数理解析研究所講究録1105 巻 1999 年 81-90

81Upper and lower bounds for the Hausdorff
dimension of the attractor forreaction-diffusion equations in. Efendiev, A. Scheel$\mathrm{I}\mathrm{R}^{n}$$\mathrm{M}.\mathrm{A}$

Abstract
We consider a reaction-diffusion equation (RDE) of the form$\partial_{t}u=\nu\Delta u-f(u)-\lambda_{0}u-g$$\{$$u|_{t=0}=u_{0}(x)$with $u=u(t, x)\in$ IR scalar and$f\in C^{1}(1\mathrm{R}\cross \mathrm{I}\mathrm{R}^{n})$$x\in \mathrm{I}\mathrm{R}^{n}$,$x\in 1\mathrm{R}^{n}$. Here $g=g(x),$$u_{0}(x),$ $\lambda_{0}>0$

and the nonlinearity

are supposed to be given. Under appropriate conditions on $f,g$ andprove both existence of a global attractor for equation (1) in the whole space$\mathrm{I}\mathrm{R}^{n}$$u_{0}$

weand lower andupper bounds (in terms of physical parameter ) for the Hausdorff dimension of the attractor.
$\nu$

Note that the case of$\mathrm{I}\mathrm{R}^{n}$has specific difficulties. First, the semigroup generated by the above

equation is not compact. The second difficulty is connected with the fact that the Laplace

operator has continuous spectrum. We overcome these difficulties by systematic use of weighted

Sobolev spaces.

Introduction and Results
One of the most important objects used to. describe large-time dynamics of infinite-

dimensional dynamical systems are global attractors and their dimensions. The global attractor
of an evolution equation is the maximal, “compact”, finite dimensional (in the sense of Hausdorff), invariant set in the phase space which attracts bounded sets. Upper and lower bounds for

82

the Hausdorff dimension of the attractor (in terms of some physical parameters) imply that eveninfinite-dimensional dynamical systems possess an asymptotic behavior determined by a finite
number of degrees of freedom. For a precise definition of the attractor and concepts of dimensionsee [2]. According to a conjecture of Landau and Ruelle-Takens, the non-trivial dynamics on
the attractors of the Navier-Stokes system determines turbulent behavior of fluids. Hence both

existence of attractors and upper and lower bounds (in terms of physical parameters) on theirdimensions, are of great interest. Note that at present the existence of global attractors as wellas estimates on their Hausdorff dimensions were obtained for many equations of mathematicalphysics in bounded domains, see [2]. In this paper, we consider a reaction-diffusion equation(RDE) of the form$\partial_{t}u=\nu\Delta u-f(u)-\lambda_{0}u-g$

(1)

$\{$

$u|_{t=0}=u_{0}(x)$where$u=u(t, x)\in \mathrm{I}\mathrm{R}$$f\in C^{1}$(IR$\cross \mathrm{I}\mathrm{R}^{n}$is scalar and$x\in \mathrm{I}\mathrm{R}^{n}$,$x\in 1\mathrm{R}^{n}$. Here $g=g(x),$) are supposed to be given. We$\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}\theta$$u_{0}(x),$$\lambda_{0}>0$

and the nonlinearity

conditions on $f,u_{0}$ , and$g$later. Our main

goal is both, to prove existence of global attractors for RDE (1), and to obtain lower and upperbounds (in terms of the Reynolds number$\nu^{-1}$) for the Hausdorff dimensions of the attractors.

For a proof of existence of the attractor in this case we follow [3]. There the following restrictions

are imposed on the nonlinearity $f=f(u)$ .

.

$\dot{\mathrm{C}}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}1$

1. $f\in C^{1}$ and $f’(u)\geq-C$

83

2. $f(u)\cdot u\geq 0$ for all$u\in \mathrm{I}\mathrm{R}$3. $|f(u)|\leq|u|^{1+\alpha}(1+|u|^{p_{2}})$ ,
where
Let

$0\leq\alpha,$$H_{l,\gamma}(\mathrm{I}\mathrm{R}^{n})$$0\leq p_{2}$and$p_{2}+ \alpha\leq p_{0}=\min\{\frac{4}{n}, \frac{2}{n-2}\}$, if $n\geq 3$ . For $n\leq 2$ we can take$p_{0}= \frac{4}{n}$.

be the weighted Sobolev spaces with norms,$||u||_{l,\gamma}^{2}.= \sum_{|\alpha|\leq l}||\partial^{\alpha}u||_{0,\gamma}^{2}$

where$||u||_{0,\gamma}^{2}= \int_{\mathrm{R}^{n}}(1+|\epsilon x|^{2})^{\gamma}|u(x)|^{2}d_{X}$and$\epsilon>0$

is a small enough, but fixed number.

