- Research
- Open Access
Global existence and uniform boundedness of smooth solutions to a parabolic-parabolic chemotaxis system with nonlinear diffusion
- Xie Li1Email author
- Received: 6 March 2015
- Accepted: 9 June 2015
- Published: 24 June 2015
Abstract
This paper is devoted to the following quasilinear chemotaxis system: \(\bigl\{\scriptsize{ \begin{array}{l} u_{t}=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(u\chi(v)\nabla v)+uf(u),\quad x\in\Omega,t>0, \\ v_{t}=\Delta v-ug(v),\quad x\in\Omega,t>0, \end{array} }\bigr. \) under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega\subset\mathbb{R}^{N}\). The given functions \(D(s)\), \(\chi(s)\), \(g(s)\), and \(f(s)\) are assumed to be sufficiently smooth for all \(s\ge0\) and such that \(f(s)\le\kappa-\mu s^{\tau}\). It is proved that the corresponding initial boundary value problem possesses a unique global classical solution for any \(\mu>0\) and \(\tau \ge1\), which is uniformly bounded in \(\Omega\times(0,+\infty)\). Moreover, when \(\kappa=0\), the decay property of the solution is also discussed in this paper.
Keywords
- global existence
- uniform boundedness
- nonlinear diffusion
- chemotaxis
MSC
- 35K55
- 35Q92
- 35Q35
- 92C17
1 Introduction
In this paper, we consider the fully parabolic chemotaxis system: where \(\Omega\subset\mathbb{R}^{N}\) is a bounded domain with smooth boundary ∂Ω, and \(\frac{\partial}{\partial\nu}\) denotes the derivative with respect to the outer normal of ∂Ω. Equations (1.1) are a generalized version of the classical Keller-Segel model [1] and describe the directed movement of cells as a response to gradients of the concentration of a chemical signal substance (e.g., oxygen) in the environment, where the chemical signal substance is consumed rather than produced by the cells themselves. In this case, \(u = u(x, t)\) denotes the density of the cells, and \(v = v(x, t)\) denotes the concentration of the oxygen, the function \(D(u)\) describes the density-dependent motility of the cells, and the logistic source \(uf(u)\) models proliferation and death of the cells, \(\chi(v)\) and \(g(v)\) denote a chemotactic sensitivity function and consumption rate of the oxygen by the cells, respectively. From a physical point of view, migration of the cells should be regarded as movement in a porous medium, and so we are led to consider the cell motility as a nonlinear function \(D(u)\) of the cell density. Precisely, we will assume that the diffusion coefficient D satisfies with \(m\ge1\) and \(c_{D}>0\). Moreover, we assume that As to the source term f we require that The initial data \(u_{0}\), \(v_{0}\) are supposed throughout this paper to satisfy
$$ \left \{ \textstyle\begin{array}{l} u_{t}=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(u\chi(v)\nabla v)+ uf(u), \quad x\in\Omega, t>0, \\ v_{t}=\Delta v-ug(v), \quad x\in\Omega, t>0, \\ \frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial\nu }=0, \quad x\in\partial\Omega, t>0, \\ u(x,0)=u_{0}(x), \qquad v(x,0)= v_{0}(x), \quad x\in\Omega, \end{array}\displaystyle \right . $$
(1.1)
$$ \left \{ \textstyle\begin{array}{l} D(s)\in C^{2} ([0,\infty) ),\qquad D(0)> 0, \\ D(s)\ge c_{D} s^{m-1} \quad \text{on } (0,+\infty) \end{array}\displaystyle \right . $$
(1.2)
$$ \left \{ \textstyle\begin{array}{l} \chi(s)\in C^{2}([0,+\infty)) \text{ is nonnegative}\quad \text{and} \quad \chi'(s)\ge0 \quad \text{on } [0,+\infty), \\ g(s)\in C^{2}([0,+\infty)) \text{ satisfies } g(0)=0 \quad \text{and} \quad g(s)\ge0\quad \text{on } (0,+\infty). \end{array}\displaystyle \right . $$
(1.3)
$$ \left \{ \textstyle\begin{array}{l} f\in C^{1}([0,\infty)), \\ f(s)\le\kappa-\mu s^{\tau}\quad \text{with } \kappa\ge0, \mu>0, \tau \ge1 \text{ on } [0,+\infty). \end{array}\displaystyle \right . $$
(1.4)
$$ \left \{ \textstyle\begin{array}{l} u_{0}\in W^{1,\theta}(\Omega) \quad \text{for some } \theta>N, u_{0}\ge0, x\in\Omega, \\ v_{0}\in W^{1,\infty}(\Omega), \qquad v_{0}\ge0, \quad x\in\Omega. \end{array}\displaystyle \right . $$
(1.5)
To motivate our study, let us first recall the following related models to (1.1): where the chemical signal substance is produced by the cells themselves. This system has been widely studied during the past decades. It was known that in the linear diffusion case \(D\equiv1\) and \(f(u) \equiv0\), solutions may blow up in finite time when \(N \ge2\) [2–4]; however, in the case \(f(u)\le a-bu\), arbitrarily small \(b > 0\) guarantee the global existence and boundedness of solutions when \(N = 2\) [5], and that appropriately large \(\frac{b}{\chi}\) preclude blow-up in the case \(N \ge3\) [6]. In the nonlinear case [7, 8] provide uniform-in-time boundedness of solutions in (1.6) under the condition that the logistic term \(f(u)\) vanishes, and that in this respect the condition \(m > 2-\frac{2}{N}\) found in work [7] is optimal, because in [9] it has been shown that if \(D(u)=(u+1)^{2-\frac{2}{N}-\varepsilon}\) for some \(\varepsilon>0\), \(f(u)\equiv0\), and Ω is a ball, then (1.7) possesses some unbounded solutions. Moreover, in [10], the authors consider a more general version of (1.6). For more related work one can refer to [11, 12]. Next we mention some results about the signal is consumed by the cells. The following chemotaxis-(Navier)-Stokes model which is a generalized version of the model proposed in[13], describes the motion of oxygen-driven swimming cells in an incompressible fluid, which is closely related to (1.1) Here, w denotes the velocity field of the fluid subject to an incompressible Navier-Stokes equation with pressure P and viscosity η, and a gravitational force ∇ϕ. In (1.7), both cells and oxygen are transported with the fluid. There are many results about the mathematical analysis on (1.7). See [14–19] etc. If the flow of the fluid is ignored (i.e., \(w\equiv0\)) or the fluid is stationary, then w is decoupled from (1.7), which yields When \(D(u)\equiv1\), the model (1.8) is originally proposed by Keller and Segel with \(g(v)=v^{\gamma}\) and \(\chi(v)=\frac{1}{v}>0\) to describe the bacterial wave propagation, where the chemical is consumed by bacterial. For this model, many progresses are made in recent years, e.g., see a review paper [20] and references therein. Moreover, when \(g(v)=v\) and \(\chi (v)=\mathrm{const.}:=\chi>0\), Tao [21] proved that if \(\|v_{0}\|_{L^{\infty}(\Omega)}\) is sufficiently small, then the corresponding initial boundary value problem possesses a unique global solution that is uniformly bounded. Moreover, the same result was obtained in [22] for the model with logistic source. For large initial data, Tao and Winkler [23] showed that the problem has a global weak solution which is eventually bounded and smooth under the assumptions that \(\chi (v)\equiv1\) and \(\Omega\subset\mathbb{R}^{3}\) is a bounded convex domain. When \(D(u)\ge c_{D} (u+1)^{m-1}\) with \(m>2-\frac{2}{N}\), Wang et al. [24] showed that the corresponding initial boundary value problem possesses a unique global classical solution that is uniformly bounded provided that \(\Omega\subset\mathbb{R}^{N}\) is a bounded convex domain and some other technical conditions are fulfilled. Recently, the result on global existence is relaxed to \(m>2-\frac{6}{n+4}\) in [25], and the result on uniformly boundedness is relaxed to \(m>2-\frac{n+2}{2n}\) in [26].
$$ \left \{ \textstyle\begin{array}{l} u_{t}=\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)+uf(u),\quad x\in\Omega, t>0, \\ v_{t}=\Delta v-\alpha_{1} v+\beta_{1} u, \quad x\in\Omega, t>0, \end{array}\displaystyle \right . $$
(1.6)
$$ \left \{ \textstyle\begin{array}{l} u_{t}+w\cdot\nabla u=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(u\chi (v)\nabla v), \quad x\in\Omega, t>0, \\ v_{t}+w\cdot\nabla v=\Delta v-ug(v),\quad x\in\Omega, t>0, \\ w_{t}+\kappa(w\cdot\nabla w)=\eta\Delta w-\nabla P+u\nabla\phi,\quad x\in\Omega, t>0, \\ \nabla\cdot w=0. \end{array}\displaystyle \right . $$
(1.7)
$$ \left \{ \textstyle\begin{array}{l} u_{t}=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(u\chi(v)\nabla v),\quad x\in\Omega, t>0, \\ v_{t}=\Delta v-ug(v), \quad x\in\Omega, t>0. \end{array}\displaystyle \right . $$
(1.8)
The aim of this paper is to study the global existence and boundedness of the solutions for the parabolic-parabolic chemotaxis system with linear or nonlinear diffusion and logistic source (1.1). Moreover, when \(\kappa=0\), i.e., cells are a priori unable to reproduce themselves, such as broadcast spawning phenomena discussed in [27, 28], the decay property of the solution component u is also obtained. Our main result is stated as follows.
