Analysis of a one-dimensional forager-exploiter model

TL;DR

This paper analyzes a one-dimensional mathematical model describing social interactions in forager-exploiter groups, proving global existence, boundedness, and exponential stabilization of solutions under certain conditions.

Contribution

It extends classical chemotaxis-consumption models by incorporating a second coupled mechanism and establishes global well-posedness and stability results.

Findings

Global bounded classical solutions exist for all regular initial data.

Solutions stabilize exponentially to a homogeneous equilibrium.

Stability holds under a smallness condition on initial data mass.

Abstract

\begin{abstract} \noindent % We consider the one-dimensional parabolic system The system \bas \left\{ \begin{array}{l} u_t= u_{xx} - \chi_1 (uw_x)_x, \\[1mm] v_t = v_{xx} - \chi_2 (vu_x)_x, \\[1mm] w_t = dw_{xx} - \lambda (u+v)w - \mu w + r, \end{array} \right. \eas % that has been proposed as a model to describe social interactions within mixed forager-exploiter groups. is considered in a bounded real interval, with positive parameters $\chi_1,\chi_2,d,\lambda$ and $\mu$, and with $r \ge 0$. Proposed to describe social interactions within mixed forager-exploiter groups, this model extends classical one-species chemotaxis-consumption systems by additionally accounting for a second axis mechanism coupled to the first in a consecutive manner. \abs % It is firstly shown that for all suitably regular initial data $(u_0, v_0, w_0)$, an associated Neumann-type initial-boundary value problem possesses a globally defined bounded classical solution. Moreover, it is asserted that this solution stabilizes to a spatially homogeneous equilibrium at an exponential rate under a smallness condition on $\min\{\io u_0, \io v_0\}$ that appears to be consistent with predictions obtained from formal stability analysis.\abs

Equations

$$\displaystyle\left\{\begin{array}[]{l}u_{t}=u_{xx}-\chi_{1}(uw_{x})_{x},\\[2.84526pt] v_{t}=v_{xx}-\chi_{2}(vu_{x})_{x},\\[2.84526pt] w_{t}=dw_{xx}-\lambda(u+v)w-\mu w+r,\end{array}\right.$$

$$\left\{\begin{array}[]{l}u_{t}=\Delta u-\chi_{1}\nabla\cdot(u\nabla w),\\[2.84526pt] v_{t}=\Delta v-\chi_{2}\nabla\cdot(v\nabla u),\\[2.84526pt] w_{t}=d\Delta v-\lambda(u+v)w-\mu w+r,\end{array}\right.$$

$$\left\{\begin{array}[]{l}u_{t}=\Delta u-\chi_{1}\nabla\cdot(u\nabla w),\\[2.84526pt] w_{t}=d\Delta v-\lambda uw-\mu w+r.\end{array}\right.$$

$$\left\{\begin{array}[]{lcll}u_{t}&=&u_{xx}-\chi_{1}(uw_{x})_{x},&x\in\Omega,\ t>0,\\[2.84526pt] v_{t}&=&v_{xx}-\chi_{2}(vu_{x})_{x},&x\in\Omega,\ t>0,\\[2.84526pt] w_{t}&=&dw_{xx}-\lambda(u+v)w-\mu w+r,&x\in\Omega,\ t>0,\\[2.84526pt] &&u_{x}=v_{x}=w_{x}=0,&x\in\partial\Omega,\ t>0,\\[2.84526pt] &&u(x,0)=u_{0}(x),\quad v(x,0)=v_{0}(x),\quad w(x,0)=w_{0}(x),&x\in\Omega,\end{array}\right.$$

$$\left\{\begin{array}[]{l}u_{0}\in W^{1,\infty}(\Omega)\mbox{ is nonnegative with }u_{0}\not\equiv 0,\\[2.84526pt] v_{0}\in W^{1,\infty}(\Omega)\mbox{ is nonnegative with }v_{0}\not\equiv 0,\qquad\mbox{and that}\\[2.84526pt] w_{0}\in W^{1,\infty}(\Omega)\mbox{ is positive in }\overline{\Omega}.\end{array}\right.$$

$$\mbox{$u,v$ and $w$ belong to }C^{0}([0,\infty);W^{1,2}(\Omega))\cap C^{2,1}(\overline{\Omega}\times(0,\infty)),$$

$$\Vert u(\cdot ,t){\Vert }_{W^{1,2}(\Omega )}+\Vert v(\cdot ,t){\Vert }_{W^{1,2}(\Omega )}+\Vert w(\cdot ,t){\Vert }_{W^{1,2}(\Omega )}\le C\text{mbox}forallt>0.$$

$$\frac{8(\lambda +\mu {)}^2(d+1)}{\lambda r{\chi }_1{\overline{u}}_0{\overline{v}}_0}+\frac{2(d+1)}{{\overline{v}}_0}\gtrsim {\chi }_2$$

$${\int }_{\Omega }{u}_0+{\int }_{\Omega }{v}_0\le M$$

$${\int }_{\Omega }{u}_0\le \epsilon (M)\text{mbox}or{\int }_{\Omega }{v}_0\le \epsilon (M),$$

$$\Vert u(\cdot ,t)-{\overline{u}}_0{\Vert }_{L^{\infty }(\Omega )}+\Vert v(\cdot ,t)-{\overline{v}}_0{\Vert }_{L^{\infty }(\Omega )}+\Vert w(\cdot ,t)-w_{\star }{\Vert }_{L^{\infty }(\Omega )}\le C{e}^{-\alpha t}\text{mbox}forallt>0,$$

$$w_{\star }:=\frac{r}{\lambda ({\overline{u}}_0+{\overline{v}}_0)+\mu }.$$

$$\mbox{$u,v$ and $w$ are elements of }C^{0}([0,T_{max});W^{1,2}(\Omega))\cap C^{2,1}(\overline{\Omega}\times(0,T_{max})),$$

$$\mbox{if $T_{max}<\infty$, then}\quad\limsup_{t\nearrow T_{max}}\Big{\{}\|u(\cdot,t)\|_{W^{1,q}(\Omega)}+\|v(\cdot,t)\|_{W^{1,q}(\Omega)}+\|w(\cdot,t)\|_{W^{1,q}(\Omega)}\Big{\}}=\infty\qquad\mbox{for all }q>1.$$

$${\int }_{\Omega }u(\cdot ,t)={\int }_{\Omega }{u}_0\text{mbox}and{\int }_{\Omega }v(\cdot ,t)={\int }_{\Omega }{v}_0\text{mbox}forallt\in (0,T_{max}).$$

$$\displaystyle\mathcal{A}(u,v,w):=\Bigg{(}\begin{array}[]{ccc}1&0&-\chi_{1}u\\ -\chi_{2}v&1&0\\ 0&0&d\end{array}\Bigg{)}$$

$$\Vert w(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}\le \frac{r}{\mu }+\Vert w_0{\Vert }_{L^{\infty }(\Omega )}{e}^{-\mu t}\text{mbox}forallt\in (0,T_{max}).$$

