The effect of noise on the chafee-infante equation: a nonlinear case study

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 135, Number 2, February 2007, Pages 373–382 S 0002-9939(06)08593-5 Article electronically published on August 1, 2006 THE EFFECT OF NOISE ON THE CHAFEE-INFANTE EQUATION: A NONLINEAR CASE STUDY TOM´ AS CARABALLO, HANS CRAUEL, JOS´ E A. LANGA, AND JAMES C. ROBINSON (Communicated by Walter Craig) Abstract. We investigate the effect of perturbing the Chafee-Infante scalar reaction diffusion equation, ut−∆u=βu−u3, by noise. While a single multiplicative Itˆo noise of sufficient intensity will stabilise the origin, its Stratonovich counterpart leaves the dimension of the attractor essentially unchanged. We then show that a collection of multiplicative Stratonovich terms can make the origin exponentially stable, while an additive noise of sufficient richness reduces the random attractor to a single point. 1. Introduction In this paper we investigate the effect of noise on the Chafee-Infante scalar reaction diffusion equation (1.1) ut−∆u=βu −u3for x∈D, with u|∂D =0, where Dis a smooth bounded domain in Rm. We show that the effect of the noise is highly dependent on the precise way in which it is included in the model. We choose the Chafee-Infante equation since it is the canonical example of those infinite-dimensional gradient systems in which the structure of the global attractor can be fully described, and we recall this theory very briefly in Section 2. We then consider the effect of adding a single multiplicative noise: in Section 3 we show that an Itˆonoise+σudWtof sufficient intensity will stabilise the origin, while in Section 4 we show that a Stratonovich noise +σu ◦dWtleaves the complexity of the system, as measured by the dimension of its attractor, essentially unchanged whatever the value of σ. The order-preserving property of the deterministic model is retained by its stochastic counterparts, and this is the key to the two new results presented here: in Section 5 we adapt the linear result of Caraballo & Robinson [10] to show that the zero solution can be made exponentially stable by the addition of a number of multiplicative noise terms (in the Stratonovich sense) and in Section 6 we give a Received by the editors August 12, 2005. 2000 Mathematics Subject Classification. Primary 37L55, 35K57; Secondary 60H15. Key words and phrases. Chafee-Infante equation, stochastic stabilisation, random attractor, random equilibrium, one-point attractor, attractor collapse. The first and third authors were supported by Ministerio de Ciencia y Tecnolog´ıa (Spain) and FEDER (European Community), project BFM2002-03068. The fourth author is a Royal Society University Research Fellow, and would like to thank the Society for all their support. c 2006 American Mathematical Society Reverts to public domain 28 years from publication 373 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 374 TOM´ AS CARABALLO ET AL. simple proof that the addition of a rich enough additive noise reduces the random attractor to a single random point (cf. Chueshov & Scheutzow [11]). 2. The deterministic Chafee-Infante equation The Chafee-Infante equation is probably the best understood deterministic parabolic PDE, and one of the few for which we have a full understanding of the structure of its attractor (see Hale [23] and Henry [24], for example). Existence and uniqueness results for the deterministic equation are proved in Marion [28] and Robinson [30]: given an initial condition u0∈L2(D)thereexists a unique weak solution u(t;u0) such that for any T>0 u(t;u0)∈L2(0,T;H1 0(D)) ∩L4((0,T)×D)∩C([0,T]; L2(D)), which we can use to define a semigroup S(t)onL2(D), via S(t)u0=u(t;u0). It is shown in both of the above references (and Temam [32]) that the equation also enjoys the existence of a global attractor A, that is, a compact invariant set that attracts the orbits of all bounded sets, i.e., S(t)A=Afor all t∈Rand dist(S(t)B,A)→0ast→∞, where Bis any