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

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]:

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

∂<!-- ? -->u∂<!-- ? -->t−<!-- - -->Δ<!-- ? -->u+f(t,u)=h(t),u|∂<!-- ? -->Ω<!-- O -->=0,u(s)=us,t><!-- > -->s.$$\begin{array}{} \displaystyle \begin{cases} \frac{\partial u}{\partial t}-\Delta u+f(t,u)=h(t),\\ u\vert _{\partial \Omega}=0,\\ u(s)=u_{s},\qquad t\gt s. \end{cases} \end{array}$$

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$\begin{array}{} \displaystyle H^1_0 \end{array}$ (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 f_u 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.

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

∂<!-- ? -->u∂<!-- ? -->t−<!-- - -->Δ<!-- ? -->u+f(t,u)=h(t),u|∂<!-- ? -->Ω<!-- O -->=0,u(s)=us,t><!-- > -->s.$$\begin{array}{} \displaystyle \begin{cases} \frac{\partial u}{\partial t}-\Delta u+f(t,u)=h(t),\\ u\vert _{\partial \Omega}=0,\\ u(s)=u_{s},\qquad t\gt s. \end{cases} \end{array}$$(1)

where f ∈ C¹(ℝ², ℝ), h(⋅) ∈ Lloc2$\begin{array}{} \displaystyle L_{loc}^2 \end{array}$ (ℝ, L²(Ω)), Ω is a bounded open subset of ℝⁿ with smooth boundary ∂ Ω, and there exists p ≥ 2, c_i > 0, i = 1, …, 6, such that

c1|u|p−<!-- - -->c2≤<!-- = -->f(t,u)u≤<!-- = -->c3|u|p+c4,$$\begin{array}{} \displaystyle c_1|u|^p-c_2\leq f(t,u)u\leq c_3|u|^p+c_4, \end{array}$$(2)

fu(t,u)≥<!-- = -->−<!-- - -->c5,$$\begin{array}{} \displaystyle f_u(t,u)\geq -c_5, \end{array}$$(3)

f(t,0)=0,∀<!-- ? -->t∈<!-- ? -->R,$$\begin{array}{} \displaystyle f(t,0)=0, \quad \forall t\in \mathbb{R}, \end{array}$$(4)

∥<!-- ? -->h(t)∥<!-- ? -->L2(Ω<!-- O -->)≤<!-- = -->c6eα<!-- a -->|t|,$$\begin{array}{} \displaystyle \|h(t)\|_{L^2(\Omega)}\leq c_6 e^{\alpha |t|}, \end{array}$$(5)

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<!-- O -->)≤<!-- = -->a|t|r+b,t∈<!-- ? -->R,a,b><!-- > -->0,r≥<!-- = -->0.$$\begin{array}{} \displaystyle \| h(t)\|_{L^2(\Omega)}\leq a|t|^r+b,\quad t\in \mathbb{R}, a,b\gt 0, r\geq 0. \end{array}$$

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)|u−<!-- - -->v|,s≤<!-- = -->t,u,v∈<!-- ? -->R.$$\begin{array}{} \displaystyle |f(s ,u)-f(s ,v)|\leq \xi (t)\,|u-v|,\quad s \leq t,\, u,v \in \mathbb{R}. \end{array}$$(6)

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

(𝒜₁)

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

Next, we assume that the process has the decomposition U(t, s) = C(t, s) + S(t, s), where, {C(t, s) : t₀ ≥ t ≥ s} and {S(t, s) : t₀ ≥ t ≥ s} are families of operators satisfying the following properties:

(𝓗₁)

There exists t̃ > 0 such that C(t, t − t̃) are contractions within the absorbing sets, where the contraction constant λ is independent of time and 0≤<!-- = -->λ<!-- ? -->≤<!-- = -->12e−<!-- - -->γ<!-- ? -->0t~<!-- ~ -->$\begin{array}{} \displaystyle 0 \leq \lambda \leq \frac{1}{2}e^{-\gamma_0 \tilde{t}} \end{array}$ with γ₀ ≥ 0 from 𝒜₂,

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$\begin{array}{} \displaystyle H^1_0 \end{array}$ (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.

