A global solution for a reaction-diffusion equation on bounded domains

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Abstract

In the literature, it has been proved the existence of a pullback global attractor for reaction-diffusion equation on a bounded domain and under some conditions, a uniform bound on the dimension of its sections. Using those results and putting a bound on the diameter of the domain, we proved that the pullback global attractor consists only of one global solution. As an application to this result, a bounded perturbation of Chafee-Infante equation has been studied.

Abstract

In the literature, it has been proved the existence of a pullback global attractor for reaction-diffusion equation on a bounded domain and under some conditions, a uniform bound on the dimension of its sections. Using those results and putting a bound on the diameter of the domain, we proved that the pullback global attractor consists only of one global solution. As an application to this result, a bounded perturbation of Chafee-Infante equation has been studied.

1 Introduction

Pullback attractors have become a suitable tool for studying non-autonomous dissipative dynamical systems generated by evolution equations arising in physical phenomena. Pullback global attractor is a family of compact subsets of the phase space which is strictly invariant under the evolution process and attracts all bounded subsets in a pullback sense. For pullback exponential attractor, we have a positively invariant family of compact subsets which have a uniformly bounded fractal dimension and pullback attract all bounded subsets at an exponential rate. We recall the definitions from [4]:

Definition 1

The two parameter family {U(t, s)|, t, s ∈ ℝ, ts} of continuous operators from the metric space X to itself is called an evolution process in X if it satisfies the properties

U(t,s)oU(s,r)=U(t,r),tsrU(t,t)=Id,tRandT×X(t,s,x)U(t,s)xXiscontinuous,

Where 𝒯 = {(t, s) ∈ ℝ × ℝ|, ts}

Definition 2

The family of non-empty subsets {𝒜(t)|, t ∈ ℝ} of X is called a pullback global attractor for the process {U(t, s)|, ts} if 𝒜(t) is compact for all t, the family {𝒜(t)|, t ∈ ℝ} is strictly invariant, that is

U(t,s)A(s)=A(t)forallts,

it pullback attracts all bounded subsets of X, that is for every bounded DX and t ∈ ℝ

limsdistH(U(t,ts)D,A(t))=0

and the family is minimal within the families of closed subsets that pullback attract all bounded subsets of X.

Definition 3

We call the family 𝓜 = {𝓜(t)|, t ∈ ℝ} a pullback exponential attractor for the process {U(t, s)|, ts} in X if

  1. the subsets 𝓜(t) ⊂ X are non-empty and compact in X for all tX,

  2. the family is positively semi-invariant, that is

    U(t,s)M(s)M(t)forallts

  3. the fractal dimension in X of the sections 𝓜(t), t ∈ ℝ, is uniformly bounded and

  4. the family {𝓜(t)|, t ∈ ℝ} exponentially pullback attracts bounded subsets of X; that is, there exists a positive constant ω > 0 such that for every bounded subset DX and t ∈ ℝ

    limseωsdistH(U(t,ts)D,M(t))=0.

For some constructions of these attractors refer for example to [1, 4, 6, 7, 8, 9]. Reaction-diffusion equation, in this regard, has been studied as an example of dissipative systems.

We consider

utΔu+f(t,u)=h(t),u|Ω=0,u(s)=us,t>s.

where the forcing term h(t) is allowed to have an exponential growth in time.

When the phase space is L2(Ω), in [1], it was obtained a uniform bound on the dimensions of sections of the pullback global attractor under a Lipschitz condition on the nonlinear term f(t, u).

In [7], under some requirements for the phase space and the process defined on it, it was constructed a uniform bound for the dimensions of sections of the pullback exponential attractor and the pullback global attractor contained in it and as an application, it was applied to the reaction-diffusion equation in the phase space H01 (Ω).

We use the bound obtained in these works for proving that the dimension of sections is zero providing a bound on the diameter of the domain Ω which brings about a unique global solution as the pullback global attractor.

