American Institute of Mathematical Sciences

This paper presents a qualitative analysis of a model describing the time and space dynamics of a virus which migrates driven by chemotaxis. The initial-boundary value problem related to applications of the model to a real biological dynamics is studied in detail. The main result consists in the proof of global existence and asymptotic stability.

We investigate the parabolic-elliptic Keller-Segel model

\begin{document}$ \begin{align*} \left\{ \begin{array}{r@{\, }l@{\quad}l@{\quad}l@{\, }c} u_{t}& = \Delta u-\, \chi\nabla\!\cdot(\frac{u}{v}\nabla v), \ &x\in\Omega, & t>0, \\ 0& = \Delta v-\, v+u, \ &x\in\Omega, & t>0, \\ \frac{\partial u}{\partial\nu}& = \frac{\partial v}{\partial\nu} = 0, &x\in\partial \Omega, & t>0, \\ u(&x, 0) = u_0(x), \ &x\in\Omega, & \end{array}\right. \end{align*} $\end{document}

in a bounded domain \begin{document}$ \Omega\subset\mathbb{R}^n $\end{document} \begin{document}$ (n\geq2) $\end{document} with smooth boundary.

We introduce a notion of generalized solvability which is consistent with the classical solution concept, and we show that whenever \begin{document}$ 0<\chi<\frac{n}{n-2} $\end{document} and the initial data satisfy only certain requirements on regularity and on positivity, one can find at least one global generalized solution.

In this paper we consider a \begin{document}$ d $\end{document}-dimensional (\begin{document}$ d = 1, 2 $\end{document}) parabolic-elliptic Keller-Segel equation with a logistic forcing and a fractional diffusion of order \begin{document}$ \alpha \in (0, 2) $\end{document}. We prove uniform in time boundedness of its solution in the supercritical range \begin{document}$ \alpha>d\left(1-c\right) $\end{document}, where \begin{document}$ c $\end{document} is an explicit constant depending on parameters of our problem. Furthermore, we establish sufficient conditions for \begin{document}$ \|u(t)-u_\infty\|_{L^\infty}\rightarrow0 $\end{document}, where \begin{document}$ u_\infty\equiv 1 $\end{document} is the only nontrivial homogeneous solution. Finally, we provide a uniqueness result.

The paper should be viewed as complement of an earlier result in [10]. In the paper just mentioned it is shown that 1d case of a quasilinear parabolic-elliptic Keller-Segel system is very special. Namely, unlike in higher dimensions, there is no critical nonlinearity. Indeed, for the nonlinear diffusion of the form \begin{document}$ 1/u $\end{document} all the solutions, independently on the magnitude of initial mass, stay bounded. However, the argument presented in [10] deals with the Jäger-Luckhaus type system. And is very sensitive to this restriction. Namely, the change of variables introduced in [10], being a main step of the method, works only for the Jäger-Luckhaus modification. It does not seem to be applicable in the usual version of the parabolic-elliptic Keller-Segel system. The present paper fulfils this gap and deals with the case of the usual parabolic-elliptic version. To handle it we establish a new Lyapunov-like functional (it is related to what was done in [10]), which leads to global existence of the initial-boundary value problem for any initial mass.

In this paper we study an anti-angiogenic therapy model that deactivates the tumor angiogenic factors. The model consists of four parabolic equations and considers the chemotaxis and a logistic law for the endothelial cells and several boundary conditions, some of them are non homogeneous. We study the parabolic problem, proving the existence of a unique global positive solution for positive initial conditions, and the stationary problem, justifying the existence of one real number, an eigenvalue of a certain problem, which determines if the semi-trivial solutions are stable or unstable and the existence of a coexistence state.

