Invariant sets and attractors for Hanusse-type chemical systems with diffusions

Mihai Nechita
(original), html, ISI/JCR, Numerical Analysis, ODEs (numerical), paper

Abstract

We are concerned with Hanusse-type chemical models with diffusions. We show that some bounded invariant sets ⊂R^{N} found for the ODE Hanusse-type models (corresponding to the case when diffusions are neglected) can be used to define invariant sets ⊂ L∞(Ω)^{N} with respect to the corresponding Hanusse-type PDE models (involving diffusions), where Ω⊂Rⁿ,n≤3, denotes the reaction domain. Simulations for both the ODE and PDE Hanussetype models are performed for suitable coefficients of the polynomials representing the reaction terms, showing that the attractors for the ODE model are also attractors, in fact the only attractors, for the PDE model

Authors

Gheorghe Moroşanu
Department of Mathematics, Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania

Mihai Nechita
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

Keywords

Hanusse-type models; Diffusions; Tangency condition; C₀-semigroup; Positively invariant sets; Attractors;

Paper coordinates

Gh. Morosanu, M. Nechita, Invariant sets and attractors for Hanusse-type chemical systems with diffusions, Comput. Math. Appl., 73 (2017) 1815–1823.
DOI: 10.1016/j.camwa.2017.02.024

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https://www.sciencedirect.com/science/article/pii/S0898122117301037?via%3Dihub

About this paper

Journal

Computers and Mathematics with Applications

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Elsevier

DOI

10.1016/j.camwa.2017.02.024

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0898-1221

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Paper in HTML form

inv sets

Invariant sets and attractors for Hanusse-type chemical systems with diffusions

Gheorghe Moroşanu a a ^(a){ }^{\mathrm{a}}a, Mihai Nechita b , a , ∗ b , a , ∗ ^(b,a,**){ }^{\mathrm{b}, \mathrm{a}, *}b,a,∗ a a ^(a){ }^{\mathrm{a}}a Department of Mathematics, Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084, Cluj-Napoca, Romania b b ^(b){ }^{\mathrm{b}}b T. Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, Cluj-Napoca, Romania

ARTICLE INFO

Article history:

Received 17 October 2016
Received in revised form 17 January 2017
Accepted 19 February 2017
Available online 9 March 2017
Dedicated to Professor Adrian Petruşel at his 54th anniversary, with thanks for his friendship and hospitality

Keywords:

Hanusse-type models
Diffusions
Tangency condition
C 0 C 0 C_(0)C_{0}C0-semigroup
Positively invariant sets
Attractors

Abstract

We are concerned with Hanusse-type chemical models with diffusions. We show that some bounded invariant sets ⊂ R N ⊂ R N subR^(N)\subset \mathbb{R}^{N}⊂RN found for the ODE Hanusse-type models (corresponding to the case when diffusions are neglected) can be used to define invariant sets ⊂ L ∞ ( Ω ) N ⊂ L ∞ ( Ω ) N subL^(oo)(Omega)^(N)\subset L^{\infty}(\Omega)^{N}⊂L∞(Ω)N with respect to the corresponding Hanusse-type PDE models (involving diffusions), where Ω ⊂ R n , n ≤ 3 Ω ⊂ R n , n ≤ 3 Omega subR^(n),n <= 3\Omega \subset \mathbb{R}^{n}, n \leq 3Ω⊂Rn,n≤3, denotes the reaction domain. Simulations for both the ODE and PDE Hanussetype models are performed for suitable coefficients of the polynomials representing the reaction terms, showing that the attractors for the ODE model are also attractors, in fact the only attractors, for the PDE model.