We, moreover, release the Lipschitz condition on f and replace it with a condition on fu which affords us to deal with such problems with a polynomial nonlinearity as Chafee-Infante equation and obtain its trivial dynamics under a bounded perturbation.

2 Statement of the problem

Let us consider the initial boundary value problem for the non-autonomous reaction-diffusion equation as follows:

utΔu+f(t,u)=h(t),u|Ω=0,u(s)=us,t>s.

where fC1(ℝ2, ℝ), h(⋅) ∈ Lloc2 (ℝ, L2(Ω)), Ω is a bounded open subset of ℝn with smooth boundary Ω, and there exists p ≥ 2, ci > 0, i = 1, …, 6, such that

c1|u|pc2f(t,u)uc3|u|p+c4,

fu(t,u)c5,

f(t,0)=0,tR,

h(t)L2(Ω)c6eα|t|,

and 0 ≤ α < λ1 where λ1 > 0 is the first eigenvalue of the operator A = −Δ where Δ is the Laplace operator in L2(Ω) with zero Dirichlet boundary condition [7].

When the phase space is L2(Ω), in [1], it was proved that the problem (1) has a pullback global attractor in L2(Ω) with f satisfying (2), (3), (4) and under the condition of a polynomial bound on the forcing term instead of (5):

h(t)L2(Ω)a|t|r+b,tR,a,b>0,r0.

and obtained a uniform bound on the fractal dimension of its sections under an additional assumption that there exists a positive and nondecreasing function ξ : ℝ → (0, ∞) such that:

|f(s,u)f(s,v)|ξ(t)|uv|,st,u,vR.

Here, we briefly state the assumptions on the process to have a pullback exponential attractor [7]:

Let we have a Banach space V and U(t, s) an evolution process on it. We assume that

  • There is a positively invariant family of closed bounded subsets of V, B(t).

  • B(t) grows exponentially in the past, i.e. diam(B(t)) < M eγ0t tt0, γ0 ≥ 0.

  • The family pullback absorbs all bounded subsets of V.

Next, we assume that the process has the decomposition U(t, s) = C(t, s) + S(t, s), where, {C(t, s) : t0ts} and {S(t, s) : t0ts} are families of operators satisfying the following properties:

  • There exists > 0 such that C(t, t) are contractions within the absorbing sets, where the contraction constant λ is independent of time and 0λ12eγ0t~ with γ0 ≥ 0 from 𝒜2,

  • There exists an auxiliary normed space (W, ∥⋅∥W) such that V is compactly embedded into W and S(t) satisfies the smoothing property within the absorbing sets,

  • The process is Lipschitz continuous within the absorbing sets.

In [7], theorem 2.2, it was shown that under these assumptions, there exists a pullback exponential attractor for the process and it was applied to the above problem in H01 (Ω) and obtained a uniform bound on the dimension of sections of its pullback global attractor under the assumption (6). Here, we state a useful corollary of it, to which our results rely upon.

Corollary 1

[7] Assume that the process {U(t, s) : ts} on a Banach space V satisfies (𝒜1)-(𝒜3), (𝓗3) and admits the above decomposition with (𝓗1) and let ν(0,12eγ0t~λ) . Assume further that

  • there exists N = Nν ∈ ℕ such that for any tt0, any R > 0 and any uB(t), there exists v1, …, vnV such that

    S(t,tt~)(B(tt~)BRV(u))i=1NBνRV(vi).

Then there exists a pullback exponential attractor {𝓜(t) = 𝓜ν(t) : t ∈ ℝ} where the bound for the fractal dimension is as follows:

suptRdimfV(M(t))lnNνln2(ν+λ)+γ0t~,

For the existence of global solutions u(t) for the problem on both L2(Ω) and H01 (Ω) refer to [7], Theorem 4.1.

Define U(t, s)us : = u(t), then we have the evolution process {U(t, s) : ts} in L2(Ω) and let us denote by ∥⋅∥ the L2(Ω) norm.

