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The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation
Construction of unique mild solution and continuity of solution for the small initial data to 1-D Keller-Segel system
Department of Mathematical Informatics, Tokyo University of Information Sciences, 4-1 Onaridai, Wakaba-ku, Chiba, 265-8501 Japan |
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.
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. |
[2] |
A. Aruchamy and J. Tyagi, Nonnegative solutions to time fractional Keller-Segel system, Math. Methods Appl. Sci, (2020), 1–19 (Online). |
[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. |
[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). |
[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. |
[6] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^nd$ edition, Cambridge University Press, 2014.
![]() |
[7] |
E. D. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989.
doi: 10.1017/CBO9780511566158.![]() ![]() |
[8] |
V. Georgiev and K. Taniguchi, Gradient estimates and their optimality for heat equation in an exterior domain, preprint, arXiv: 1710.00592 (2017). |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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.
|
[16] |
K. Osaki and A. Yagi,
Finite Dimensional Attractor for one-dimensional Keller-Segel Equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[17] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, 1972.
![]() |
[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. |
[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. |
[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.
|
[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. |
[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. |
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. |
[2] |
A. Aruchamy and J. Tyagi, Nonnegative solutions to time fractional Keller-Segel system, Math. Methods Appl. Sci, (2020), 1–19 (Online). |
[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. |
[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). |
[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. |
[6] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^nd$ edition, Cambridge University Press, 2014.
![]() |
[7] |
E. D. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989.
doi: 10.1017/CBO9780511566158.![]() ![]() |
[8] |
V. Georgiev and K. Taniguchi, Gradient estimates and their optimality for heat equation in an exterior domain, preprint, arXiv: 1710.00592 (2017). |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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.
|
[16] |
K. Osaki and A. Yagi,
Finite Dimensional Attractor for one-dimensional Keller-Segel Equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[17] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, 1972.
![]() |
[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. |
[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. |
[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.
|
[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. |
[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. |
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