$$y(t):=y_0{e}^{-\mu t}+\frac{r}{\mu }(1-e^{-\mu t}),t\ge 0,$$

$${\overline{w}}_t-d{\overline{w}}_{xx}+\lambda (u+v)w+\mu \overline{w}-r$$

$$\displaystyle w(x,t)\leq y_{0}e^{-\mu t}+\frac{r}{\mu}\qquad\mbox{for all $x\in\Omega$ and }t\in(0,T_{max}),$$

$$\Vert w_x(\cdot ,t){\Vert }_{L^q(\Omega )}\le K(M,q)+C{e}^{-\alpha t}\text{mbox}forallt\in (0,T_{max}).$$

$$\Vert {\partial }_x{e}^{dt\Delta }\phi {\Vert }_{L^q(\Omega )}\le c_1(q)\Vert \phi {\Vert }_{W^{1,\infty }(\Omega )}\text{mbox}forall\phi \in W^{1,\infty }(\Omega )$$

$$\Vert {\partial }_x{e}^{dt\Delta }\phi {\Vert }_{L^q(\Omega )}\le c_2(q)(1+t^{-1+\frac{1}{2q}})\Vert \phi {\Vert }_{L^1(\Omega )}\text{mbox}forall\phi \in C^0(\overline{\Omega }).$$

$$\displaystyle w(\cdot,t)=e^{t(d\Delta-\mu)}w_{0}+\int_{0}^{t}e^{(t-s)(d\Delta-\mu)}\Big{\{}-\lambda(u(\cdot,s)+v(\cdot,s))w(\cdot,s)+r\Big{\}}ds,\qquad t\in(0,T_{max}),$$

$$\Vert w_x(\cdot ,t){\Vert }_{L^q(\Omega )}$$

$$\displaystyle c_{2}(q)\lambda\int_{0}^{t}e^{-\mu(t-s)}\Big{(}1+(t-s)^{-1+\frac{1}{2q}}\Big{)}\cdot\Big{\{}\|u(\cdot,s)\|_{L^{1}(\Omega)}+\|v(\cdot,s)\|_{L^{1}(\Omega)}\Big{\}}\|w(\cdot,s)\|_{L^{\infty}(\Omega)}ds$$

$$\Vert w_x(\cdot ,t){\Vert }_{L^q(\Omega )}$$

$${\int }_{\Omega }{u}^2(\cdot ,t)\le K(M){\overline{u}}_0^2+C{e}^{-\alpha t}\text{mbox}forallt\in (0,T_{max})$$

$${\int }_t^{t+\tau }{\int }_{\Omega }{u}_x^2\le K(M){\overline{u}}_0^2+C{e}^{-\alpha t}\text{mbox}forallt\in (0,T_{max}-\tau ),$$

$$\Vert w_x(\cdot ,t){\Vert }_{L^2(\Omega )}\le k_1(M)+c_1{e}^{-\alpha t}\text{mbox}forallt\in (0,T_{max}),$$

Full text

[TABLE]

is considered in a bounded real interval, with positive parameters $\chi_{1},\chi_{2},d,\lambda$ and $\mu$, and with $r\geq 0$. Proposed to describe social interactions within mixed forager-exploiter groups, this model extends classical one-species chemotaxis-consumption systems by additionally accounting for a second taxis mechanism coupled to the first in a consecutive manner.

It is firstly shown that for all suitably regular initial data $(u_{0},v_{0},w_{0})$, an associated Neumann-type initial-boundary value problem possesses a globally defined bounded classical solution. Moreover, it is asserted that this solution stabilizes to a spatially homogeneous equilibrium at an exponential rate under a smallness condition on $\min\{\int_{\Omega}u_{0},\int_{\Omega}v_{0}\}$ that appears to be consistent with predictions obtained from formal stability analysis.

**Key words: ** chemotaxis; global existence; large time behavior

MSC 2010: 35Q92, 35B40 (primary); 35K55, 35Q91, 92C17 (secondary)

1 Introduction

It is well-known that social interactions in mixed-species groups may lead to rich spatial patterns, already in some cases of populations consisting of merely two fractions, and the ambition to understand fundamental principles for such complex dynamics has been attracting considerable interest during the past decade in the literature near the borderline regions between, inter alia, biology, behavioral sciences and mathematics ([9], [18], [11], [3]). With the macroscopic formation of shearwater flocks through attraction to kittiwake foragers in Alaska forming a paradigmatic example, a rather clearly traceable type of social interplay, in the comparatively simple case of only two involved species, is constituted by so-called forager-exploiter interaction: The members of a first population, the “foragers”, search for food by directly moving upward gradients of the nutrient concentration; contrary to this, as “exploiters” the individuals of a second population pursue a more indirect strategy by rather orienting their movement toward regions of higher forager population densities.

As a macroscopic model for the spatio-temporal evolution of the population densities $u=u(x,t)$ and $v=v(x,t)$ of foragers and scroungers in such contexts, additionally accounting for the nutrient density $w=w(x,t)$ as a third unknown the authors in [18] propose the parabolic PDE system

[TABLE]

with positive parameters $\chi_{1},\chi_{2},d$ and $\lambda$, and with $\mu\geq 0$ and $r\geq 0$. Besides presupposing random diffusion of all quantities, this model assumes the food resources to be degraded by both foragers and scroungers upon contact, and to possibly be supplied by an external source and spontaneously decaying. In its most characteristic part, however, (1.1) accounts for the group-specific strategies of directed motion by postulating taxis-type cross-diffusion mechanisms to be responsible in both cases.

As a particular feature thereby generated, (1.1) contains a sequential coupling of two taxis processes, which may be expected to considerably increase the mathematical complexity of (1.1) when compared, for instance, to the corresponding one-species chemotaxis-consumption system, depending on the application context sometimes also referred to as prey-taxis system ([15], [16], [28]), that is obtained on letting $v\equiv 0$ in (1.1), and that is hence given by

[TABLE]

In the prototypical case when $\mu=r=0$, namely, the structure of the latter is artless enough so as to allow for a meaningful energy structure that can be used as a technical basis for a comprehensive theory of global classical well-posedness in low-dimensional boundary value problems in which the spatial dimension $n$ satisfies $n\leq 2$, of global weak solvability when $n=3$, and even of asymptotic stabilization toward spatially homogeneous equilibria whenever $n\leq 3$ ([19]). Adaptations of such approaches have been utilized to address several variants of (1.2), partially even involving further components and interaction mechanisms ([14], [28], [25], [23]), but capturing the additional intricacy induced by the second taxis interaction in (1.1) seems beyond the abilities of such methods.

In particular, to the best of our knowledge it is yet left completely open by the analytical literature how far the coupling of the nutrient taxis mechanism from (1.2) to a further cross-diffusion process sensitive to the gradient of the first population may lead to substantial destabilization of the tendency toward homogeneity, as known to occur in (1.2); only under some restrictive assumptions on $r$ and the initial data $w|_{t=0}$, in essence ensuring that $w$ remains below a suitably small threshold throughout evolution, it has recently been possible to achieve some results on global existence of certain generalized solutions, as well as on their large time stabilization toward constants when moreover $r=r(x,t)$ decays suitably in time ([27]). In light of the strong singularity-supporting potential of chemotactic cross-diffusion, well-known as a striking feature e.g. of the classical Keller-Segel system ([12], [22]), already answers to questions from basic theory of global solvability in (1.1) thus seem far from obvious, even in the simplest case in which $n=1$; in fact, the numerical experiments reported in [18] indicate quite a rich dynamical potential of (1.1) already in such one-dimensional frameworks.

