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Global unbounded solutions of the Fujita equation in the intermediate range

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Abstract

We consider the Fujita equation \(u_t={\varDelta } u+u^{p}\) on \({\mathbb {R}}^N\) with \(N\ge 3\). We prove the existence of global, unbounded solutions for \(p_S<p<p_{JL}\), where \(p_S:={(N+2)}/{(N-2)}\) and

$$\begin{aligned} p_{JL}:=\left\{ \begin{array}{l@{\quad }l} \,\frac{(N-2)^2-4N+8\sqrt{N-1}}{(N-2)(N-10)} &{} \text {if } N>10,\\ \,\infty &{} \text {if } N\le 10. \end{array}\right. \end{aligned}$$

Previously, it was known that global, unbounded solutions exist for \(p\ge p_{JL}\), whereas for \(p<p_S\) there are no such solutions.

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Acknowledgments

This research was carried out during a visit of P. Poláčik to the Tokyo Institute of Technology. P. Poláčik was upported in part by NSF Grant DMS–1161923. Eiji Yanagida was upported in part by JSPS Grant-in-Aid for Scientific Research (A) (No. 24244012).

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN , 55455, USA

    Peter Poláčik

  2. Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo , 152-8551, Japan

    Eiji Yanagida

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  1. Peter Poláčik
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  2. Eiji Yanagida
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Correspondence to Eiji Yanagida.

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Poláčik, P., Yanagida, E. Global unbounded solutions of the Fujita equation in the intermediate range. Math. Ann. 360, 255–266 (2014). https://doi.org/10.1007/s00208-014-1038-2

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  • DOI: https://doi.org/10.1007/s00208-014-1038-2

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