This paper presents global existence and asymptotic behavior of solutions to the chemotaxis-growth system

\begin{document}$ \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) + ru -\mu u^\alpha, \qquad x\in \Omega, \ t>0, \\ \ v_t = \Delta v + wz, \qquad x\in \Omega, \ t>0, \\ \ w_t = -wz, \qquad x\in \Omega, \ t>0, \\ \ z_t = \Delta z - z + u, \qquad x\in \Omega, \ t>0, \end{array} \right. $\end{document}

in a smoothly bounded domain \begin{document}$ \Omega \subset \mathbb{R}^n $\end{document}, \begin{document}$ n \le 3 $\end{document}, where \begin{document}$ r>0 $\end{document}, \begin{document}$ \mu>0 $\end{document} and \begin{document}$ \alpha>1 $\end{document}. Without the logistic source \begin{document}$ ru-\mu u^\alpha $\end{document}, the stabilization of this system has been shown by Fujie, Ito, Winkler and Yokota (2016), whereas especially about asymptotic behavior, the logistic source disturbs applying this method directly. In the present paper, a way out of this difficulty is introduced and the asymptotic behavior of solutions to the system with logistic source is precisely determined.

This paper deals with the quasilinear Keller-Segel system

\begin{document}$ \begin{align*} \begin{cases} u_t = \nabla\cdot(D(u)\nabla u)-\nabla\cdot(S(u)\nabla v), &x \in \Omega, \ t>0, \\ \ v_t = \Delta v - v +u, &x \in \Omega, \ t>0 \end{cases} \end{align*} $\end{document}

in \begin{document}$ \Omega = \mathbb{R}^N $\end{document} or in a smoothly bounded domain \begin{document}$ \Omega\subset \mathbb{R}^N $\end{document}, with nonnegative initial data \begin{document}$ u_0\in L^1(\Omega) \cap L^\infty(\Omega) $\end{document}, and \begin{document}$ v_0\in L^1(\Omega) \cap W^{1, \infty}(\Omega) $\end{document}; in the case that \begin{document}$ \Omega $\end{document} is bounded, it is supplemented with homogeneous Neumann boundary condition. The diffusivity \begin{document}$ D(u) $\end{document} and the sensitivity \begin{document}$ S(u) $\end{document} are assumed to fulfill \begin{document}$ D(u)\ge u^{m-1}\ (m\geq1) $\end{document} and \begin{document}$ S(u)\leq u^{q-1}\ (q\geq 2) $\end{document}, respectively. This paper derives uniform-in-time boundedness of nonnegative solutions to the system when \begin{document}$ q<m+\frac{2}{N} $\end{document}. In the case \begin{document}$ \Omega = \mathbb{R}^N $\end{document} the result says boundedness which was not attained in a previous paper (J. Differential Equations 2012; 252:1421-1440). The proof is based on the maximal Sobolev regularity for the second equation. This also simplifies a previous proof given by Tao-Winkler (J. Differential Equations 2012; 252:692-715) in the case of bounded domains.

We consider a parabolic-elliptic chemotaxis system generalizing

\begin{document}$ \begin{align} & {{u}_{t}}=\nabla \cdot ({{(u+1)}^{m-1}}\nabla u)-\nabla \cdot (u{{(u+1)}^{\sigma -1}}\nabla v) \\ & \ 0=\Delta v-v+u \\ \end{align} $\end{document}

in bounded smooth domains \begin{document}$ \Omega \subset \mathbb{R}^N $\end{document}, \begin{document}$ N\ge 3 $\end{document}, and with homogeneous Neumann boundary conditions. We show that

● solutions are global and bounded if \begin{document}$ \sigma<m-\frac{N-2}{N} $\end{document}

● solutions are global if \begin{document}$ \sigma\le 0 $\end{document}

● close to given radially symmetric functions there are many initial data producing unbounded solutions if \begin{document}$ \sigma>m-\frac{N-2}{N} $\end{document}.

In particular, if \begin{document}$ \sigma\le 0 $\end{document} and \begin{document}$ \sigma>m-\frac{N-2}{N} $\end{document}, there are many initial data evolving into solutions that blow up after infinite time.

