Existence and global behavior of the solution to a parabolic equation with nonlocal diffusion

Fengfei Jin; Baoqiang Yan; Fengfei Jin; Baoqiang Yan

doi:10.3934/math.2021313

AIMS Mathematics

2021, Volume 6, Issue 5: 5292-5315. doi: 10.3934/math.2021313

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Research article

Existence and global behavior of the solution to a parabolic equation with nonlocal diffusion

Fengfei Jin ,
Baoqiang Yan ^,

School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China

Received: 09 December 2020 Accepted: 22 February 2021 Published: 12 March 2021
MSC : 35K57, 35B35

In this paper, we are concerned with the existence, uniqueness and long-time behavior of the solutions for a parabolic equation with nonlocal diffusion even if the reaction term is not Lipschitz-continuous at $ 0 $ and grows superlinearly or exponentially at $ +\infty $. First, we give a special sub-supersolution pair for some parabolic equations with nonlocal diffusion and establish the method of sub-supersolution. Second, using the sub-supersolution method, we prove the existence, uniqueness and long-time behavior of positive solutions. Finally, some one-dimensional numerical experiments are presented.
- parabolic equation with nonlocal diffusion,
- sub-supersolution method, existence,
- uniqueness,
- long-time behavior
Citation: Fengfei Jin, Baoqiang Yan. Existence and global behavior of the solution to a parabolic equation with nonlocal diffusion[J]. AIMS Mathematics, 2021, 6(5): 5292-5315. doi: 10.3934/math.2021313

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Abstract

In this paper, we are concerned with the existence, uniqueness and long-time behavior of the solutions for a parabolic equation with nonlocal diffusion even if the reaction term is not Lipschitz-continuous at $ 0 $ and grows superlinearly or exponentially at $ +\infty $. First, we give a special sub-supersolution pair for some parabolic equations with nonlocal diffusion and establish the method of sub-supersolution. Second, using the sub-supersolution method, we prove the existence, uniqueness and long-time behavior of positive solutions. Finally, some one-dimensional numerical experiments are presented.

References

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