Theorem 1.1
Let
\(m\ge1\), \(\tau\ge1\), \(\Omega\subset \mathbb{R}^{2}\)
be a bounded domain with smooth boundary. Suppose that
\(D(s)\), \(\chi(s)\), \(f(s)\), and
\(g(s)\)
satisfy (1.2)-(1.4). Then for each
\((u_{0},v_{0})\)
fulfilling (1.5), the problem (1.1) possesses a unique classical solution which is global in time and uniformly bounded in
\(\Omega\times(0,\infty)\). Furthermore, if
\(\kappa =0\), then the component
u
of global classical solution has the following decay property:
$$ \bigl\Vert u(\cdot,t)\bigr\Vert _{L^{\infty}(\Omega)}\rightarrow0 \quad \textit{as } t\nearrow \infty. $$
(1.9)
2 Preliminaries
We first state the local existence of classical solutions of (1.1). The proof is based on an appropriate fixed point argument. One can refer to [18, 22, 29] etc. for more details.
Lemma 2.1
Let
\(\Omega\subset\mathbb{R}^{N}\) (\(N\ge1\)) be a bounded domain with smooth boundary. Suppose that
\(D(s)\), \(\chi(s)\), \(f(s)\), and
\(g(s)\)
satisfy (1.2)-(1.4), and assume that the initial data
\((u_{0},v_{0})\in(W^{1,\theta}(\Omega))^{2}\) (for some
\(\theta>N\)). Then the model (1.1) has a unique local-in-time nonnegative classical solution
\((u,v)\in(C([0,T^{*});W^{1,\theta}(\Omega))\cap C^{2,1}(\bar{\Omega}\times(0,T^{*})))^{2}\). Here, \(T^{*}\)
denotes the maximal existence time. Moreover, if
\(T^{*}<\infty\), then
$$ \bigl\Vert u(\cdot,t)\bigr\Vert _{L^{\infty}(\Omega)}\rightarrow \infty \quad \textit{as } t\nearrow T^{*}. $$
(2.1)
The following \(L^{1}\) estimate can easily be checked.
Lemma 2.2
Assume that
\(v_{0}\in W^{1,\infty}(\Omega)\). Then the solution
\((u,v)\)
of (1.1) satisfies the following properties:
and
$$ \bigl\Vert u(\cdot,t)\bigr\Vert _{L^{1}(\Omega)}\le M_{1}:=\max \biggl\{ \|u_{0}\|_{L^{1}(\Omega )}, \biggl( \frac{\kappa}{\mu} \biggr)^{\frac{1}{\tau}}|\Omega| \biggr\} \quad \textit{for all } t \in \bigl[0,T^{*} \bigr) $$
(2.2)
$$ \bigl\Vert v(\cdot,t)\bigr\Vert _{L^{\infty}(\Omega)}\le \|v_{0}\|_{L^{\infty}(\Omega)}\quad \textit{for all } t\in \bigl[0,T^{*} \bigr). $$
(2.3)
Proof
Integrating the first equation in (1.1) over Ω, and utilizing the Hölder inequality, yields Then \(y(t):=\int_{\Omega}u \, dx\) satisfies Thus, a standard ODE argument implies that \(y(t)\le\max \{\|u_{0}\| _{L^{1}(\Omega)}, (\frac{\kappa}{\mu} )^{\frac{1}{\tau}}|\Omega| \}\), for all \(t\in(0,T^{*})\). Since \(g\ge0\), (2.3) immediately results from the parabolic maximum principle applied to the second equation in (1.1). □
$$\begin{aligned} \frac{d}{dt}\int_{\Omega}u \, dx&\le\kappa\int _{\Omega}u \, dx-\mu\int_{\Omega}u^{1+\tau} \, dx \\ &\le\kappa\int_{\Omega}u \, dx- \frac{\mu}{|\Omega|^{\tau}} \biggl(\int_{\Omega}u \, dx \biggr)^{1+\tau}\quad \text{for all } t\in\bigl(0,T^{*}\bigr). \end{aligned}$$
$$y'(t)\le\kappa y(t)-\frac{\mu}{|\Omega|^{\tau}} y(t)^{1+\tau}\quad \text{for all } t\in\bigl(0,T^{*}\bigr). $$
Next we establish an elementary inequality, which will be used in Lemma 3.2.
Lemma 2.3
Let
\(\mu>0\), \(\tau\ge1\), \(\kappa\ge0\), and
\(A>0\). Then there exists a positive constant
\(C^{*}:=C(\kappa,\mu,A)\)
such that
$$ Az^{2}+\kappa(1+z)\ln(1+z)-\mu z^{1+\tau} \ln(1+z)+(1+z)\ln(1+z)-z\le C^{*}\quad \textit{for all } z>0. $$
(2.4)
Proof
Define Then it satisfies Thus for some \(z_{0} > 0\) we have \(\varphi< 0\) on \((z_{0},\infty)\). In view of the continuity of φ on \([0,\infty)\), we have which implies (2.4). □
$$\varphi(z):=Az^{2}+\kappa(1+z)\ln(1+z)-\mu z^{1+\tau} \ln(1+z)+(1+z)\ln(1+z)-z. $$
$$ \varphi(0)=0 \quad \text{and}\quad \lim_{z\rightarrow\infty} \frac{\varphi (z)}{z^{1+\tau}\ln(1+z)}=-\mu. $$
$$\varphi(z)\le\max_{z\in[0,z_{0}]}\varphi(z) \quad \text{for all } z>0, $$
3 Proof of Theorem 1.1
3.1 Proof of global existence
In this section, we focus on proof of global existence in Theorem 1.1. Inspired by [30], we will first establish estimate on \(\int_{\Omega}u^{m+1}\, dx+\int_{\Omega}|\nabla v|^{4}\, dx\) to obtain estimate on \(\int_{\Omega}u^{p} \, dx\) for any \(p>1\). To achieve this purpose, we need a series of a priori estimates.