This paper is concerned with a general fully parabolic Keller-Segel system, defined in a convex bounded and smooth domain \begin{document}$Ω$\end{document} of \begin{document}$\mathbb{R}^N, $\end{document} for N∈{2, 3}, with coefficients depending on the chemical concentration, perturbed by a logistic source and endowed with homogeneous Neumann boundary conditions. For each space dimension, once a suitable energy function in terms of the solution is defined, we impose proper assumptions on the data and an exponential decay of such energies is established.

This paper deals with the two-species chemotaxis-competition system

\begin{document}$\begin{equation*} \begin{cases} u_t = d_1Δ u - \nabla · (u χ_1(w)\nabla w) +μ_1 u(1-u-a_1 v)&{\rm in} \ Ω × (0, ∞), \\ v_t = d_2Δ v - \nabla · (v χ_2(w)\nabla w) +μ_2 v(1-a_2u-v)&{\rm in} \ Ω × (0, ∞), \\ w_t = d_3Δ w + α u + β v - γ w&{\rm in} \ Ω × (0, ∞), \end{cases} \end{equation*}$ \end{document}

where \begin{document}$Ω$\end{document} is a bounded domain in \begin{document}$\mathbb{R}^n$\end{document} with smooth boundary \begin{document}$\partial Ω$\end{document}, \begin{document}$n≥ 2$\end{document}; \begin{document}$χ_i$\end{document} are functions satisfying some conditions. About this problem, Bai-Winkler [1] first obtained asymptotic stability in (1) under some conditions in the case that \begin{document}$a_1, a_2∈ (0, 1)$\end{document}. Recently, the conditions assumed in [1] were improved ([6]); however, there is a gap between the conditions assumed in [1] and [6]. The purpose of this work is to improve the conditions assumed in the previous works for asymptotic behavior in the case that \begin{document}$a_1, a_2∈ (0, 1)$\end{document}.

In this article we consider a coupled system of differential equations to describe the evolution of a biological species. The system consists of two equations, a second order parabolic PDE of nonlinear type coupled to an ODE. The system contains chemotactic terms with constant chemotaxis coefficient describing the evolution of a biological species "\begin{document}$u$\end{document}" which moves towards a higher concentration of a chemical species "\begin{document}$v$\end{document}" in a bounded domain of \begin{document}$ \mathbb{R}^n$\end{document}. The chemical "\begin{document}$v$\end{document}" is assumed to be a non-diffusive substance or with neglectable diffusion properties, satisfying the equation

\begin{document}$v_t = h(u, v).$ \end{document}

We obtain results concerning the bifurcation of constant steady states under the assumption

\begin{document} $ h_v+χ u h_u>0 $ \end{document}

with growth terms \begin{document}$g$\end{document}. The Parabolic-ODE problem is also considered for the case \begin{document}$h_v+χ u h_u = 0$\end{document} without growth terms, i.e. \begin{document}$g \equiv 0$\end{document}. Global existence of solutions is obtained for a range of initial data.

The current paper is devoted to the study of traveling wave solutions of the following parabolic-elliptic-elliptic chemotaxis systems,

\begin{document}$\begin{equation}\label{main-eq-abstract}\begin{cases}u_{t} = Δ u- \nabla · ({ χ_1 u} \nabla v_1)+ \nabla · ({ χ_2 u} \nabla v_2) + u(a-bu), \;\;\;\;x∈\mathbb{R}^N, \\0 = Δ v_1-λ_1v_1+μ_1u, \;\;\;\;x∈\mathbb{R}^N, \\0 = Δ v_2-λ_2v_2+μ_2u, \;\;\;\; x∈\mathbb{R}^N, \end{cases}\;\;\;\;\;\;\;\;(0.1)\end{equation}$ \end{document}