Main results.  The purpose of the present work now consists in making sure that the evident mathematical challenges notwithstanding, and especially despite the apparent lack of any favorable energy structure, at least the one-dimensional version of (1.1) does not only allow for a rather comprehensive theory of classical solvability, but beyond this is even accessible to an essentially exhaustive qualitative analysis in parameter constellations for which formal considerations predict asymptotic homogenization: In particular, we shall develop an analytical approach which firstly enables us to assert global existence of bounded classical solutions for widely arbitrary initial data, and which secondly is subtle enough so as to allow for a conclusion on large time stabilization toward constant steady states under an additional smallness assumption on the total population sizes of foragers and scroungers that quite precisely seems to match a corresponding condition formally identified as essentially necessary and sufficient therefor by means of a linear stability analysis in [18].

To take this more precise, in a bounded open interval $\Omega\subset\mathbb{R}$ let us consider the initial-boundary value problem for (1.1) given by

[TABLE]

where $\chi_{1},\chi_{2},d,\lambda$ and $\mu$ are positive constants and $r$ is nonnegative, and where the initial data are such that

[TABLE]

Then the first of our main results asserts global existence of bounded classical solutions in the following flavor.

Theorem 1.1

Let $\Omega\subset\mathbb{R}$ be a bounded open interval, and let $\chi_{1},\chi_{2},d,\lambda$ and $\mu$ be positive and $r$ be nonnegative. Then for any choice of $(u_{0},v_{0},w_{0})$ fulfilling (1.4), the problem (1.3) possesses a global classical solution $(u,v,w)$ which is uniquely determined by the properties that

[TABLE]

and which is such that $u>0$ and $v>0$ in $\overline{\Omega}\times(0,\infty)$ as well as $w>0$ in $\overline{\Omega}\times[0,\infty)$. Moreover, this solution is bounded in the sense that there exists $C>0$ such that

[TABLE]

Next intending to identify circumstances under which the diffusion processes in (1.3) are sufficiently strong so as to warrant relaxation into homogeneous states, we note that in light of corresponding results on asymptotically diffusive behavior in chemotaxis systems with essentially superlinear nonlinear ingredients ([4], [5], [21]) it seems far from audacious to conjecture that such dynamics can always be observed if solutions remain small in all their components, and with respect to suitably fine topologies, throughout evolution.

In order to approach a more subtle picture in this regard, we recall that a linear stability analysis detailed in [18] suggests to expect, in the normalized case when $\Omega=(0,1)$ and the spatial averages $\overline{u}_{0}:=\frac{1}{|\Omega|}\int_{\Omega}u_{0}$ and $\overline{v}_{0}:=\frac{1}{|\Omega|}v_{0}$ satisfy $\overline{u}_{0}+\overline{v}_{0}=1$, the relation

[TABLE]

as the decisive condition for prevalence of homogeneity. In particular, this condition is entirely independent of $w_{0}$, and moreover it involves $u_{0}$ and $v_{0}$ exclusively through their $L^{1}$ norms as their biologically best interpretable derivate. If, more generally, $\overline{u}_{0}+\overline{v}_{0}$ is supposed to remain below a given number in an arbitrary bounded $\Omega$, then assuming (1.7) evidently becomes equivalent to imposing a certain smallness hypothesis on $\min\{\overline{u}_{0},\overline{v}_{0}\}$. The second of our main results now provides a rigorous mathematical counterpart of this formal consideration.

Theorem 1.2

Suppose that $\Omega\subset\mathbb{R}$ is a bounded open interval, that $\chi_{1},\chi_{2},d,\lambda$ and $\mu$ are positive, and that $r\geq 0$. Then for all $M>0$ one can find $\varepsilon(M)>0$ with the property that whenever $u_{0},v_{0}$ and $w_{0}$ are such that besides (1.4) we have

[TABLE]

as well as

[TABLE]

there exist $C=C(u_{0},v_{0},w_{0})>0$ and $\alpha=\alpha(u_{0},v_{0},w_{0})>0$ such that the solution $(u,v,w)$ of (1.3) satisfies

[TABLE]

where $w_{\star}$ is the nonnegative constant given by

[TABLE]

2 Local existence and an explicit $L^{\infty}$ bound for $w$

The following basic statement on local existence and extensibility can be obtained from standard theory on evolution systems of parabolic type.

Lemma 2.1

Suppose that $\Omega\subset\mathbb{R}$ is a bounded open interval, that $\chi_{1},\chi_{2},d,\lambda$ and $\mu$ are positive and $r\geq 0$, and that (1.4) holds. Then there exist $T_{max}\in(0,\infty]$ and nonnegative functions $u,v,w$, uniquely determined by the requirement that

[TABLE]

which solve (1.3) in the classical sense in $\Omega\times(0,T_{max})$, and which are such that

[TABLE]

Moreover, $u>0$ and $v>0$ in $\overline{\Omega}\times(0,T_{max})$ and $w>0$ in $\overline{\Omega}\times[0,T_{max})$, and we have

[TABLE]

Proof.  Since all eigenvalues of the diffusion matrix

[TABLE]

are positive, the general theory on local existence and maximal extension from [1] is applicable to (1.3); in particular, the existence of a uniquely determined maximal classical solution follows from [1, Theorems 14.4 and 14.6], and the extensibility criterion (2.2) is ensured by [1, Theorem 15.5]. Moreover, the strong maximum principle along with our assumptions $u_{0}\not\equiv 0$ and $v_{0}\not\equiv 0$ in (1.4) yield positivity of $u$ and $v$ in $\overline{\Omega}\times(0,T_{max})$, whereas a simple comparison argument shows that $w(x,t)\geq\min_{x\in\overline{\Omega}}w_{0}(x)e^{-\mu t}>0$ in $\overline{\Omega}\times[0,T_{max})$ thanks to our assumption that $w_{0}>0$. Finally, the mass conservation properties in (2.3) immediately result from integration in the first equation and the second equation in (1.3), respectively. $\Box$ Constituting another basic but important feature of (1.3), the following pointwise bound on $w$ is an immediate consequence of the maximum principle.

