No Access

Global radial renormalized solution to a producer–scrounger model with singular sensitivities

Xinru Cao

College of Science, Donghua University, Shanghai 201620, P. R. China

https://doi.org/10.1142/S0218202520400084Cited by:21 (Source: Crossref)

This article is part of the issue:

Special Issue on "Chemotaxis Systems in Complex Frameworks:Pattern Formation, Qualitative Analysis and Blowup Prevention"
Guest Editors: N. Bellomo, Y. Tao and M. Winkler

Abstract

This paper is concerned with the parabolic system

{ut=Δu−∇⋅(uw∇w),(x,t)∈Ω×(0,T),vt=Δv−∇⋅(vu∇u),(x,t)∈Ω×(0,T),wt=Δw−(u+v)w−λw+g(x,t),(x,t)∈Ω×(0,T),<math display="block" altimg="eq-00001.gif"><mfenced separators="" open="{" close=""><mrow><mtable equalrows="false" columnlines="none" equalcolumns="false" class="array"><mtr><mtd class="array" columnalign="left"><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo class="MathClass-rel">=</mo><mi mathvariant="normal">Δ</mi><mi>u</mi><mo stretchy="false" class="MathClass-bin">−</mo><mo class="MathClass-op">∇</mo><mo stretchy="false" class="MathClass-bin">⋅</mo><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><mi>u</mi></mrow><mrow><mi>w</mi></mrow></mfrac><mo class="MathClass-op">∇</mo><mi>w</mi></mrow></mfenced><mo class="MathClass-punc">,</mo><mspace width="1em" class="quad"></mspace></mtd><mtd class="array" columnalign="left"><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-punc">,</mo><mi>t</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">∈</mo><mi mathvariant="normal">Ω</mi><mo stretchy="false" class="MathClass-bin">×</mo><mo class="MathClass-open" stretchy="false">(</mo><mn>0</mn><mo class="MathClass-punc">,</mo><mi>T</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo></mtd></mtr><mtr><mtd class="array" columnalign="left"><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo class="MathClass-rel">=</mo><mi mathvariant="normal">Δ</mi><mi>v</mi><mo stretchy="false" class="MathClass-bin">−</mo><mo class="MathClass-op">∇</mo><mo stretchy="false" class="MathClass-bin">⋅</mo><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><mi>v</mi></mrow><mrow><mi>u</mi></mrow></mfrac><mo class="MathClass-op">∇</mo><mi>u</mi></mrow></mfenced><mo class="MathClass-punc">,</mo><mspace width="1em" class="quad"></mspace></mtd><mtd class="array" columnalign="left"><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-punc">,</mo><mi>t</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">∈</mo><mi mathvariant="normal">Ω</mi><mo stretchy="false" class="MathClass-bin">×</mo><mo class="MathClass-open" stretchy="false">(</mo><mn>0</mn><mo class="MathClass-punc">,</mo><mi>T</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo></mtd></mtr><mtr><mtd class="array" columnalign="left"><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo class="MathClass-rel">=</mo><mi mathvariant="normal">Δ</mi><mi>w</mi><mo stretchy="false" class="MathClass-bin">−</mo><mo class="MathClass-open" stretchy="false">(</mo><mi>u</mi><mo stretchy="false" class="MathClass-bin">+</mo><mi>v</mi><mo class="MathClass-close" stretchy="false">)</mo><mi>w</mi><mo stretchy="false" class="MathClass-bin">−</mo><mi>λ</mi><mi>w</mi><mo stretchy="false" class="MathClass-bin">+</mo><mi>g</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-punc">,</mo><mi>t</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo><mspace width="1em" class="quad"></mspace></mtd><mtd class="array" columnalign="left"><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-punc">,</mo><mi>t</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">∈</mo><mi mathvariant="normal">Ω</mi><mo stretchy="false" class="MathClass-bin">×</mo><mo class="MathClass-open" stretchy="false">(</mo><mn>0</mn><mo class="MathClass-punc">,</mo><mi>T</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo></mtd></mtr></mtable></mrow></mfenced></math>(0.1)

in a bounded ball

$Ω = B_{R} (0) \subset ℝ^{n}$ (

$n \geq 2$ ) with

$R > 0$ . Where

$λ \geq 0$ and

$0 \leq g \in C^{1} (\bar{Ω} \times (0, \infty))$ . It is shown that for arbitrarily radially symmetric initial data

$(u_{0}, v_{0}, w_{0})$ , which are nonnegative and suitably regular, the corresponding Neumann initial-boundary problem admits a global renormalized solution, which is moreover smooth in

$B_{R} ∖ {0} \times (0, \infty)$ .