bounded subset of L2(D) and dist(A, B) is the Hausdorff semidistance between Aand B, dist(A, B)=sup a∈A inf b∈B|a−b|. Since the equation defines a gradient system, the attractor consists of the collection of all the stationary points and their unstable manifolds (see [23], [30], or [32]). For the case of a one-dimensional domain it is known that as βpasses through each successive eigenvalue λnof the Laplacian on [0,L] another direction becomes unstable and two new stationary points appear in a pitchfork bifurcation. It follows that for λn<β<λ n+1 the dimension of the attractor is n.Sinceλn∼n2, it follows that d(A)∼β1/2. (Here we use the upper box-counting or ‘fractal’ dimension.) In a smooth m-dimensional domain Da similar result is valid for the dimension of the attractor, namely that d(A)∼βm/2; see Temam [32] for the upper bound and Babin & Vishik [5] for the lower bound. 3. Linear stabilisation via a multiplicative Itˆ onoise The effect of perturbing the equation by a multiplicative noise in the Itˆosense, (3.1) du=[∆u+βu −u3]dt+σudWt, is to introduce a somewhat ‘artificial’ stabilising effect. This is most easily seen by considering the equivalent Stratonovich equation (3.2) du=[∆u+(β−1 2σ2)u−u3]dt+σu ◦dWt, where the linear stability has clearly been enhanced by the new −1 2σ2uterm. Making the change of variable v=ue−σWtproduces a family of nonautonomous equations parametrised by ω: vt=∆v+(β−1 2σ2)v−e2σWt(ω)v3. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THE EFFECT OF NOISE ON THE CHAFEE-INFANTE EQUATION 375 Taking the inner product of this equation with vin L2(D)weobtain 1 2 d dt|v|2=−|Dv|2+(β−1 2σ2)|v|2−e2σWtv4 L4 ≤−λ1|v|2+(β−1 2σ2)|v|2. It follows that we have |v(t)|2≤|v(0)|2e−γt,where γ=2(λ1−β)+σ2, and hence |u(t)|2≤|u(0)|2e2σWte−γt. Since P-almost surely (P-a.s.) lim t→∞ |Wt| t=0, it follows that for σ2>2(β−λ1) the origin becomes (pathwise) exponentially stable. We note here that the addition of nonlinear multiplicative Itˆonoisecanalso have a stabilising effect (see e.g. Caraballo, Liu, & Mao [9]). However, here and in Section 5 our aim is to demonstrate that stabilisation can be obtained using a very simple multiplicative noise, namely a linear one. 4. A single multiplicative Stratonovich noise The above observation that an Itˆo noise produces a somewhat artificial stabilising effect led us to study the effect of a single multiplicative Stratonovich term on the Chafee-Infante equation (4.1) du=[∆u+βu −u3]dt+σu ◦dWt (Wtis a two-sided one-dimensional Brownian motion) using the framework of random dynamical systems, which we now recall. 4.1. Random dynamical systems and random attractors. In the interest of brevity we only state the definitions here (for more background on random dynamical systems see Arnold [2]). Let (Ω,F,P) be a probability space and {ϑt:Ω→Ω,t ∈R}a family of measure preserving transformations such that (t, ω)→ ϑtωis measurable, ϑ0=id, and ϑt+s=ϑtϑsfor all s, t ∈R. The flow ϑttogether with the corresponding probability space (Ω,F,P,(ϑt)t∈R) is called a (measurable) dynamical system. A continuous random dynamical system (RDS) on a Polish space (X, d)with Borel σ-algebra Bover ϑon (Ω,F,P) is a measurable map ϕ:R+×Ω×X→X (t, ω, x)→ ϕ(t, ω)x such that P-a.s. i) ϕ(0,ω)=id onX, ii) ϕ(t+s, ω)=ϕ(t, ϑsω)◦ϕ(s, ω) for all t, s ∈R+(cocycle property), and iii) ϕ(t, ω):X→Xis continuous. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 376 TOM´ AS CARABALLO ET AL. Arandom attractor for an RDS ϕis a random set ω→A(ω) such that (i) Ais a random compact set, that is, P-a. s., A(ω) is compact, and for all x∈Xthe map ω→ distx, A(ω)is measurable with respect to F, (ii) P-a. s. ϕ(t, ω)A(ω)=A(ϑtω) for all t≥0, and (iii) for every D⊂Xbounded, P-a. s., lim t→∞ distϕ(t, ϑ−tω)D, A(ω)=0. 