Lemma 3.1
Let the same assumptions as in Theorem
1.1
hold. Then the solution of (1.1) satisfies
for any
\(p>m\)
and each
\(t\in(0,T^{*})\). Particularly, we have
$$\begin{aligned}& \frac{d}{dt}\int_{\Omega}u^{p} \,dx+\frac{p(p-1)c_{D}}{2}\int_{\Omega}u^{p+m-3}|\nabla u|^{2} \,dx \\& \quad \le\frac{p(p-1)\|\chi\|_{L^{\infty}(\Omega )}^{2}}{2c_{D}}\int_{\Omega}u^{p-m+1}|\nabla v|^{2} \,dx \\& \qquad {}+\kappa p\int_{\Omega}u^{p} \,dx-\mu p\int_{\Omega}u^{p+\tau}\,dx \end{aligned}$$
(3.1)
$$\begin{aligned}& \frac{d}{dt}\int_{\Omega}u^{m+1} \,dx+\frac{m(m+1)c_{D}}{2}\int_{\Omega}u^{2m-2}|\nabla u|^{2} \,dx \\& \quad \le\frac{m(m+1)\|\chi\|_{L^{\infty}(\Omega )}^{2}}{2c_{D}}\int_{\Omega}u^{2}|\nabla v|^{2} \,dx \\& \qquad {}+\kappa(m+1)\int_{\Omega}u^{m+1} \,dx-\mu(m+1)\int_{\Omega}u^{m+1+\tau}\,dx. \end{aligned}$$
(3.2)
Proof
Multiplying (1.1)1 by \(u^{p-1}\) and integrating the resulted equation over Ω, we obtain We now estimate the last three terms of the above equality. Indeed, by (1.2) we have and by the Young inequality, we get For the last term, in view of (1.5), we have Summarily, we obtain Taking \(p=m+1\), one can easily deduce (3.2). This completes the proof of Lemma 3.1. □
$$\begin{aligned}& \frac{1}{p}\frac{d}{dt}\int_{\Omega}u^{p} \,dx+(p-1)\int_{\Omega}u^{p-2}D(u)| \nabla u|^{2} \,dx \\& \quad =(p-1)\int_{\Omega}u^{p-1} \chi(v)\nabla u\cdot \nabla v \,dx+\int_{\Omega}u^{p}f(u)\,dx. \end{aligned}$$
(3.3)
$$(p-1)\int_{\Omega}u^{p-2}D(u)|\nabla u|^{2} \,dx\ge(p-1)c_{D}\int_{\Omega}u^{p+m-3}|\nabla u|^{2} \,dx, $$
$$\begin{aligned}& (p-1)\int_{\Omega}u^{p-1}\chi(v)\nabla u\cdot\nabla v \,dx \\& \quad \le(p-1)\|\chi\| _{L^{\infty}(\Omega)}\int_{\Omega}u^{p-1}|\nabla u||\nabla v| \,dx \\& \quad \le\frac{(p-1)c_{D}}{2}\int_{\Omega}u^{p+m-3}|\nabla u|^{2} \,dx+\frac{(p-1)\| \chi\|_{L^{\infty}(\Omega)}^{2}}{2c_{D}}\int_{\Omega}u^{p-m+1}|\nabla v|^{2} \,dx. \end{aligned}$$
$$\int_{\Omega}u^{p}f(u)\,dx\le\kappa\int _{\Omega}u^{p}\,dx-\mu\int_{\Omega}u^{p+\tau}\,dx. $$
$$\begin{aligned}& \frac{d}{dt}\int_{\Omega}u^{p} \,dx+ \frac{p(p-1)c_{D}}{2}\int_{\Omega}u^{p+m-3}|\nabla u|^{2} \,dx \\& \quad \le\frac{p(p-1)\|\chi\|_{L^{\infty}(\Omega )}^{2}}{2c_{D}}\int_{\Omega}u^{p-m+1}|\nabla v|^{2} \,dx \\& \qquad {}+\kappa p\int_{\Omega}u^{p} \,dx-\mu p\int_{\Omega}u^{p+\tau}\,dx. \end{aligned}$$
We then establish a coupled estimate on \(\int_{\Omega}[(1+u)\ln (1+u)-u ]\,dx+\int_{\Omega}|\nabla v|^{2} \,dx\).
Lemma 3.2
Let the same assumptions as in Theorem
1.1
hold. Then there exists a positive constant
C
such that
and
$$ \int_{\Omega}\bigl[(1+u)\ln(1+u)-u \bigr]\,dx\le C \quad \textit{for all } t\in\bigl(0,T^{*}\bigr) $$
(3.4)
$$ \int_{\Omega}|\nabla v|^{2} \,dx\le C \quad \textit{for all } t\in\bigl(0,T^{*}\bigr). $$
(3.5)
Proof
We shall divide the proof into three steps.
Step 1. First testing (1.1)1 against \(\ln(1+u)\), we have for all \(t\in(0,T^{*})\). To estimate each term on the right side of (3.6), we first note that Then utilizing the Young inequality and integration by parts, we get where we have used \(\chi'(v)\ge0\). For the last term on the right of (3.6), according to (1.4), we obtain Then combining (3.7) and (3.8) along with (3.6), we have
$$\begin{aligned}& \frac{d}{dt}\int_{\Omega}\bigl[(1+u) \ln(1+u)-u \bigr] \,dx+\int_{\Omega}\frac {D(u)}{1+u}|\nabla u|^{2}\,dx \\& \quad =\int_{\Omega}\frac{u}{1+u}\chi(v)\nabla u\cdot \nabla v \,dx +\int_{\Omega}uf(u)\ln(1+u)\,dx \end{aligned}$$
(3.6)
$$ \ln(1+u)\le u \quad \text{for all } u\ge0. $$
$$\begin{aligned}& \int_{\Omega}\frac{u}{1+u}\chi(v)\nabla u \cdot\nabla v\,dx \\& \quad =\int_{\Omega}\bigl[\ln(1+u)-u \bigr]\chi(v) \Delta v\,dx+\int_{\Omega}\bigl[\ln(1+u)-u \bigr]\chi '(v)|\nabla v|^{2}\,dx \\& \quad \le\frac{\|\chi\|_{L^{\infty}}^{2}}{2}\int_{\Omega}\bigl[\ln(1+u)-u \bigr]^{2}\,dx+\frac{1}{2}\int_{\Omega}|\Delta v|^{2}\,dx \\& \quad \le\frac{\|\chi\|_{L^{\infty}}^{2}}{2}\int_{\Omega}u^{2}\,dx+ \frac{1}{2}\int_{\Omega}|\Delta v|^{2}\,dx\quad \text{for all } t\in\bigl(0,T^{*}\bigr), \end{aligned}$$
(3.7)
$$\begin{aligned}& \int_{\Omega}uf(u)\ln(1+u)\,dx \\& \quad \le\int _{\Omega}u\bigl(\kappa-\mu u^{\tau}\bigr)\ln (1+u)\,dx \\& \quad = \int_{\Omega}\bigl[\kappa u\ln(1+u)-\mu u^{1+\tau} \ln(1+u)\bigr]\,dx\quad \text{for all } t\in\bigl(0,T^{*}\bigr). \end{aligned}$$
(3.8)
$$\begin{aligned}& \frac{d}{dt} \int_{\Omega}\bigl[(1+u) \ln(1+u)-u \bigr]\,dx+\int_{\Omega}\frac {D(u)}{1+u}|\nabla u|^{2} \,dx \\& \quad \le\frac{\|\chi\|_{L^{\infty}}^{2}}{2}\int_{\Omega}u^{2} \,dx+ \frac{1}{2}\int_{\Omega}|\Delta v|^{2}\,dx \\& \qquad {}+ \int _{\Omega}\bigl[ \kappa u\ln(1+u)-\mu u^{1+\tau}\ln (1+u) \bigr]\,dx \quad \text{for all } t\in\bigl(0,T^{*}\bigr). \end{aligned}$$
(3.9)
Step 2. In order to cancel \(\frac{1}{2}\int_{\Omega}|\Delta v|^{2}\,dx\) on the right of (3.9), we first test (1.1)2 against \(-\Delta v\) to find that Then by integration by parts and the Young inequality, we have where (2.3) has been used.