Communicated by N. Bellomo, Y. Tao and M. Winkler

Keywords:

AMSC: 35K55, 35B40, 35Q35, 92C17, 35B35

References

1. H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis (Springer, 1993), pp. 9–126. Crossref, Google Scholar
2. N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues, Math. Mod. Methods Appl. Sci. 25 (2015) 1663–1763. Link, Web of Science, Google Scholar
3. T. Black, J. Lankeit and M. Mizukami, Singular sensitivity in a Keller–Segel-fluid system, J. Evolution Equations 18 (2018) 561–581. Crossref, Web of Science, Google Scholar
4. W. Hoffman, D. Heinemann and J. A. Wiens, The ecology of seabird feeding flocks in Alaska, Auk 98 (1981) 437–456. Web of Science, Google Scholar
5. H.-Y. Jin and Z.-A. Wang, Global stability of prey-taxis systems, J. Differential Equations 262 (2017) 1257–1290. Crossref, Web of Science, Google Scholar
6. E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol. 30 (1971) 235–248. Crossref, Web of Science, Google Scholar
7. O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type: Translated from the Russian by S. Smith, Translations of Mathematical Monographs, Vol. 23 (American Mathematical Society, 1968). Crossref, Google Scholar
8. J. Lankeit, Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion, J. Differential Equations 262 (2017) 4052–4084. Crossref, Web of Science, Google Scholar
9. J. Lankeit and M. Winkler, A generalized solution concept for the Keller–Segel system with logarithmic sensitivity: Global solvability for large nonradial data, Nonlinear Differential Equations Appl. NoDEA 24 (2017) 49. Crossref, Web of Science, Google Scholar
10. J. Lee, T. Hillen and M. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dyn. 3 (2019) 551–573. Crossref, Google Scholar
11. Y. Liu, Global existence and boundedness of classical solutions to a forager-exploiter model with volume-filling effects, Nonlinear Anal. Real World Appl. 50 (2019) 519–531. Crossref, Web of Science, Google Scholar
12. M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations 103 (1993) 146–178. Crossref, Web of Science, Google Scholar
13. N. Tania, B. Vanderlei, J. P. Heath and L. Edelstein-Keshet, Role of social interactions in dynamic patterns of resource patches and forager aggregation, Proc. Natl. Acad. Sci. 109 (2012) 11228–11233. Crossref, Web of Science, Google Scholar
14. Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations 252 (2012) 2520–2543. Crossref, Web of Science, Google Scholar
15. Y. Tao and M. Winkler, Large time behavior in a forager-exploiter model with different taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci. 29 (2019) 2151–2182. Link, Web of Science, Google Scholar
16. Y. Tao and M. Winkler, Global smooth solvability of a parabolic-elliptic nutrient taxis system in domains of arbitrary dimension, J. Differential Equations 267 (2019) 388–406. Crossref, Web of Science, Google Scholar
17. Z.-A. Wang, Z. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations 260 (2016) 2225–2258. Crossref, Web of Science, Google Scholar
18. M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier–Stokes system, Arch. Ration. Mech. Anal. 211 (2014) 455–487. Crossref, Web of Science, Google Scholar
19. M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal. 47 (2015) 3092–3115. Crossref, Web of Science, Google Scholar
20. M. Winkler, The two-dimensional Keller–Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties, Math. Models Methods Appl. Sci. 26 (2016) 987–1024. Link, Web of Science, Google Scholar
21. M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: Eventual smoothness and equilibration of small-mass solutions, preprint (2016). Google Scholar
22. M. Winkler, Asymptotic homogenization in a three-dimensional nutrient taxis system involving food-supported proliferation, J. Differential Equations 263 (2017) 4826–4869. Crossref, Web of Science, Google Scholar
23. M. Winkler, Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption, J. Differential Equations 264 (2018) 2310–2350. Crossref, Web of Science, Google Scholar
24. M. Winkler, Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions, Math. Models Methods Appl. Sci. 29 (2019) 373–418. Link, Web of Science, Google Scholar
25. S. Wu, J. Shi and B. Wu, Global existence of solutions and uniform persistence of a diffusive predator–prey model with prey-taxis, J. Differential Equations 260 (2016) 5847–5874. Crossref, Web of Science, Google Scholar