On Schrödinger type evolution equations with non-Lipschitz coefficients

Abstract

We prove results of well-posedness of the global Cauchy problem in Sobolev spaces for a class of evolution equations with real characteristics that contains an EulerBernoulli vibrating beam model. We consider non-Lipschitz coefficients with respect to the time variable t and study the sharp rate of their oscillations. This is coupled with some necessary decay conditions as the spatial variable x → ∞.

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

“…Well-posedness results in Sobolev spaces for higher order 2-evolution equations are also available (see, for instance, [18,5,10]). …”

Semilinear p-evolution equations in Sobolev spaces

“…Results of H ∞ well-posedness of the Cauchy problem for linear p-evolution equations of the first order D t u + a p (t)D p x u + p−1 j=0 a j (t, x)D j x u = f (t, x), (1.4) or for linear p-evolution equations of higher order, have already been obtained, first for real valued (or complex valued, with imaginary part not depending on x) coefficients (see for instance [Ag, AC] and the references therein), then for complex valued coefficients depending on the space variable x under suitable decay conditions on the coefficients as |x| → +∞ (see [I2,KB,CC1,ABZ1,AB1] for equations of the form (1.4), [AB2,ACC,CR1,CR2,T1] for higher order equations, and [ACa] for equation (1.4) in a different framework). Among all these results, in the present paper we shall need an extension of the following theorem of [ABZ1] (see Theorem 2.1 below):…”

A necessary condition for $H^\infty$ well-posedness of p-evolution equations

“…A necessary condition is given also in [20] for p = 2. Sufficient conditions for well-posedness in H ∞ and/or Gevrey classes for 2 or 3−evolution equations have been given by many authors (see, for instance, [21], [27], [13], [24], [12], [18], [19], [15]). The general case p ≥ 2 has been recently considered in [8], proving H ∞ well-posedness of the Cauchy problem (1.2) under suitable decay conditions, as |x| → +∞, on Im D β x a j , for j ≤ p − 1 and [β/2] ≤ j − 1.…”