In this paper we study the local and global existence of solutions for a class of heat equation in whole $ \mathbb{R}^{N} $ where the nonlinearity has a critical growth for $ N \geq 2 $. In order to prove the global existence, we will use the potential well theory combined with the Nehari manifold, and also with the Pohozaev manifold that is a novelty for this type of problem. Moreover, the blow-up phenomena of local solutions is investigated by combining the subdifferential approach with the concavity method.

中文翻译：

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一元热方程在整体$ \ mathbb {R} ^ N $中的解的存在性

在本文中，我们研究了整体$ \ mathbb {R} ^ {N} $中一类热方程解的局部和全局存在性，其中非线性对于$ N \ geq 2 $具有临界增长。为了证明整体存在，我们将使用势阱理论与Nehari流形以及Pohozaev流形相结合，这是此类问题的新颖之处。此外，通过将亚微分方法与凹面方法相结合，研究了局部解的爆炸现象。