Volume 12, Issue 4
Boundedness in a Forager-Exploiter Model Accounting for Gradient-Dependent Flux-Limitation

Qian Zhao & Bin Liu

East Asian J. Appl. Math., 12 (2022), pp. 848-873.

Published online: 2022-08

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  • Abstract

The forager-exploiter model with gradient-dependent flux-limitation $$u_t=\Delta u-\chi\nabla \cdot (uk_f(1+|\nabla w|^2)^{-\frac{\alpha}{2}})\nabla w,$$$$v_t=\Delta v-\xi\nabla \cdot (vk_g(1+|\nabla u|^2)^{-\frac{\beta}{2}})\nabla u,$$$$w_t=\Delta w-(u+v)w-\mu w+r(x,t)$$is considered in smooth bounded domains $Ω ⊂ \mathbb{R}^N,$ $N ≥ 2.$ It is shown that if $α > (N − 2)/N(N − 1),$ $β > 0,$ then for any nonnegative functions $u_0,$ $v_0,$ $w_0∈ W^{2,∞}(Ω)$ such that $u_0 \not\equiv 0$ and $v_0 \not\equiv 0,$ the problem has a global classical solution $(u, v, w) ∈ (C^0 (\overline{Ω} × [0,∞))\cap C^{2,1}(\overline{Ω} × (0,∞)))^3$.

  • Keywords

Chemotaxis, forager-exploiter model, boundedness, flux-limitation.

  • AMS Subject Headings

35A09, 35K65, 92C17

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-12-848, author = {Qian and Zhao and and 24359 and and Qian Zhao and Bin and Liu and and 24360 and and Bin Liu}, title = {Boundedness in a Forager-Exploiter Model Accounting for Gradient-Dependent Flux-Limitation}, journal = {East Asian Journal on Applied Mathematics}, year = {2022}, volume = {12}, number = {4}, pages = {848--873}, abstract = {

The forager-exploiter model with gradient-dependent flux-limitation $$u_t=\Delta u-\chi\nabla \cdot (uk_f(1+|\nabla w|^2)^{-\frac{\alpha}{2}})\nabla w,$$$$v_t=\Delta v-\xi\nabla \cdot (vk_g(1+|\nabla u|^2)^{-\frac{\beta}{2}})\nabla u,$$$$w_t=\Delta w-(u+v)w-\mu w+r(x,t)$$is considered in smooth bounded domains $Ω ⊂ \mathbb{R}^N,$ $N ≥ 2.$ It is shown that if $α > (N − 2)/N(N − 1),$ $β > 0,$ then for any nonnegative functions $u_0,$ $v_0,$ $w_0∈ W^{2,∞}(Ω)$ such that $u_0 \not\equiv 0$ and $v_0 \not\equiv 0,$ the problem has a global classical solution $(u, v, w) ∈ (C^0 (\overline{Ω} × [0,∞))\cap C^{2,1}(\overline{Ω} × (0,∞)))^3$.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.291021.140222 }, url = {http://global-sci.org/intro/article_detail/eajam/20887.html} }
TY - JOUR T1 - Boundedness in a Forager-Exploiter Model Accounting for Gradient-Dependent Flux-Limitation AU - Zhao , Qian AU - Liu , Bin JO - East Asian Journal on Applied Mathematics VL - 4 SP - 848 EP - 873 PY - 2022 DA - 2022/08 SN - 12 DO - http://doi.org/10.4208/eajam.291021.140222 UR - https://global-sci.org/intro/article_detail/eajam/20887.html KW - Chemotaxis, forager-exploiter model, boundedness, flux-limitation. AB -

The forager-exploiter model with gradient-dependent flux-limitation $$u_t=\Delta u-\chi\nabla \cdot (uk_f(1+|\nabla w|^2)^{-\frac{\alpha}{2}})\nabla w,$$$$v_t=\Delta v-\xi\nabla \cdot (vk_g(1+|\nabla u|^2)^{-\frac{\beta}{2}})\nabla u,$$$$w_t=\Delta w-(u+v)w-\mu w+r(x,t)$$is considered in smooth bounded domains $Ω ⊂ \mathbb{R}^N,$ $N ≥ 2.$ It is shown that if $α > (N − 2)/N(N − 1),$ $β > 0,$ then for any nonnegative functions $u_0,$ $v_0,$ $w_0∈ W^{2,∞}(Ω)$ such that $u_0 \not\equiv 0$ and $v_0 \not\equiv 0,$ the problem has a global classical solution $(u, v, w) ∈ (C^0 (\overline{Ω} × [0,∞))\cap C^{2,1}(\overline{Ω} × (0,∞)))^3$.

Qian Zhao & Bin Liu. (2022). Boundedness in a Forager-Exploiter Model Accounting for Gradient-Dependent Flux-Limitation. East Asian Journal on Applied Mathematics. 12 (4). 848-873. doi:10.4208/eajam.291021.140222
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