Theorem 1 ([3]) Let $\gamma>0,$$g\in H_{0,\gamma},$$u_{0}\in H_{1,\gamma}$

and let the nonlinearity $f=f(u)$ satisfy

Condition 1. Then there exists a unique solution $u(t, x)$ of$RDE(\mathit{1})$$L_{2}([0,T], H_{2,\gamma})\cap L_{\infty}([0, T], H_{1,\gamma})$Moreover the mappings$A\subset H_{0,\gamma}$$S_{t}$, which belongs to.

: $u_{0}(x)\mapsto u(t, x)$ form a semigroup which possesses a global attractor.

Let us consider$\mathrm{t}\mathrm{I}_{1}\mathrm{e}$following special cases of Theorem 1.

Proposition 1 The global attractor $A$ consists of only one point in the following cases
1. $g=0$ and

2.

$f$

saiisfies Condition 1.

$g\in H_{0,\gamma}(\mathrm{I}\mathrm{R}^{n})$and $f’(u)\geq-\lambda_{0}$ .

84

Proof: First consider the case 1), that is,$\partial_{t}u=\nu\Delta u-f(u)-\lambda_{0}\mathrm{u}$

(2)

$\{$

$u|_{t=0}=u_{0}(x)$The condition $f(u)\cdot u\geq 0$ implies thatboth sides of (2) by$\varphi\cdot u$, where$u\equiv 0$.

is an equilibrium. On the other hand, multiplying$\varphi(x)=(1+|\epsilon x|^{2})^{\gamma}$that any solution of (2) tends to$u\equiv 0$

as$tarrow\infty$

and integrating with respect to , we obtain$x$

incase 2). We start with the special case of 2), where$H_{0,\gamma}$$f\equiv 0$. Hence $A=\{0\}$ . Let us next consider
and$g\in H_{0,\gamma}$. Then (1) takes the form$\partial_{t}u=\nu\Delta u-\lambda_{0u-g}(x)$$\{$$u|_{t=0}=u_{0}(x)$Let$u_{*}(x)$be the unique solution of

(3)

As

.

$\nu\Delta u_{*}-\lambda_{0}u_{*}=g$$g\in H_{0,\gamma}$, we have$u_{*}\in H_{2,\gamma}(\mathrm{I}\mathrm{R}^{n})$. On the other hand, equation (3) can be rewritten as$\frac{\partial u_{*}}{\partial t}=\nu\triangle u_{*}-\lambda_{0}u_{*}-g$

and as a result we btain$\mathit{0}$$\frac{\partial(u-u_{*})}{\partial t}=\nu\Delta(u-u_{*})-\lambda_{0}(u-u_{*})$,

which in turn implies $A=\{u_{*}\}=\{(\nu\Delta-\lambda_{0}I)^{-1}g\}$ . We next proof that in the general case 2)

the attractor also consists of only one single point.

85

Let

let

$u_{1}$$S$

be the set of all equilibria of (1). We prove thatand$u_{2}$$S$

consists of a single point. Indeed,

be two distinct equilibria, that is,(4)$\nu\Delta u_{1}-f(u_{1})-\lambda_{0}u_{1}-g(x)=0$(5)$\nu\Delta u_{2}-f(u_{2})-\lambda_{0}u_{2}-g(x)=0$.

Subtracting (5) ffom (4) we obtain(6)$\nu\Delta(u_{1}-u_{2})-(f(u_{1})-f(u_{2}))-\lambda_{0}(u_{1}-u_{2})=0$.

Multiplying both sides of (6) by $(u_{1}-u_{2})$ and integrating with respect to , we obtain, using$x$$u_{i}\in H0_{\gamma},,$$\gamma>0$(7), that$\nu\int|\nabla(u_{1}-u_{2})|^{2}dx+\int(\Psi(u_{1})-\Psi(u_{2}))(u_{1}-u_{2})dx\geq 0$

where $\Psi(u):=f(u)-\lambda_{0}u$ . Note that $\Psi’(u)\geq 0$ . Therefore the integrand in (7) is positive and$u_{1}\equiv u_{2}$. Then the assertion of Proposition 1, Case 2 is an easy consequence of the gradientstructure of (1).

$\mathrm{N}$

Proposition lleads to the following natural question: How rich is the global attractor $A$ for

reaction-diffusion equation (1). Our main result gives a partial answer to this question. In orderto formulate it, we introduce a class of nonlinearities$\mathcal{M}$: we say$f\in \mathcal{M}$, iff there exists$\xi\in$IRsuch that $f’(\xi)<-\lambda_{0}$ .