3 Main Results

A priori estimate method is widely used to capture the asymptotic behavior of PDE systems. In the following lemma, using some standard estimates and a new trick, we obtain a useful estimate of the solutions suitable for our purpose.

Lemma 2

Consider the initial boundary value problem (1) with the assumptions (2)-(5) and let {U(t, s) : ts} be the process of it in L2(Ω). let us, vsL2(Ω) are two initial values and u(t), v(t) are the solutions corresponding to them. we have:

u(t)v(t)e(λ1c5)(ts)usvs.

Proof

We write (1) for u and v and consider their difference. Let ω = uv then we have

tωΔω+(f(t,u)f(t,v))=0,

We multiply it by ω and integrate over Ω. Using the boundary condition and the Green formula we have

12tω(t)2+ω2+Ω(f(t,u)f(t,v))ωdx=0,

Let Ω1 = {x ∈ Ω:u(t, x) ≠ v(t, x)}, since fC1 as a function of u, by Mean value theorem ∀x ∈ Ω1z := z(t,x)R:f(t,u)f(t,v)uv=fu(z,t) . Note that z is between u and v as values in ℝ and Ω1 is a Lebesgue measurable set. Note also that fu(t, z) as a function of x is not necessarily even measurable but fu(t, z)|ω|2 = (f(t, u) − f(t, v))ω is an integrable function on Ω1. Hence, we have

12tω(t)2+ω2+Ω1fu(t,z)|ω|2dx=0,

Then by (3)

c5Ω|ω|2c5Ω1|ω|2Ω1fu(t,z)|ω|2

and then

12tω(t)2+ω2c5ω20,

Due to the Poincaré inequality

12tω(t)2+(λ1c5)ω(t)20ts,

Thus, using the Gronwall lemma, the relation (9) follows.

Let Vn = span {e1, …, en} be the linear space spanned by the first n eigenfunctions of A = −Δ in L2(Ω) and let Pn : L2(Ω) → Vn denotes the orthogonal projection and Qn its complementary projection. For uL2(Ω) we write u = Pn(u) + Qn(u) = u1 + u2. We consider the difference ω = uv of two solutions of (1) with us, vsL2(Ω).Taking the inner product in L2(Ω) with Qn(ω) = ω2, we obtain

12tω22+ω22+Ω(f(t,u)f(t,v))ω2dx=0

As in [7] (section 4), we can obtain the absorbing sets B(t) for the problem so that diam(B(t)) ≤ Leα2t tt0 ≤ 0.

As refers to the uniform bound on the dimension of sections of pullback global attractor, in [2, 3], the Lipschitz condition on f is as a sufficient condition.

Here, we make our new assumption which replaces the Lipschitz condition on f with an assumption on fu.

As before in the Lemma 2, set Ω1 = {x ∈ Ω:u(t, x) ≠ v(t, x)} and z = z(t, x). we have

Ω(f(t,u)f(t,v))ω2dx=Ω1f(t,u)f(t,v)uvωω2dx=Ω1fu(t,z)ωω2dx

Let > 0. B(t) is absorbing, Hence, for the solutions starting at t in B(t), according to [11] (Theorem 11.6 and lines after its proof), we can get so large that u(t), v(t) ∈ L(Ω) and thus |u(t)| ≤ Leα2t and |v(t)| ≤ Leα2t almost everywhere( needs to be further adjusted in the following lines). Since z is between u and v, then, |z(t)| ≤ Leα2t a.e. Since fu is continuous w.r.t. u, so let

g(t)=supu{fu(t,u):|u|Leα2t}tt0,

Now, suppose that:

g(t)is a bounded function(tt0)

This is our new assumption which takes the place of the Lipschitz condition in dealing with (10). According to (12), there exists η0 ≥ 0 such that |fu(t, z)| ≤ η0 a.e. for tt0.

Let Ω1+={xΩ1:ωω20}andΩ1={xΩ1:ωω2<0}.

