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doi: 10.3934/cpaa.2021194
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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 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 & Applied Analysis, 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.  Google Scholar

[2]

M. DelgadoI. GayteC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

K. KawasakiA. MochizukiT. 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.  Google Scholar

[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.  Google Scholar

[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 doi: 978-0-8218-1573-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. F. LeyvaC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Z. A. WangZ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

M. Winkler, Elliptic Harnack inequalities in linear parabolic equations and application to the asymptotics in a doubly degenerate nutrient taxis system, preprint. Google Scholar

[20]

M. Winkler, Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion model for bacterial motion on a nutrient-poor agar, preprint. Google Scholar

[21]

M. Winkler, Persistent localization vs. eventual positivity in a doubly degenerate reaction-diffusion(- chemotaxis) system modeling bacterial motion in nutrient-poor environments, preprint. Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

M. DelgadoI. GayteC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

K. KawasakiA. MochizukiT. 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.  Google Scholar

[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.  Google Scholar

[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 doi: 978-0-8218-1573-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. F. LeyvaC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Z. A. WangZ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

M. Winkler, Elliptic Harnack inequalities in linear parabolic equations and application to the asymptotics in a doubly degenerate nutrient taxis system, preprint. Google Scholar

[20]

M. Winkler, Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion model for bacterial motion on a nutrient-poor agar, preprint. Google Scholar

[21]

M. Winkler, Persistent localization vs. eventual positivity in a doubly degenerate reaction-diffusion(- chemotaxis) system modeling bacterial motion in nutrient-poor environments, preprint. Google Scholar

[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.  Google Scholar

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