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doi: 10.3934/dcdsb.2021099
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Construction of unique mild solution and continuity of solution for the small initial data to 1-D Keller-Segel system

Yumi Yahagi

Department of Mathematical Informatics, Tokyo University of Information Sciences, 4-1 Onaridai, Wakaba-ku, Chiba, 265-8501 Japan

Received  October 2020 Revised  January 2021 Early access March 2021

Fund Project: The author is supported by JSPS KAKENHI Grant Number JP19K14558

In this paper, a one-dimensional Keller-Segel system of parabolic-parabolic type which is defined on the bounded interval with the Dirichlet boundary condition is considered. Under the assumption that initial data is sufficiently small, a unique mild solution to the system is constructed and the continuity of solution for the initial data is shown, by using an argument of successive approximations.

Keywords: Keller-Segel system, mild solution, Dirichlet boundary condition, successive approximations, small initial data.
Mathematics Subject Classification: Primary: 35A01, 35A02; Secondary: 35B30.
Citation: Yumi Yahagi. Construction of unique mild solution and continuity of solution for the small initial data to 1-D Keller-Segel system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021099
References:
[1]

S. Albeverio, Y. Yahagi and M. W. Yoshida, An explicit time asymptotics of a solution to Keller-Segel system on bounded interval, preprint. Google Scholar

[2]

A. Aruchamy and J. Tyagi, Nonnegative solutions to time fractional Keller-Segel system, Math. Methods Appl. Sci, (2020), 1–19 (Online). Google Scholar

[3]

X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Conti. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.  Google Scholar

[4]

J. A. Carrillo, J. Li and Z. A. Wang, Boundary spike-layer solutions of the singular Keller-Segel system: Existence and stability, Proc. London Math. Soc., (2020). Google Scholar

[5]

L. Corrias, M. Escobedo and J. Matos, Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane, J. Differential Equations, 257 (2014), 1840-1878.  doi: 10.1016/j.jde.2014.05.019.  Google Scholar

[6] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^nd$ edition, Cambridge University Press, 2014.   Google Scholar
[7] E. D. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989.  doi: 10.1017/CBO9780511566158.  Google Scholar
[8]

V. Georgiev and K. Taniguchi, Gradient estimates and their optimality for heat equation in an exterior domain, preprint, arXiv: 1710.00592 (2017). Google Scholar

[9]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math.Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[10]

Q. Hou and Z. A. Wang, Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane, J. Math. Pures Appl., 130 (2019), 251-287.  doi: 10.1016/j.matpur.2019.01.008.  Google Scholar

[11]

Q. Q. Hou, C. J. Liu, Y. G. Wang and Z. A. Wang, Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis: One dimensional case, SIAM J. Math. Anal., 50 (2018), 3058-3091.  doi: 10.1137/17M112748X.  Google Scholar

[12]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[13]

H. Kozono, Y. Sugiyama and T. Wachi, Existence and uniqueness theorem on mild solutions to the Keller-Segel system in the scaling invariant space, J. Differential Equations, 252 (2012), 1213-1228.  doi: 10.1016/j.jde.2011.08.025.  Google Scholar

[14]

C. C. Lee, Z. A. Wang and W. Yang, Boundary-layer profile of a singularly perturbed non-local semi-linear problem arising in chemotaxis, Nonlinearity, 33 (2020), 5111-5141.   Google Scholar

[15]

Y. Miura, Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions, Discrete Conti. Dyn. Syst., 37 (2017), 1603-1630.   Google Scholar

[16]

K. Osaki and A. Yagi, Finite Dimensional Attractor for one-dimensional Keller-Segel Equations, Funkcial. Ekvac., 44 (2001), 441-469.   Google Scholar

[17] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, 1972.   Google Scholar
[18]

Y. Sugiyama and Y. Yahagi, Uniqueness and continuity of solution for the initial data in the scaling invariant class of degenerate Keller-Segel system, J. Evol. Equ., 11 (2011), 319-337.  doi: 10.1007/s00028-010-0093-8.  Google Scholar

[19]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[20]

M. Winkler, Small-Mass Solutions in the Two-Dimensional Keller–Segel System Coupled to the Navier–Stokes Equations, SIAM J. Math. Anal., 52 (2020), 2041-2080.   Google Scholar

