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 Scr\"odinger type models, ...). We show that, for $p\geq2$, a lack of regularity in $t$ can be balanced by a damping of the too fast oscillations as the space variable $x\to\infty$. This can not happen in the hyperbolic case $p=1$ because of the finite speed of propagation.