In the sequel, we try to separate the third term in (10), that is, (11) into positive and negative parts and drop the positive one to remain the expression negative. We have

12tω22+ω22c5Ω1+ωω2dx+η0Ω1ωω2dx0tt0

We have ∫ωω2 dx = ∫|ω2|2 dx + ∫ω1ω2 dx, Hence,

12tω22+ω22c5ω22c5Ω1+ω1ω2dx+η0Ω1ω1ω2dx0tt0

We have Ω1+ω1ω2dx+Ω1ω1ω2dx=Ω1ω1ω2dx=ΩΩ1ω120,

As a consequence,

12tω22+ω22c5ω22+(η0+c5)Ω1ω1ω2dx0tt0

Using Cauchy-Schwartz inequality, (9) and ∥ω1∥, ∥ω2∥ ≤ ∥ω∥, we have

12tω22+ω22c5ω22(η0+c5)e(λ1c5)(ts)ω(s)2s<tt0

By Poincaré inequality and Gronwall lemma and for n large enough, we have:

ω2(t)2(e2(λn+1c5)(ts)+(η0+c5)e(λ1c5)(ts)(λn+1c5)1)ω(s)2

Setting s = t, we obtain for u, vB(t) and tt0

QnU(t,tt~)uQnU(t,tt~)v(e2(λn+1c5)t~+(η0+c5)e(λ1c5)t~(λn+1c5)1)12uv,tt0,u,vB(tt~)

We can choose n ∈ ℕ so large that

λ:=(e2(λn+1c5)t~+(η0+c5)e(λ1c5)t~(λn+1c5)1)12<12eα2t~.

Then (𝓗1) is satisfied with C(t, t) = QnU(t, t).

Hence, in fulfilling the requirements of Corollary 1 to obtain a pullback exponential attractor, for meeting (𝓗1), we can replace the assumption (6) with the assumption (12).

Now, we state the main result in L2(Ω):

Theorem 3

Let {U(t, s) : ts} be the process specified in the Lemma 2 and {𝒜(t) : t ∈ ℝ} in L2(Ω) be the pullback global attractor related to it with a uniform bound on the dimension of its sections. if

λ1>c5+α2,

then, for any t ∈ ℝ the section 𝒜(t) has a zero fractal dimension.

Proof

In [7], by checking the requirements of Corollary 1, it was established a pullback global attractor with a uniform bound on its sections. We again check (H2) using Lemma 2. we proceed as follows: From (9) we have:

ω1(t)e(λ1c5)(ts)ω(s)

If λ1c5>α2 , taking s so negative that, fixing = ts, we have

ν=e(λ1c5)t~<12eα2t~λ. Now, for this ν, any R > 0 and uB(t)

PnU(t,tt~)(B(tt~)BRL2(Ω)(u))BνRVn(PnU(t,tt~)u)

Thus, (H2) is satisfied with Nν = 1. Moreover, by (8), the fractal dimension of its sections is 0. Hence, the pullback global attractor has zero dimensional sections.

We know that attractors are connected sets and zero dimensional connected sets are singletons. Then, we have the following corollary.

Corollary 4

The pullback global attractor under the assumption (13) consists only of one global solution.

The assumption (13), is interpreted as a restriction on the bounded domain Ω, since λ1 is only depended on the geometry of the bounded domain. We know from the classic work [10] that if Ω is a bounded, convex, Lipschitz domain with diameter d, then the Poincaré constant is at most dπ forp=2(inW01,p(Ω)) . It is also well-known that for a smooth, bounded domain Ω, since the Rayleigh quotient for the Laplace operator in H01 (Ω) is minimized by the eigenfunction corresponding to the first eigenvalue λ1 of −Δ, we have uλ11u. Hence, for our case, If Ω is a bounded, smooth and convex domain, we have λ11dπorλ1πd.

Now, (13) says λ1>c5+α2 , Hence if c5+α2<πd , the assumption (13) automatically holds and then the zero dimensionality of 𝓜(t) holds true for the convex domains with d<π(c5+α2)1 and for a convex Ω with larger diameters, we must check (13) to be held for assuring zero dimensionality of sections of pullback global attractor of reaction-diffusion equation defined on it.

