Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRenormalized solutions to a chemotaxis system with consumption of chemoattractant
72
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
This paper studies the following system of differential equations modeling tumor angiogenesis in a bounded smooth domain $Omega subset mathbb{R}^N$ ($N=1,2$): $$label{0} left{begin{array}{ll} p_t=Delta p- ablacdotp p(displaystylefrac alpha {1+c} abla c+rho abla w)+lambda p(1-p),,& xin Omega, t>0, c_t=Delta c-c-mu pc,, &xin Omega, t>0, w_t= gamma p(1-w),,& xin Omega, t>0, end{array}right. $$ where $alpha, rho, lambda, mu$ and $gamma$ are positive parameters. For any reasonably regular initial data $(p_0, c_0, w_0)$, we prove the global boundedness ($L^infty$-norm) of $p$ via an iterative method. Furthermore, we investigate the long-time behavior of solutions to the above system under an additional mild condition, and improve previously known results. In particular, in the one-dimensional case, we show that the solution $(p,c,w)$ converges to $(1,0,1)$ with an explicit exponential rate as time tends to infinity.
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.
We study a doubly tactic resource consumption model bess left{begin{array}{lll} u_t=tr u- ablacd(u abla w),[1mm] v_t=tr v- ablacd(v abla u)+v(1-v^{beta-1}),[1mm] w_t=tr w-(u+v)w-w+r end{array}right. eess in a smooth bounded domain $ooinR^2$ with homogeneous Neumann boundary conditions, where $rin C^1(barOmegatimes[0,infty))cap L^infty(Omegatimes(0,infty))$ is a given nonnegative function fulfilling bess int_t^{t+1}ii| nsqrt{r}|^2<yy for all t>0. eess It is shown that, firstly, if $beta>2$, then the corresponding Neumann initial-boundary problem admits a global bounded classical solution. Secondly, when $beta=2$, the Neumann initial-boundary problem admits a global generalized solution.
This paper investigates an incompressible chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion begin{eqnarray} left{begin{array}{lll} n_t+ucdot abla n= ablacdot(| abla n|^{p-2} abla n)- ablacdot(nchi(c) abla c),& xinOmega, t>0, c_t+ucdot abla c=Delta c-nf(c),& xinOmega, t>0, u_t+(ucdot abla) u=Delta u+ abla P+n ablaPhi,& xinOmega, t>0, ablacdot u=0,& xinOmega, t>0 end{array}right. end{eqnarray} under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a bounded convex domain $Omegasubset mathbb{R}^3$ with smooth boundary. Here, $Phiin W^{1,infty}(Omega)$, $0<chiin C^2([0,infty))$ and $0leq fin C^1([0,infty))$ with $f(0)=0$. It is proved that if $p>frac{32}{15}$ and under appropriate structural assumptions on $f$ and $chi$, for all sufficiently smooth initial data $(n_0,c_0,u_0)$ the model possesses at least one global weak solution.
This work considers a chemotaxis system for multi-species that includes birth or death rate terms, which implies no mass preservation of the populations. We aim to show the convergence to a $L^{infty} - $weak solutions, that is local in time, of the JKO - scheme arising from the Optimal Transport Theory, in the spirit of [35,14]. Currently, $L^{infty}$ solutions have shown to be important in order to get uniqueness. Since death rate case does not ensure global solutions, for arbitrary initial data, in this framework, it could be interest to analyze the Blowing-up phenomenon of this system. Therefore, in the last section, we get sufficient conditions that implies blowing-up phenomenon in finite time and we draw several stages where this occurs. This last part can be seen as a partial generalization of the blowing-up results in [16].
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
