This paper is concerned with a reaction-diffusion logistic model. In \cite{L06}, Lou observed that a heterogeneous environment with diffusion makes the total biomass greater than the total carrying capacity. Regarding the ratio of biomass to carrying capacity, Ni \cite{HN16} raised a conjecture that the ratio has a upper bound depending only on the spatial dimension. For the one-dimensional case, Bai, He, and Li \cite{BHL16} proved that the optimal upper bound is $3$. Recently, Inoue and Kuto \cite{IK20} showed that the supremum of the ratio is infinity when the domain is a multi-dimensional ball. In this paper, we generalized the result of \cite{IK20} to an arbitrary smooth bounded domain in $\mathbb{R}^n, n \geq 2$. We use the sub-solution and super-solution method. The idea of the proof is essentially the same as the proof of \cite{IK20} but we have improved the construction of sub-solutions. This is the complete answer to the conjecture of Ni.
This paper is concerned with a reaction-diffusion logistic model. In \cite{L06}, Lou observed that a heterogeneous environment with diffusion makes the total biomass greater than the total carrying capacity. Regarding the ratio of biomass to carrying capacity, Ni \cite{HN16} raised a conjecture that the ratio has a upper bound depending only on the spatial dimension. For the one-dimensional case, Bai, He, and Li \cite{BHL16} proved that the optimal upper bound is $3$. Recently, Inoue and Kuto \cite{IK20} showed that the supremum of the ratio is infinity when the domain is a multi-dimensional ball. In this paper, we generalized the result of \cite{IK20} to an arbitrary smooth bounded domain in $\mathbb{R}^n, n \geq 2$. We use the sub-solution and super-solution method. The idea of the proof is essentially the same as the proof of \cite{IK20} but we have improved the construction of sub-solutions. This is the complete answer to the conjecture of Ni.
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