© 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Consider the differential equation in R N R N R^(N)\mathbb{R}^{N}RN
(1.1) u ′ ( t ) = f ( u ( t ) ) , t ∈ R , (1.1) u ′ ( t ) = f ( u ( t ) ) , t ∈ R , {:(1.1)u^(')(t)=f(u(t))","quad t inR",":}\begin{equation*} u^{\prime}(t)=f(u(t)), \quad t \in \mathbb{R}, \tag{1.1} \end{equation*}(1.1)u′(t)=f(u(t)),t∈R,
where f : R N → R N f : R N → R N f:R^(N)rarrR^(N)f: \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}f:RN→RN is assumed to be a locally Lipschitz function. Thus, from the general existence theory, we know that for any u 0 ∈ R N u 0 ∈ R N u_(0)inR^(N)u_{0} \in \mathbb{R}^{N}u0∈RN there exists a unique solution u = u ( t ) u = u ( t ) u=u(t)u=u(t)u=u(t) of Eq. (1.1) that satisfies the initial condition
(1.2) u ( 0 ) = u 0 , (1.2) u ( 0 ) = u 0 , {:(1.2)u(0)=u_(0)",":}\begin{equation*} u(0)=u_{0}, \tag{1.2} \end{equation*}(1.2)u(0)=u0,
being defined on an open interval containing t 0 = 0 t 0 = 0 t_(0)=0t_{0}=0t0=0 (i.e., u u uuu is a local solution of the Cauchy problem (1.1), (1.2)).
Definition 1. A subset D ⊂ R N D ⊂ R N D subR^(N)D \subset \mathbb{R}^{N}D⊂RN is said to be positively invariant with respect to Eq. (1.1) if for all u 0 ∈ D u 0 ∈ D u_(0)in Du_{0} \in Du0∈D the trajectory of the solution of problem (1.1), (1.2) remains in D D DDD as long as the solution exists in the future: u ( t ) ∈ D u ( t ) ∈ D u(t)in Du(t) \in Du(t)∈D for all t > 0 , t ∈ D ( u ) t > 0 , t ∈ D ( u ) t > 0,t in D(u)t>0, t \in D(u)t>0,t∈D(u) (the domain of u u uuu ).
Let us recall the well-known Brezis-Nagumo theorem (see [1] and [2, p. 25]):
Theorem 1. A closed set D ⊂ R N D ⊂ R N D subR^(N)D \subset \mathbb{R}^{N}D⊂RN is positively invariant with respect to Eq. (1.1) if and only if the following tangency condition is satisfied
(1.3) lim h ↓ 0 1 h d ( z + h f ( z ) , D ) = 0 , ∀ z ∈ D (1.3) lim h ↓ 0   1 h d ( z + h f ( z ) , D ) = 0 , ∀ z ∈ D {:(1.3)lim_(h darr0)(1)/(h)d(z+hf(z)","D)=0","quad AA z in D:}\begin{equation*} \lim _{h \downarrow 0} \frac{1}{h} d(z+h f(z), D)=0, \quad \forall z \in D \tag{1.3} \end{equation*}(1.3)limh↓01hd(z+hf(z),D)=0,∀z∈D
where d ( x , D ) := min y ∈ D ‖ x − y ‖ d ( x , D ) := min y ∈ D   ‖ x − y ‖ d(x,D):=min_(y in D)||x-y||d(x, D):=\min _{y \in D}\|x-y\|d(x,D):=miny∈D‖x−y‖, with ‖ ⋅ ‖ ‖ ⋅ ‖ ||*||\|\cdot\|‖⋅‖ a norm of R N R N R^(N)\mathbb{R}^{N}RN, say the Euclidean norm.
Remark 1. In fact Theorem 1 is valid in any general real Banach space X X XXX, under the condition that f : X → X f : X → X f:X rarr Xf: X \rightarrow Xf:X→X is a locally Lipschitz function (obviously, if X X XXX is finite dimensional, then this condition is equivalent to the Lipschitz continuity of f f fff on every bounded subset of X X XXX ).
Remark 2. If D ⊂ R N D ⊂ R N D subR^(N)D \subset \mathbb{R}^{N}D⊂RN is a compact set satisfying condition (1.3), then for every u 0 ∈ D u 0 ∈ D u_(0)in Du_{0} \in Du0∈D there exists a unique solution u u uuu of problem (1.1), (1.2) defined on [ 0 , + ∞ ) [ 0 , + ∞ ) [0,+oo)[0,+\infty)[0,+∞), with u ( t ) ∈ D u ( t ) ∈ D u(t)in Du(t) \in Du(t)∈D for all t ∈ [ 0 , + ∞ ) t ∈ [ 0 , + ∞ ) t in[0,+oo)t \in[0,+\infty)t∈[0,+∞). Indeed, the trajectory of the unique local solution u u uuu of problem (1.1), (1.2) remains in the compact set D D DDD (cf. Theorem 1) and so the solution exists on the whole positive half-line: D ( u ) = [ 0 , + ∞ ) D ( u ) = [ 0 , + ∞ ) D(u)=[0,+oo)D(u)=[0,+\infty)D(u)=[0,+∞).
