Ill-posedness issues for a class of parabolic equations

Source abstract

We prove that the Cauchy problem for the one-dimensional parabolic equations , with initial data in H s (R), cannot be solved by an iterative scheme based on the Duhamel formula for s < −1 if ( k, d ) = (2, 0) and s < s c ( k, d ) = ½ − (2 − d )/( k − 1) otherwise. This exactly completes the positive results on the Cauchy problem in H s (R) for these equations and shows the particularity of the case ( k, d ) = (2, 0), for which we prove that the critical space H s c (R) = H −3/2 (R), by standard scaling arguments, cannot be reached. Our results also hold in the periodic setting.