For the existence of global solutions u(t) for the problem on both L²(Ω) and H01$\begin{array}{} \displaystyle H^1_0 \end{array}$ (Ω) refer to [7], Theorem 4.1.

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

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.

Let V_n = span {e₁, …, e_n} be the linear space spanned by the first n eigenfunctions of A = −Δ in L²(Ω) and let P_n : L²(Ω) → V_n denotes the orthogonal projection and Q_n its complementary projection. For u ∈ L²(Ω) we write u = P_n(u) + Q_n(u) = u₁ + u₂. We consider the difference ω = u − v of two solutions of (1) with u_s, v_s ∈ L²(Ω).Taking the inner product in L²(Ω) with Q_n(ω) = ω₂, we obtain

12∂<!-- ? -->t∥<!-- ? -->ω<!-- ? -->2∥<!-- ? -->2+∥<!-- ? -->∇<!-- ? -->ω<!-- ? -->2∥<!-- ? -->2+∫<!-- ? -->Ω<!-- O -->(f(t,u)−<!-- - -->f(t,v))ω<!-- ? -->2dx=0$$\begin{array}{} \displaystyle \frac{1}{2}\partial_t\| \omega_2\|^2+\|\nabla\omega_2\|^2+\int_\Omega (f(t,u)-f(t,v))\omega_2\,dx=0 \end{array}$$(10)

As in [7] (section 4), we can obtain the absorbing sets B(t) for the problem so that diam(B(t)) ≤ Le−<!-- - -->α<!-- a -->2t$\begin{array}{} \displaystyle Le^{-\frac{\alpha}{2}t} \end{array}$t ≤ t₀ ≤ 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 f_u.

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

∫<!-- ? -->Ω<!-- O -->(f(t,u)−<!-- - -->f(t,v))ω<!-- ? -->2dx=∫<!-- ? -->Ω<!-- O -->1f(t,u)−<!-- - -->f(t,v)u−<!-- - -->vω<!-- ? -->ω<!-- ? -->2dx=∫<!-- ? -->Ω<!-- O -->1fu(t,z)ω<!-- ? -->ω<!-- ? -->2dx$$\begin{array}{} \displaystyle \int_\Omega (f(t,u)-f(t,v))\omega_2 \, dx=\int_{\Omega_1} \frac{f(t,u)-f(t,v)}{u-v}\omega \omega_2\,dx =\int_{\Omega_1}f_u(t,z)\omega \omega_2\, dx \end{array}$$(11)

Let t̃ > 0. B(t − t̃) is absorbing, Hence, for the solutions starting at t − t̃ in B(t − t̃), according to [11] (Theorem 11.6 and lines after its proof), we can get t̃ so large that u(t), v(t) ∈ L^∞(Ω) and thus |u(t)| ≤ Le−<!-- - -->α<!-- a -->2t$\begin{array}{} \displaystyle Le^{-\frac{\alpha}{2}t} \end{array}$ and |v(t)| ≤ Le−<!-- - -->α<!-- a -->2t$\begin{array}{} \displaystyle Le^{-\frac{\alpha}{2}t} \end{array}$ almost everywhere(t̃ needs to be further adjusted in the following lines). Since z is between u and v, then, |z(t)| ≤ Le−<!-- - -->α<!-- a -->2t$\begin{array}{} \displaystyle Le^{-\frac{\alpha}{2}t} \end{array}$ a.e. Since f_u is continuous w.r.t. u, so let

g(t)=supu{fu(t,u):|u|≤<!-- = -->Le−<!-- - -->α<!-- a -->2t}t≤<!-- = -->t0,$$\begin{array}{} \displaystyle g(t)=\sup_u\{f_u(t,u):|u|\leq Le^{-\frac{\alpha}{2}t}\}\qquad t\leq t_0, \end{array}$$

Now, suppose that:

g(t)is a bounded function(t≤<!-- = -->t0)$$\begin{array}{} \displaystyle g(t) \quad \text{is a bounded function} \quad (t\leq t_0) \end{array}$$(12)

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 |f_u(t, z)| ≤ η₀ a.e. for t ≤ t₀.