Lemma 2.2

We have

[TABLE]

Proof.  We let $\overline{w}(x,t):=y(t)$ for $x\in\overline{\Omega}$ and $t\geq 0$, where

[TABLE]

denotes the solution of $y^{\prime}(t)+\mu y(t)=r$, $t>0$, with $y(0)=y_{0}:=\|w_{0}\|_{L^{\infty}(\Omega)}$. Then $\overline{w}(\cdot,0)=y_{0}\geq w(\cdot,0)$ in $\Omega$ as well as $\frac{\partial\overline{w}}{\partial\nu}=0$ on $\partial\Omega\times(0,\infty)$. As moreover, by nonnegativity of $\lambda,u$ and $v$,

[TABLE]

by means of a comparison argument we conclude that $w\leq\overline{w}$ in $\Omega\times(0,T_{max})$, and that thus

[TABLE]

which is equivalent to (2.5). $\Box$

3 Further estimates. Linking regularity to the size of $\int_{\Omega}u_{0}+\int_{\Omega}v_{0}$

The goal of this section is to reveal further regularity properties of the above local solution which on the one hand will allow for its global extension, but which on the other will also prepare our subsequent qualitative analysis. For this purpose, the dependence of the obtained estimates on the averages $\overline{u}_{0}$ and $\overline{v}_{0}$ as well as on their sum, as appearing in (1.8), will carefully be traced throughout this section.

3.1 A bound for $w_{x}$ in $L^{q}$

In a first step we shall employ parabolic smoothing estimates to see that the mere mass conservation properties in (2.3) entail $L^{q}$ bounds for the chemotactic gradient acting in the first equation from (1.3). In not requiring any restriction other than that $q$ be finite, we here already make essential use of our assumption on the spatial framework to be one-dimensional.

Lemma 3.1

There exists $\alpha>0$ such that for all $M>0$ and any $q>1$ one can find $K(M,q)>0$ with the property that whenever $u_{0},v_{0}$ and $w_{0}$ satisfy (1.4) as well as (1.8), there exists $C=C(u_{0},v_{0},w_{0})>0$ such that

[TABLE]

Proof.  Relying on known regularization features of the Neumann heat semigroup $(e^{\sigma\Delta})_{\sigma\geq 0}$ on $\Omega$ with $\Delta:=(\cdot)_{xx}$ ([21]), let us fix $c_{1}(q)>0$ and $c_{2}(q)>0$ such that for all $t>0$,

[TABLE]

and

[TABLE]

Then representing $w$ according to

[TABLE]

we can combine (3.2) with (3.3) to estimate

[TABLE]

Here using (2.3) along with (1.8) and Lemma 2.2, we find that

[TABLE]

because \int_{0}^{t}e^{-\mu(t-s)}\Big{(}1+(t-s)^{-1+\frac{1}{2q}}\Big{)}ds=\int_{0}^{t}e^{-\mu s}(1+s^{-1+\frac{1}{2q}})ds\leq\int_{0}^{\infty}e^{-\mu s}(1+s^{-1+\frac{1}{2q}})ds\leq\frac{2}{\mu}+2q and \int_{0}^{t}e^{-\mu t}\Big{(}1+(t-s)^{-1+\frac{1}{2q}}\Big{)}ds=te^{-\mu t}+2qt^{\frac{1}{2q}}e^{-\mu t}, whence moreover estimating $te^{-\mu t}\leq\frac{2}{\mu}e^{-\frac{\mu}{2}t}$ as well as $2qt^{\frac{1}{2q}}e^{-\mu t}\leq\frac{2}{\mu}e^{-\frac{\mu}{2}t}$ for $t>0$, from (3.4) we infer that

[TABLE]

which is precisely of the claimed form with suitably chosen $K(M,q)>0$ and $C=C(u_{0},v_{0},w_{0})>0$, and with $\alpha:=\frac{\mu}{2}$. $\Box$

3.2 Estimating $u$ in $L^{\infty}$ and $u_{x}$ in a spatio-temporal $L^{2}$ norm

Through a standard testing procedure performed using the first equation in (1.3), when applied to $q:=2$ the latter has a first consequence on regularity of $u$ as well as its gradient.

Lemma 3.2

There exists $\alpha>0$ with the property that given any $M>0$ one can choose $K(M)>0$ such that if (1.4) and (1.8) hold, there exists $C=C(u_{0},v_{0},w_{0})>0$ such that

[TABLE]

and

[TABLE]

where $\tau:=\min\{1,\frac{1}{2}T_{max}\}$.

Proof.  Let us first apply Lemma 3.1 to $q:=2$ to fix positive constants $\alpha$ and $k_{1}(M)$ such that whenever (1.4) and (1.8) hold, one can find $c_{1}=c_{1}(u_{0},v_{0},w_{0})>0$ such that

[TABLE]

where without loss of generality we may assume that $k_{1}(M)\geq 1$ and $\alpha<\frac{1}{6}$. Apart from that, by means of the Gagliardo-Nirenberg inequality and Young’s inequality we can choose $c_{2}>0$, $c_{3}>0$ and $c_{4}>0$ such that

[TABLE]

that

[TABLE]

and that

[TABLE]

Now assuming (1.4) and (1.8), we test the first equation in (1.3) by $u$ to see using the Cauchy-Schwarz inequality as well as (3.7), (3.8), (3.9) and (2.3), that

[TABLE]

where by (3.10),

[TABLE]

and where once more by Young’s inequality,

[TABLE]

for all $t\in(0,T_{max})$. Since $k_{1}(M)\geq 1$ and thus

[TABLE]

from (3.11) we altogether obtain that

[TABLE]

with

[TABLE]

and

[TABLE]

Using that $6\alpha<1$, we may invoke Lemma 6.1 to firstly conclude from (3.12) that

[TABLE]

and that thus (3.5) holds with evident choices of the constants therein. After that, by direct integration of (3.12) we see that since $\tau\leq 1$,

[TABLE]

which shows that (3.13) also entails (3.6). $\Box$ By again going back to Lemma 3.1, in light of the outcome of Lemma 3.2 we can now once more employ heat semigroup estimates to actually improve the topological setting in (3.5) so as to involve the respective $L^{\infty}$ norm.

Lemma 3.3

There exists $\alpha>0$ such that if $M>0$, then one can fix $K(M)>0$ such that under the assumptions (1.4) and (1.8) it is possible to find $C=C(u_{0},v_{0},w_{0})>0$ such that

[TABLE]

Proof.  We begin by employing Lemma 3.2 and Lemma 3.1 to take $\alpha_{1}\in(0,1)$ and $\alpha_{2}\in(0,1)$ such that given $M>0$ we can find $k_{1}(M)>0$ and $k_{2}(M)>0$ with the property that if (1.4) and (1.8) hold, then with some $c_{i}=c_{i}(u_{0},v_{0},w_{0})>0$, $i\in\{1,2\}$, we have

[TABLE]

and

[TABLE]

and hence, by the Hölder inequality,

[TABLE]

with $\alpha:=\min\{\alpha_{1},\alpha_{2}\}$ and $c_{3}=c_{3}(u_{0},v_{0},w_{0}):=c_{1}k_{2}(M)+c_{2}k_{1}(M)\overline{u}_{0}+c_{1}c_{2}$. Next, parabolic smoothing estimates ([21], [8]) provide $c_{4}>0$ and $c_{5}>0$ such that for all $t>0$,

[TABLE]

and

[TABLE]

so that henceforth assuming (1.4) and (1.8) for some $M>0$, and rewriting the first equation in (1.3) in the form

[TABLE]

by means of an associated variation-of-constants representation we may estimate

[TABLE]

Here by the maximum principle and the fact that $\alpha\leq 1$,

[TABLE]

whereas (2.3) ensures that

[TABLE]

with $c_{6}:=\int_{0}^{\infty}e^{-\sigma}(1+\sigma^{-\frac{1}{2}})d\sigma<\infty$. Moreover, thanks to (3.15) we have

[TABLE]

where

[TABLE]

and where

[TABLE]

with $c_{8}:=\int_{0}^{\infty}e^{-(1-\alpha)\sigma}(1+\sigma^{-\frac{7}{8}})d\sigma$ being finite thanks to our restriction that $\alpha<1$.