Proposition 1 showed that

Corollary 1 If

$f$satisfies Condition 1, but $f\not\in M$ , then the attractor

singleton. In particular, $\dim A=0$ .

$A$

consists of a

86

Next we formulate our main result.

Theorem 2 Let$L_{0}>0$$u_{0}\in H_{1,\gamma}(\mathrm{I}\mathrm{R}^{n}),$$\gamma>0$and$f\in \mathcal{M}$satisfy Condition 1. Then there exists

such that for all $L\geq L_{0}$ there is (an explicitly given)

attractor$A_{L}$of$RDE(\mathit{1})$$C_{1}$

and$C_{2}$

depend on

Proof: We start with $n=1$ . Let$L$$\lambda_{0}$but not on

$\nu$

and

$L$

.

.

be any given positive number andthat $z(x)=\xi \mathrm{i}\mathrm{f}-L\leq x\leq L$ and $z(x)=0$ if $|x|\geq L+1$ . Here$f’(\xi)<-\lambda_{0}$$\xi\in$$z(x)\in C_{0}^{\infty}(1\mathrm{R})$such

IR is chosen such that. Define$g_{L}(x):=\nu\triangle z(x)-f(z(x))-\lambda_{0^{Z}}(x)$Note that, and the global

with $g=g_{L}(x)$ admits the following double-sided estimates$C_{1}\nu^{-n/2}L^{n}\leq\dim A_{L}\leq C_{2^{\mathcal{U}^{-n/2}}}L^{n}$

Here the constants$g_{L}(x)\in H_{0,\gamma}$$g_{L}(x)\in H_{0,\gamma}(\mathrm{I}\mathrm{R}^{n}),$ $\gamma>0$,. Consider now$\partial_{t}u=\nu\Delta u-f(u)-\lambda_{0}u-g_{L}(x)$

(8)

$\{$

$u|_{t=0}=u_{0}(x)$Obviously $u_{*}(x):=z(x)$ is an equilibrium for (8). Let $M_{+}(z)$ be the unstable manifold at thisequilibrium (see [2]). Note that the unstable manifold exists, and is finite dimensional, since

the essential spectrum of the linearization, determined by $\nu\triangle-f’(z(x))-\lambda_{0}$ , is strictly left of
the imaginary axis. Since $M_{+}(z)\subset A$ , a lower bound for $\dim M_{+}(z)$ yields a lower bound forthe dimension of $A$ as well. Therefore we have to estimate $\dim M_{+}(z)$ . To this end, we have to87

study the linearized equation at $z(x)$ , that is, the linearized operator,$A’(z(x))w=\nu\Delta w-f’(z(x))w-\lambda_{0}w$

Since$z\in C_{0}^{\infty}(\mathrm{I}\mathrm{R})$it is not difficult to see that

1. $\langle A’(z(x))w, h\rangle=\langle w, A’(z(x))h\rangle$ for2.

$\langle A’(z(x))w, w\rangle\leq\beta\langle w, w\rangle$

Here by$\langle\cdot, \cdot\rangle$$H_{1,\gamma}(\mathrm{I}\mathrm{R}^{n})$.

$\mathrm{a}\mathrm{U}w,$$h\in H_{2,\gamma}$, for some $\beta>0$ and for all

we denote the scalar product in$H_{0,\gamma}(\mathrm{I}\mathrm{R}^{n})$$w\in H_{2,\gamma}$. Let$R_{k}$.

be a -dimensional subspace of$k$, defined as$R_{k}:=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}[w_{1}(x), \cdots,w_{k}(x)]$where $w_{j}(x)$ are eigenvalues$\mathrm{o}\mathrm{f}-\Delta$, with Dirichlet boundary conditions, that is$\int-\Delta w_{j}=\lambda_{j}w_{j}$(9)$\downarrow w_{j}(-L)=w_{j}(L)=0$with$0<\lambda_{1}<\lambda_{2}\leq\cdots\leq\lambda_{k}\leq\cdots$

computation shows$\lambda_{j}=(_{L}^{\pi}-i)^{2}$

, and

and

$k$

is chosen such that $\lambda_{k}<-f’(\xi)-\lambda_{0}$ . An explicit$w_{j}(x)=\sin_{L}^{\pi}\lrcorner x,$$|x|\leq L$

real line by $w_{j}(x)\equiv 0$ in $|x|\geq L$ . Thus, in order to fulfill(10). We continue $w_{j}(x)$ on the whole$\lambda_{k}<-f’(\xi)-\lambda_{0}$$k=[L(-f’(\xi)-\lambda_{0}.)^{1/2}\cdot\pi^{-1}]$

where $[x]$ denotes the integer part of$x$. We next show that$\langle A’(z(x))w, w\rangle|_{w\in R_{k}}>0$.

