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Some monotone properties for solutions to a reaction-diffusion model

  • * Corresponding author: Rui Li

    * Corresponding author: Rui Li 
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  • Motivated by the recent investigation of a predator-prey model in heterogeneous environments [20], we show that the maximum of the unique positive solution of the scalar equation

    $ \begin{equation} \begin{cases} \mu\Delta\theta+(m(x)-\theta)\theta = 0 \quad &\text{in} \quad \Omega,\\ \frac{\partial \theta}{\partial n} = 0 \quad &\text{on} \quad \partial\Omega \end{cases} \end{equation} $

    is a strictly monotone decreasing function of the diffusion rate $ \mu $ for several classes of function $ m $, which substantially improves a result in [20]. However, the minimum of the positive solution of (1) is not always monotone increasing in the diffusion rate [15].

    Mathematics Subject Classification: 34D23, 92D25.


    \begin{equation} \\ \end{equation}
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