Corollary 5

Consider the problem (1) on a bounded, smooth and convex domain Ω with the assumptions (2)-(5) and (12) and let {𝒜(t) : t ∈ ℝ} be the pullback global attractor in L2(Ω) associated to it. If diam(Ω) < π(c5+α2)1 , then 𝒜(t) = {u(t)} where u is a global solution of the problem.

In what follows, As an application to Theorem 3, we examine the asymptotic behavior of a new version of non-autonomous Chafee-Infante equation respecting the new assumption (12). Some other versions of this equation in stronger spaces have arisen in [5], [3], [2] and [7]. In [3], the unperturbed version of the following equation has been studied and the trivial asymptotic dynamics has been shown when λ < 1. We show that under a bounded perturbation, this trivial dynamics remains valid.

Example 6

Consider the non-autonomous Chafee-Infante equation on the domain (0, π) as follows:

utuxx=λuβ(t)u3+h(t),u(t,0)=u(t,π)=0,

with the initial condition u(s, x) = u0(x), x ∈ (0, π) and 0 < b0β(t) ≤ B0 is C1 [3] andh(t)∥ ≤ M. Obviously, this equation meets the conditions (2)-(5) with f(t, u) = β(t)u3λ u, α = 0 and c5 = λ. For checking (12), We have fu(t, u) = 3β(t)u2λ and so obviously, g(t) is bounded in the past. Hence, Corollary 5 shows that its pullback global attractor in L2(0, π) consists only of one global solution if λ < 1. Note that f does not meet the Lipschitz condition (6) and as a result, we could not prove the uniform bound on the dimensions of the sections of its pullback global attractor by the results in [1] or [7].

Also, [3] has obtained trivial attractor {0} for the unperturbed equation when λ < 1, while here the perturbed version has a global solution {u(t)} as its pullback global attractor when λ < 1 which generalizes that result.

Moreover, this result is consistent with the result about this problem in [12], Section 4, that is, if h(t) = ϵ h1(t)r(x) where h1 : ℝ → ℝ is an almost periodic function and r(x) ∈ L2(0, π) then λ ∈ (0, 1) implies n = 0, and all solutions converge to its unique almost periodic solution u(t), so we have the trivial pullback global attractor as {u(t)} which its sections are one-point sets [[12], Theorem 4.1]. In fact, our result shows that not only for almost periodic perturbations but also for all bounded perturbations, the pullback global attractor only contains a global solution which, in some sense, improves the result of [12] for λ ∈ (0, 1).

Remark 1

When the phase space is H01 (Ω), according to [7], Theorem 4.1, the assumption (6) needs for the existence of solutions and hence, for defining U(t, s) on H01 (Ω). Thus, here we could not replace it with the new assumption (12).

Remark 2

In the case of H01 (Ω), the results of theorem 3 and its corollaries are valid with some modifications for the expressions (9) and (13) and a new bound on the diameter of the domain.

Communicated by Juan L.G. Guirao

References

  • [1]

    Caraballo, T., Langa, JA & Valero, J. (2003) “The dimension of attractors of nonautonomous partial differential equations,” The ANZIAM Journal, 45, pp. 207–222,

  • [2]

    Carvalho, A. N., Langa, JA & Robinson, J. (2012) “Structure and bifurcation of pullback attractors in a non-autonomous Chafee-Infante equation,” Proc. Amer. Math. Soc. 140, pp. 2357–2373,

  • [3]

    Carvalho, A. N., Langa, JA & Robinson, J. (2012) Attractors for infinite-dimensional non-autonomous dynamical systems, 1st Ed. (Springer Science & Business Media, USA).