4.2. Our equation as a random dynamical system, and its attractor. In order to cast our equation as a random dynamical system we let (Ω,F,P)denote the probability space generating the two-sided Wiener process Wt, and define a shift ϑton Ω by Wt(ϑsω)=Wt+s(ω)−Ws(ω), the additional subtracted term ensuring that W·(ϑsω) is still a Brownian motion. Existence and uniqueness results due to Pardoux [29] guarantee that for each initial condition u0∈L2(D)andT>0,there exists a unique strong solution u(t;u0)∈L2(Ω ×(0,T); H1 0(D)) ∩L4(Ω ×(0,T)×D)∩L2(Ω; C(0,T;L2(D))). We can use the resulting solution to define a random dynamical system ϕon the phase space L2(D) by setting ϕ(t, ω)u0=u(t;u0). The computations presented in Caraballo et al. [7], which are relatively standard, show that (4.1) has a random attractor. We then used the stochastic extension of the deterministic theory (due to Debussche [20]) to show that the Hausdorff dimension of the attractor (which is P-a.s. constant; see Crauel & Flandoli [16]) is bounded by dwhen β<1 d d  j=1 λj, where λjare the eigenvalues of the Laplacian arranged in increasing order. It follows from recent work of Langa & Robinson [26] that these calculations also provide the same upper bound on the upper box-counting dimension of the attractor (which is also constant P-a.s. [26]). Since λn∼n2/m, this implies that d(A)≤cβm/2. In a subsequent paper [8] we adapted a proof of Da Prato & Debussche [17] to show that provided m≤5 (a technical condition) the unstable manifold near the origin has dimension at least dwhen β>λ d. This leads to a lower bound on the dimension of the same order as the upper bound, and hence shows that the dimension of the random attractor is of the same order as its deterministic counterpart, namely d(A(ω)) ∼βm/2. In this sense the addition of a single multiplicative Stratonovich noise has no effect on the asymptotic complexity of the dynamics. 5. Exponential stability of the zero solution via a number of Stratonovich multiplicative noise terms Generalising the finite-dimensional result of Arnold, Crauel, & Wihstutz [4], Caraballo and Robinson [10] recently showed that a linear PDE ut=Au can be stabilised by a collection of multiplicative Stratonovich noisy terms if and only if the trace of the linear partial differential operator Ais negative. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THE EFFECT OF NOISE ON THE CHAFEE-INFANTE EQUATION 377 In this section we show that the nonlinear equation (1.1) can be stabilised by adding a similar collection of noisy terms: (5.1) du=[∆u+βu −u3]dt+ d  i=1 Biu◦dWi t. Essentially we show that solutions of (5.1) can be bounded using appropriate positive solutions of the linear equation (5.2) du=[∆u+βu]dt+ d  i=1 Biu◦dWi t. Since (5.2) can be stabilised via a suitable choice of {Bi}, so can (5.1). The proof makes essential and continual use of the order-preserving properties of (5.1). To begin with, we recall the stabilisation result for linear equations given in [10]: we suppose that Ais a linear operator with a sequence of eigenvalues µjwith corresponding eigenfunctions ejthat form an orthonormal basis of a Hilbert space H. We also assume that the eigenvalues µjare bounded above, and order them so that µ1≥µ2≥.... Theorem 5.1. If the trace of Ais negative, then there exist bounded linear operators Bk:H→H,k=1,...,d, such that the zero solution of du=Audt+ d  j=1 Bku◦dWk is exponentially stable with probability one. The proof is simple: choose Nsuch that N j=1 µj<0, and consider the projection of ut=Auonto the two complementary subspaces Pand Qspanned by {ej}N j=1 and {ej}∞ j=N+1 respectively. The Qcomponents of the equation converge exponentially to zero, while the solutions of the finite-dimensional ODE ˙p=−Ap (where pdenotes the orthogonal projection of uonto P) can be stabilised [4] by adding a collection of noisy terms +Bkp◦dWk,wheretheBkare N×Nskewsymmetric matrices. These matrices correspond to linear operators Bk:H→H. In our case we will choose H=L2(D); we will also denote by −Athe linear operator in Hassociated to the Laplacian. We then take A=−A+βI,which clearly satisfies the conditions of Theorem 5.1, and let Nbe the smallest integer such that N j=1(β−λj)<0. It follows that there exist linear operators Bk:H→H such that the zero solution of (5.3) du=[−Au +βu]dt+ d  j=1 Bku◦dWk t is exponentially stable with probability one. We now show how to use this to deduce stabilisation of the nonlinear equation via the addition of the same noisy terms. Theorem 5.2. There exist bounded linear operators Bk:H→H, and independent real Wiener processes Wk t,k=1,...d, such that the zero solution of (5.4) du=(−Au +βu −u3)dt+ d  j=1 Bku◦dWk t is exponentially stable with probability one. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 378 TOM´ AS CARABALLO ET AL. Before giving the proof, we note that by considering the equivalent Itˆoformof (5.4) it follows from results of Pardoux [29] that for each u0∈L2(D)andT>0, there exists a unique strong solution satisfying u(t;u0)∈L2(Ω ×(0,T); H1 0(D)) ∩L2(Ω; C(0,T;L2(D))). Proof. The stabilisation of the zero solution follows once we show that solutions of (5.4) can be bounded by appropriate solutions of the linear equation (5.3). The key observation is that we can bound from above an arbitrary solution of (5.4) pointwise by a positive solution of the same equation (and from below by a negative one). Indeed, if u0(x)≤U0(x), then the order-preserving property of (5.4) (see Kotelenez [25] or Chueshov & Vuillermot [12]) guarantees that (5.5) u(t, x, ω;u0)≤u(t, x, ω;U0) for almost all (t, x, ω)∈[0,+∞)×[0,L]×Ω, where with the obvious notation u(t, x, ω;u0) denotes the solution of (5.4) with u(0,x)=u0(x) and noise ω. It follows that u(t, x, ω;−|u0|)≤u(t, x, ω;u0)≤u(t, x, ω;|u0|). Now, positive solutions of the nonlinear equation enjoy a comparison principle with those of the linear equation (see Chueshov & Vuillermot [12] again): if u0≥0, then u(t, x, ω;u0)≤uL(t, x, ω;u0), where uL(t, x, ω) is the solution of the corresponding linear stochastic PDE du=(−Au +βu)dt+ d  j=1 Bku◦dWk t. It follows that uL(t, x, ω;−|u0|)≤u(t, x, ω;u0)≤uL(t, x, ω;|u0|), which can be rewritten, since uLsolves a linear equation, as |u(t, x, ω;u0)|≤uL(t, x, ω;|u0|). Since solutions of (5.3) tend exponentially to zero with probability one, so do all solutions of (5.4).  6. Collapse of the random attractor produced by additive noise In this final section we show that the addition of a sufficiently rich additive white noise will reduce the random attractor of the equation to a single (random) point. Such behaviour was originally demonstrated for the one-dimensional ordinary differential equation dx=[αx −x3]dt+dWtwith α>0 by Crauel & Flandoli [15], and has recently been shown by Tearne [31] for a general gradient ODE of the form dx=−∇V(x)+dWt, where x∈Rm,Wtis an m-dimensional Brownian motion, and is sufficiently small (note that this is not in general an order-preserving system). Here we prove a similar result for the equation (6.1) du=[∆u+βu −u3]dt+√CdWt,x∈D=[0,L], License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THE EFFECT OF NOISE ON THE CHAFEE-INFANTE EQUATION 379 where Wt,t∈R,is a two-sided cylindrical Wiener process on H=L2(D)andCis a bounded linear operator with bounded inverse on H.Notethatherewehaveto restrict ourselves to a one-dimensional domain. Our argument, which we will make precise below, could be generalised to treat more abstract problems (cf. Chueshov & Scheutzow [11]) but the underlying idea is simple: Results of Arnold & Chueshov [3] on the structure of random attractors in order-preserving systems guarantee the existence of two random fixed points a and athat are contained in the attractor and are such that a(ω)≤u≤a(ω) for all u∈A(ω). Corresponding to these random fixed points there are invariant measures δa(ω)and δa(ω). Since the noise in (6.1) is sufficiently rich to guarantee that the equation has aunique invariant measure (e.g., Da Prato, Debussche, & Goldys [18]) it follows that the laws of a(ω)anda(ω) must coincide. It is only a small step from this, using the fact that a(ω)≤a(ω), to the deduction that a(ω)=a(ω)=a(ω), and hence that A(ω)={a(ω)}, i.e., the attractor is a single point. We now recall