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The modified Camassa-Holm equation in Lagrangian coordinates
A new criterion to a two-chemical substances chemotaxis system with critical dimension
Department of Applied Mathematics, Northwestern Polytechnical University, 127 West Youyi Road, Xi'an 710072, Shaanxi, China |
$\begin{align*}\begin{cases}u_t = Δ u-χ\nabla·(u\nabla v) &\text{ in }Ω×\mathbb R^+,\\v_t = Δ v-v+w &\text{ in }Ω×\mathbb R^+,\\w_t = Δ w-w+u &\text{ in }Ω×\mathbb R^+,\end{cases}\end{align*}$ |
References:
[1] |
N. D. Alikakos,
Lp-bounds of solutions of reaction diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[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. Models Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[3] |
X. Cao, An interpolation inequality and its application in Keller-Segel model, preprint, arXiv:1707.09235. |
[4] |
X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model Zeitschrift Für Angewandte Mathematik Und Physik, 67 (2016).
doi: 10.1007/s00033-015-0601-3. |
[5] |
K. Fujie and T. Senba,
Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differetial Equations, 37 (2017), 61-83.
doi: 10.1016/j.jde.2017.02.031. |
[6] |
M. Hieber and J. Prüss,
Heat kernels and maximal lp-lq estimate for parabolic evolution equations, Comm. Partial Differential Equaitons, 22 (1997), 1647-1669.
doi: 10.1080/03605309708821314. |
[7] |
T. Hillen and K. J. Painter,
A user's guide to PDE models in a chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[8] |
D. Horstmann,
From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I. Jber. DMV, 105 (2003), 103-165.
|
show all references
References:
[1] |
N. D. Alikakos,
Lp-bounds of solutions of reaction diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[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. Models Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[3] |
X. Cao, An interpolation inequality and its application in Keller-Segel model, preprint, arXiv:1707.09235. |
[4] |
X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model Zeitschrift Für Angewandte Mathematik Und Physik, 67 (2016).
doi: 10.1007/s00033-015-0601-3. |
[5] |
K. Fujie and T. Senba,
Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differetial Equations, 37 (2017), 61-83.
doi: 10.1016/j.jde.2017.02.031. |
[6] |
M. Hieber and J. Prüss,
Heat kernels and maximal lp-lq estimate for parabolic evolution equations, Comm. Partial Differential Equaitons, 22 (1997), 1647-1669.
doi: 10.1080/03605309708821314. |
[7] |
T. Hillen and K. J. Painter,
A user's guide to PDE models in a chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[8] |
D. Horstmann,
From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I. Jber. DMV, 105 (2003), 103-165.
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