  • [4]

    Carvalho, A. N. & Sonner, S. (2013) “Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results,” Commun. Pure Appl. Anal 12, pp. 3047–3071,

  • [5]

    Carvalho, A. N. & Sonner, S. (2014) “Pullback exponential attractors for evolution processes in Banach spaces: properties and applications,” Commun. Pure Appl. Anal 3, pp. 1141–1165,

  • [6]

    Cholewa, J., Czaja, R., & Mola, G. (2008) “Remarks on the fractal dimension of bi-space global and exponential attractors,” Bollettino UMI 9, pp. 121–145.

  • [7]

    Czaja, R. (2014) “Pullback exponential attractors with admissible exponential growth in the past,” Nonlinear Anal. Theor. 104, pp. 90–108,

  • [8]

    Czaja, R. & Efendiev, M. (2011) “Pullback exponential attractors for nonautonomous equations Part I: Semilinear parabolic problems,” J. Math. Anal. Appl. 381, pp. 748–765,

  • [9]

    Czaja, R. & Efendiev, M. (2011) “Pullback exponential attractors for nonautonomous equations Part II: Applications to reaction–diffusion systems,” J. Math. Anal. Appl. 381, pp. 766–780,

  • [10]

    Payne, L. E. & Weinberger, H. F. (1960) “An optimal Poincaré inequality for convex domains,” Arch. Ration. Mech. Anal.”, 5, pp. 286–292,

  • [11]

    Robinson, J. C. (2001) Infinite-dimensional dynamical systems: an introduction to dissipative parabolic PDEs and the theory of global attractors, 1st Ed. (Cambridge University Press, UK).

  • [12]

    Wang, B. (2011) “Almost periodic dynamics of perturbed infinite-dimensional dynamical systems,” Nonlinear Anal. Theor. 74, pp. 7252–7260,

[1]

Caraballo, T., Langa, JA & Valero, J. (2003) “The dimension of attractors of nonautonomous partial differential equations,” The ANZIAM Journal, 45, pp. 207–222,

[2]

Carvalho, A. N., Langa, JA & Robinson, J. (2012) “Structure and bifurcation of pullback attractors in a non-autonomous Chafee-Infante equation,” Proc. Amer. Math. Soc. 140, pp. 2357–2373,

[3]

Carvalho, A. N., Langa, JA & Robinson, J. (2012) Attractors for infinite-dimensional non-autonomous dynamical systems, 1st Ed. (Springer Science & Business Media, USA).

[4]

Carvalho, A. N. & Sonner, S. (2013) “Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results,” Commun. Pure Appl. Anal 12, pp. 3047–3071,

[5]

Carvalho, A. N. & Sonner, S. (2014) “Pullback exponential attractors for evolution processes in Banach spaces: properties and applications,” Commun. Pure Appl. Anal 3, pp. 1141–1165,

[6]

Cholewa, J., Czaja, R., & Mola, G. (2008) “Remarks on the fractal dimension of bi-space global and exponential attractors,” Bollettino UMI 9, pp. 121–145.

[7]

Czaja, R. (2014) “Pullback exponential attractors with admissible exponential growth in the past,” Nonlinear Anal. Theor. 104, pp. 90–108,

[8]

Czaja, R. & Efendiev, M. (2011) “Pullback exponential attractors for nonautonomous equations Part I: Semilinear parabolic problems,” J. Math. Anal. Appl. 381, pp. 748–765,

[9]

Czaja, R. & Efendiev, M. (2011) “Pullback exponential attractors for nonautonomous equations Part II: Applications to reaction–diffusion systems,” J. Math. Anal. Appl. 381, pp. 766–780,

[10]

Payne, L. E. & Weinberger, H. F. (1960) “An optimal Poincaré inequality for convex domains,” Arch. Ration. Mech. Anal.”, 5, pp. 286–292,

[11]

Robinson, J. C. (2001) Infinite-dimensional dynamical systems: an introduction to dissipative parabolic PDEs and the theory of global attractors, 1st Ed. (Cambridge University Press, UK).

[12]

Wang, B. (2011) “Almost periodic dynamics of perturbed infinite-dimensional dynamical systems,” Nonlinear Anal. Theor. 74, pp. 7252–7260,

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