The Cauchy problem for Schrödinger type equations with variable coefficients

Weighted energy estimates for p-evolution equations in SG classes

“…This phenomenon, which is usual in the theory of degenerate hyperbolic equations, appears so also in the theory of non-degenerate p-evolution equations for p ≥ 2 and has been yet observed in [13,21,22]. Notice that assumptions (1.6)-(1.9) are consistent with the conditions in (1.3), (1.4).…”

“…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.…”

“…When the coefficients a j (t, x) are complex valued for some 1 ≤ j ≤ p−1, then we know from [8,20] that some decay conditions for |x| → ∞ must be required on the imaginary part of the coefficients in order to obtain H ∞ well posedness. In the papers [20,21], Ichinose has given necessary and sufficient conditions for the case p = 2, x ∈ R. Kajitani and Baba [22] then proved that, for p = 2 and a 2 (t) constant, x ∈ R n , the Cauchy problem (1.1) is H ∞ well posed if Im a 1 (t, x) = O(|x| −σ ), σ ≥ 1, as |x| → ∞, (1.3) uniformly with respect to t ∈ [0, T ]. Second-order equations with p = 2 and decay conditions as |x| → ∞ have been considered, for example, in [12,16].…”

The Cauchy problem for 𝑝-evolution equations

“…see, for example, [7] for p = 2 and Section 2 of this paper for p ≥ 2. Indeed, in the Schrödinger case p = 2, the necessity of the condition (1.8) for the well-posedness in H ∞ has been fully proved; see, e.g., [6].…”

Schrödinger equations of higher order

“…in [3], [14], [15], [18], [19], [20], [22], and [25]. Several results of well-posedness in Sobolev spaces H µ and in Gevrey classes have been obtained for operators with coefficients in the principal part independent of the time variable t. On the other hand, starting from the pioneering results of [9], we know that a low regularity, less than Lipschitz continuity, of the coefficients with respect to the time variable has a very strong influence on the well-posedness of the hyperbolic Cauchy problem.…”