Archive | 2021
A semilinear heat equation with initial data in negative Sobolev spaces
Abstract
We give a sufficient conditions for the existence, locally in time, of solutions to semilinear heat equations with nonlinearities of type \\begin{document}$ |u|^{p-1}u $\\end{document} , when the initial datas are in negative Sobolev spaces \\begin{document}$ H_q^{-s}(\\Omega) $\\end{document} , \\begin{document}$ \\Omega \\subset \\mathbb{R}^N $\\end{document} , \\begin{document}$ s \\in [0,2] $\\end{document} , \\begin{document}$ q \\in (1,\\infty) $\\end{document} . Existence is for instance proved for \\begin{document}$ q>\\frac{N}{2}\\left(\\frac{1}{p-1}-\\frac{s}{2}\\right)^{-1} $\\end{document} . This is an extension to \\begin{document}$ s \\in (0,2] $\\end{document} of previous results known for \\begin{document}$ s = 0 $\\end{document} with the critical value \\begin{document}$ \\frac{N(p-1)}{2} $\\end{document} . We also observe the uniqueness of solutions in some appropriate class.