Finite-time blow-up in a repulsive chemotaxis-consumption system
Y. Wang, M. Winkler, Proceedings of the Royal Society of Edinburgh: Section A Mathematics 153 (2022) 1150–1166.
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Author
Wang, Yulan;
Winkler, Michael
Abstract
<jats:p>In a ball <jats:inline-formula><jats:alternatives><jats:tex-math>$\Omega \subset \mathbb {R}^{n}$</jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210522000397_inline1.png" /></jats:alternatives></jats:inline-formula> with <jats:inline-formula><jats:alternatives><jats:tex-math>$n\ge 2$</jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210522000397_inline2.png" /></jats:alternatives></jats:inline-formula>, the chemotaxis system
<jats:disp-formula><jats:alternatives><jats:tex-math>\[ \left\{ \begin{array}{@{}l} u_t = \nabla \cdot \big( D(u)\nabla u\big) + \nabla\cdot \big(\dfrac{u}{v} \nabla v\big), \\ 0=\Delta v - uv \end{array} \right. \]</jats:tex-math><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" position="float" xlink:href="S0308210522000397_eqnU1.png" /></jats:alternatives></jats:disp-formula>is considered along with no-flux boundary conditions for <jats:inline-formula><jats:alternatives><jats:tex-math>$u$</jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210522000397_inline3.png" /></jats:alternatives></jats:inline-formula> and with prescribed constant positive Dirichlet boundary data for <jats:inline-formula><jats:alternatives><jats:tex-math>$v$</jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210522000397_inline4.png" /></jats:alternatives></jats:inline-formula>. It is shown that if <jats:inline-formula><jats:alternatives><jats:tex-math>$D\in C^{3}([0,\infty ))$</jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210522000397_inline5.png" /></jats:alternatives></jats:inline-formula> is such that <jats:inline-formula><jats:alternatives><jats:tex-math>$0< D(\xi ) \le {K_D} (\xi +1)^{-\alpha }$</jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210522000397_inline6.png" /></jats:alternatives></jats:inline-formula> for all <jats:inline-formula><jats:alternatives><jats:tex-math>$\xi >0$</jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210522000397_inline7.png" /></jats:alternatives></jats:inline-formula> with some <jats:inline-formula><jats:alternatives><jats:tex-math>${K_D}>0$</jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210522000397_inline8.png" /></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$\alpha >0$</jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210522000397_inline9.png" /></jats:alternatives></jats:inline-formula>, then for all initial data from a considerably large set of radial functions on <jats:inline-formula><jats:alternatives><jats:tex-math>$\Omega$</jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210522000397_inline10.png" /></jats:alternatives></jats:inline-formula>, the corresponding initial-boundary value problem admits a solution blowing up in finite time.</jats:p>
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Publishing Year
Journal Title
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Volume
153
Issue
4
Page
1150-1166
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Cite this
Wang Y, Winkler M. Finite-time blow-up in a repulsive chemotaxis-consumption system. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 2022;153(4):1150-1166. doi:10.1017/prm.2022.39
Wang, Y., & Winkler, M. (2022). Finite-time blow-up in a repulsive chemotaxis-consumption system. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 153(4), 1150–1166. https://doi.org/10.1017/prm.2022.39
@article{Wang_Winkler_2022, title={Finite-time blow-up in a repulsive chemotaxis-consumption system}, volume={153}, DOI={10.1017/prm.2022.39}, number={4}, journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, publisher={Cambridge University Press (CUP)}, author={Wang, Yulan and Winkler, Michael}, year={2022}, pages={1150–1166} }
Wang, Yulan, and Michael Winkler. “Finite-Time Blow-up in a Repulsive Chemotaxis-Consumption System.” Proceedings of the Royal Society of Edinburgh: Section A Mathematics 153, no. 4 (2022): 1150–66. https://doi.org/10.1017/prm.2022.39.
Y. Wang and M. Winkler, “Finite-time blow-up in a repulsive chemotaxis-consumption system,” Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol. 153, no. 4, pp. 1150–1166, 2022, doi: 10.1017/prm.2022.39.
Wang, Yulan, and Michael Winkler. “Finite-Time Blow-up in a Repulsive Chemotaxis-Consumption System.” Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol. 153, no. 4, Cambridge University Press (CUP), 2022, pp. 1150–66, doi:10.1017/prm.2022.39.
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