Inserting (3.17)-(3.19) into (3.16) thus shows that for all $t\in(0,T_{max})$,

[TABLE]

and therefore establishes (3.14) upon the observation that $c_{4},c_{5},c_{6}$ and $c_{7}$ do not depend on our particular choice of $u_{0},v_{0}$ and $w_{0}$. $\Box$

3.3 Space-time $L^{2}$ bounds for $v$ and for $w_{xx}$

Now unlike in the analysis of (1.2), for globally extending our solution the bounds obtained in Lemma 3.3 and Lemma 3.1 seem yet insufficient: In view of the second equation in (1.3) it seems that for the detection of appropriate estimates for the second solution component, further information on the respectively relevant cross-diffusive gradient $u_{x}$ seems in order. To prepare our derivation thereof in the next section, let us here provide some preliminary bounds on $v$, $v_{x}$ and $w_{xx}$ useful for that purpose.

We begin with a basic space-time integrability feature of $(\ln(v+1))_{x}$ which by another straightforward testing procedure can be seen to be quite a direct consequence of our present knowledge on $u_{x}$ from Lemma 3.2.

Lemma 3.4

One can find $\alpha>0$ with the property that to any $M>0$ there corresponds some $K(M)>0$ such that if (1.4) and (1.8) hold, then there exists $C=C(u_{0},v_{0},w_{0})>0$ fulfilling

[TABLE]

where $\tau:=\min\{1,\frac{1}{2}T_{max}\}$.

Proof.  We multiply the second equation in (1.3) by $\frac{1}{v+1}$ and integrate by parts to see that due to Young’ inequality,

[TABLE]

As $0\leq\ln(\xi+1)\leq\xi$ for all $\xi\geq 0$, further integration shows that thanks to (2.3),

[TABLE]

so that (3.20) becomes a consequence of Lemma 3.2. $\Box$ Again thanks to the fact that the considered setting is one-dimensional, a simple interpolation argument shows that the above entails a space time bound on $v$ itself, rather than on the quantity $\ln(v+1)$ addressed in Lemma 3.4.

Lemma 3.5

There exists $\alpha>0$ such that whenever $M>0$, one can pick $K(M)>0$ such that if (1.4) and (1.8) hold, then with some $C=C(u_{0},v_{0},w_{0})>0$ we have

[TABLE]

where again $\tau:=\min\{1,\frac{1}{2}T_{max}\}$.

Proof.  According to the one-dimensional Gagliardo-Nirenberg inequality, we can fix $c_{1}>0$ such that

[TABLE]

which when applied to $\sqrt{v(\cdot,t)+1}$, $t\in(0,T_{max})$, shows that since $\|\sqrt{v+1}\|_{L^{2}(\Omega)}^{2}=\int_{\Omega}v_{0}+|\Omega|$ for all $t\in(0,T_{max})$ by (2.3),

[TABLE]

for all $t\in(0,T_{max})$. Once more in view of (2.3), using the Cauchy-Schwarz inequality we see that herein

[TABLE]

whence altogether, after a time integration,

[TABLE]

due to the fact that $\tau\leq 1$. The claimed statement thus readily results from Lemma 3.4. $\Box$ Thus having at hand spatio-temporal integral estimates for $v$ and, through e.g. Lemma 3.3, also for $u$, we have collected sufficient regularity information on all the source terms in the third equation from (1.3), when considered as a semilinear heat equation, so as to obtain the following as the outcome of a further standard testing process.

Lemma 3.6

One can find $\alpha>0$ in such a way that for each $M>0$ there exists $K(M)>0$ such that assuming (1.4) and (1.8) entails that with some $C=C(u_{0},v_{0},w_{0})>0$,

[TABLE]

where once more $\tau:=\min\{1,\frac{1}{2}T_{max}\}$.

Proof.  By means of Lemma 3.2 and Lemma 3.5, we can find $\alpha_{1}>0$ and $\alpha_{2}>0$ such that given any $M>0$ one can pick $k_{1}(M)>0$ and $k_{2}(M)>0$ such that whenever (1.4) and (1.8) hold, for the corresponding solution of (1.3) we have

[TABLE]

and

[TABLE]

with some $c_{i}=c_{i}(u_{0},v_{0},w_{0})>0$, $i\in\{1,2\}$, and $\tau=\min\{1,\frac{1}{2}T_{max}\}$.

Now indeed assuming (1.4) and (1.8), we use $w_{xx}$ as a test function for the third equation in (1.3) to see that by Young’s inequality,

[TABLE]

for all $t\in(0,T_{max})$ and hence

[TABLE]

Here aiming at an application of Lemma 6.2, we pick any $\alpha>0$ such that $\alpha<2\mu$ and $\alpha\leq\min\{\alpha_{1},\alpha_{2}\}$, and combine (3.23) and (3.24) with the outcome of Lemma 2.2 and Young’s inequality to estimate

[TABLE]

for all $t\in(0,T_{max}-\tau)$, where we have used that $\tau\leq 1$ and that hence $\int_{t}^{t+\tau}e^{-\beta s}ds\leq e^{-\beta t}$ for all $t>0$ and any $\beta>0$. Since $\alpha\leq\min\{\alpha_{1},\alpha_{2},2\mu\}$, this readily implies that

[TABLE]

with k_{3}(M):=\frac{4\lambda^{2}r^{2}}{d{\color[rgb]{1,0,0}\mu^{2}}}(k_{1}(M)+k_{2}(M)) and

[TABLE]

Upon employing Lemma 6.2, we thus obtain that (3.25) firstly entails the inequality

[TABLE]

if we let

[TABLE]

Thereupon, directly integrating (3.25) shows that again due to (3.26),

[TABLE]

which yields (3.22) upon again recalling that $\tau\leq 1$. $\Box$

3.4 Estimating $u_{x}$ in $L^{2}$

Thanks to Lemma 3.6, we now have appropriate information on the coefficient functions $a(x,t):=-\chi_{1}w_{x}$ and $b(x,t):=-\chi_{1}w_{xx}$ in the identity $u_{t}=u_{xx}+a(x,t)u_{x}+b(x,t)u$ to see that again due to a variational argument, $u_{x}$ indeed enjoys the following integrability features which go substantially beyond those obtained in Lemma 3.2.