In [3] the Hanusse mathematical model [4] was revisited and additional similar models have been proposed to describe chemical reactions involving more species (see also Section 3). In fact, all these models (systems of ODEs whose right-hand sides are some polynomial functions of degree 2 of the unknown variables (concentrations)) can be regarded as differential equations in R N , N ≥ 3 R N , N ≥ 3 R^(N),N >= 3\mathbb{R}^{N}, N \geq 3RN,N≥3, of the form (1.1). For certain coefficients of these differential systems, some positively invariant sets of the form
(1.4) D = [ a 1 , a ¯ 1 ] × [ a 2 , a ¯ 2 ] × ⋯ × [ a N , a ¯ N ] ⊂ R N , 0 < a i < a ¯ i < ∞ ( i = 1 , N ― ) , (1.4) D = a 1 , a ¯ 1 × a 2 , a ¯ 2 × ⋯ × a N , a ¯ N ⊂ R N , 0 < a i < a ¯ i < ∞ ( i = 1 , N ¯ ) , {:(1.4)D=[a_(1), bar(a)_(1)]xx[a_(2), bar(a)_(2)]xx cdots xx[a_(N), bar(a)_(N)]subR^(N)","quad0 < a_(i) < bar(a)_(i) < oo(i= bar(1,N))",":}\begin{equation*} D=\left[a_{1}, \bar{a}_{1}\right] \times\left[a_{2}, \bar{a}_{2}\right] \times \cdots \times\left[a_{N}, \bar{a}_{N}\right] \subset \mathbb{R}^{N}, \quad 0<a_{i}<\bar{a}_{i}<\infty(i=\overline{1, N}), \tag{1.4} \end{equation*}(1.4)D=[a1,a¯1]×[a2,a¯2]×⋯×[aN,a¯N]⊂RN,0<ai<a¯i<∞(i=1,N―),
have been identified, based on the tangency condition (1.3). Indeed, for such coefficients the following inequalities were established in [3] for all 1 ≤ i ≤ N 1 ≤ i ≤ N 1 <= i <= N1 \leq i \leq N1≤i≤N
f i ( x 1 , … , x i − 1 , a i , x i + 1 , … , x N ) > 0 , f i ( x 1 , … , x i − 1 , a ¯ i , x i + 1 , … , x N ) < 0 , (1.5) ∀ x j ∈ [ a j , a ¯ j ] , j = 1 , N ― , j ≠ i , f i x 1 , … , x i − 1 , a i , x i + 1 , … , x N > 0 , f i x 1 , … , x i − 1 , a ¯ i , x i + 1 , … , x N < 0 , (1.5) ∀ x j ∈ a j , a ¯ j , j = 1 , N ¯ , j ≠ i , {:[f_(i)(x_(1),dots,x_(i-1),a_(i),x_(i+1),dots,x_(N)) > 0","quadf_(i)(x_(1),dots,x_(i-1), bar(a)_(i),x_(i+1),dots,x_(N)) < 0","],[(1.5)quad AAx_(j)in[a_(j), bar(a)_(j)]","j= bar(1,N)","j!=i","]:}\begin{align*} & f_{i}\left(x_{1}, \ldots, x_{i-1}, a_{i}, x_{i+1}, \ldots, x_{N}\right)>0, \quad f_{i}\left(x_{1}, \ldots, x_{i-1}, \bar{a}_{i}, x_{i+1}, \ldots, x_{N}\right)<0, \\ & \quad \forall x_{j} \in\left[a_{j}, \bar{a}_{j}\right], j=\overline{1, N}, j \neq i, \tag{1.5} \end{align*}fi(x1,…,xi−1,ai,xi+1,…,xN)>0,fi(x1,…,xi−1,a¯i,xi+1,…,xN)<0,(1.5)∀xj∈[aj,a¯j],j=1,N―,j≠i,
which lead to (1.3) (see, e.g., [5, Lemma 4.1, p.72]). Based on the invariance of the D's (following from Theorem 1), a closed trajectory (periodic solution) was supposed to exist for each of such ODE models. This was indeed the case, even more, the existence of an attractive closed curve (attractor) was established by numerical simulations in [3] for each of the Hanussetype models investigated there, for suitable sets of coefficients.
In this paper we consider the same Hanusse-type models, but this time we also take into account the diffusions of species inside the reaction domain Ω ⊂ R n , n ≤ 3 Ω ⊂ R n , n ≤ 3 Omega subR^(n),n <= 3\Omega \subset \mathbb{R}^{n}, n \leq 3Ω⊂Rn,n≤3. So we have PDE systems of the form
(1.6) ∂ u i ∂ t = α i Δ u i + f i ( u ) , t ≥ 0 , x ∈ Ω , i = 1 , N ― , (1.6) ∂ u i ∂ t = α i Δ u i + f i ( u ) , t ≥ 0 , x ∈ Ω , i = 1 , N ¯ , {:(1.6)(delu_(i))/(del t)=alpha_(i)Deltau_(i)+f_(i)(u)","quad t >= 0","x in Omega","i= bar(1,N)",":}\begin{equation*} \frac{\partial u_{i}}{\partial t}=\alpha_{i} \Delta u_{i}+f_{i}(u), \quad t \geq 0, x \in \Omega, i=\overline{1, N}, \tag{1.6} \end{equation*}(1.6)∂ui∂t=αiΔui+fi(u),t≥0,x∈Ω,i=1,N―,