Let Ω<!-- O -->1+={x∈<!-- ? -->Ω<!-- O -->1:ω<!-- ? -->ω<!-- ? -->2≥<!-- = -->0}andΩ<!-- O -->1−<!-- - -->={x∈<!-- ? -->Ω<!-- O -->1:ω<!-- ? -->ω<!-- ? -->2<<!-- ? -->0}$\begin{array}{} \displaystyle \Omega_1^+=\{x\in \Omega_1:\omega \omega_2\geq 0\}~~\text{and}~ ~\Omega_1^-=\{x\in \Omega_1:\omega \omega_2\lt 0\} \end{array}$.

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

12∂<!-- ? -->t∥<!-- ? -->ω<!-- ? -->2∥<!-- ? -->2+∥<!-- ? -->∇<!-- ? -->ω<!-- ? -->2∥<!-- ? -->2−<!-- - -->c5∫<!-- ? -->Ω<!-- O -->1+ω<!-- ? -->ω<!-- ? -->2dx+η<!-- ? -->0∫<!-- ? -->Ω<!-- O -->1−<!-- - -->ω<!-- ? -->ω<!-- ? -->2dx≤<!-- = -->0t≤<!-- = -->t0$$\begin{array}{} \displaystyle \frac{1}{2}\partial_t\| \omega_2\|^2+\|\nabla\omega_2\|^2-c_5\int_{\Omega_1^+}\omega\omega_2 \,dx+\eta_0\int_{\Omega_1^-}\omega\omega_2 \, dx \leq 0 \qquad t\leq t_0 \end{array}$$

We have ∫ωω₂dx = ∫|ω₂|²dx + ∫ω₁ω₂dx, Hence,

12∂<!-- ? -->t∥<!-- ? -->ω<!-- ? -->2∥<!-- ? -->2+∥<!-- ? -->∇<!-- ? -->ω<!-- ? -->2∥<!-- ? -->2−<!-- - -->c5∥<!-- ? -->ω<!-- ? -->2∥<!-- ? -->2−<!-- - -->c5∫<!-- ? -->Ω<!-- O -->1+ω<!-- ? -->1ω<!-- ? -->2dx+η<!-- ? -->0∫<!-- ? -->Ω<!-- O -->1−<!-- - -->ω<!-- ? -->1ω<!-- ? -->2dx≤<!-- = -->0t≤<!-- = -->t0$$\begin{array}{} \displaystyle \frac{1}{2}\partial_t\| \omega_2\|^2+\|\nabla\omega_2\|^2-c_5\|\omega_2\|^2-c_5\int_{\Omega_1^+}\omega_1\omega_2 \,dx +\eta_0 \int_{\Omega_1^-}\omega_1\omega_2 \, dx \leq 0\quad t\leq t_0 \end{array}$$

We have ∫<!-- ? -->Ω<!-- O -->1+ω<!-- ? -->1ω<!-- ? -->2dx+∫<!-- ? -->Ω<!-- O -->1−<!-- - -->ω<!-- ? -->1ω<!-- ? -->2dx=∫<!-- ? -->Ω<!-- O -->1ω<!-- ? -->1ω<!-- ? -->2dx=−<!-- - -->∫<!-- ? -->Ω<!-- O -->−<!-- - -->Ω<!-- O -->1ω<!-- ? -->12≤<!-- = -->0,$\begin{array}{} \int_{\Omega_1^+}\omega_1\omega_2\, dx+\int_{\Omega_1^-}\omega_1\omega_2\, dx=\int_{\Omega_1}\omega_1\omega_2\, dx=-\int_{\Omega-\Omega_1}\omega_1^2\leq 0, \end{array}$