Lemma 3.7

There exists $\alpha>0$ such that for arbitrary $M>0$ it is possible to choose $K(M)>0$ in such a way that whenever (1.4) and (1.8) are satisfied, there exists $C=C(u_{0},v_{0},w_{0})>0$ such that writing $\tau:=\min\{1,\frac{1}{2}T_{max}\}$ we have

[TABLE]

and

[TABLE]

Proof.  As a consequence of Lemma 3.3 and Lemma 3.6, we may pick $\alpha_{1}>0$ and $\alpha_{2}>0$ such that for all $M>0$ we can find $k_{1}(M)>0$ and $k_{2}(M)>0$ with the property that under the assumptions (1.4) and (1.8) one may fix $c_{1}=c_{1}(u_{0},v_{0},w_{0})>0$ and $c_{2}=c_{2}(u_{0},v_{0},w_{0})>0$ fulfilling

[TABLE]

and

[TABLE]

Apart from that, we combine the Gagliardo-Nirenberg inequality with Young’s inequality to obtain $c_{3}>0$ and $c_{4}>0$ such that

[TABLE]

and that

[TABLE]

Now assuming (1.4) and (1.8) to be valid for some $M>0$, we integrate by parts in the first equation from (1.3) and use the Cauchy-Schwarz inequality along with (3.31), Young’s inequality and (3.32) to see that for all $t\in(0,T_{max})$,

[TABLE]

because of (2.3). In view of the hypothesis (1.8), this shows that abbreviating $c_{5}:=\frac{(c_{3}+2)^{2}\chi_{1}^{2}}{2}$ we have

[TABLE]

where thanks to (3.29) and (3.30), fixing any $\alpha\in(0,1)$ such that $\alpha\leq\min\{\alpha_{1},\alpha_{2}\}$ we can estimate

[TABLE]

with $k_{3}(M):=c_{5}k_{1}(M)k_{2}(M)+c_{4}M^{2}$ and $c_{6}\equiv c_{6}(u_{0},v_{0},w_{0}):=c_{2}c_{5}k_{1}(M)+\frac{c_{1}c_{5}k_{2}(M)}{\tau}+c_{1}c_{2}c_{5}$. As a consequence of Lemma 6.2 and other restriction that $\alpha<1$, from (3.33) we thus infer that writing c_{7}\equiv c_{7}(u_{0},v_{0},w_{0}):=\frac{e^{\alpha}}{\tau}\cdot\Big{\{}\int_{\Omega}u_{0x}^{2}+c_{6}+\frac{c_{6}}{1-\alpha}+\frac{k_{3}(M)}{\tau}\Big{\}}+c_{6}e^{\alpha} we have

[TABLE]

and that hence, by integration of (3.33) and again using (3.34),

[TABLE]

Since $\tau\leq 1$, the claimed properties directly result from (3.35) and (3.36). $\Box$

3.5 An $L^{2}$ bound for $v_{x}$

We can thereby gradually improve our knowledge on the second solution component, firstly addressing $v$ itself in the course of a further testing procedure:

Lemma 3.8

There exists $\alpha>0$ such that for all $M>0$ one can fix $K(M)>0$ having the property that whenever (1.4) and (1.8) hold, with some $C=C(u_{0},v_{0},w_{0})>0$ we have

[TABLE]

Proof.  On the basis of Lemma 3.7, it is possible to pick $\alpha_{1}\in(0,1)$ in such a way that given $M>0$ we can choose $k_{1}(M)>0$ which is such that if (1.4) and (1.8) hold,

[TABLE]

with some $c_{1}=c_{1}(u_{0},v_{0},w_{0})>0$. Once more relying on the Gagliardo-Nirenberg inequality and Young’s inequality, we furthermore fix $c_{2}>0$, $c_{3}>0$ and $c_{4}>0$ such that

[TABLE]

that

[TABLE]

and that

[TABLE]

Then supposing that (1.4) and (1.8) to be satisfied, we use the second equation in (1.3) to see that due to the Cauchy-Schwarz inequality and (2.3), applications of (3.39), (3.41), (3.40) and Young’s inequality show that for all $t\in(0,T_{max})$,

[TABLE]

Since here, by using Young’s inequality and relying on the fact that $\tau\leq 1$, we can estimate

[TABLE]

with k_{2}(M):=\Big{\{}(c_{4}+9c_{2}^{2}\chi_{2}^{2})k_{1}(M)+9c_{2}^{2}\chi_{2}^{2}+c_{3}\Big{\}}\cdot|\Omega|^{4} and c_{5}\equiv c_{5}(u_{0},v_{0},w_{0}):=c_{1}\cdot(9c_{2}^{2}\chi_{2}^{2}+{\color[rgb]{1,0,0}c_{4}})\|v_{0}\|_{L^{1}(\Omega)}^{4}, from (3.42) we thus infer that

[TABLE]

Through Lemma 6.1, applicable here since $\alpha_{1}<1$, this entails that

[TABLE]

and hence completes the proof. $\Box$ Yet concentrating on $v$ itself, we next resort to a semigroup-based argument once more to turn the above into an esimate involving the norm in $L^{\infty}(\Omega)$.

Lemma 3.9

One can find $\alpha>0$ in such a manner that for each $M>0$ there exists $K(M)>0$ such that if (1.4) and (1.8) are satisfied, then

[TABLE]

with some $C=C(u_{0},v_{0},w_{0})>0$.

Proof.  A verification of this can be achieved in a way quite similar to that in Lemma 3.3: By the Hölder inequality as well as Lemma 3.8 and Lemma 3.7, we see that with some $\alpha\in(0,1)$, given any $M>0$ we can find $k_{1}(M)>0$ such that if (1.4) and (1.8) hold, there exists $c_{1}=c_{1}(u_{0},v_{0},w_{0})>0$ fulfilling

[TABLE]

Henceforth assuming (1.4) and (1.8), we combine this with known regularization features of the Neumann heat semigroup and (2.3) to see that with some positive constants $c_{2}$ and $c_{3}$ independent of $u_{0},v_{0}$ and $w_{0}$ we have

[TABLE]

which readily yields (3.43) due to the inequalities $\alpha<1$ and $\tau\leq 1$. $\Box$ As we are now in a position quite identical to that encountered immediately before Lemma 3.7, we can repeat the argument thereof to finally derive the following gradient estimate for the crucial second solution component.

Lemma 3.10

There exists $\alpha>0$ such that for all $M>0$ one can choose $K(M)>0$ with the property that if (1.4) and (1.8) hold, the with some $C=C(u_{0},v_{0},w_{0})>0$, we have

[TABLE]

Proof.  The claimed inequality can be derived by means of an essentially verbatim copy of the argument from Lemma 3.7, instead of referring to Lemma 3.3 and Lemma 3.6 now relying on Lemma 3.9 and (3.28); we may therefore refrain from giving details here. $\Box$

4 Global existence and boundedness. Proof of Theorem 1.1

Now asserting global extensibility of our solution actually reduces to a mere collection of our previously obtained estimates, where at this stage neither any knowledge on the precise dependence thereof on $M$ or on $\overline{u}_{0}$ and $\overline{v}_{0}$ is needed, nor do we rely on the exponentially decaying contributions to the above inequalities.

Lemma 4.1

For all $u_{0},v_{0}$ and $w_{0}$ fulfilling (1.4), we have $T_{max}=\infty$, and furthermore we can find $C=C(u_{0},v_{0},w_{0})>0$ such that (1.6) holds.

