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February  2022, 21(2): 687-704. doi: 10.3934/cpaa.2021194

## Nonnegative solutions to a doubly degenerate nutrient taxis system

 1 College of Information and Technology, Donghua University, Shanghai 201620, China 2 Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

* Corresponding author

Received  May 2021 Revised  October 2021 Published  February 2022 Early access  November 2021

Fund Project: The first author was funded by the China Scholarship Council (No. 202006630070). The second author acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the project Emergence of structures and advantages in cross-diffusion systems (No. 411007140, GZ: WI 3707/5-1)

This paper deals with the doubly degenerate nutrient taxis system
 $\begin{eqnarray*} \left\{ \begin{array}{ll} u_t = (uv u_x)_x - (u^2 vv_x)_x + \ell uv, \qquad & x\in \Omega, \ t>0, \\ v_t = v_{xx} -uv, \qquad & x\in \Omega, \ t>0, \end{array} \right. \end{eqnarray*}$
in an open bounded interval
 $\Omega\subset \mathbb{R}$
, with
 $\ell \ge0$
, which has been proposed to model the formation of diverse morphological aggregation patterns observed in colonies of Bacillus subtilis growing on the surface of thin agar plates.
It is shown that under the mere assumption that
 $\begin{eqnarray*} \left\{ \begin{array}{l} u_0\in W^{1,\infty}( \Omega) \mbox{ is nonnegative with } u_0\not\equiv 0 \qquad \mbox{and} \\ v_0\in W^{1,\infty}( \Omega) \mbox{ is positive in } \overline{\Omega}, \end{array} \right. \qquad \qquad (\star) \end{eqnarray*}$
an associated no-flux initial boundary value problem possesses a globally defined and continuous weak solution
 $(u,v)$
, where
 $u\ge 0$
and
 $v>0$
in
 $\overline{\Omega}\times [0,\infty)$
, and that moreover there exists
 $u_\infty\in C^0( \overline{\Omega})$
such that the solution
 $(u,v)$
approaches the pair
 $(u_\infty,0)$
in the large time limit with respect to the topology
 $(L^{\infty}( \Omega)) ^2$
. This extends comparable results recently obtained in [17], the latter crucially relying on the additional requirement that
 $\int_\Omega \ln u_0>-\infty$
, to situations involving nontrivially supported initial data
 $u_0$
, which seems to be of particular relevance in the addressed application context of sparsely distributed populations.
Citation: Genglin Li, Michael Winkler. Nonnegative solutions to a doubly degenerate nutrient taxis system. Communications on Pure and Applied Analysis, 2022, 21 (2) : 687-704. doi: 10.3934/cpaa.2021194
##### References:
 [1] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value probems, in Function Spaces, Differential Operators and Nonlinear Analysis, Teubner-Texte Math, vol 133 (eds. H. Schmeisser and H. Triebel), Teubner, Stuttgart, (1993), 9–126. doi: 10.1007/978-3-663-11336-2_1. [2] M. Delgado, I. Gayte, C. Morales-Rodrigo and A. Suárez, An angiogenesis model with nonlinear chemotactic response and flux at the tumor boundary, Nonlinear Anal., 72 (2010), 330-347.  doi: 10.1016/j.na.2009.06.057. [3] H. Fujikawa, Periodic growth of Bacillus subtilis colonies on agar plates, Phys. A, 189 (1992), 15-21.  doi: 10.1016/0378-4371(92)90123-8. [4] H. Fujikawa and M. Matsushita, Fractal growth of Bacillus subtilis on agar plates, J. Phys. Soc. Japan, 47 (1989), 2764-2767.  doi: 10.1143/JPSJ.58.3875. [5] K. Kawasaki, A. Mochizuki, T. Umeda and N. Shigesada, Modeling spatio-temporal patterns generated by Bacillus subtilis, J. Math. Biol., 188 (1997), 177-185.  doi: 10.1006/jtbi.1997.0462. [6] E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Math. Biol, 26 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8. [7] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Trans., Providence, RI, 1968 [8] J. Lankeit, Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity, J. Differ. Equ., 262 (2017), 4052-4084.  doi: 10.1016/j.jde.2016.12.007. [9] Ph. Laurençot and D. Wrzosek, A chemotaxis model with threshold density and degenerate diffusion, in Nonlinear Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 64 (eds. H. Brezis, M. Chipot, J. Escher), Birkhäuser Basel, (2005), 273–290. doi: https://doi.org/10.1007/3-7643-7385-7_16. [10] M. Matsushita and H. Fujikawa, Diffusion-limited growth in bacterial colony formation, Phys. A, 168 (1990), 498-506.  doi: 10.1016/0378-4371(90)90402-E. [11] J. F. Leyva, C. Málaga and R. G. Plaza, The effects of nutrient chemotaxis on bacterial aggregation patterns with non-linear degenerate cross diffusion, Phys. A, 392 (2013), 5644-5662.  doi: 10.1016/j.physa.2013.07.022. [12] R. G. Plaza, Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process, J. Math. Biol, 78 (2019), 1681-1711.  doi: 10.1007/s00285-018-1323-x. [13] M. M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Equ., 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045. [14] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equ., 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019. [15] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008. [16] Z. A. Wang, Z. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differ. Equ., 260 (2016), 2225-2258.  doi: 10.1016/j.jde.2015.09.063. [17] M. Winkler, Does spatial homogeneity ultimately prevail in nutrient taxis systems? A paradigm for structure support by rapid diffusion decay in an autonomous parabolic flow, Trans. Amer. Math. Soc., 374 (2021), 219-268.  doi: 10.1090/tran/8163. [18] M. Winkler, Small-signal solutions of a two-dimensional doubly degenerate taxis system modeling bacterial motion in nutrient-poor environments, Nonlinear Anal. Real World Appl., 63 (2022), 103407.  doi: 10.1016/j.nonrwa.2021.103407. [19] M. Winkler, Elliptic Harnack inequalities in linear parabolic equations and application to the asymptotics in a doubly degenerate nutrient taxis system, preprint. [20] M. Winkler, Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion model for bacterial motion on a nutrient-poor agar, preprint. [21] M. Winkler, Persistent localization vs. eventual positivity in a doubly degenerate reaction-diffusion(- chemotaxis) system modeling bacterial motion in nutrient-poor environments, preprint. [22] D. Wrzosek, Global attractor for a chemotaxis model with prevention of overcrowding, Nonlinear Anal., 59 (2004), 1293-1310.  doi: 10.1016/j.na.2004.08.015.

