三维趋化系统全局弱解的存在性和渐近稳定性

doi:10.12052/gdutxb.200148

Abstract

Abstract: The well-posedness and large-time behavior for a parabolic-hyperbolic system with discontinuous initial value is mainly considered. The system is transformed from a three dimensional PDE-ODE hybrid chemotaxis system describing the initiation of tumor angiogenesis by a Cole-Hopf type transformation. With discontinuous data, it is proved that the solution of the transformed system converges to a constant equilibrium as time tends to infinity. Lastly, the results for the pre-transformed PDE-ODE hybrid system are obtained by reversing the Cole-Hopf transformation. Compared with the relevant results on continuous initial datum, the asymptotic stability of discontinuous initial values with large oscillations is proved. In the proof, the so-called “effective viscous flux” is used for the analysis to obtain the desired energy estimates and regularity. The "effective viscous flux" technique is rarely used in the literature of chemotaxis model. But it has proved to be a useful tool for studying chemotaxis systems with initial datum having low regularity.

Key words: chemotaxis, asymptotic stability, discontinuous initial data, effective viscous flux

CLC Number:

O175

Tang Hao-yi, Peng Hong-yun. Global Weak Solutions and Asymptotics of a 3D Chemotaxis System[J].Journal of Guangdong University of Technology, 2022, 39(01): 93-98.

References

[1] DENG C, LI T. Well-posedness of a 3D parabolic-hyperbolic Keller-Segel system in the Sobolev space framework [J]. Journal of Differential Equations, 2014, 257(5): 1311-1332.
[2] WANG Z A, XIANG Z, YU P. Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis [J]. Journal of Differential Equations, 2016, 260(3): 2225-2258.
[3] GRANERO-BELINCHÓN R. Global solutions for a hyperbolic-parabolic system of chemotaxis [J]. Journal of Mathematical Analysis and Applications, 2017, 449(1): 872-883.
[4] HOU Q Q, LIU C J, WANG Y G, et al. Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis: one dimensional case [J]. SIAM Journal on Mathematical Analysis, 2018, 50(3): 3058-3091.
[5] CORRIAS L, PERTHAME B, ZAAG H. A chemotaxis model motivated by angiogenesis [J]. Comptes Rendus Mathématique Académie des Sciences Paris, 2003, 336(2): 141-146.
[6] LEVINE H A, SLEEMAN B D. A system of reaction diffusion equations arising in the theory of reinforced random walks [J]. SIAM Journal on Applied Mathematics, 1997, 57(3): 683-730.
[7] MEI M, PENG H, WANG Z A. Asymptotic profile of a parabolic-hyperbolic system with boundary effect arising from tumor angiogenesis [J]. Journal of Differential Equations, 2015, 259(10): 5168-51,91.
[8] GUO J, XIAO J X, ZHAO H J, et al. Global solutions to a hyperbolic-parabolic coupled system with large initial data[J]. Acta Mathematica Scientia, 2009, 29(3): 629-641.
[9] LI D, PAN R, ZHAO K. Quantitative decay of a hybrid type chemotaxis model with large data [J]. Nonlinearity, 2015, 28: 2181-2210.
[10] MARTINEZ V, WANG Z A, ZHAO K. Asymptotic and viscous stability of large-amplitude solutions of a hyperbolic system arising from biology [J]. Indiana University Mathematics Journal, 2018, 67(4): 1383-1424.
[11] LI T, PAN R H, ZHAO K. Global dynamics of a chemotaxis model on bounded domains with large data [J]. SIAM Journal on Applied Mathematics, 2012, 72: 417-443.
[12] CHAE M, CHOI K, KANG K, et al. Stability of planar traveling waves in a Keller-Segel equation on an infinite strip domain [J]. Journal of Differential Equations, 2018, 265(1): 237-279.
[13] LI D, LI T, ZHAO K. On a hyperbolic-parabolic system modeling chemotaxis [J]. Mathematical Models and Methods in Applied Sciences, 2011, 21(8): 1631-1650.
[14] FAN J, ZHAO K. Blow up criteria for a hyperbolic-parabolic system arising from chemotaxis [J]. Journal of Mathematical Analysis and Applications, 2012, 394: 687-695.
[15] PENG H Y, WANG Z A. On a parabolic-hyperbolic chemotaxis system with discontinuous data: well- posedness, stability and regularity[J]. Journal of Differential Equations, 2020, 268(8): 4374-4 415.
[16] PENG H Y, WANG Z A, ZHU C J. Global weak solutions and asymptotics of a singular PDE-ODE chemotaxis system with discontinuous data[J/OL]. Science China-Mathematics, 2020[2020- 10-10]. https://engine.scichina.com/doi/10.1007/s11425-019-1754-0.
[17] HOFF D. Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data [J]. Archive for Rational Mechanics and Analysis, 1995, 132(1): 1-14.
[18] FRIEDMAN A. Partial Differential Equations[M]. New York: Holt, Rinehart Winston, 1969: 2-11.

Metrics

Viewed

Full text

Abstract

Cited

Shared

Discussed

Comments

Recommended 0

No Suggested Reading articles found!

Global Weak Solutions and Asymptotics of a 3D Chemotaxis System

HTML

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 5

Metrics

Comments

Recommended 0

[1]	HUANG Ming-Hui. Asymptotic Stability of Nonlinear Differential Equation with Time Delays [J]. Journal of Guangdong University of Technology, 2016, 33(01): 62-66.
[2]	DONG Chao, GAO Xue-Jun, ZHOU Min. Global Asymptotic Stability and Boundedness of Solutions to-a Kind of Non-Linear Third-Order Delay Differential Equations [J]. Journal of Guangdong University of Technology, 2015, 32(2): 98-103.
[3]	XU Lin, SONG Chang-Xiu. Stability and Boundedness of a Class of Thirdorder Delay Differential Equations [J]. Journal of Guangdong University of Technology, 2015, 32(1): 128-132.
[4]	XIE Shu-Fang. An Analysis of the Stability of Timelag Neutral Delay Differential Systems [J]. Journal of Guangdong University of Technology, 2011, 28(3): 77-82.
[5]	Xie Zu-wei,Peng Shi-guo,Luo Ying-hua . The Asymptotic Stability of Networks with Leakage Delays [J]. Journal of Guangdong University of Technology, 2009, 26(1): 13-.