Proof.  In view of the extensibility criterion (2.2) from Lemma 2.1, for any fixed $(u_{0},v_{0},w_{0})$ satisfying (1.4) we may apply Lemma 3.7, Lemma 3.10 and Lemma 3.1 to $M:=\int_{\Omega}u_{0}+\int_{\Omega}v_{0}$ and $q:=2$ and thereby readily obtain that indeed $T_{max}$ cannot be finite, and that hence moreover (1.6) is a consequence of (3.27), (3.44) and (3.1). $\Box$ In other words, we thereby already have derived our main result on global classical solvability in (1.3):

Proof  of Theorem 1.1.  We only need to combine Lemma 4.1 with Lemma 2.1. $\Box$

5 Convergence for small values of $\min\{\overline{u}_{0},\overline{v}_{0}\}$. Proof of Theorem 1.2

Next, in contrast to our development of the above existence statement, our investigation of the large time behavior in (1.3), as forming the objective of this section, will considerably benefit from the more detailed information provided by our estimates from Section 3.

5.1 Identifying a conditional energy functional

The following lemma basically only collects the essence of what will be needed from Section 3 for our subsequent qualitative analysis.

Lemma 5.1

Let $M>0$. Then there exists $K(M)>0$ with the property that if (1.4) and (1.8) hold, then one can find $t_{0}=t_{0}(u_{0},v_{0},w_{0})\geq 0$ such that

[TABLE]

and

[TABLE]

as well as

[TABLE]

Proof.  From Lemma 3.3, Lemma 3.9 and Lemma 2.2 we infer the existence of $\alpha>0$ such that whenever $M>0$, one can find $k_{1}(M)>0$ such that if (1.4) and (1.8) hold, we have

[TABLE]

and

[TABLE]

as well as

[TABLE]

where we note that in light of the fact that $T_{max}=\infty$ we now know that the number $\tau$ in Lemma 3.9 actually satisfies $\tau=1$. For $t_{0}:=\frac{1}{\alpha}\cdot\ln_{+}\frac{1}{k_{1}(M)\cdot\min\{\overline{u}_{0},\overline{v}_{0},1\}}$ and with $K(M):=2k_{1}(M)$, the claimed inequalities now directly result from (5.4)-(5.6). $\Box$ Now a key toward our proof of stabilization can be found in the following observation on a genuine energy-type structure in (1.3) when restricted to trajectories corresponding to initial data compatible with (1.8) and (1.9). The presence of such conditional energy functionals, interpretable as a rigorous mathematical manifestation of superlinear dependence on the unknown in the crucial nonlinearities, has been used in several studies on asymptotic behavior in related chemotaxis problems in the recent few years (see e.g. [20], [6], [28], [23], [29] or also [26] for an incomplete collection); in comparison to most of these, the seemingly most unique feature of the present situation consists in that here it is possible to relax the smallness condition appearing therein in such a substantial manner that in its remaining part it merely reduces to a smallness assumption essentially equivalent to (1.8)-(1.9):

Lemma 5.2

Let $M>0$. Then there exists $\delta(M)>0$ such that if $u_{0},v_{0}$ and $w_{0}$ are such that if beyond (1.4) and (1.8) we have

[TABLE]

then it is possible to find $b=b(u_{0},v_{0},w_{0})>0$ and $t_{0}=t_{0}(u_{0},v_{0},w_{0})>0$ with the property that

[TABLE]

and

[TABLE]

satisfy

[TABLE]

Proof.  Given $M>0$, we first apply Lemma 5.1 to fix $k_{1}(M)>0$ with the property that for any choice of $(u_{0},v_{0},w_{0})$ complying with (1.4) and (1.8) we can find $t_{0}(u_{0},v_{0},w_{0})\geq 0$ such that for all $t>t_{0}$,

[TABLE]

and we thereupon claim that the intended conclusion holds if we let

[TABLE]

To see this, we fix any $(u_{0},v_{0},w_{0})$ fulfilling (1.4) and (1.8) as well as (5.7), and abbreviating $t_{0}:=t_{0}(u_{0},v_{0},w_{0})$, $L_{u}:=k_{1}(M)\overline{u}_{0},L_{v}:=k_{1}(M)\overline{v}_{0}$ and $L_{w}:=k_{1}(M)$ we infer from (5.12) that it is possible to pick $b=b(u_{0},v_{0},w_{0})>0$ such that

[TABLE]

We then let ${\mathcal{F}}$ and ${\mathcal{D}}$ be as accordingly defined through (5.8) and (5.9), and in order to verify (5.10) we integrate by parts in (1.3) and use (2.3) to compute

[TABLE]

and

[TABLE]

as well as

[TABLE]

for $t>0$. Here once more integrating by parts we see that

[TABLE]

which by the Cauchy-Schwarz inequality firstly entails that

[TABLE]

and hence

[TABLE]

and which, as a consequence, secondly shows that thus in (5.16) we can estimate

[TABLE]

Upon combining (5.14)-(5.16) and neglecting two further well-signed summands, we therefore obtain that

[TABLE]

Here by Young’s inequality and (5.11),

[TABLE]

because thanks to the right inequality in (5.13) we know that

[TABLE]

Likewise, Young’s inequality together with (5.11) moreover shows that

[TABLE]

since by the left restriction in (5.13),

[TABLE]

It thus remains to insert (5.18) and (5.19) into (5.17) to end up with (5.10). $\Box$

5.2 Exponential convergence. Proof of Theorem 1.2

A first and rather immediate consequence of (5.10) when combined with well-known inequalities of logarithmic Sobolev and Csiszár-Kullback type yields convergence already at algebraic rates, but yet with respect to spatial $L^{1}$ norms only.

Lemma 5.3

Given $M>0$, let $\delta(M)>0$ be as in Lemma 5.2, and suppose that (1.4), (1.8) and (5.7) hold. Then there exist $C=C(u_{0},v_{0},w_{0})>0$ and $\alpha=\alpha(u_{0},v_{0},w_{0})>0$ such that

[TABLE]

Proof.  Given $(u_{0},v_{0},w_{0})$ such that (1.4), (1.8) and (5.7) hold, we take $b=b(u_{0},v_{0},w_{0})>0$ and $t_{0}=t_{0}(u_{0},v_{0},w_{0})>0$ as provided by Lemma 5.2, and recall that according to a logarithmic Sobolev inequality ([10], [17]) and (2.3) there exists $c_{1}>0$ such that

[TABLE]

Writing $c_{2}:=\max\{c_{1},\frac{2}{\mu}\}$, for ${\mathcal{F}}$ and ${\mathcal{D}}$ as in (5.8) and (5.9) we thus obtain that

[TABLE]

so that (5.10) implies the autonomous ODI

[TABLE]

Upon integration, this entails that

[TABLE]

and thereby establishes (5.20) with suitably large $C>0$ and $\alpha:=\frac{1}{2c_{2}}$, because according to a Csiszár-Kullback inequality ([7], [2]) and (2.3) we can find $c_{3}>0$ fulfilling

[TABLE]

and

[TABLE]

for all $t>0$. $\Box$ Thanks to the temporally uniform $H^{1}$ bounds for both $u$ and $v$ known from Theorem 1.1, a straightforward interpolation finally asserts exponential convergence also with respect to $L^{\infty}$ norms.