$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla v|^{2}\,dx =&-\int_{\Omega}|\Delta v|^{2}\,dx+ \int_{\Omega}ug(v)\Delta v\,dx+\int_{\Omega}v\Delta v\,dx \\ &{}-\int_{\Omega}v\Delta v\,dx \quad \text{for all } t\in \bigl(0,T^{*}\bigr). \end{aligned}$$
$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\int_{\Omega}| \nabla v|^{2}\,dx \le&-\int_{\Omega}|\Delta v|^{2}\,dx+\|g\|_{L^{\infty}}\int_{\Omega}u|\Delta v|\,dx \\ &{}-\int_{\Omega}|\nabla v|^{2}\,dx+ \frac{\int_{\Omega}|\Delta v|^{2}\,dx}{4}+\int_{\Omega}| v|^{2}\,dx \\ \le&-\frac{\int_{\Omega}|\Delta v|^{2}\,dx}{2}+\|g\|_{L^{\infty}}^{2}\int _{\Omega}u^{2}\,dx-\int_{\Omega}|\nabla v|^{2}\,dx \\ &{}+\|v_{0}\|_{L^{\infty}(\Omega)}^{2}|\Omega | \quad \text{for all } t\in\bigl(0,T^{*}\bigr), \end{aligned}$$
(3.10)
Step 3. Adding (3.10) to (3.9) yields with \(C_{2}=\frac{\|\chi\|_{L^{\infty}(\Omega)}^{2}+2\|g\|_{L^{\infty}(\Omega )}^{2}}{2}\), which is a positive constant according to (1.3) and (2.3). Adding \(\int_{\Omega}[(1+u)\ln(1+u)-u ]\,dx\) to both sides of this and dropping the nonnegative term \(\int_{\Omega}\frac {D(u)}{1+u}|\nabla u|^{2} \,dx\) on the left, we obtain for all \(t\in(0,T^{*})\). In view of Lemma 2.3, there exists a positive constant \(C^{*}\) such that Thus \(y(t):=\int_{\Omega}[(1+u)\ln(1+u)-u ]\,dx+\frac{1}{2}\int_{\Omega}|\nabla v|^{2}\,dx\) satisfies By standard ODE argument, we can derive which implies (3.4) and (3.5). □
$$\begin{aligned}& \frac{d}{dt} \biggl\{ \int_{\Omega}\bigl[(1+u)\ln(1+u)-u \bigr]\,dx+\frac{1}{2}\int_{\Omega}|\nabla v|^{2} \,dx \biggr\} \\& \qquad {}+\int_{\Omega}\frac {D(u)}{1+u}|\nabla u|^{2} \,dx+\int_{\Omega}|\nabla v|^{2}\,dx \\& \quad \le C_{2}\int_{\Omega}u^{2} \,dx+\int _{\Omega}\bigl[\kappa u\ln(1+u)-\mu u^{1+\tau }\ln(1+u)\bigr] \,dx \\& \qquad {}+\|v_{0}\|_{L^{\infty}(\Omega)}^{2}|\Omega| \quad \text{for all } t\in\bigl(0,T^{*}\bigr) \end{aligned}$$
$$\begin{aligned}& \frac{d}{dt} \biggl\{ \int_{\Omega}\bigl[(1+u)\ln(1+u)-u \bigr]\,dx+\frac{1}{2}\int_{\Omega}| \nabla v|^{2}\,dx \biggr\} \\& \qquad {}+\int_{\Omega}|\nabla v|^{2}\,dx+\int_{\Omega}\bigl[(1+u)\ln(1+u)-u \bigr] \,dx \\& \quad \le C_{2}\int_{\Omega}u^{2} \,dx+\int _{\Omega}\bigl[(1+u)\ln(1+u)-u \bigr]\,dx \\& \qquad {}+ \int _{\Omega}\bigl[\kappa u\ln(1+u)-\mu u^{1+\tau}\ln(1+u)\bigr] \,dx+\|v_{0}\|_{L^{\infty}(\Omega)}^{2}|\Omega| \end{aligned}$$
(3.11)
$$\int_{\Omega}\bigl\{ C_{2} u^{2}+\kappa u \ln(1+u)-\mu u^{1+\tau}\ln (1+u)+(1+u)\ln(1+u)-u \bigr\} \,dx\le C^{*}|\Omega|. $$
$$\frac{dy}{dt}+y\le C^{*}|\Omega|+\|v_{0}\|_{L^{\infty}(\Omega)}^{2}| \Omega | \quad \text{for all } t\in\bigl(0,T^{*}\bigr). $$
$$y(t)\le\max \bigl\{ y(0), C^{*}|\Omega|+\|v_{0}\|_{L^{\infty}(\Omega)}^{2}| \Omega | \bigr\} \quad \text{for all } t\in\bigl(0,T^{*}\bigr), $$
In order to obtain the coupled estimate on \(\int_{\Omega}u^{m+1} \,dx+\int_{\Omega}|\nabla v|^{4}\,dx\), we then derive the following energy estimate on \(\int_{\Omega}|\nabla v|^{4}\,dx\).
Lemma 3.3
Let the same assumptions as in Theorem
1.1
hold. Then there exists a positive constant
\(C_{1}\)
such that the solution of (1.1) satisfies
$$\begin{aligned}& \frac{d}{dt}\int_{\Omega}|\nabla v|^{4}\,dx+\frac{1}{2}\int_{\Omega}\bigl\vert \nabla|\nabla v|^{2}\bigr\vert ^{2}\,dx \\& \quad \le(4+N)\|g \|_{L^{\infty}(\Omega)}^{2}\int_{\Omega}u^{2}| \nabla v|^{2} \,dx+C_{1}\quad \textit{for all } t\in \bigl(0,T^{*}\bigr). \end{aligned}$$
(3.12)
Proof
Differentiating equation (1.1)2, we obtain which, together with the point-wise identity \(2\nabla v\cdot\nabla \Delta v=\Delta|\nabla v|^{2}-2|D^{2} v|^{2}\), yields Multiplying both sides of (3.13) by \(2|\nabla v|^{2}\) and integrating over Ω, we have For the first term on the right-hand side of (3.14), we can use integration by parts and the Young inequality to obtain where we have used the fact \(|\Delta v|^{2}\le N|D^{2} v|^{2}\). For the second term on the right-hand side of (3.14), thanks to the boundedness of \(\int_{\Omega}|\nabla v|^{2} \,dx\), by the same procedure as (3.7)-(3.10) in [31], we deduce that there exists a positive constant \(C_{1}\) such that Combining (3.15) and (3.16) with (3.14), we obtain (3.12) immediately. □
$$\bigl(|\nabla v|^{2} \bigr)_{t}=2\nabla v\cdot\nabla\Delta v-2\nabla v\cdot \nabla\bigl(ug(v)\bigr), $$
$$ \bigl(|\nabla v|^{2} \bigr)_{t}=\Delta|\nabla v|^{2}-2\bigl\vert D^{2} v\bigr\vert ^{2}-2 \nabla v\cdot \nabla\bigl(ug(v)\bigr). $$
(3.13)
$$\begin{aligned}& \frac{d}{dt}\int_{\Omega}|\nabla v|^{4}\,dx+2\int_{\Omega}\bigl\vert \nabla |\nabla v|^{2}\bigr\vert ^{2}\,dx+4\int_{\Omega}| \nabla v|^{2}\bigl|D^{2} v\bigr|^{2}\,dx \\& \quad =-4\int_{\Omega}|\nabla v|^{2}\nabla v\cdot \nabla\bigl(ug(v)\bigr)\,dx+2\int_{\partial\Omega}|\nabla v|^{2}\frac{\partial|\nabla v|^{2}}{\partial\nu }\,dx\quad \text{for all } t\in\bigl(0,T^{*} \bigr). \end{aligned}$$
(3.14)
$$\begin{aligned}& -4\int_{\Omega}|\nabla v|^{2}\nabla v \cdot\nabla\bigl(ug(v)\bigr)\,dx \\& \quad =4\int_{\Omega}ug(v)|\nabla v|^{2}\Delta v \,dx+4\int_{\Omega}ug(v)\nabla v\cdot \nabla |\nabla v|^{2}\,dx \\& \quad \le\frac{4}{N}\int_{\Omega}|\nabla v|^{2}| \Delta v|^{2}\,dx+\int_{\Omega}\bigl\vert \nabla| \nabla v|^{2}\bigr\vert ^{2}\,dx+(4+N)\int _{\Omega}u^{2}g^{2}(v)|\nabla v|^{2} \,dx \\& \quad \le4\int_{\Omega}|\nabla v|^{2}\bigl\vert D^{2} v\bigr\vert ^{2}\,dx+\int_{\Omega}\bigl\vert \nabla |\nabla v|^{2}\bigr\vert ^{2}\,dx+(4+N) \|g\|_{L^{\infty}(\Omega)}^{2}\int_{\Omega}u^{2}| \nabla v|^{2} \,dx, \end{aligned}$$
(3.15)
$$ 2\int_{\partial\Omega}|\nabla v|^{2} \frac{\partial|\nabla v|^{2}}{\partial \nu}\,dx\le\frac{1}{2}\int_{\Omega}\bigl\vert \nabla|\nabla v|^{2} \bigr\vert ^{2} \,dx+C_{1}\quad \text{for all } t\in\bigl(0,T^{*}\bigr). $$
(3.16)
Corollary 3.1
Let the same assumptions as in Theorem
1.1
hold. Then for the same constant
\(C_{1}\)
as in Lemma
3.3, the solution of (1.1) carries the property
