Efendiev, Messoud A.

Scheel, A.

Kyoto University Research Information Repository

TitleUpper and lower bounds for the Hausdorff dimension of theattractor for reaction-diffusion equations in $\mathbb{R}^n$(Nonlinear Evolution Equations and Applications)Author(s)Efendiev, Messoud A.; Scheel, A.Citation数理解析研究所講究録 (1999), 1105: 81-90Issue Date1999-07URL http://hdl.handle.net/2433/63230RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityUpper and lower bounds for the Hausdorffdimension of the attractor forreaction-diffusion equations in $\mathrm{I}\mathrm{R}^{n}$$\mathrm{M}.\mathrm{A}$. Efendiev, A. ScheelAbstractWe consider a reaction-diffusion equation (RDE) of the form$\{$$\partial_{t}u=\nu\Delta u-f(u)-\lambda_{0}u-g$$u|_{t=0}=u_{0}(x)$ , $x\in 1\mathrm{R}^{n}$with $u=u(t, x)\in$ IR scalar and $x\in \mathrm{I}\mathrm{R}^{n}$ . Here $g=g(x),$ $u_{0}(x),$ $\lambda_{0}>0$ and the nonlinearity$f\in C^{1}(1\mathrm{R}\cross \mathrm{I}\mathrm{R}^{n})$ are supposed to be given. Under appropriate conditions on $f,g$ and $u_{0}$ weprove both existence of a global attractor for equation (1) in the whole space $\mathrm{I}\mathrm{R}^{n}$ and lower andupper bounds (in terms of physical parameter $\nu$) for the Hausdorff dimension of the attractor.Note that the case of $\mathrm{I}\mathrm{R}^{n}$ has specific difficulties. First, the semigroup generated by the aboveequation is not compact. The second difficulty is connected with the fact that the Laplaceoperator has continuous spectrum. We overcome these difficulties by systematic use of weightedSobolev spaces.Introduction and ResultsOne of the most important objects used to. describe large-time dynamics of infinite-dimensional dynamical systems are global attractors and their dimensions. The global attractorof an evolution equation is the maximal, “compact”, finite dimensional (in the sense of Haus-dorff), invariant set in the phase space which attracts bounded sets. Upper and lower bounds for1105 1999 81-90 81the Hausdorff dimension of the attractor (in terms of some physical parameters) imply that eveninfinite-dimensional dynamical systems possess an asymptotic behavior determined by a finitenumber 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 onthe attractors of the Navier-Stokes system determines turbulent behavior of fluids. Hence bothexistence 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(1) $\{$$\partial_{t}u=\nu\Delta u-f(u)-\lambda_{0}u-g$$u|_{t=0}=u_{0}(x)$ , $x\in 1\mathrm{R}^{n}$where $u=u(t, x)\in \mathrm{I}\mathrm{R}$ is scalar and $x\in \mathrm{I}\mathrm{R}^{n}$ . Here $g=g(x),$ $u_{0}(x),$ $\lambda_{0}>0$ and the nonlinearity$f\in C^{1}$ (IR $\cross \mathrm{I}\mathrm{R}^{n}$ ) are supposed to be given. We $\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}\theta$ conditions on $f,u_{0}$ , and $g$ later. Our maingoal 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 restrictionsare 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$822. $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 $0\leq\alpha,$ $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}$ .Let $H_{l,\gamma}(\mathrm{I}\mathrm{R}^{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)$ satisfyCondition 1. Then there exists a unique solution $u(t, x)$ of $RDE(\mathit{1})$ , which belongs to$L_{2}([0,T], H_{2,\gamma})\cap L_{\infty}([0, T], H_{1,\gamma})$.Moreover the mappings $S_{t}$ : $u_{0}(x)\mapsto u(t, x)$ form a semigroup which possesses a global attractor$A\subset H_{0,\gamma}$ .