A well-posed Cauchy problem for an evolution equation with coefficients of low regularity

In the hyperbolic Cauchy problem, the well-posedness in Sobolev spaces is strictly related to the modulus of continuity of the coefficients. This holds true for p-evolution equations with real characteristics (p = 1 hyperbolic equations, p = 2 vibrating plate and Schrodinger type models, ...). We show that, for p >= 2, a lack of regularity in t can be balanced by a damping of the too fast oscillations as the space variable x -> infinity. This cannot happen in the hyperbolic case p = 1 because of the finite speed of propagation. (C) 2013 Elsevier Inc. All rights reserved.