where \begin{document}$a>0, \ b>0, $\end{document} \begin{document}$u(x, t)$\end{document} represents the population density of a mobile species, \begin{document}$v_1(x, t), $\end{document} represents the population density of a chemoattractant, \begin{document}$v_2(x, t)$\end{document} represents the population density of a chemorepulsion, the constants \begin{document}$χ_1≥ 0$\end{document} and \begin{document}$χ_2≥ 0$\end{document} represent the chemotaxis sensitivities, and the positive constants \begin{document}$λ_1, λ_2, μ_1$\end{document}, and \begin{document}$μ_2$\end{document} are related to growth rate of the chemical substances. It was proved in an earlier work by the authors of the current paper that there is a nonnegative constant \begin{document}$K$\end{document} depending on the parameters \begin{document}$χ_1, μ_1, λ_1, χ_2, μ_2$\end{document}, and \begin{document}$λ_2$\end{document} such that if \begin{document}$b+χ_2μ_2>χ_1μ_1+K$\end{document}, then the positive constant steady solution \begin{document}$(\frac{a}{b}, \frac{aμ_1}{bλ_1}, \frac{aμ_2}{bλ_2})$\end{document} of (0.1) is asymptotically stable with respect to positive perturbations. In the current paper, we prove that if \begin{document}$b+χ_2μ_2>χ_1μ_1+K$\end{document}, then there exists a positive number \begin{document}$c^{*}(χ_1, μ_1, λ_1, χ_2, μ_2, λ_2)≥ 2\sqrt{a}$\end{document} such that for every \begin{document}$ c∈ ( c^{*}(χ_1, μ_1, λ_1, χ_2, μ_2, λ_2)\ , \ ∞)$\end{document} and \begin{document}$ξ∈ S^{N-1}$\end{document}, the system has a traveling wave solution \begin{document}$(u(x, t), v_1(x, t), v_2(x, t)) = (U(x·ξ-ct), V_1(x·ξ-ct), V_2(x·ξ-ct))$\end{document} with speed \begin{document}$c$\end{document} connecting the constant solutions \begin{document}$(\frac{a}{b}, \frac{aμ_1}{bλ_1}, \frac{aμ_2}{bλ_2})$\end{document} and \begin{document}$(0, 0, 0)$\end{document}, and it does not have such traveling wave solutions of speed less than \begin{document}2\sqrt a $\end{document}. Moreover we prove that

\begin{document}$\begin{equation*}\lim\limits_{(χ_{1}, χ_2)?(0^+, 0^+)}c^{*}(χ_1, μ_1, λ_1, χ_2, μ_2, λ_2) = \begin{cases}\ 2\sqrt{a} \;\;\text{if}\;\; a≤ \min\{λ_1, λ_2\}\\\frac{a+λ_1}{\sqrt{λ_1}} \;\;\text{if}\;\; λ_1≤ \min\{a, λ_2\}\\\frac{a+λ_2}{\sqrt{λ_2}} \;\;\text{if}\;\; λ_2≤ \min\{a, λ_1\}\end{cases}\end{equation*}$ \end{document}

for every \begin{document}$ λ_1, λ_2, μ_1, μ_2>0$\end{document}, and

\begin{document}$\begin{equation*}\lim\limits_{x?∞}\frac{U(x)}{e^{-\sqrt a μ x}} = 1, \end{equation*}$ \end{document}

where \begin{document}$μ$\end{document} is the only solution of the equation \begin{document}$μ+\frac{1}{μ} = \frac{c}{\sqrt{a}}$\end{document} in the interval \begin{document}$(0\ , \ \min\{1, \sqrt{\frac{λ_1}{a}}, \sqrt{\frac{λ_2}{a}}\})$\end{document}.

