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Register a new userGlobal existence, uniform boundedness, and stabilization in a chemotaxis system with density-suppressed motility and nutrient consumption
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Well-posedness and uniform-in-time boundedness of classical solutions are investigated for a three-component parabolic system which describes the dynamics of a population of cells interacting with a chemoattractant and a nutrient. The former induces a chemotactic bias in the diffusive motion of the cells and is accounted for by a density-suppressed motility. Well-posedness is first established for generic positive and non-increasing motility functions vanishing at infinity. Growth conditions on the motility function guaranteeing the uniform-in-time boundedness of solutions are next identified. Finally, for sublinearly decaying motility functions, convergence to a spatially homogeneous steady state is shown, with an exponential rate for consumption rates behaving linearly near zero.
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This paper deals with a boundary-value problem in three-dimensional smooth bounded convex domains for the coupled chemotaxis-Stokes system with slow $p$-Laplacian diffusion begin{equation} onumber left{ begin{aligned} &n_t+ucdot abla n= ablacdotleft(| abla n|^{p-2} abla nright)- ablacdot(n abla c), &xinOmega, t>0, &c_t+ucdot abla c=Delta c-nc,&xinOmega, t>0, &u_t=Delta u+ abla P+n ablaphi ,&xinOmega, t>0, & ablacdot u=0, &xinOmega, t>0, end{aligned} right. end{equation} where $phiin W^{2,infty}(Omega)$ is the gravitational potential. It is proved that global bounded weak solutions exist whenever $p>frac{23}{11}$ and the initial data $(n_0,c_0,u_0)$ are sufficiently regular satisfying $n_0geq 0$ and $c_0geq 0$.
This paper investigates a high-dimensional chemotaxis system with consumption of chemoattractant begin{eqnarray*} left{begin{array}{l} u_t=Delta u- ablacdot(u abla v), v_t=Delta v-uv, end{array}right. end{eqnarray*} under homogeneous boundary conditions of Neumann type, in a bounded convex domain $Omegasubsetmathbb{R}^n~(ngeq4)$ with smooth boundary. It is proved that if initial data satisfy $u_0in C^0(overline{Omega})$ and $v_0in W^{1,q}(Omega)$ for some $q>n$, the model possesses at least one global renormalized solution.
In this paper, we study a semilinear system involving the curl operator in a bounded and convex domain in $R^3$, which comes from the steady-state approximation for Bean critical-state model for type-II superconductors. We show the existence and the $L^{infty}$ estimate for weak solutions to this system.
In this paper, we study the active hydrodynamics, described in the Q-tensor liquid crystal framework. We prove the existence of global weak solutions in dimension two and three, with suitable initial datas. By using Littlewood-Paley decomposition, we also get the higher regularity of the weak solutions and the uniqueness of weak-strong solutions in dimension two.
We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension $dgeq 3$ and find two critical curves intersecting at one same point which separate the global existence and blow up of weak solutions to the problem. More precisely, above these curves (i.e. subcritical case), the problem admits a global weak solution obtained by the limits of strong solutions to an approximated system. Based on the second moment of solutions, initial data are constructed to make sure blow up occurs in finite time below these curves (i.e. critical and supercritical cases). In addition, the existence or non-existence of minimizers of free energy functional is discussed on the critical curves and the solutions exist globally in time if the size of initial data is small. We also investigate the crossing point between the critical lines in which a refined criteria in terms of the masses is given again to distinguish the dichotomy between global existence and blow up. We also show that the blow ups is simultaneous for both species.
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