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Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$

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  • This paper gives the gradient estimate for solutions to the quasilinear non-degenerate parabolic-parabolic Keller-Segel system (KS) on the whole space $\mathbb{R}^N$. The gradient estimate for (KS) on bounded domains is known as an application of Amann's existence theory in [1]. However, in the whole space case it seems necessary to derive the gradient estimate directly. The key to the proof is a modified Bernstein's method. The result is useful to obtain the whole space version of the global existence result by Tao-Winkler [13] except for the boundedness.
    Mathematics Subject Classification: Primary: 35K57; Secondary: 35B33.


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  • [1]

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    Y. Sugiyama, Global existence in the sub-critical cases and finite time blow-up in the super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.


    Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180.


    Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.doi: 10.1016/j.jde.2006.03.003.


    Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.doi: 10.1016/j.jde.2011.08.019.

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