the formal existence and uniqueness results for (6.1), and give a rigorous proof that the random attractor is a point. We take (Ω,F,P) to be the probability space that generates the cylindrical Wiener process Wt, and define a shift ϑton Ω by Wt(ϑsω)=Wt+s(ω)−Ws(ω)as in Section 4. Under these assumptions, it is known (Da Prato and Zabczyk [19]) that for each u0∈L2(D)andT>0 there exists a unique solution u(t;u0)for(6.1),with u(t;u0)∈L2Ω×(0,T); H1 0(D)∩L4Ω×(0,T)×D∩L2Ω; C(0,T;L2(D)). It follows that the solutions of (6.1) generate a random dynamical system on L2(D) if we define ϕ(t, ω)u0=u(t;ω, u0), where u(t;ω,u0) is the solution of (6.1) with noise ωand initial condition u(0) = u0. Theorem 6.1. The random attractor for (6.1) consists of a single point, i.e., there exists a random variable a:Ω→Hwith ϕ(t, ω)a(ω)=a(ϑtω)for every t≥0P-a.s. such that A(ω)={a(ω)}. Proof. The existence of a random attractor A(ω)for(6.1)canbeprovedusing standard techniques (see, e. g., Allouba and Langa [1], Debussche [20] or Yuhong Li [27]). From Theorem 3.3 in Crauel [13] we know that ω→A(ω)ismeasurable with respect to the past F−,whichistheσ-algebra F−=σ{ω→ ϕ(s, ϑ−tω)x:x∈X, 0≤s≤t}. Theorem 5.8 in Chueshov and Vuillermot [12] guarantees that the random dynamical system associated with (6.1) is order-preserving, i.e., if u≤v,thenP-a. s., for every t≥0, ϕ(t, ω)u≤ϕ(t, ω)v. As a consequence (Theorem 2 in Arnold & Chueshov [3]) there exist equilibria aand asuch that, P-a.s., a(ω)≤u≤a(ω) for all u∈A(ω). We now wish to associate invariant measures with the two extremal equilibria. In order to do this we first show that they are measurable with respect to the past F−: it will then follow from Crauel [14] that for each equilibrium a(ω), ρ=E(δa(ω))is License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 380 TOM´ AS CARABALLO ET AL. aPt-invariant Markov measure, i.e., ρ∈Pr(H) (it is a probability measure on H) such that Ptρ=ρfor every t≥0,where (Pt)t≥0is the canonical Markov semigroup on Pr(H) associated with ϕ. Since A(ω) is a closed random set that is measurable with respect to the past F−, there exists a countable family {an}n∈Nof F−-measurable selections, such that A(ω)=cl{an:n∈N}a. s. We now define a sequence (bn)n∈N, which need not be selections of A(ω), by b1=a1,andbn+1 =max{an+1,b n}, or, more explicitly, bn+1(ω)=an+1(ω)ifan+1(ω)≥bn(ω), bn(ω)otherwise, for n≥1. Then every bnis F−-measurable, and the sequence is increasing. Furthermore we have ak≤bn≤aP-a. s. for all k≤n,andn∈N. We claim that bnconverges to aP-a. s. In fact, if bnwould be bounded away from awith positive probability, we would get that anis bounded away from awith positive probability, hence sup an< a with positive probability, which would contradict A=cl{an:n∈N}P-a. s., and a∈A.Consequently,a= lim bnis F−-measurable. A similar argument implies that ais F−-measurable as well. We therefore obtain two Pt-invariant Markov measures E(δa)andE(δa). These are just the laws of the equilibria, i.e., for any Borel set B∈B(H) E(δa)(B)=L(a)(B)=P(u∈B). Now, Da Prato et al. [18] (Section 6) showed that the Markov semigroup associated with (6.1) has a unique invariant measure,1which means that L(a)=L(a), andsoinparticularwemusthaveE(a)=E(a). Since P-a.s. a≤aand aand aare real functions defined on [0,L], we must have a=a,P-a.s. Setting a(ω)=a(ω)=a(ω), it follows that P-a.s. A(ω)={a(ω)}, i.e., the attractor consists of a single random point.  7. Conclusion Here we have aimed to draw attention to the very different effects that different types of noise can have on the asymptotic behaviour of deterministic systems. Of course, all the above analysis could be carried out for more general systems, but we believe that treating a simple canonical model helps to clarify the arguments. In particular, although elementary, it seems worthwhile to emphasise the somewhat artificial stabilisation effect produced by the multiplicative Itˆonoise+σu dWt discussed in Section 3. 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