As a consequence,

12∂<!-- ? -->t∥<!-- ? -->ω<!-- ? -->2∥<!-- ? -->2+∥<!-- ? -->∇<!-- ? -->ω<!-- ? -->2∥<!-- ? -->2−<!-- - -->c5∥<!-- ? -->ω<!-- ? -->2∥<!-- ? -->2+(η<!-- ? -->0+c5)∫<!-- ? -->Ω<!-- O -->1−<!-- - -->ω<!-- ? -->1ω<!-- ? -->2dx≤<!-- = -->0t≤<!-- = -->t0$$\begin{array}{} \displaystyle \frac{1}{2}\partial_t\| \omega_2\|^2+\|\nabla\omega_2\|^2-c_5\|\omega_2\|^2 +( \eta_0+c_5)\int_{\Omega_1^-}\omega_1\omega_2 \, dx \leq 0\qquad t\leq t_0 \end{array}$$

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

12∂<!-- ? -->t∥<!-- ? -->ω<!-- ? -->2∥<!-- ? -->2+∥<!-- ? -->∇<!-- ? -->ω<!-- ? -->2∥<!-- ? -->2−<!-- - -->c5∥<!-- ? -->ω<!-- ? -->2∥<!-- ? -->2≤<!-- = -->(η<!-- ? -->0+c5)e−<!-- - -->(λ<!-- ? -->1−<!-- - -->c5)(t−<!-- - -->s)∥<!-- ? -->ω<!-- ? -->(s)∥<!-- ? -->2s<<!-- ? -->t≤<!-- = -->t0$$\begin{array}{} \displaystyle \frac{1}{2}\partial_t\| \omega_2\|^2+\|\nabla\omega_2\|^2-c_5\|\omega_2\|^2\leq ( \eta_0+c_5)e^{-(\lambda_1-c_5)(t-s)}\|\omega (s)\|^2\qquad s\lt t\leq t_0 \end{array}$$

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

∥<!-- ? -->ω<!-- ? -->2(t)∥<!-- ? -->2≤<!-- = -->(e−<!-- - -->2(λ<!-- ? -->n+1−<!-- - -->c5)(t−<!-- - -->s)+(η<!-- ? -->0+c5)e−<!-- - -->(λ<!-- ? -->1−<!-- - -->c5)(t−<!-- - -->s)(λ<!-- ? -->n+1−<!-- - -->c5)−<!-- - -->1)∥<!-- ? -->ω<!-- ? -->(s)∥<!-- ? -->2$$\begin{array}{} \displaystyle \|\omega_2(t)\|^2\leq (e^{-2(\lambda_{n+1}-c_5)(t-s)}+ ( \eta_0+c_5)e^{-(\lambda_1-c_5)(t-s)}(\lambda_{n+1}-c_5)^{-1})\|\omega(s)\|^2 \end{array}$$

