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A semilinear heat equation with initial data in negative Sobolev spaces

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

    Mathematics Subject Classification: Primary: 35K91, 49K40; Secondary: 35B65.


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