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##### References:
 [1] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value probems, in Function Spaces, Differential Operators and Nonlinear Analysis, Teubner-Texte Math, vol 133 (eds. H. Schmeisser and H. Triebel), Teubner, Stuttgart, (1993), 9–126. doi: 10.1007/978-3-663-11336-2_1. [2] M. Delgado, I. Gayte, C. Morales-Rodrigo and A. Suárez, An angiogenesis model with nonlinear chemotactic response and flux at the tumor boundary, Nonlinear Anal., 72 (2010), 330-347.  doi: 10.1016/j.na.2009.06.057. [3] H. Fujikawa, Periodic growth of Bacillus subtilis colonies on agar plates, Phys. A, 189 (1992), 15-21.  doi: 10.1016/0378-4371(92)90123-8. [4] H. Fujikawa and M. Matsushita, Fractal growth of Bacillus subtilis on agar plates, J. Phys. Soc. Japan, 47 (1989), 2764-2767.  doi: 10.1143/JPSJ.58.3875. [5] K. Kawasaki, A. Mochizuki, T. Umeda and N. Shigesada, Modeling spatio-temporal patterns generated by Bacillus subtilis, J. Math. Biol., 188 (1997), 177-185.  doi: 10.1006/jtbi.1997.0462. [6] E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Math. Biol, 26 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8. [7] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Trans., Providence, RI, 1968 [8] J. Lankeit, Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity, J. Differ. Equ., 262 (2017), 4052-4084.  doi: 10.1016/j.jde.2016.12.007. [9] Ph. Laurençot and D. Wrzosek, A chemotaxis model with threshold density and degenerate diffusion, in Nonlinear Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 64 (eds. H. Brezis, M. Chipot, J. Escher), Birkhäuser Basel, (2005), 273–290. doi: https://doi.org/10.1007/3-7643-7385-7_16. [10] M. Matsushita and H. Fujikawa, Diffusion-limited growth in bacterial colony formation, Phys. A, 168 (1990), 498-506.  doi: 10.1016/0378-4371(90)90402-E. [11] J. F. Leyva, C. Málaga and R. G. Plaza, The effects of nutrient chemotaxis on bacterial aggregation patterns with non-linear degenerate cross diffusion, Phys. A, 392 (2013), 5644-5662.  doi: 10.1016/j.physa.2013.07.022. [12] R. G. Plaza, Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process, J. Math. Biol, 78 (2019), 1681-1711.  doi: 10.1007/s00285-018-1323-x. [13] M. M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Equ., 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045. [14] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equ., 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019. [15] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008. [16] Z. A. Wang, Z. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differ. Equ., 260 (2016), 2225-2258.  doi: 10.1016/j.jde.2015.09.063. [17] M. Winkler, Does spatial homogeneity ultimately prevail in nutrient taxis systems? A paradigm for structure support by rapid diffusion decay in an autonomous parabolic flow, Trans. Amer. Math. Soc., 374 (2021), 219-268.  doi: 10.1090/tran/8163. [18] M. Winkler, Small-signal solutions of a two-dimensional doubly degenerate taxis system modeling bacterial motion in nutrient-poor environments, Nonlinear Anal. Real World Appl., 63 (2022), 103407.  doi: 10.1016/j.nonrwa.2021.103407. [19] M. Winkler, Elliptic Harnack inequalities in linear parabolic equations and application to the asymptotics in a doubly degenerate nutrient taxis system, preprint. [20] M. Winkler, Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion model for bacterial motion on a nutrient-poor agar, preprint. [21] M. Winkler, Persistent localization vs. eventual positivity in a doubly degenerate reaction-diffusion(- chemotaxis) system modeling bacterial motion in nutrient-poor environments, preprint. [22] D. Wrzosek, Global attractor for a chemotaxis model with prevention of overcrowding, Nonlinear Anal., 59 (2004), 1293-1310.  doi: 10.1016/j.na.2004.08.015.
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