Setting s = t − t̃, we obtain for u, v ∈ B(t − t̃) and t ≤ t₀

∥<!-- ? -->QnU(t,t−<!-- - -->t~<!-- ~ -->)u−<!-- - -->QnU(t,t−<!-- - -->t~<!-- ~ -->)v∥<!-- ? -->≤<!-- = -->(e−<!-- - -->2(λ<!-- ? -->n+1−<!-- - -->c5)t~<!-- ~ -->+(η<!-- ? -->0+c5)e−<!-- - -->(λ<!-- ? -->1−<!-- - -->c5)t~<!-- ~ -->(λ<!-- ? -->n+1−<!-- - -->c5)−<!-- - -->1)12∥<!-- ? -->u−<!-- - -->v∥<!-- ? -->,t≤<!-- = -->t0,u,v∈<!-- ? -->B(t−<!-- - -->t~<!-- ~ -->)$$\begin{array}{} \displaystyle \|Q_nU(t,t-\tilde{t})u-Q_nU(t,t-\tilde{t})v\|\ \leq \\ \displaystyle (e^{-2(\lambda_{n+1}-c_5)\tilde{t}}+(\eta_0+c_5) e^{-(\lambda_1-c_5)\tilde{t}}(\lambda_{n+1}-c_5)^{-1})^{\frac{1}{2}}\|u-v\|, \quad t\leq t_0,\, u,\,v\in B(t-\tilde{t}) \end{array}$$

We can choose n ∈ ℕ so large that

λ<!-- ? -->:=(e−<!-- - -->2(λ<!-- ? -->n+1−<!-- - -->c5)t~<!-- ~ -->+(η<!-- ? -->0+c5)e−<!-- - -->(λ<!-- ? -->1−<!-- - -->c5)t~<!-- ~ -->(λ<!-- ? -->n+1−<!-- - -->c5)−<!-- - -->1)12<<!-- ? -->12e−<!-- - -->α<!-- a -->2t~<!-- ~ -->.$$\begin{array}{} \displaystyle \lambda := (e^{-2(\lambda_{n+1}-c_5)\tilde{t}}+(\eta_0+c_5) e^{-(\lambda_1-c_5)\tilde{t}}(\lambda_{n+1}-c_5)^{-1})^{\frac{1}{2}}\lt \frac{1}{2}e^{-\frac{\alpha}{2}\tilde{t}}. \end{array}$$

Then (𝓗₁) is satisfied with C(t, t − t̃) = Q_nU(t, t − t̃).

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):

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

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<!-- p --> forp=2(inW01,p(Ω<!-- O -->))$\begin{array}{} \frac{d}{\pi}~\text{ for}~~ p = 2 ~(\text{in}~~ W_0^{1,p}(\Omega)) \end{array}$ . It is also well-known that for a smooth, bounded domain Ω, since the Rayleigh quotient for the Laplace operator in H01$\begin{array}{} \displaystyle H^1_0 \end{array}$ (Ω) is minimized by the eigenfunction corresponding to the first eigenvalue λ₁ of −Δ, we have ∥<!-- ? -->u∥<!-- ? -->≤<!-- = -->λ<!-- ? -->1−<!-- - -->1∥<!-- ? -->∇<!-- ? -->u∥<!-- ? -->$\begin{array}{} \displaystyle \|u\|\leq \lambda_1^{-1} \|\nabla u\| \end{array}$. Hence, for our case, If Ω is a bounded, smooth and convex domain, we have λ<!-- ? -->1−<!-- - -->1≤<!-- = -->dπ<!-- p -->orλ<!-- ? -->1≥<!-- = -->π<!-- p -->d$\begin{array}{} \displaystyle \lambda_1^{-1}\leq \frac{d}{\pi}~~ \text{or}~~ \lambda_1\geq \frac{\pi}{d} \end{array}$.

Now, (13) says λ<!-- ? -->1><!-- > -->c5+α<!-- a -->2$\begin{array}{} \displaystyle \lambda_1\gt c_5+\frac{\alpha}{2} \end{array}$ , Hence if c5+α<!-- a -->2<<!-- ? -->π<!-- p -->d$\begin{array}{} \displaystyle c_5+\frac{\alpha}{2}\lt \frac{\pi}{d} \end{array}$ , the assumption (13) automatically holds and then the zero dimensionality of 𝓜(t) holds true for the convex domains with d<<!-- ? -->π<!-- p -->(c5+α<!-- a -->2)−<!-- - -->1$\begin{array}{} \displaystyle d\lt \pi(c_5+\frac{\alpha}{2})^{-1} \end{array}$ 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.

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.