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.

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具有负Sobolev空间中初始数据的半线性热方程

当初始数据位于负Sobolev空间$ H_q ^ {-中时，我们为局部具有非线性类型$ | u | ^ {p-1} u $的半线性热方程解提供了充分的条件。 s}（\ Omega）$，$ \ Omega \ subset \ mathbb {R} ^ N $，$ s \ in [0,2] $，$ q \ in（1，\ infty）$。例如，证明存在$ q> \ frac {N} {2} \ left（\ frac {1} {p-1}-\ frac {s} {2} \ right）^ {-1} $。这是对$ s \ in（0,2] $先前以$ s = 0 $已知的结果的临界值$ \ frac {N（p-1）} {2} $的扩展。我们还观察到了唯一性一些适当的类中的解决方案。