LARGE-TIME BEHAVIOR OF A PARABOLIC-PARABOLIC CHEMOTAXIS MODEL WITH LOGARITHMIC SENSITIVITY IN ONE DIMENSION

By Youshan Tao, Lihe Wang, Zhian Wang. 2012

95 citations recorded by the source corpus.

This paper deals with the chemotaxis system $$ \left\{ \begin{array}{ll} u_t ={D} u_{xx}-\chi [u(\ln v)_x]_x, & x\in (0, 1), \ t>0,\\ v_t =\varepsilon v_{xx} +uv-\mu v, & x\in (0, 1), \ t>0, \end{array} \right. $$ under Neumann boundary condition, where $\chi<0$, $D>0$, $\varepsilon>0$ and $\mu>0$ are constants. It is shown that for any sufficiently smooth initial data $(u_0, v_0)$ fulfilling $u_0\ge 0$, $u_0 \not\equiv 0$ and $v_0>0$, the system possesses a unique global smooth solution that enjoys exponential convergence properties in $L^\infty(\Omega)$ as time goes to infinity, which depend on the sign of $\mu-\bar{u}_0$, where $\bar{u}_0 :=\int_0^1 u_0 dx$. Moreover, we prove that the constant pair $(\mu, (\frac{\mu}{\lambda})^{\frac{D}{\chi}})$ (where $\lambda>0$ is an arbitrary constant) is the only positive stationary solution. The biological implications of our results will be given in the paper.

Subfields: Computational Theory and Mathematics, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/7541c8d669c9c8a133505d5d7e4a8e4f2b53359a

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Citations in the corpus

The two-dimensional Keller–Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties (2016, background): “1) with widely arbitrary initial data have been derived in [39] and [28].”
Mathematics of traveling waves in chemotaxis --Review paper-- (2012, background): “2) on other aspects, such as self-similar solutions and collapse behavior [67], the global existence and large time behavior of solutions [90, 26, 46, 84, 79], and Cauchy problem [45].”
Boundary spike‐layer solutions of the singular Keller–Segel system: existence and stability (2019, background): “[3, 4, 6, 13, 21, 23–26]), global well-posedness and/or asymptotic behavior of solutions (see [5, 8, 19, 20, 22, 28, 33, 37, 42, 43] in one dimensional bounded or unbounded space and [7, 9, 18, 22, 32, 35, 39, 41] in multidimensional spaces) and boundary layer solutions [10–12].”
Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D (2017, background): “1) are very sparse, with widely arbitrary initial data only being treated for the one-dimensional case ([28],[16]).”