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 cases1. $g=0$ and $f$ saiisfies Condition 1.2. $g\in H_{0,\gamma}(\mathrm{I}\mathrm{R}^{n})$ and $f’(u)\geq-\lambda_{0}$ .83Proof: First consider the case 1), that is,(2) $\{$$\partial_{t}u=\nu\Delta u-f(u)-\lambda_{0}\mathrm{u}$$u|_{t=0}=u_{0}(x)$ .The condition $f(u)\cdot u\geq 0$ implies that $u\equiv 0$ is an equilibrium. On the other hand, multiplyingboth sides of (2) by $\varphi\cdot u$, where $\varphi(x)=(1+|\epsilon x|^{2})^{\gamma}$ and integrating with respect to $x$ , we obtainthat any solution of (2) tends to $u\equiv 0$ as $tarrow\infty$ in $H_{0,\gamma}$ . Hence $A=\{0\}$ . Let us next considercase 2). We start with the special case of 2), where $f\equiv 0$ and $g\in H_{0,\gamma}$ . Then (1) takes the form$\{$$\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) $\nu\Delta u_{*}-\lambda_{0}u_{*}=g$As $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 $\mathit{0}$btain$\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.84Let $S$ be the set of all equilibria of (1). We prove that $S$ consists of a single point. Indeed,let $u_{1}$ and $u_{2}$ 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 $x$ , we obtain, using$u_{i}\in H0_{\gamma},,$ $\gamma>0$ , that(7) $\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$ forreaction-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 thatCorollary 1 If $f$ satisfies Condition 1, but $f\not\in M$ , then the attractor $A$ consists of asingleton. In particular, $\dim A=0$ .85Next we formulate our main result.Theorem 2 Let $u_{0}\in H_{1,\gamma}(\mathrm{I}\mathrm{R}^{n}),$ $\gamma>0$ and $f\in \mathcal{M}$ satisfy Condition 1. Then there exists$L_{0}>0$ such that for all $L\geq L_{0}$ there is (an explicitly given) $g_{L}(x)\in H_{0,\gamma}$ , and the globalattractor $A_{L}$ of $RDE(\mathit{1})$ 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 $C_{1}$ and $C_{2}$ depend on $\lambda_{0}$ but not on $\nu$ and $L$ .Proof: We start with $n=1$ . Let $L$ be any given positive number and $z(x)\in C_{0}^{\infty}(1\mathrm{R})$ suchthat $z(x)=\xi \mathrm{i}\mathrm{f}-L\leq x\leq L$ and $z(x)=0$ if $|x|\geq L+1$ . Here $\xi\in$ IR is chosen such that$f’(\xi)<-\lambda_{0}$ . Define$g_{L}(x):=\nu\triangle z(x)-f(z(x))-\lambda_{0^{Z}}(x)$ ,Note that $g_{L}(x)\in H_{0,\gamma}(\mathrm{I}\mathrm{R}^{n}),$ $\gamma>0$ . Consider now(8) $\{$$\partial_{t}u=\nu\Delta u-f(u)-\lambda_{0}u-g_{L}(x)$$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, sincethe essential spectrum of the linearization, determined by $\nu\triangle-f’(z(x))-\lambda_{0}$ , is strictly left ofthe 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 to86study 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 that1. $\langle A’(z(x))w, h\rangle=\langle w, A’(z(x))h\rangle$ for $\mathrm{a}\mathrm{U}w,$ $h\in H_{2,\gamma}$2. $\langle A’(z(x))w, w\rangle\leq\beta\langle w, w\rangle$, for some $\beta>0$ and for all $w\in H_{2,\gamma}$ .Here by $\langle\cdot, \cdot\rangle$ we denote the scalar product in $H_{0,\gamma}(\mathrm{I}\mathrm{R}^{n})$ . Let $R_{k}$ be a $k$-dimensional subspace of$H_{1,\gamma}(\mathrm{I}\mathrm{R}^{n})$ , 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$ , and $k$ is