Lemma 5.4

Under the assumptions of Lemma 5.3, one can find $C=C(u_{0},v_{0},w_{0})>0$ and $\alpha=\alpha(u_{0},v_{0},w_{0})>0$ such that

[TABLE]

and

[TABLE]

Proof.  By means of a Gagliardo-Nirenberg interpolation, we can find $c_{1}>0$ such that

[TABLE]

whereas Theorem 1.1 warrants the existence of $c_{2}>0$ such that

[TABLE]

In view of (2.3), we therefore obtain that for all $t>0$,

[TABLE]

which due to Lemma 5.3 implies both (5.21) and (5.22). $\Box$ On the basis of a final testing procedure independent from the above, again combined with an interpolation argument of the above flavor, we can lastly derive exponential and spatially uniform stabilization also in the third solution component.

Lemma 5.5

Suppose that the hypotheses of Lemma 5.3 are satisfied. Then there exist $C=C(u_{0},v_{0},w_{0})>0$ and $\alpha=\alpha(u_{0},v_{0},w_{0})>0$ such that with $w_{\star}\geq 0$ given by (1.11) we have

[TABLE]

Proof.  Using that by definition of $w_{\star}$ we have $\{\lambda(\overline{u}_{0}+\overline{v}_{0})+\mu\}w_{\star}=r$, upon testing the third equation in (1.3) by $w-w_{\star}$ we obtain that for all $t>0$,

[TABLE]

with $c_{1}:=\lambda(\overline{u}_{0}+\overline{v}_{0})+\mu>0$. Here since $\|w(\cdot,t)\|_{L^{\infty}(\Omega)}\leq c_{2}:=\frac{r}{\mu}+\|w_{0}\|_{L^{\infty}(\Omega)}$ for all $t>0$ by Lemma 2.2, Young’s inequality shows that

[TABLE]

and similarly estimating the rightmost summand in (5.24) we altogether infer that

[TABLE]

if we let $c_{3}:=\frac{2c_{2}^{2}\lambda^{2}}{c_{1}}$. As Lemma 5.4 provides $c_{4}>0$ and $\alpha_{1}\in(0,c_{1})$ fulfilling

[TABLE]

through e.g. Lemma 6.1 we readily conclude from (5.25) that

[TABLE]

with $c_{5}:=\int_{\Omega}(w_{0}+w_{\star})^{2}+\frac{c_{4}}{c_{1}-\alpha_{1}}$.

Now since Lemma 4.1 in conjunction with Lemma 3.1 asserts the existence of $c_{6}>0$ such that $\|w(\cdot,t)-w_{\star}\|_{W^{1,2}(\Omega)}\leq c_{6}$ for all $t>0$, and hence by the Gagliardo-Nirenberg inequality we can find $c_{7}>0$ such that

[TABLE]

this implied that

[TABLE]

and hence entails (5.23). $\Box$ Our main statements on temporal asymptotics thereby become almost evident:

Proof  of Theorem 1.2.  For $M>0$ we take $\delta(M)>0$ as provided by Lemma 5.2 and let $\varepsilon(M)>0$ be small enough such that $\varepsilon(M)\leq\frac{\delta(M)}{M^{2}}$ and $\varepsilon(M)\leq\sqrt{\frac{\delta}{M)}{M}}$. It can then readily be verified that assuming (1.8) together with (1.9) entails (5.7), so that for completing the proof it is sufficient to collect the statements from Lemma 5.4 and Lemma 5.5. $\Box$

6 Appendix: Two statements on ODE comparison

Let us finally state two elementary results of quite straightforward ODE comparison arguments, in view of our above applications with particular focus on the respective dependence on the parameters appearing therein. We begin with a simple observation that has been used in the proofs of Lemma 3.2, Lemma 3.8 and Lemma 5.5.

Lemma 6.1

Let $\kappa>0,a\geq 0,b\geq 0$ and $\alpha>0$ be such that $\alpha<\kappa$, and suppose that $y\in C^{0}([0,T))\cap C^{1}((0,T))$ is a nonnegative function satisfying

[TABLE]

with some $T\in(0,\infty]$. Then

[TABLE]

Proof.  We let $c_{1}:=y(0)+\frac{a}{\kappa-\alpha}$ and $\overline{y}(t):=c_{1}e^{-\alpha t}+\frac{b}{\kappa}$ for $t\geq 0$. Then clearly $\overline{y}(0)>y(0)$, and since moreover $\overline{y}^{\prime}(t)+\kappa\overline{y}(t)-ae^{-\alpha t}-b=\{(\kappa-\alpha)c_{1}-a\}e^{-\alpha t}=(\kappa-\alpha)y(0)e^{-\alpha t}\geq 0$ for all $t>0$ thanks to the nonnegativity of $y$ and our assumption that $\alpha<\kappa$, the inequality in (6.2) results from a comparison argument. $\Box$ Our second statement in this direction, as used in crucial places in Lemma 3.6 and Lemma 3.7, merely imposes some hypothesis on temporal averages of the respective force term, and therefore requires a slightly more subtle argument:

Lemma 6.2

Let $\kappa>0,a\geq 0,b\geq 0$ and $\alpha\in(0,\kappa)$, and assume that with some $T\in(0,\infty]$ and $\tau\in(0,T)$ such that $\tau\leq 1$, the nonnegative functions $y\in C^{0}([0,T))\cap C^{1}((0,T))$ and $f\in L^{1}_{loc}([0,T))$ are such that

[TABLE]

and

[TABLE]

Then

[TABLE]

Proof.  Letting $z(t):=\int_{t}^{t+\tau}y(s)ds$ for $t\in[0,T-\tau)$, by two integrations of (6.3), from (6.4) we obtain that firstly

[TABLE]

and that secondly,

[TABLE]

whence in particular

[TABLE]

As thus

[TABLE]

due to our assumption that $\tau\leq 1$, by applying Lemma 6.1 to (6.6) we see that

[TABLE]

which implies that

[TABLE]

In view of the definition of $z(t-\tau)$, this especially means that whenever $t\in(\tau,T)$, we can find $t_{0}(t)\in(t-\tau,t)$ fulfilling

[TABLE]

Again employing (6.7), and moreover using the that $e^{-\alpha t_{0}}<e^{-\alpha(t-\tau)}$ due to the inclusion $t_{0}\in(t-\tau,t)$, we conclude that for any such $t$,

[TABLE]

from which (6.5) immediately follows once more due to the inequality $\tau\leq 1$.

If $t\in(0,\tau]$, however, we infer (6.5) directly from (6.8), because clearly

[TABLE]

for all $t\in(0,\tau]$. $\Box$

Acknowledgment. Y. Tao acknowledges support of the National Natural Science Foundation of China (No. 11571070). M. Winkler was supported by the Deutsche Forschungsgemeinschaft within the project Analysis of chemotactic cross-diffusion in complex frameworks.