On the Flow Map for a Class of Parabolic Equations

We consider the Cauchy problem for the one-dimensional parabolic equations $$\partial _t u - \partial _{xx} u \pm \partial _x^d u^k = 0,\;k \in \mathbb{N}^* ,\;d \in \{ 0,1\} ,$$ , with initial data in $$H^s (\mathbb{R}).$$ . We study the flow map corresponding to the integral equation. Our results complete the known results on ill-posedness in $$H^s (\mathbb{R}).$$ and show the particularity of the case (k, d) = (2, 0) for which we prove that the critical space $$H^{s_c } (\mathbb{R}) = H^{ - 3/2} (\mathbb{R})$$ suggesting by standard scaling arguments cannot be reached. Our results hold also in the periodic setting.