On the unboundedness of the ratio of species and resources for the diffusive logistic equation

Jumpei Inoue; Kousuke Kuto

doi:10.3934/dcdsb.2020186

Article Contents

2021, Volume 26, Issue 5: 2441-2450. Doi: 10.3934/dcdsb.2020186

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On the unboundedness of the ratio of species and resources for the diffusive logistic equation

Jumpei Inoue^1, , and
Kousuke Kuto^2,

1.
Department of Pure and Applied Mathematics, Graduate School of Fundamental Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 164-8555, Japan
2.
Department of Applied Mathematics, School of Fundamental Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 164-8555, Japan

^* Corresponding author: Jumpei Inoue
^* Corresponding author: Jumpei Inoue

Dedicated to Professor Yoshio Yamada on the occasion of his 70th birthday

Received: January 2020

Revised: April 2020

Early access: June 2020

Published: May 2021

The second author is supported by JSPS KAKENHI Grant-in-Aid Grant Number 19K03581

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Abstract

Concerning a class of diffusive logistic equations, Ni [1,Abstract] proposed an optimization problem to consider the supremum of the ratio of the $ L^1 $ norms of species and resources by varying the diffusion rates and the profiles of resources, and moreover, he gave a conjecture that the supremum is $ 3 $ in the one-dimensional case. In [1], Bai, He and Li proved the validity of this conjecture. The present paper shows that the supremum is infinity in a case when the habitat is a multi-dimensional ball. Our proof is based on the sub-super solution method. A key idea of the proof is to construct an $ L^1 $ unbounded sequence of sub-solutions.

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Mathematics Subject Classification: Primary: 35Q92, 35B30; Secondary: 35B09, 35B40.

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References

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