In this paper we consider the Cauchy boundary value problem for the abstract
Kirchhoff equation with a continuous nonlinearity m : [0,+\infty) -->
[0,+\infty). It is well known that a local solution exists provided that the
initial data are regular enough. The required regularity depends on the
continuity modulus of m. In this paper we present some counterexamples in order
to show that the regularity required in the existence results is sharp, at
least if we want solutions with the same space regularity of initial data. In
these examples we construct indeed local solutions which are regular at t = 0,
but exhibit an instantaneous (often infinite) derivative loss in the space
variables.Comment: 33 pages, some remarks and appendix adde
