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A note on the Fujita exponent in fractional heat equation involving the Hardy potential

1 Laboratoire d'Analyse Nonlinéaire et Mathématiques Appliquées. Département de Mathématiques, Université Abou Bakr Belkaïd, Tlemcen, Tlemcen 13000, Algeria
2 Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

This contribution is part of the Special Issue: Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday
Guest Editor: Gianmaria Verzini
Link: https://www.aimspress.com/newsinfo/1429.html

Special Issues: Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday

In this work, we are interested on the study of the Fujita exponent and the meaning of the blow-up for the fractional Cauchy problem with the Hardy potential, namely,\begin{equation*}u_t+(-\Delta)^s u=\lambda\dfrac{u}{|x|^{2s}}+u^{p}\;{\rm in}\;{{\boldsymbol R}^N}, u(x,0)=u_{0}(x)\;{\rm in}\;{{\boldsymbol R}^N},\end{equation*}where $N> 2s$, $0<s<1$, $(-\Delta)^s$ is the fractional laplacian of order $2s$, $\lambda >0$, $u_0\ge 0$, and $1<p<p_{+}(s,\lambda)$, where $p_{+}(\lambda, s)$ is the critical existence power to be given subsequently.
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Keywords Fujita exponent; fractional Cauchy heat equation with Hardy potential; blow-up; global solution

Citation: Boumediene Abdellaoui, Ireneo Peral, Ana Primo. A note on the Fujita exponent in fractional heat equation involving the Hardy potential. Mathematics in Engineering, 2020, 2(4): 639-656. doi: 10.3934/mine.2020029


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