[21]

Y. Yahagi, Construction of a unique mild solution of one-dimensional Keller-Segel systems with uniformly elliptic operators having variable coefficients, Math. Slovaca, 68 (2018), 845-866.  doi: 10.1515/ms-2017-0150.  Google Scholar

[22]

Y. Yahagi, Asymptotic behavior of solutions to the one-dimensional Keller-Segel system with small chemotaxis, Tokyo J. Math., 41 (2018), 175-191.  doi: 10.3836/tjm/1502179267.  Google Scholar

show all references

References:
[1]

S. Albeverio, Y. Yahagi and M. W. Yoshida, An explicit time asymptotics of a solution to Keller-Segel system on bounded interval, preprint. Google Scholar

[2]

A. Aruchamy and J. Tyagi, Nonnegative solutions to time fractional Keller-Segel system, Math. Methods Appl. Sci, (2020), 1–19 (Online). Google Scholar

[3]

X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Conti. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.  Google Scholar

[4]

J. A. Carrillo, J. Li and Z. A. Wang, Boundary spike-layer solutions of the singular Keller-Segel system: Existence and stability, Proc. London Math. Soc., (2020). Google Scholar

[5]

L. Corrias, M. Escobedo and J. Matos, Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane, J. Differential Equations, 257 (2014), 1840-1878.  doi: 10.1016/j.jde.2014.05.019.  Google Scholar

[6] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^nd$ edition, Cambridge University Press, 2014.   Google Scholar
[7] E. D. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989.  doi: 10.1017/CBO9780511566158.  Google Scholar
[8]

V. Georgiev and K. Taniguchi, Gradient estimates and their optimality for heat equation in an exterior domain, preprint, arXiv: 1710.00592 (2017). Google Scholar

[9]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math.Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[10]

Q. Hou and Z. A. Wang, Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane, J. Math. Pures Appl., 130 (2019), 251-287.  doi: 10.1016/j.matpur.2019.01.008.  Google Scholar

[11]

Q. Q. Hou, C. J. Liu, Y. G. Wang and Z. A. Wang, Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis: One dimensional case, SIAM J. Math. Anal., 50 (2018), 3058-3091.  doi: 10.1137/17M112748X.  Google Scholar

[12]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[13]

H. Kozono, Y. Sugiyama and T. Wachi, Existence and uniqueness theorem on mild solutions to the Keller-Segel system in the scaling invariant space, J. Differential Equations, 252 (2012), 1213-1228.  doi: 10.1016/j.jde.2011.08.025.  Google Scholar

[14]

C. C. Lee, Z. A. Wang and W. Yang, Boundary-layer profile of a singularly perturbed non-local semi-linear problem arising in chemotaxis, Nonlinearity, 33 (2020), 5111-5141.   Google Scholar

[15]

Y. Miura, Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions, Discrete Conti. Dyn. Syst., 37 (2017), 1603-1630.   Google Scholar

[16]

K. Osaki and A. Yagi, Finite Dimensional Attractor for one-dimensional Keller-Segel Equations, Funkcial. Ekvac., 44 (2001), 441-469.   Google Scholar

[17] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, 1972.   Google Scholar
[18]

Y. Sugiyama and Y. Yahagi, Uniqueness and continuity of solution for the initial data in the scaling invariant class of degenerate Keller-Segel system, J. Evol. Equ., 11 (2011), 319-337.  doi: 10.1007/s00028-010-0093-8.  Google Scholar

[19]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[20]

M. Winkler, Small-Mass Solutions in the Two-Dimensional Keller–Segel System Coupled to the Navier–Stokes Equations, SIAM J. Math. Anal., 52 (2020), 2041-2080.   Google Scholar

[21]

Y. Yahagi, Construction of a unique mild solution of one-dimensional Keller-Segel systems with uniformly elliptic operators having variable coefficients, Math. Slovaca, 68 (2018), 845-866.  doi: 10.1515/ms-2017-0150.  Google Scholar

[22]

Y. Yahagi, Asymptotic behavior of solutions to the one-dimensional Keller-Segel system with small chemotaxis, Tokyo J. Math., 41 (2018), 175-191.  doi: 10.3836/tjm/1502179267.  Google Scholar

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