# American Institute of Mathematical Sciences

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

 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

Received  January 2020 Revised  April 2020 Published  June 2020

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

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.

Citation: Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020186
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