,, we can$\mathrm{c}\mathrm{h}o$

ose

88

Since $(A’(z(x))w_{j}, w_{k})=0$ if $j\neq k$ , it is sufficient to che&(10) at $w=w_{j)}j=1,$$\cdots k$. Indeed,$\langle A’(z(x))w_{j},w_{j}\rangle=-\int_{-L}^{L}((w_{j}’(x))^{2}+(f’(\xi)+\lambda_{0})w_{j}^{2}(x))dx=$$=- \int_{-L}^{L}(\frac{\pi^{2}j^{2}}{L^{2}}w_{j}^{2}+(f’(\xi)+\lambda_{0})w_{j}^{2})dx=$$=-( \frac{\pi^{2}j^{2}}{L^{2}}+f’(\xi)+\lambda_{0})\int_{-L}^{L}w_{j}^{2}(x)dx>0$

for all $j=1,$$\cdots,$$k$according to (10). Due to Courant’s Minimax Principle, we have$\dim M_{+}(z)\geq k=[L(-f’(\xi)-\lambda_{0})^{1/2}\cdot\pi^{-1}]$.

Hence in case of $n=1$ we obtain$\dim A\geq LC_{1}(\lambda_{0})$for sufficiently large L.
(11)$\mathrm{R}\mathrm{e}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{n}\mathrm{g}-\Delta \mathrm{b}\mathrm{y}-\nu\Delta$in (9) yields$\dim A\geq L\nu^{-1/2}C_{1}(\lambda_{0})$.

It remains to show that(12)for sufficiently large$\dim A\leq C_{2}(\lambda_{0})L\cdot\nu^{-1/2}$$L$. A proof of (12) is based on the estimate (see [3])$\dim A\leq C(\lambda_{0})\nu^{-1/2}||g||_{L_{2}(\mathbb{R})}^{2}$

where $g(x)$ is the forcing function in (1). If we set $g=g_{L}(x)$ , then(13)

for sufficently large$\dim A\leq C_{2}(\lambda_{0})\nu^{-1/2}\cdot L$$L$. This proves Theorem 2 in the case $n=1$ .

$||g||_{L_{2}(\mathbb{R})}^{2}=O(L)$

and

89

In the case $n>1$ , the proof of Theorem 2 is based on the estimate of eigenvalues

a ball. Consider, instead of (9), the Dirchlet problem in a ball$\mathrm{o}\mathrm{f}-\Delta$in$B_{L}=\{x\in 1\mathrm{R}^{n}|||x||\leq L\}$$-\Delta w_{j}=\lambda_{j}w_{j}$$\{$$w_{j}|_{\partial B_{L}}=0$

It is$\mathrm{w}\mathrm{e}\mathrm{U}$-known that$\lambda_{j}\sim j^{\frac{2}{\mathfrak{n}}}$for $jarrow\infty$ , so that the estimates (12) and (13) take the form

$C_{1}(\lambda_{0})\nu^{-n/2}L^{n}\leq\dim A\leq C_{2}(\lambda_{0})\nu^{-n/2}L^{n}$for sufficiently large$L$.

Corollary 2 The dimension of the aitractor can be made arbitrarily large.

Acknowledgement. We would like to express ur gratitude to A. Babin and B. Fiedler for$\mathit{0}$inspiring discussions and helpful remarks. We are also grateful to J. F. Toland and S. Zelick for
useful comments during their visit at the Ree University of Berlin.

References

1. F. Abergel. Existence and Finte dimensionality of the Global Attractor for Evolution
Equations on Unbounded Domains. J. Diff. Eq, 83 (1990), 85-108.

2. A.V. Babin, M. Vishik. Attractors of Evolution Equations. Studies in Mathematics andits Applications, 25. North Holland 1992.

3. A.V. Babin, M. Vishik. Attractors of partial differential evolution equations in an unbounded domain. Proc. Royal Soc. Edinburgh 116A (1990), 221-243, 1990.

90

4. S. Merino. On the existence of the compact global attractor for semilinear reaction-diffusion
equation in$1\mathrm{R}^{n}$. J. Diff. Eq 132 (1996), 87-106.

5. D. Daners, S. Merino. Gradient-like Parabolic Semiflows onsity of Sydney 1997. Rep. 97-20.

$BUC(\mathrm{I}\mathrm{R}^{N})$. Preprint Univer-