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Asymptotic behavior of the nonlocal diffusion equation with localized source


Authors: Jinge Yang, Shuangshuang Zhou and Sining Zheng
Journal: Proc. Amer. Math. Soc. 142 (2014), 3521-3532
MSC (2010): Primary 35K55, 35B40, 35B33
DOI: https://doi.org/10.1090/S0002-9939-2014-12138-1
Published electronically: June 23, 2014
MathSciNet review: 3238427
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Abstract: In this paper we study the nonlocal diffusion equation with localized source: $ u_t= J*u-u+a(x)u^p$ in $ \mathbb{R}^N\times (0,T)$, with $ a(x)$ nonnegative, continuous, and compactly supported. It is found that the localized source $ a(x)$ drastically changes the asymptotic behavior of the nonlocal diffusion equation that the Fujita phenomenon happens only if $ N=1$. That is to say, the solutions must be global provided the initial data are small if $ N>1$. Furthermore, we determine the second critical exponent $ b_c=\frac {1}{p-1}$ with $ N=1$, and $ b_c=0$ with $ N>1$, rather than $ b_c=\frac {2}{p-1} $ for the case of homogeneous source with all $ N\ge 1$. This implies that the scope of initial data for global solutions determined by the second critical exponent $ b_c$ is enlarged due to the localized factor $ a(x)$. Finally, the time-decay profile of the global solutions is also studied for slow-decay initial data. It is mentioned that we need some new techniques to deal with the nonlocal diffusion in the model. For example, different from local diffusion equations, because of a lack of regularity mechanism from the nonlocal diffusion, we employ the moving plane method in integral form to deal with the mild solutions instead of the maximum principle. In addition, due to the localization of the source, we have to use precise weighted $ L^1$ estimates for the critical situation to replace the general test function method.


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Additional Information

Jinge Yang
Affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China
Address at time of publication: Department of Science, Nanchang Institute of Technology, Nanchang 330099, People’s Republic of China – and – School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China

Shuangshuang Zhou
Affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China – and – School of Mathematics and Computational Science, Hunan City University, Yiyang 413000, People’s Republic of China

Sining Zheng
Affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China
Email: snzheng@dlut.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2014-12138-1
Keywords: Nonlocal diffusion, critical Fujita exponent, the critical second exponent, global existence, blow-up, asymptotic profile
Received by editor(s): October 27, 2012
Published electronically: June 23, 2014
Additional Notes: This work was supported by the National Natural Science Foundation of China (11171048, 11326146) and the Education Department Program of Jiangxi Province (GJJ14759)
The third author is the corresponding author
Communicated by: Walter Craig
Article copyright: © Copyright 2014 American Mathematical Society