Loss of regularity for p-evolution type models

Abstract

The goal of the paper is to study the loss of regularity for special p-evolution type models with bounded coefficients in the principal part. The obtained loss of regularity is related in an optimal way to some unboundedness conditions for the derivatives of coefficients up to the second-order with respect to t.

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

“…Literature about well-posedness in Sobolev spaces of the Cauchy problem for hyperbolic operators is really wide; coming up to p 2, many results of well-posedness in Sobolev spaces are available under the assumption that the coefficients a j of (1.1) are real (see, for instance, [1][2][3][4]7,9]). On the contrary, when the coefficients a j (t, x) for 1 j p − 1 are not real, we only know results for p = 2, 3; all these results show that, in order to have a well-posed Cauchy problem in Sobolev spaces, a suitable decay in x for the imaginary part of the coefficients is needed.…”

On Schrödinger type evolution equations with non-Lipschitz coefficients

“…and, for a real coefficient a 3 = a 3 (t, x), −4 , β ≥ 0, (1.10) then we may conclude the L 2 well-posedness of the Cauchy problem (1.6) (see [4]). …”

Regularity of hyperbolic magnetic Schrödinger equation with oscillating coefficients

“…As for the difference of regularity for initial Cauchy data, [5] considered the typical p-evolution model in 1-Dimension. In effect, the principal operator determined the difference p when its coefficient is Log-Lipschitz continuous with respect to the time [2,6,7].…”