This paper considers the following parabolic-elliptic chemotaxis-growth system with nonlinear diffusion

\begin{document}$\left\{ \begin{array}{l}{u_t} = \nabla (D(u)\nabla u) - \nabla (\chi {u^q}\nabla v) + \mu u(1 - {u^\alpha }),\;\;\;\;\;\;\;\;& x \in \Omega ,{\mkern 1mu} {\mkern 1mu} t > 0,\\0 = \Delta v - v + {u^\gamma },\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&x \in \Omega ,{\mkern 1mu} {\mkern 1mu} t > 0\end{array} \right.$ \end{document}

under homogeneous Neumann boundary conditions for some constants \begin{document}$q≥ 1$\end{document}, \begin{document}$α>0$\end{document} and \begin{document}$γ≥ 1$\end{document}, where \begin{document}$D(u)≥ c_D u^{m-1}$\end{document}\begin{document}$(m≥ 1)$\end{document} for all \begin{document}$u>0$\end{document} and \begin{document}$D(u)>0$\end{document} for all \begin{document}$u≥ 0$\end{document}, and \begin{document}$Ω\subset\mathbb{R}^N$\end{document} \begin{document}$(N≥ 1)$\end{document} is a bounded domain with smooth boundary. It is shown that when \begin{document}$ m>q+γ-\frac{2}{N}, \, \, \mathbf{or}$\end{document}\begin{document}$ α>q+γ-1, \, \, \mathbf{or}$\end{document}\begin{document}$α = q+γ-1\, \, {\rm{and}}\, \, μ>μ^*$\end{document}, where

\begin{document}$ {\mu ^*} = \left\{ \begin{array}{l}\begin{array}{*{20}{l}}{\frac{{(\alpha + 1 - m)N - 2}}{{(\alpha + 1 - m)N + 2(\alpha - \gamma )}}\chi ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{if}}~~{\mkern 1mu} {\mkern 1mu} m \le q + \gamma - \frac{2}{N},}\end{array}\\0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{if}}~~{\mkern 1mu} {\mkern 1mu} m > q + \gamma - \frac{2}{N},\end{array} \right.$ \end{document}

then the above system possesses a global bounded classical solution for any sufficiently smooth initial data. The results improve the results by Wang et al. (J. Differential Equations 256 (2014)) and generalize the results of Zheng (J. Differential Equations 259 (2015)) and Galakhov et al. (J. Differential Equations 261 (2016)).

In this paper we deal with the initial-boundary value problem for chemotaxis-fluid model involving more complicated nonlinear coupling term, precisely, the following self-consistent system

\begin{document}$\left\{ \begin{array}{l}{n_t} + u \cdot \nabla n = \Delta {n^m} - \nabla \cdot (n\nabla c) + \nabla \cdot (n\nabla \phi ), \;\;\;\;\;\;(x, t) \in \Omega \times (0, T), \\{c_t} + u \cdot \nabla c = \Delta c - nc, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(x, t) \in \Omega \times (0, T), \\{u_t} + (u \cdot \nabla )u + \nabla P = \Delta u - n\nabla \phi + n\nabla c, \;\;\;\;\;\;\;\;\;\;\;\;\;(x, t) \in \Omega \times (0, T), \\\nabla \cdot u = 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(x, t) \in \Omega \times (0, T), \end{array} \right.$ \end{document}

where \begin{document}$Ω\subset \mathbb{R}^2$\end{document} is a bounded domain with smooth boundary.

The novelty here is that both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid is considered, which leads to the stronger coupling than usual chemotaxis-fluid model studied in the most existing literatures. To the best of our knowledge, there is no global solvability result on this chemotaxis-Navier-Stokes system in the past works. It is proved in this paper that global weak solutions exist whenever \begin{document}$m>1$\end{document} and the initial data is suitably regular. This extends a result by Di Francesco, Lorz and Markowich (Discrete Cont. Dyn. Syst. A 28 (2010)) which asserts global existence of weak solutions under the constraint \begin{document}$m∈(\frac{3}{2}, 2]$\end{document} in the corresponding Stokes-type simplified system.