$$\begin{aligned}& \frac{d}{dt} \biggl\{ \int_{\Omega}u^{m+1} \,dx+\int_{\Omega}|\nabla v|^{4}\,dx \biggr\} + \biggl\{ \int_{\Omega}u^{m+1} \,dx+\int _{\Omega}|\nabla v|^{4}\,dx \biggr\} \\& \qquad {}+\frac{m(m+1)c_{D}}{2}\int_{\Omega}u^{2m-2}| \nabla u|^{2} \,dx+\frac{1}{2}\int_{\Omega}\bigl\vert \nabla|\nabla v|^{2}\bigr\vert ^{2}\,dx \\& \quad \le \biggl(\frac{m(m+1)\|\chi\|_{L^{\infty}(\Omega)}^{2}}{2c_{D}}+(4+N)\|g\| _{L^{\infty}(\Omega)}^{2} \biggr)\int_{\Omega}u^{2}|\nabla v|^{2} \,dx \\& \qquad {}+ \bigl(1+\kappa(m+1) \bigr)\int_{\Omega}u^{m+1} \,dx-\mu(m+1)\int_{\Omega}u^{m+1+\tau}\,dx+ \int_{\Omega}|\nabla v|^{4}\,dx+C_{1}. \end{aligned}$$
(3.17)
Proof
Adding (3.12) to (3.2) yields Adding \(\int_{\Omega}u^{m+1} \,dx+\int_{\Omega}|\nabla v|^{4}\,dx\) to both sides of this, then yields (3.17). □
$$\begin{aligned}& \frac{d}{dt} \biggl\{ \int_{\Omega}u^{m+1} \,dx+\int_{\Omega}|\nabla v|^{4}\,dx \biggr\} \\& \qquad {}+\frac{m(m+1)c_{D}}{2}\int_{\Omega}u^{2m-2}|\nabla u|^{2} \,dx+\frac{1}{2}\int_{\Omega}\bigl\vert \nabla|\nabla v|^{2}\bigr\vert ^{2}\,dx \\& \quad \le \biggl(\frac{m(m+1)\|\chi\|_{L^{\infty}(\Omega)}^{2}}{2c_{D}}+(4+N)\|g\| _{L^{\infty}(\Omega)}^{2} \biggr)\int_{\Omega}u^{2}|\nabla v|^{2} \,dx \\& \qquad {}+\kappa (m+1)\int_{\Omega}u^{m+1} \,dx -\mu(m+1)\int_{\Omega}u^{m+1+\tau} \,dx+C_{1}. \end{aligned}$$
(3.18)
Now we are ready to establish the estimate on \(\int_{\Omega}u^{m+1} \,dx+\int_{\Omega}|\nabla v|^{4}\,dx\).
Lemma 3.4
Let
\(m>2-\frac{4}{N+2}\), \(N\ge1\). Then there exists a positive constant
C
such that
and
$$ \int_{\Omega}u^{m+1} \,dx\le C \quad \textit{for all } t\in\bigl(0,T^{*}\bigr) $$
(3.19)
$$ \int_{\Omega}|\nabla v|^{4}\,dx\le C \quad \textit{for all } t\in\bigl(0,T^{*}\bigr). $$
(3.20)
Proof
In order to obtain the estimate on the couple of \(\int_{\Omega}u^{m+1} \,dx+\int_{\Omega}|\nabla v|^{4}\,dx\), we need to estimate the integrals on the right of inequality (3.17). We first utilize the Gagliardo-Nirenberg inequality and (3.5) to estimate \(\int_{\Omega}|\nabla v|^{4}\,dx\): Here in the last inequality we have used the Young inequality, because \(\lambda=\frac{N}{2+N}\in(0,1)\). Similarly, invoking the Young inequality with \(\epsilon_{2}>0\) to estimate \(\int_{\Omega}u^{2}|\nabla v|^{2} \,dx\), we have Further utilize the Gagliardo-Nirenberg inequality and (3.5) to estimate Since \(m+\tau+1>\frac{4N+4}{N+2}\), utilizing the Young inequality we have for all \(t\in(0,T^{*})\). Taking \(\epsilon_{2}=\frac{1}{4c_{2}}\), then from (3.22)-(3.24), we have for all \(t\in(0,T^{*})\). For the third term \([\kappa(m+1)+1]\int_{\Omega}u^{m+1} \,dx\), utilizing the Young inequality we have Since \(\frac{m(m+1)c_{D}}{2}\int_{\Omega}u^{2m-2}|\nabla u|^{2} \,dx\) is nonnegative, combining (3.24), (3.25), and (3.26) with (3.17) yields for all \(t\in(0,T^{*})\), where \(C_{4}=C_{1}+C_{2}+C_{3}\). Thus an ODE comparison shows that \(y(t):=\int_{\Omega}u^{m+1} \,dx+\int_{\Omega}|\nabla v|^{4}\,dx\) satisfies which implies (3.19) and (3.20). □
$$\begin{aligned} \int_{\Omega}|\nabla v|^{4} \,dx &=\bigl\Vert |\nabla v|^{2}\bigr\Vert ^{2}_{L^{2}(\Omega)} \\ &\le c_{1}\bigl\Vert \nabla|\nabla v|^{2}\bigr\Vert ^{2\lambda}_{L^{2}(\Omega)}\bigl\Vert |\nabla v|^{2}\bigr\Vert ^{2(1-\lambda)}_{L^{1}(\Omega)}+c_{1}\bigl\Vert |\nabla v|^{2}\bigr\Vert ^{2}_{L^{1}(\Omega)} \\ &\le c_{2}\bigl\Vert \nabla|\nabla v|^{2}\bigr\Vert ^{2\lambda}_{L^{2}(\Omega)}+c_{2} \\ &\le\frac{1}{4}\bigl\Vert \nabla|\nabla v|^{2}\bigr\Vert ^{2}_{L^{2}(\Omega)}+c_{3}\quad \text{for all } t\in \bigl(0,T^{*}\bigr). \end{aligned}$$
(3.21)
$$\begin{aligned}& \biggl(\frac{m(m+1)\|\chi\|_{L^{\infty}(\Omega)}^{2}}{2c_{D}}+(4+N)\|g\| _{L^{\infty}(\Omega)}^{2} \biggr)\int_{\Omega}u^{2}|\nabla v|^{2} \,dx \\& \quad \le\epsilon_{2}\int_{\Omega}\bigl(|\nabla v|^{2}\bigr)^{\frac{2N+2}{N}} \,dx+ \biggl( \frac {m(m+1)\|\chi\|_{L^{\infty}(\Omega)}^{2}}{2c_{D}}+(4+N)\|g\|_{L^{\infty}(\Omega )}^{2} \biggr)^{\frac{2N+2}{N+2}} \\& \qquad {}\times\epsilon_{2}^{-\frac{N}{N+2}}\int_{\Omega}u^{\frac{4N+4}{N+2}}\,dx \quad \text{for all } t\in\bigl(0,T^{*}\bigr). \end{aligned}$$
(3.22)
$$\begin{aligned} \epsilon_{2}\int_{\Omega}\bigl(|\nabla v|^{2}\bigr)^{\frac{2N+2}{N}} \,dx &=\epsilon_{2} \bigl\Vert |\nabla v|^{2}\bigr\Vert ^{\frac{2N+2}{N}}_{L^{\frac{2N+2}{N}}(\Omega)} \\ &\le\epsilon_{2} c_{1}\bigl\Vert \nabla|\nabla v|^{2}\bigr\Vert ^{2}_{L^{2}(\Omega)}\bigl\Vert |\nabla v|^{2}\bigr\Vert ^{\frac{2}{N}}_{L^{1}(\Omega)}+c_{1}\bigl\Vert |\nabla v|^{2} \bigr\Vert ^{\frac{2N+2}{N}}_{L^{1}(\Omega)} \\ &\le\epsilon_{2} c_{2}\bigl\Vert \nabla|\nabla v|^{2}\bigr\Vert ^{2}_{L^{2}(\Omega )}+c_{2} \quad \text{for all } t\in\bigl(0,T^{*}\bigr). \end{aligned}$$
(3.23)
$$\begin{aligned}& \biggl(\frac{m(m+1)\|\chi\|_{L^{\infty}(\Omega)}^{2}}{2c_{D}}+(4+N)\|g\| _{L^{\infty}(\Omega)}^{2} \biggr)^{\frac{2N+2}{N+2}}\epsilon_{2}^{-\frac {N}{N+2}}\int _{\Omega}u^{\frac{4N+4}{N+2}}\,dx \\& \quad \le\frac{\mu(m+1)}{2}\int _{\Omega}u^{m+1+\tau}\,dx+c(\epsilon_{2}) \end{aligned}$$
(3.24)
$$\begin{aligned}& \biggl(\frac{m(m+1)\|\chi\|_{L^{\infty}(\Omega)}^{2}}{2c_{D}}+(4+N)\|g\| _{L^{\infty}(\Omega)}^{2} \biggr)\int_{\Omega}u^{2}|\nabla v|^{2} \,dx \\& \quad \le\frac{1}{4}\bigl\Vert \nabla|\nabla v|^{2}\bigr\Vert ^{2}_{L^{2}(\Omega)}+\frac{\mu(m+1)}{2}\int_{\Omega}u^{m+1+\tau}\,dx+C_{2} \end{aligned}$$
(3.25)
$$ \bigl[\kappa(m+1)+1\bigr]\int_{\Omega}u^{m+1} \,dx\le\frac{\mu(m+1)}{2}\int_{\Omega}u^{m+1+\tau}\,dx+C_{3}\quad \text{for all } t\in\bigl(0,T^{*}\bigr). $$
(3.26)
$$ \frac{d}{dt} \biggl\{ \int_{\Omega}u^{m+1} \,dx+ \int_{\Omega}|\nabla v|^{4}\,dx \biggr\} + \biggl\{ \int _{\Omega}u^{m+1} \,dx+\int_{\Omega}| \nabla v|^{4}\,dx \biggr\} \le C_{4} $$
(3.27)
$$ y(t)\le\max\bigl\{ y(0),C_{4}\bigr\} \quad \text{for all } t\in\bigl(0,T^{*}\bigr), $$
(3.28)
We note \(m>1\) if \(N=2\) in Lemma 3.4. However, if \(m=1\), \(N=2\), and \(\tau=1\), i.e., \(m+\tau+1=\frac{4N+4}{N+2}=3\), the Young inequality will fail to lead to (3.24). In this case, we need to utilize the following generalization of the Gagliardo-Nirenberg inequality for the general case when \(r > 0\) (cf. [32], Lemma A.5, for a detailed proof), which extends the standard case when \(r \ge1\) in [33], to build a bound for \(\int_{\Omega}u^{3} \,dx\).
