Introduction

In this paper, we study the large time behavior of bounded solutions to the Cauchy problem for the following system of partial differential equations in ${\mathbb{R}}^n$, $n\ge 2$:

$$\frac{\partial u}{\partial t}=\nabla\cdot(\nabla u-u\nabla v), x\in{\mathbb{R}}^n, \ t>0,$$ (1.1)

$$\frac{\partial v}{\partial t}=\Delta v-v+u, x\in{\mathbb{R}}^n, \ t>0,$$ (1.2)

$$u(x,0)=u_0,\quad v(x,0)=v_0, x\in{\mathbb{R}}^n.$$ (1.3)

The system (1.1), (1.2) is a mathematical model describing chemotaxis, that is, the directed movement of an organism in response to gradients of a chemical attractant(see [2,10,17]). $u(x,t)$ corresponds to the population of the organism at place $x$ and time $t$, and $v(x,t)$ to the concentration of the chemical.

Throughout this paper, it is always assumed that

$$u_0, \ v_0, \ \partial_jv_0 \in L^1({\mathbb{R}}^n)\cap{\cal B}({\mathbb{R}}^n) \ (1\le j\le n).$$

Here, we use the notation $\partial_j=\partial/\partial x_j$ for simplicity, and ${\cal B}({\mathbb{R}}^n)$ is the Banach space of all bounded and uniformly continuous functions on ${\mathbb{R}}^n$ with the usual supremum norm. We write (1.1)-(1.3) in the form of the integral equation:

$$u(t)= e^{t\Delta}u_0 - \int_0^t \nabla\cdot e^{(t-s)\Delta}(u\nabla v)(s)ds,$$ (1.4)

$$v(t)= e^{-t}e^{t\Delta}v_0 + \int_0^t e^{-t+s}e^{(t-s)\Delta}u(s)ds,$$ (1.5)

where

$$(e^{t\Delta}f)(x)=\int_{{\mathbb{R}}^n} G(x-y,t)f(y)dy$$

and $G(x,t)$ is the heat kernel

$$G(x,t)=\frac{1}{(4\pi t)^{\frac{n}{2}}}\exp (-\frac{\vert x\vert^2 }{4t}).$$

A function $(u(x,t), v(x,t))$ on ${\mathbb{R}}^n\times [0, T] \ (0<T<\infty)$ is said to be a solution of (1.1)-(1.3) on ${\mathbb{R}}^n\times [0, T]$ if

$$u, \ v, \ \partial_jv \in C([0, T] : L^1({\mathbb{R}}^n))\cap C([0, T] : {\cal B}({\mathbb{R}}^n)) \quad (1\le j\le n)$$

and $(u, v)$ satisfies (1.4), (1.5) on $[0, T]$. It is also said that $(u, v)$ is a solution of (1.1)-(1.3) on ${\mathbb{R}}^n \times [0,\infty)$ if $(u, v)$ is a solution of (1.1)-(1.3) on ${\mathbb{R}}^n\times [0, T]$ for every $0<T<\infty$. Using standard regularity arguments for the heat equation(see Chapter IV of [11]), we see that $(u, v)$ is a classical solution of (1.1)-(1.3), which satisfies

$$u, v \in C((0, T): W^{2,p}({\mathbb{R}}^n))\cap C^1((0, T): L^p({\mathbb{R}}^n))$$   for every $$\ 1<p<\infty.$$

It is shown in [14] that every bounded solution of (1.1)-(1.3) on ${\mathbb{R}}^2\times [0, \infty)$ decays to zero as $t\to\infty$ and behaves like the heat kernel. We give the large time behavior for higher dimensional case in the following theorem. In what follows, $\Vert\cdot\Vert _p$ represents the usual $L^p$-norm.

Theorem 1.1   Let $(u, v)$ be a solution of (1.1)-(1.3) on ${\mathbb{R}}^n \times [0,\infty)$ and $n\ge 2$. Suppose that

$$\sup_{t>0} \bigl( \Vert u(t)\Vert _p + \Vert v(t)\Vert _p \bigr)<\infty p=1, \infty.$$ (1.6)

Then, for every $1<p\le \infty$,

$$\sup_{t>0} (1+t)^{n(1-1/p)/2}\bigl( \Vert u(t)\Vert _p + \Vert v(t)\Vert _p \bigr)< \infty,$$

$$\lim_{t\to\infty} t^{n(1-1/p)/2}\Big\Vert u(t)- \int_{{\mathbb{R}}^n}u_0\,dy G(t)\Big\Vert _p=0,$$

$$\lim_{t\to\infty} t^{n(1-1/p)/2}\Big\Vert v(t)- \int_{{\mathbb{R}}^n}u_0\,dy G(t)\Big\Vert _p=0.$$

For nonnegative solutions of (1.1)-(1.3), we need (1.6) only for $p=\infty$, because integrating (1.4) and (1.5) on ${\mathbb{R}}^n$ respectively, we observe that

$$\Vert u(t)\Vert _1= \Vert u_0\Vert _1,\quad \Vert v(t)\Vert _1= e^{-t}\Vert v_0\Vert _1 + (1- e^{-t})\Vert u_0\Vert _1.$$

Concerning the existence of bounded solutions, it is possible for a nonnegative solution of (1.1)-(1.3) in ${\mathbb{R}}^n (n\ge 2)$ to blow up in finite time(see [2]). For a simplified version of (1.1)-(1.2) replaced (1.2) by $0=\Delta v -v + u$ in ${\mathbb{R}}^2$, it is shown that the nonnegative solution exists globally in time under the condition $\int_{{\mathbb{R}}^2} u_0\,dx<8\pi$(see [3]), and that the finite-time blowup of nonnegative solutions may occur under the condition $\int_{{\mathbb{R}}^2} u_0\,dx>8\pi$(see [3,12]). It is also shown in [12] that the finite-time blowup of nonnegative solutions in ${\mathbb{R}}^n (n\ge 3)$ may occur even if $\int_{{\mathbb{R}}^n} u_0\,dx$ is small. For blowup problems to (1.1), (1.2) in a bounded domain, we refer to [1,4,5,7,8,9,13,15,16,18,19] and the references therein.

The following theorem gives the existence of bounded solutions to (1.1)-(1.3) in ${\mathbb{R}}^n, n\ge 2$, when $\Vert u_0\Vert _1, \Vert\nabla v_0\Vert _1, \Vert\nabla v_0\Vert _\infty$ are small but $\Vert u_0\Vert _\infty$ is not necessarily small. The uniqueness of solutions is also given in the theorem. The nonnegativity of solutions is not assumed.

Theorem 1.2   (i) The uniqueness of solutions to (1.1)-(1.3) holds.

(ii) Let $n\ge 2$. For any given $K>0$ let $(u_0, v_0)$ be an initial function satisfying

$$\Vert u_0\Vert _\infty \le K.$$

Then there exists a small $\delta>0$, depending on $K$, such that (1.1)-(1.3) admits a solution on ${\mathbb{R}}^n \times [0,\infty)$ satisfying (1.6), provided that

$$\Vert u_0\Vert _1 \le\delta, \ \Vert\nabla v_0\Vert _1 \le\delta, \ \Vert\nabla v_0\Vert _\infty \le\delta.$$