where u i = u i ( t , x ) u i = u i ( t , x ) u_(i)=u_(i)(t,x)u_{i}=u_{i}(t, x)ui=ui(t,x) denote the variable concentrations of the corresponding intermediate species, Δ := ∑ i = 1 n ∂ 2 ∂ x i 2 Δ := ∑ i = 1 n   ∂ 2 ∂ x i 2 Delta:=sum_(i=1)^(n)(del^(2))/(delx_(i)^(2))\Delta:=\sum_{i=1}^{n} \frac{\partial^{2}}{\partial x_{i}^{2}}Δ:=∑i=1n∂2∂xi2, and α i α i alpha_(i)\alpha_{i}αi are positive constants (diffusion coefficients). We associate with (1.6) the following natural boundary conditions
(1.7) ∂ u i ∂ v = 0 , t ≥ 0 , x ∈ ∂ Ω , i = 1 , N ― , (1.7) ∂ u i ∂ v = 0 , t ≥ 0 , x ∈ ∂ Ω , i = 1 , N ¯ , {:(1.7)(delu_(i))/(del v)=0","quad t >= 0","x in del Omega","i= bar(1,N)",":}\begin{equation*} \frac{\partial u_{i}}{\partial v}=0, \quad t \geq 0, x \in \partial \Omega, i=\overline{1, N}, \tag{1.7} \end{equation*}(1.7)∂ui∂v=0,t≥0,x∈∂Ω,i=1,N―,
where ∂ u i ∂ v ∂ u i ∂ v (delu_(i))/(del v)\frac{\partial u_{i}}{\partial v}∂ui∂v denotes the outward normal derivative of u i u i u_(i)u_{i}ui, and initial conditions
(1.8) u i ( 0 , x ) = u i 0 ( x ) , x ∈ Ω , i = 1 , N ― . (1.8) u i ( 0 , x ) = u i 0 ( x ) , x ∈ Ω , i = 1 , N ¯ . {:(1.8)u_(i)(0","x)=u_(i)^(0)(x)","quad x in Omega","i= bar(1,N).:}\begin{equation*} u_{i}(0, x)=u_{i}^{0}(x), \quad x \in \Omega, i=\overline{1, N} . \tag{1.8} \end{equation*}(1.8)ui(0,x)=ui0(x),x∈Ω,i=1,N―.
In the next section of this paper we show that the D D DDD invariant sets ⊂ R N ⊂ R N subR^(N)\subset \mathbb{R}^{N}⊂RN with respect to system (1.6) without diffusions can be used to define invariant sets ⊂ L ∞ ( Ω ) N ⊂ L ∞ ( Ω ) N subL^(oo)(Omega)^(N)\subset L^{\infty}(\Omega)^{N}⊂L∞(Ω)N with respect to system (1.6) with diffusions. The last section of the paper (Section 3) is devoted to simulations for both the ODE and PDE Hanusse-type models for different values of the data (i.e., different suitable values of the coefficients involved in the polynomial reaction terms as well as different values of the α i α i alpha_(i)\alpha_{i}αi 's), which will show that the attractive closed orbits (attractors) of the ODE Hanusse-type systems are also attractors for the PDE Hanusse-type systems. In fact there are no other attractors of these PDE systems. According to the usual terminology (see [6]), a Hanusse-type model may become self-organized (i.e., its trajectory gets closer and closer to a limit cycle as t t ttt goes to infinity) for suitable coefficients of the polynomials f i f i f_(i)f_{i}fi. Note that the existence of these attractors is based only on numerical simulations (we only show rigorously the existence of bounded, positively invariant sets). It is well-known that the existence of periodic solutions (in particular the existence of attractive closed orbits) in the case N ≥ 3 N ≥ 3 N >= 3N \geq 3N≥3 is an open problem even for ODE systems so performing simulations is currently the only possibility to investigate the temporal behavior of these models.
Besides the Hanusse-type models investigated in this paper, there have also been reported other self-organized chemical models, including the Brusselator (see [7]) and the Oregonator which is the simplest model of the Belousov-Zhabotinsky
reaction (see [8]). Our analysis could be extended to these models and similar theoretical and numerical results are expected in these cases. We plan to investigate these models in the near future.