chosen such that $\lambda_{k}<-f’(\xi)-\lambda_{0}$ . An explicitcomputation shows $\lambda_{j}=(_{L}^{\pi}-i)^{2}$ and $w_{j}(x)=\sin_{L}^{\pi}\lrcorner x,$ $|x|\leq L$ . We continue $w_{j}(x)$ on the wholereal line by $w_{j}(x)\equiv 0$ in $|x|\geq L$ . Thus, in order to fulfill $\lambda_{k}<-f’(\xi)-\lambda_{0}$ , we can $\mathrm{c}\mathrm{h}o$ose(10) $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$ .87Since $(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=$$=-( \frac{\pi^{2}j^{2}}{L^{2}}+f’(\xi)+\lambda_{0})\int_{-L}^{L}w_{j}^{2}(x)dx>0$for all $j=1,$ $\cdots,$ $k$ according to (10). Due to Courant’s Minimax Principle, we have$\dim M_{+}(z)\geq k=[L(-f’(\xi)-\lambda_{0})^{1/2}\cdot\pi^{-1}]$ .Hence in case of $n=1$ we obtain$\dim A\geq LC_{1}(\lambda_{0})$for sufficiently large L. $\mathrm{R}\mathrm{e}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{n}\mathrm{g}-\Delta \mathrm{b}\mathrm{y}-\nu\Delta$ in (9) yields(11) $\dim A\geq L\nu^{-1/2}C_{1}(\lambda_{0})$ .It remains to show that(12) $\dim A\leq C_{2}(\lambda_{0})L\cdot\nu^{-1/2}$for sufficiently large $L$ . A proof of (12) is based on the estimate (see [3])$\dim A\leq C(\lambda_{0})\nu^{-1/2}||g||_{L_{2}(\mathbb{R})}^{2}$where $g(x)$ is the forcing function in (1). If we set $g=g_{L}(x)$ , then $||g||_{L_{2}(\mathbb{R})}^{2}=O(L)$ and(13) $\dim A\leq C_{2}(\lambda_{0})\nu^{-1/2}\cdot L$for sufficently large $L$ . This proves Theorem 2 in the case $n=1$ .88In the case $n>1$ , the proof of Theorem 2 is based on the estimate of eigenvalues $\mathrm{o}\mathrm{f}-\Delta$ ina ball. Consider, instead of (9), the Dirchlet problem in a ball $B_{L}=\{x\in 1\mathrm{R}^{n}|||x||\leq L\}$$\{$$-\Delta w_{j}=\lambda_{j}w_{j}$$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 $\mathit{0}$ur gratitude to A. Babin and B. Fiedler forinspiring discussions and helpful remarks. We are also grateful to J. F. Toland and S. Zelick foruseful comments during their visit at the Ree University of Berlin.References1. F. Abergel. Existence and Finte dimensionality of the Global Attractor for EvolutionEquations 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 un-bounded domain. Proc. Royal Soc. Edinburgh 116A (1990), 221-243, 1990.894. S. Merino. On the existence of the compact global attractor for semilinear reaction-diffusionequation in $1\mathrm{R}^{n}$ . J. Diff. Eq 132 (1996), 87-106.5. D. Daners, S. Merino. Gradient-like Parabolic Semiflows on $BUC(\mathrm{I}\mathrm{R}^{N})$ . Preprint Univer-sity of Sydney 1997. Rep. 97-20.90

Upper and lower bounds for the Hausdorff dimension of the attractor for reaction-diffusion equations in $\mathbb{R}^n$ (Nonlinear Evolution Equations and Applications)

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Upper and lower bounds for the Hausdorff dimension of the attractor for reaction-diffusion equations in $\mathbb{R}^n$ (Nonlinear Evolution Equations and Applications)

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Upper and lower bounds for the Hausdorff dimension of the attractor for reaction-diffusion equations in Rn\mathbb{R}^nRn (Nonlinear Evolution Equations and Applications)

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Upper and lower bounds for the Hausdorff dimension of the attractor for reaction-diffusion equations in $\mathbb{R}^n$ (Nonlinear Evolution Equations and Applications)