Lemma 3.5
Let
\(\Omega\subset\mathbb{R}^{2}\)
be a bounded domain with smooth boundary, and let
\(p\in(1,\infty)\)
and
\(r\in(0, p)\). Then there exists
\(C > 0\)
such that for each
\(\eta> 0\)
one can pick
\(C_{\eta}> 0\)
with the property that
holds for all
\(u\in W^{1,2}(\Omega)\).
$$ \|u\|^{p}_{L^{p}(\Omega)}\le\eta\|\nabla u \|^{p-r}_{L^{2}(\Omega)}\bigl\Vert u\ln \vert u\vert \bigr\Vert ^{r}_{L^{r}(\Omega)}+C\|u\|^{p}_{L^{r}(\Omega)}+C_{\eta}$$
(3.29)
Lemma 3.6
Let
\(m=1\), \(N=2\). Then there exists a positive constant
C
such that
and
$$ \int_{\Omega}u^{2} \,dx\le C \quad \textit{for all } t\in\bigl(0,T^{*}\bigr) $$
(3.30)
$$ \int_{\Omega}|\nabla v|^{4}\,dx\le C \quad \textit{for all } t\in\bigl(0,T^{*}\bigr). $$
(3.31)
Proof
By the same procedure as Lemma 3.4, we only need to handle in (3.22). Invoking Lemma 3.5 along with (3.4) and Lemma 2.2, we deduce Taking \(\epsilon_{2}\le\min \{\frac{1}{4c_{2}},\frac{c_{D}^{2}}{4c_{3}^{2}} \}\), from (3.22), (3.23), and (3.32) we then deduce Combing (3.21), (3.26), (3.33), with (3.17) yields for all \(t\in(0,T^{*})\), which implies (3.30) and (3.31). □
$$\biggl(\frac{m(m+1)\|\chi\|_{L^{\infty}(\Omega)}^{2}}{2c_{D}}+(4+N)\|g\| _{L^{\infty}(\Omega)}^{2} \biggr)^{\frac{2N+2}{N+2}}\epsilon_{2}^{-\frac {N}{N+2}}\int _{\Omega}u^{\frac{4N+4}{N+2}}\,dx $$
$$\begin{aligned}& \biggl(\frac{m(m+1)\|\chi\|_{L^{\infty}(\Omega)}^{2}}{2c_{D}}+(4+N)\|g\| _{L^{\infty}(\Omega)}^{2} \biggr)^{\frac{2N+2}{N+2}}\epsilon_{2}^{-\frac {N}{N+2}}\int _{\Omega}u^{\frac{4N+4}{N+2}}\,dx \\& \quad = \biggl(\frac{\|\chi\|_{L^{\infty}(\Omega)}^{2}}{c_{D}}+6\|g\|_{L^{\infty}(\Omega)}^{2} \biggr)^{\frac{3}{2}}\epsilon_{2}^{-\frac{1}{2}}\int _{\Omega}u^{3}\,dx \\& \quad \le \biggl(\frac{\|\chi\|_{L^{\infty}(\Omega)}^{2}}{c_{D}}+6\|g\|_{L^{\infty}(\Omega)}^{2} \biggr)^{\frac{3}{2}}\epsilon_{2}^{-\frac{1}{2}}\int _{\Omega}(u+1)^{3}\,dx \\& \quad \le \biggl(\frac{\|\chi\|_{L^{\infty}(\Omega)}^{2}}{c_{D}}+6\|g\|_{L^{\infty}(\Omega)}^{2} \biggr)^{\frac{3}{2}} \\& \qquad {}\times\epsilon_{2}^{-\frac{1}{2}} \bigl[\epsilon _{2}\|\nabla u\|^{2}_{L^{2}(\Omega)}\bigl\Vert (u+1) \ln(u+1)\bigr\Vert _{L^{1}(\Omega)}+C\|u+1\| ^{3}_{L^{1}(\Omega)}+C_{\epsilon_{2}} \bigr] \\& \quad \le c_{3}\sqrt{\epsilon_{2}}\|\nabla u \|^{2}_{L^{2}(\Omega)}+c_{4}(\epsilon _{2})\quad \text{for all } t\in\bigl(0,T^{*}\bigr). \end{aligned}$$
(3.32)
$$\begin{aligned}& \biggl(\frac{m(m+1)\|\chi\|_{L^{\infty}(\Omega)}^{2}}{2c_{D}}+(4+N)\|g\| _{L^{\infty}(\Omega)}^{2} \biggr)\int_{\Omega}u^{2}|\nabla v|^{2} \,dx \\& \quad = \biggl(\frac{\|\chi\|_{L^{\infty}(\Omega)}^{2}}{c_{D}}+6\|g\|_{L^{\infty}(\Omega )}^{2} \biggr) \int_{\Omega}u^{2}|\nabla v|^{2} \,dx \\& \quad \le\frac{1}{4} \bigl\| \nabla|\nabla v|^{2} \bigr\| ^{2}_{L^{2}(\Omega)}+\frac {c_{D}}{2}\|\nabla u\|^{2}_{L^{2}(\Omega)}+c_{5} \quad \text{for all } t\in\bigl(0,T^{*}\bigr). \end{aligned}$$
(3.33)
$$ \frac{d}{dt} \biggl\{ \int_{\Omega}u^{2} \,dx+ \int_{\Omega}|\nabla v|^{4}\,dx \biggr\} + \biggl\{ \int _{\Omega}u^{2} \,dx+\int_{\Omega}| \nabla v|^{4}\,dx \biggr\} \le C_{5} $$
(3.34)
Lemma 3.4 and Lemma 3.6 result in the following useful corollary that will be used in the proof of Lemma 3.7 below.
