A semilinear heat equation with initial data in negative Sobolev spaces

We give a sufficient conditions for the existence, locally in time, of solutions to semilinear heat equations with nonlinearities type \begin{document}$ |u|^{p-1}u $\end{document}, when initial datas are negative Sobolev spaces id="M2">\begin{document}$ H_q^{-s}(\Omega) id="M3">\begin{document}$ \Omega \subset \mathbb{R}^N id="M4">\begin{document}$ s \in [0,2] id="M5">\begin{document}$ q (1,\infty) $\end{document}. Existence is instance proved id="M6">\begin{document}$ q>\frac{N}{2}\left(\frac{1}{p-1}-\frac{s}{2}\right)^{-1} This an extension id="M7">\begin{document}$ (0,2] $\end{document} previous results known id="M8">\begin{document}$ = 0 critical value id="M9">\begin{document}$ \frac{N(p-1)}{2} We also observe uniqueness some appropriate class.