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

  • * Corresponding author: Jumpei Inoue

    * Corresponding author: Jumpei Inoue 

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

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

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


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