2. Transfer of invariance from ODE to PDE systems

Assume that Ω ⊂ R n Ω ⊂ R n Omega subR^(n)\Omega \subset \mathbb{R}^{n}Ω⊂Rn is a bounded domain with smooth boundary ∂ Ω , n ≤ 3 ∂ Ω , n ≤ 3 del Omega,n <= 3\partial \Omega, n \leq 3∂Ω,n≤3. In what follows X X XXX will denote the space L 2 ( Ω ) N L 2 ( Ω ) N L^(2)(Omega)^(N)L^{2}(\Omega)^{N}L2(Ω)N equipped with the norm
‖ v ‖ X := ( ∫ Ω ‖ v ( x ) ‖ 2 d x ) 1 / 2 = ( ∑ i = 1 N ‖ v i ‖ L 2 ( Ω ) 2 ) 1 / 2 ‖ v ‖ X := ∫ Ω   ‖ v ( x ) ‖ 2 d x 1 / 2 = ∑ i = 1 N   v i L 2 ( Ω ) 2 1 / 2 ||v||_(X):=(int_(Omega)||v(x)||^(2)dx)^(1//2)=(sum_(i=1)^(N)||v_(i)||_(L^(2)(Omega))^(2))^(1//2)\|v\|_{X}:=\left(\int_{\Omega}\|v(x)\|^{2} d x\right)^{1 / 2}=\left(\sum_{i=1}^{N}\left\|v_{i}\right\|_{L^{2}(\Omega)}^{2}\right)^{1 / 2}‖v‖X:=(∫Ω‖v(x)‖2dx)1/2=(∑i=1N‖vi‖L2(Ω)2)1/2
for all v = ( v 1 , … , v N ) ∈ X v = v 1 , … , v N ∈ X v=(v_(1),dots,v_(N))in Xv=\left(v_{1}, \ldots, v_{N}\right) \in Xv=(v1,…,vN)∈X. For a set D ⊂ R N D ⊂ R N D subR^(N)D \subset \mathbb{R}^{N}D⊂RN denote
(2.1) D ~ = { v ∈ X : v ( x ) ∈ D for a.a. x ∈ Ω } . (2.1) D ~ = { v ∈ X : v ( x ) ∈ D  for a.a.  x ∈ Ω } . {:(2.1) tilde(D)={v in X:v(x)in D" for a.a. "x in Omega}.:}\begin{equation*} \tilde{D}=\{v \in X: v(x) \in D \text { for a.a. } x \in \Omega\} . \tag{2.1} \end{equation*}(2.1)D~={v∈X:v(x)∈D for a.a. x∈Ω}.
If D D DDD is a closed subset of R N R N R^(N)\mathbb{R}^{N}RN, then obviously D ~ D ~ tilde(D)\tilde{D}D~ is a closed subset of X X XXX. In the next part of this paper, we will assume that D D DDD is compact, so f | D f D f|_(D)\left.f\right|_{D}f|D (the restriction of f f fff to D D DDD ) is a Lipschitz function on D D DDD, with a Lipschitz constant L = L ( D ) > 0 L = L ( D ) > 0 L=L(D) > 0L=L(D)>0L=L(D)>0. In fact, f | D f D f|_(D)\left.f\right|_{D}f|D can be extended to a Lipschitz function on R N R N R^(N)\mathbb{R}^{N}RN (see [9]), again denoted f f fff. So, without any restriction of generality, one can assume that f f fff is Lipschitz on R N R N R^(N)\mathbb{R}^{N}RN with a Lipschitz constant L > 0 L > 0 L > 0L>0L>0.