Corollary 3.2
Let
\(N=2\), \(m\ge1\), and assume that the initial data
\((u_{0},v_{0})\)
satisfies (1.5). Then there exists a positive constant
C
such that for any constant
\(k>1\)
there exists
\(C(k)>0\)
such that
$$ \int_{\Omega}|\nabla v|^{k} \le C(k) \quad \textit{for all } t\in\bigl(0,T^{*}\bigr). $$
(3.35)
Proof
Since \(m+1\ge N=2\), then by (3.19), (3.20), (3.30), (3.31), and the standard parabolic regularity theory (cf. [12], Lemma 4.1 or [34], Lemma 1), we can immediately obtain (3.35). □
Lemma 3.7
Let the same assumptions as in Theorem
1.1
hold. Then for any
\(p>1\)
there exists a positive constant
\(C(p)\)
such that
$$ \int_{\Omega}u^{p} \,dx\le C(p)\quad \textit{for all } t\in\bigl(0,T^{*}\bigr). $$
(3.36)
Proof
Let us recall to (3.1) once again. Adding \(\int_{\Omega}u^{p} \,dx\) to the both sides of (3.1) and neglecting the nonnegative term \(\frac{p(p-1)c_{D}}{2}\int_{\Omega}u^{p+m-3}|\nabla u|^{2} \,dx\) on the left, we arrive at for any \(p>m\) and each \(t\in(0,T^{*})\). Utilizing (3.35) and the Young inequality we have with some \(c_{2}>0\) and \(C(\frac{2(p+\tau)}{\tau+m-1})\) defined by Corollary 3.2, as well as Collecting (3.37)-(3.39), we thus deduce that \(y(t):=\int_{\Omega}u^{p} \,dx\) satisfies the differential inequality with \(C_{6}=c_{2}C(\frac{2(p+\tau)}{\tau+m-1})+\tilde{C}_{1}\). Upon an ODE comparison, this implies Thus (3.36) holds for any \(p>m\). Since \(\int_{\Omega}u \,dx\le M_{1}\), utilizing the interpolation inequality, (3.36) holds also for any \(1< p\le m\). This completes the proof of Lemma 3.7. □
$$\begin{aligned}& \frac{d}{dt}\int_{\Omega}u^{p} \,dx+\int_{\Omega}u^{p} \,dx \\& \quad \le\frac{p(p-1)\|\chi \|_{L^{\infty}(\Omega)}^{2}}{2c_{D}}\int _{\Omega}u^{p-m+1}|\nabla v|^{2} \,dx \\& \qquad {} +(\kappa p+1)\int_{\Omega}u^{p} \,dx-\mu p\int _{\Omega}u^{p+\tau}\,dx \end{aligned}$$
(3.37)
$$\begin{aligned}& \frac{p(p-1)\chi^{2}}{2c_{D}}\int_{\Omega}u^{p-m+1}| \nabla v|^{2} \,dx \\& \quad \le\frac {\mu p}{2}\int_{\Omega}u^{p+\tau} \,dx+c_{4}\int_{\Omega}|\nabla v|^{\frac {2(p+\tau)}{\tau+m-1}} \\& \quad \le\frac{\mu p}{2}\int_{\Omega}u^{p+\tau} \,dx+c_{2}C\biggl(\frac{2(p+\tau)}{\tau +m-1}\biggr)\quad \text{for all } t\in \bigl(0,T^{*}\bigr), \end{aligned}$$
(3.38)
$$ [\kappa p+1]\int_{\Omega}u^{p} \,dx \le\frac{\mu p}{2}\int_{\Omega}u^{p+\tau } \,dx+ \tilde{C}_{1} \quad \text{for all } t\in\bigl(0,T^{*}\bigr). $$
(3.39)
$$ \frac{d}{dt}\int_{\Omega}u^{p} \,dx+\int_{\Omega}u^{p} \,dx\le C_{6}\quad \text{for all } t\in\bigl(0,T^{*}\bigr), $$
(3.40)
$$ y(t)\le\max\bigl\{ y(0),C_{6}\bigr\} \quad \text{for all } t\in \bigl(0,T^{*}\bigr). $$
Once the uniform estimate of \(\|u(\cdot, t)\|_{L^{k}(\Omega)}\) has been established, we can use the classical Alikakos iteration method to obtain the uniform bound of \(\|u(\cdot, t)\|_{L^{\infty}(\Omega)}\).
Lemma 3.8
We are now in the position to prove the global existence in Theorem 1.1.
3.2 Proof of decay property in Theorem 1.1
In this short subsection, we discuss the decay property in the limit case \(\kappa=0\). Our approach is inspired by that in [35].
Lemma 3.9
Suppose that
\(f(s)\)
satisfies (1.4) with
\(\kappa=0\). Then
and
$$ \int^{\infty}_{0}\int _{\Omega}u^{\tau+1} \, dx\, dt< \frac{1}{\mu} \|u_{0}\|_{L^{1}(\Omega)} $$
(3.42)
$$ \int_{\Omega}u(x,t)\,dx\le\|u_{0} \|_{L^{1}(\Omega)}|\Omega| \bigl( |\Omega|^{\tau}+\mu\tau t \|u_{0}\|^{\tau}_{L^{1}(\Omega)} \bigr)^{-\frac{1}{\tau}}\quad \textit{for all } t>0. $$
(3.43)
Proof
Integrating (1.1) in space we obtain under the assumption \(\kappa=0\), which implies both (3.42) and (3.43). □
$$\frac{d}{dt}\int_{\Omega}u\, dx=\int_{\Omega}uf(u)\,dx\le-\mu\int_{\Omega}u^{\tau+1}\,dx\le- \frac{\mu}{|\Omega|^{\tau}} \biggl(\int_{\Omega}u \,dx \biggr)^{1+\tau} \quad \text{for all } t>0, $$
Lemma 3.10
Let
\(\kappa=0\), \(\mu>\frac{\|u_{0}\|_{L^{1}(\Omega)}}{(\tau+1)\|v_{0}\| _{L^{1}(\Omega)}}\), and assume that
Then there exists a positive constant
C
appropriately small such that
$$ \int^{\infty}_{0}\int _{\Omega}g(x,t)^{\frac{\tau+1}{\tau}}\,dx< \frac{\tau +1}{\tau} \biggl( \|v_{0}\|_{L^{1}(\Omega)}-\frac{1}{(\tau+1)\mu} \|u_{0}\| _{L^{1}(\Omega)} \biggr). $$
(3.44)
$$ \int_{\Omega}v(x,t)\,dx\ge C \quad \textit{for all } t>0. $$
(3.45)
Proof
Integrating the second equation in (1.1) in space and applying the Young inequality, yield In view of (3.42), (3.46) implies Therefore, (3.44) asserts the positivity of the right-hand side of (3.47). We thus obtain the desired result (3.45). □
$$ \frac{d}{dt}\int_{\Omega}v\,dx=-\int _{\Omega}ug(v)\,dx\ge-\frac{1}{\tau+1}\int_{\Omega}u^{\tau+1}\,dx-\frac{\tau}{\tau+1}\int_{\Omega}g(v)^{\frac{\tau +1}{\tau}}\,dx. $$
(3.46)
$$ \int_{\Omega}v\,dx\ge\|v_{0} \|_{L^{1}(\Omega)}-\frac{1}{(\tau+1)\mu} \|u_{0}\| _{L^{1}(\Omega)}- \frac{\tau}{\tau+1}\int^{\infty}_{0}\int _{\Omega}g(v)^{\frac {\tau+1}{\tau}}\,dx \quad \text{for all } t>0. $$
(3.47)
Lemma 3.11
There exist
\(\alpha\in(0,1)\)
and
\(C>0\)
such that
$$ \|u\|_{C^{\alpha,\frac{\alpha}{2}}(\bar{\Omega}\times[t,t+1])}\le C\quad \textit{for all } t>0. $$
(3.48)
Proof
Rewriting the first equation of (1.1) in the form Utilizing the Young inequality we can estimate and evidently in \(\Omega\times(0,\infty)\). As Lemma 3.8 and Corollary 3.2 imply u and ∇v are bounded in \(L^{\infty}((0,\infty);L^{k}(\Omega ))\) for any \(k\in(1,\infty)\), and that u is a bounded solution of (3.49), the Hölder continuity of u, i.e., (3.48), immediately results from a known result on parabolic Hölder regularity ([36], Theorem 1.3). □
$$ u_{t}=\nabla\cdot \bigl(D(u)\nabla u-u\chi(v)\nabla v \bigr)+uf(u),\quad x\in\Omega,t>0. $$
(3.49)
$$ \bigl(D(u)\nabla u-u\chi(v)\nabla v \bigr)\cdot\nabla u\ge\frac {D(u)}{2}| \nabla u|^{2}-\frac{u^{2}\chi^{2}(v)|\nabla v|^{2}}{2D(u)} $$
$$ \bigl\vert D(u)\nabla u-u\chi(v)\nabla v\bigr\vert \le D(u)|\nabla u|+u\chi (v)|\nabla v| $$
Proof of decay property in Theorem 1.1
In view of the Hölder continuity of u, i.e., (3.48), Arzelà-Ascoli theorem asserts \((u(\cdot,t))_{t>1}\) is relatively compact in \(C^{0}(\bar{\Omega})\). Thus, the decay property (3.43) implies that \(\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\rightarrow0\) as \(t\rightarrow\infty\). □
Remark 3.1
The decay property in Theorem 1.1 reveals the fact that if the proliferation of cells is ignored, then the cells will not survive. However, Lemma 3.10 shows that if the death rate of the cells is large and the consumption rate of oxygen is small enough, then there is always oxygen remaining.
