial growth with respect to the space variable. On the other hand, the assumptions on the coefficients given in our paper are stronger than the ones used in [5,6].…”

“…The proof of Theorem 1.2 is in part inspired by [6], but it takes advantage of the fact that, in the new framework we are considering, we can admit initial data with polynomial growth with respect to the space variable. On the other hand, the assumptions on the coefficients given in our paper are stronger than the ones used in [5,6]. Hence, one can modify the approach and define the functions λ p−k in the following by (2.14) for 1 ≤ k ≤ p − 1 (hence also for k = 1) as in [6] and repeat readily the argument of the proof using the estimates of Lemma 2.1 in [6] instead of Lemma 2.5 in the case k = 1.…”