Denote by f ~ f ~ tilde(f)\tilde{f}f~ the realization of f f fff on X X XXX, i.e., ( f ~ ( v ) ) ( x ) := f ( v ( x ) ) ( f ~ ( v ) ) ( x ) := f ( v ( x ) ) ( tilde(f)(v))(x):=f(v(x))(\tilde{f}(v))(x):=f(v(x))(f~(v))(x):=f(v(x)) for a.a. x ∈ Ω x ∈ Ω x in Omegax \in \Omegax∈Ω. Obviously, f ~ f ~ tilde(f)\tilde{f}f~ is a Lipschitz operator with the same Lipschitz constant L L LLL. We have
Proposition 1. If D ⊂ R N D ⊂ R N D subR^(N)D \subset \mathbb{R}^{N}D⊂RN is a compact set and condition (1.3) is satisfied, then for every u 0 ∈ D ~ u 0 ∈ D ~ u_(0)in tilde(D)u_{0} \in \tilde{D}u0∈D~ (defined by (2.1) above) there exists a unique solution u = u ( t , x ) ∈ C 1 ( [ 0 , ∞ ) ; L ∞ ( Ω ) N ) ⊂ C 1 ( [ 0 , ∞ ) ; X ) N u = u ( t , x ) ∈ C 1 [ 0 , ∞ ) ; L ∞ ( Ω ) N ⊂ C 1 ( [ 0 , ∞ ) ; X ) N u=u(t,x)inC^(1)([0,oo);L^(oo)(Omega)^(N))subC^(1)([0,oo);X)^(N)u=u(t, x) \in C^{1}\left([0, \infty) ; L^{\infty}(\Omega)^{N}\right) \subset C^{1}([0, \infty) ; X)^{N}u=u(t,x)∈C1([0,∞);L∞(Ω)N)⊂C1([0,∞);X)N to the problem
(2.2) d d t u ( t , ⋅ ) = f ~ ( u ( t , ⋅ ) ) , t ≥ 0 (2.3) u ( 0 , ⋅ ) = u 0 ( ⋅ ) (2.2) d d t u ( t , ⋅ ) = f ~ ( u ( t , ⋅ ) ) , t ≥ 0 (2.3) u ( 0 , ⋅ ) = u 0 ( ⋅ ) {:[(2.2)(d)/(dt)u(t","*)= tilde(f)(u(t","*))","quad t >= 0],[(2.3)u(0","*)=u_(0)(*)]:}\begin{align*} \frac{d}{d t} u(t, \cdot) & =\tilde{f}(u(t, \cdot)), \quad t \geq 0 \tag{2.2}\\ u(0, \cdot) & =u_{0}(\cdot) \tag{2.3} \end{align*}(2.2)ddtu(t,⋅)=f~(u(t,⋅)),t≥0(2.3)u(0,⋅)=u0(⋅)
such that u ( t , x ) ∈ D u ( t , x ) ∈ D u(t,x)in Du(t, x) \in Du(t,x)∈D for all t ≥ 0 t ≥ 0 t >= 0t \geq 0t≥0 and a.a. x ∈ Ω x ∈ Ω x in Omegax \in \Omegax∈Ω.
Proof. Denote by X ∞ X ∞ X_(oo)X_{\infty}X∞ the product space L ∞ ( Ω ) N . X ∞ L ∞ ( Ω ) N . X ∞ L^(oo)(Omega)^(N).X_(oo)L^{\infty}(\Omega)^{N} . X_{\infty}L∞(Ω)N.X∞ is a real Banach space with respect to the norm
(2.4) ‖ v ‖ X ∞ = esssup x ∈ Ω ‖ v ( x ) ‖ , ∀ v = ( v 1 , … , v N ) ∈ X ∞ (2.4) ‖ v ‖ X ∞ = esssup x ∈ Ω ‖ v ( x ) ‖ , ∀ v = v 1 , … , v N ∈ X ∞ {:(2.4)||v||_(X_(oo))=esssup_(x in Omega)||v(x)||","quad AA v=(v_(1),dots,v_(N))inX_(oo):}\begin{equation*} \|v\|_{X_{\infty}}=\operatorname{esssup}_{x \in \Omega}\|v(x)\|, \quad \forall v=\left(v_{1}, \ldots, v_{N}\right) \in X_{\infty} \tag{2.4} \end{equation*}(2.4)‖v‖X∞=esssupx∈Ω‖v(x)‖,∀v=(v1,…,vN)∈X∞