Since the global existence and decay property both have been proved, we then have completed the proof of Theorem 1.1.
Declarations
Acknowledgements
The author thanks the reviewers for their valuable suggestions, which greatly improved the paper. This work is supported by the Natural Science Project of Sichuan Province Department of Education (No. 15ZB0145).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
(1)
College of Mathematics and Information, China West Normal University, Nanchong, 637002, China
References
- Keller, EF, Segel, LA: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399-415 (1970) MATHView ArticleGoogle Scholar
- Herrero, MA, Velázquez, JJL: A blow-up mechanism for a chemotaxis model. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 24, 633-683 (1997) MATHGoogle Scholar
- Nagai, T: Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains. J. Inequal. Appl. 6, 37-55 (2001) MATHMathSciNetGoogle Scholar
- Winkler, M: Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system. J. Math. Pures Appl. 100, 748-767 (2013). arXiv:1112.4156v1 MathSciNetView ArticleGoogle Scholar
- Osaki, K, Tsujikawa, T, Yagi, A, Mimura, M: Exponential attractor for a chemotaxis-growth system of equations. Nonlinear Anal., Theory Methods Appl. 51, 119-144 (2002) MATHMathSciNetView ArticleGoogle Scholar
- Winkler, M: Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source. Commun. Partial Differ. Equ. 35, 1516-1537 (2010) MATHView ArticleGoogle Scholar
- Tao, Y, Winkler, M: Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity. J. Differ. Equ. 252, 692-715 (2012) MATHMathSciNetView ArticleGoogle Scholar
- Sugiyama, Y: Time global existence and asymptotic behavior of solutions to degenerate quasilinear parabolic systems of chemotaxis. Differ. Integral Equ. 20, 133-180 (2007) MATHMathSciNetGoogle Scholar
- Winkler, M: Does a ‘volume-filling effect’ always prevent chemotactic collapse? Math. Methods Appl. Sci. 33, 12-24 (2010) MATHMathSciNetGoogle Scholar
- Li, X, Xiang, Z: Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete Contin. Dyn. Syst., Ser. A 35(8), 3503-3531 (2015). doi:10.3934/dcds.2015.35.3503 MATHMathSciNetView ArticleGoogle Scholar
- Hillen, T, Painter, KJ: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58, 183-217 (2009) MATHMathSciNetView ArticleGoogle Scholar
- Horstmann, D, Winkler, M: Boundedness vs. blow-up in a chemotaxis system. J. Differ. Equ. 215, 52-107 (2005) MATHMathSciNetView ArticleGoogle Scholar
- Tuval, I, Cisneros, L, Dombrowski, C, Wolgemuth, CW, Kessler, JO, Goldstein, RE: Bacterial swimming and oxygen transport near contact lines. Proc. Natl. Acad. Sci. USA 102, 2277-2282 (2005) MATHView ArticleGoogle Scholar
- Duan, R, Xiang, Z: A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion. Int. Math. Res. Not. 2014, 1833-1852 (2014) MathSciNetGoogle Scholar
- Francesco, M, Lorz, A, Markowich, PA: Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior. Discrete Contin. Dyn. Syst., Ser. A 28, 1437-1453 (2010) MATHView ArticleGoogle Scholar
- Liu, JG, Lorz, A: A coupled chemotaxis-fluid model: global existence. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 28, 643-652 (2011) MATHMathSciNetView ArticleGoogle Scholar
- Lorz, A: Coupled chemotaxis fluid model. Math. Models Methods Appl. Sci. 20, 987-1004 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Tao, Y, Winkler, M: Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion. Discrete Contin. Dyn. Syst., Ser. A 32(5), 1901-1914 (2012) MATHMathSciNetView ArticleGoogle Scholar
- Tao, Y, Winkler, M: Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 30, 157-178 (2013) MATHMathSciNetView ArticleGoogle Scholar
- Wang, ZA: Mathematics of traveling waves in chemotaxis. Discrete Contin. Dyn. Syst., Ser. B 18(3), 601-641 (2013) MathSciNetView ArticleGoogle Scholar
- Tao, Y: Boundedness in a chemotaxis model with oxygen consumption by bacteria. J. Math. Anal. Appl. 381, 521-529 (2011) MATHMathSciNetView ArticleGoogle Scholar
- Wang, L, Khan, SUD, Khan, SUD: Boundedness in a chemotaxis system with consumption of chemoattractant and logistic source. Electron. J. Differ. Equ. 2013, 209 (2013) MathSciNetView ArticleGoogle Scholar
- Tao, Y, Winkler, M: Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant. J. Differ. Equ. 252, 2520-2543 (2012) MATHMathSciNetView ArticleGoogle Scholar
- Wang, L, Mu, C, Zhou, S: Boundedness in a parabolic-parabolic chemotaxis system with nonlinear diffusion. Z. Angew. Math. Phys. 65(6), 1137-1152 (2014) MATHMathSciNetView ArticleGoogle Scholar
- Wang, L, Mu, C, Lin, K, Zhao, J: Global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant. Z. Angew. Math. Phys. (2015). doi:10.1007/s00033-014-0491-9 Google Scholar
- Wang, Y, Xiang, Z: Global existence and boundedness in a higher-dimensional quasilinear chemotaxis system. Preprint Google Scholar
- Espejo, E, Suzuki, T: Reaction terms avoiding aggregation in slow fluids. Nonlinear Anal., Real World Appl. 21, 110-126 (2015) MATHMathSciNetView ArticleGoogle Scholar
- Kiselev, A, Ryzhik, L: Biomixing by chemotaxis and enhancement of biological reactions. Commun. Partial Differ. Equ. 37, 298-312 (2012) MATHMathSciNetView ArticleGoogle Scholar
- Tao, Y, Winkler, M: A chmotaxis-haptotaxis model: the roles of nonlinear diffusion and logistic source. SIAM J. Math. Anal. 43, 685-704 (2011) MATHMathSciNetView ArticleGoogle Scholar
- Liu, D, Tao, Y: Global boundedness in a fully parabolic attraction-repulsion chemotaxis model. Math. Methods Appl. Sci. (2014). doi:10.1002/mma.3240 Google Scholar
- Ishida, S, Seki, K, Yokota, T: Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains. J. Differ. Equ. 256, 2993-3010 (2014) MATHMathSciNetView ArticleGoogle Scholar
- Tao, Y, Winkler, M: Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant. J. Differ. Equ. 257, 784-815 (2014) MATHMathSciNetView ArticleGoogle Scholar
- Biler, P, Hebisch, W, Nadzieja, T: The Debye system: existence and large time behavior of solutions. Nonlinear Anal. 23, 1189-1209 (1994) MATHMathSciNetView ArticleGoogle Scholar
- Kowalczyk, R, Szymanska, Z: On the global existence of solutions to an aggregation model. J. Math. Anal. Appl. 343, 379-398 (2008) MATHMathSciNetView ArticleGoogle Scholar
- Tao, Y, Winkler, M: Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system. Z. Angew. Math. Phys. (2015). doi:10.1007/s00033-015-0541-y Google Scholar
- Porzio, MM, Vespri, V: Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J. Differ. Equ. 103(1), 146-178 (1993) MATHMathSciNetView ArticleGoogle Scholar
Copyright
© Li 2015
