On the ratio of biomass to total carrying capacity in high dimensions. (English) Zbl 1476.35122
Summary: This paper is concerned with a reaction-diffusion logistic model. In [J. Differ. Equations 223, No. 2, 400–426 (2006; Zbl 1097.35079)], Y. 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, X. He and W.-M. Ni [Commun. Pure Appl. Math. 69, No. 5, 981–1014 (2016; Zbl 1338.92105)] raised a conjecture that the ratio has a upper bound depending only on the spatial dimension. For the one-dimensional case, X. Bai et al. [Proc. Am. Math. Soc. 144, No. 5, 2161–2170 (2016; Zbl 1381.35185)] proved that the optimal upper bound is \(3\). Recently, J. Inoue and K. Kuto [Discrete Contin. Dyn. Syst., Ser. B 26, No. 5, 2441–2450 (2021; Zbl 1471.35276)] 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 [loc. cit.] 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 [loc. cit.] but we have improved the construction of sub-solutions. This is the complete answer to the conjecture of Ni.
MSC:
| 35K58 | Semilinear parabolic equations |
| 35B09 | Positive solutions to PDEs |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
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