Obviously, f ~ f ~ tilde(f)\tilde{f}f~ is a Lipschitz operator from X ∞ X ∞ X_(oo)X_{\infty}X∞ into itself, and consequently there exists a unique solution u ∈ C 1 ( [ 0 , ∞ ) ; X ∞ ) u ∈ C 1 [ 0 , ∞ ) ; X ∞ u inC^(1)([0,oo);X_(oo))u \in C^{1}\left([0, \infty) ; X_{\infty}\right)u∈C1([0,∞);X∞) to problem (2.2) and (2.3) Therefore, for a.a. x ∈ Ω , u ( ⋅ , x ) ∈ C 1 ( [ 0 , ∞ ) ; R N ) x ∈ Ω , u ( ⋅ , x ) ∈ C 1 [ 0 , ∞ ) ; R N x in Omega,u(*,x)inC^(1)([0,oo);R^(N))x \in \Omega, u(\cdot, x) \in C^{1}\left([0, \infty) ; \mathbb{R}^{N}\right)x∈Ω,u(⋅,x)∈C1([0,∞);RN) is a solution of problem (1.1), (1.2) with u 0 := u 0 ( x ) ∈ D u 0 := u 0 ( x ) ∈ D u_(0):=u_(0)(x)in Du_{0}:=u_{0}(x) \in Du0:=u0(x)∈D, and u ( t , x ) ∈ D u ( t , x ) ∈ D u(t,x)in Du(t, x) \in Du(t,x)∈D for all t ≥ 0 t ≥ 0 t >= 0t \geq 0t≥0 (cf. Theorem 1).
Now, let us consider the following differential equation in X X XXX
(2.5) u ′ ( t ) = A u ( t ) + f ~ ( u ( t ) ) , t ≥ 0 (2.5) u ′ ( t ) = A u ( t ) + f ~ ( u ( t ) ) , t ≥ 0 {:(2.5)u^(')(t)=Au(t)+ tilde(f)(u(t))","quad t >= 0:}\begin{equation*} u^{\prime}(t)=A u(t)+\tilde{f}(u(t)), \quad t \geq 0 \tag{2.5} \end{equation*}(2.5)u′(t)=Au(t)+f~(u(t)),t≥0
with the initial condition
(2.6) u ( 0 ) = u 0 ∈ D ~ (2.6) u ( 0 ) = u 0 ∈ D ~ {:(2.6)u(0)=u_(0)in tilde(D):}\begin{equation*} u(0)=u_{0} \in \tilde{D} \tag{2.6} \end{equation*}(2.6)u(0)=u0∈D~
where A : D ( A ) ⊂ X → X A : D ( A ) ⊂ X → X A:D(A)sub X rarr XA: D(A) \subset X \rightarrow XA:D(A)⊂X→X is the infinitesimal generator of a C 0 C 0 C_(0)C_{0}C0-semigroup of linear operators { S ( t ) : X → X , t ≥ 0 } { S ( t ) : X → X , t ≥ 0 } {S(t):X rarr X,t >= 0}\{S(t): X \rightarrow X, t \geq 0\}{S(t):X→X,t≥0}. For the basic theory of C 0 C 0 C_(0)C_{0}C0-semigroups see, e.g., [10-12] or [13].
Recall that u ∈ C ( [ 0 , T ) ; X ) , 0 < T ≤ + ∞ u ∈ C ( [ 0 , T ) ; X ) , 0 < T ≤ + ∞ u in C([0,T);X),0 < T <= +oou \in C([0, T) ; X), 0<T \leq+\inftyu∈C([0,T);X),0<T≤+∞, is said to be a mild solution of problem (2.5), (2.6) if
(2.7) u ( t ) = S ( t ) u 0 + ∫ 0 t S ( t − s ) f ~ ( u ( s ) ) d s , 0 ≤ t < T (2.7) u ( t ) = S ( t ) u 0 + ∫ 0 t   S ( t − s ) f ~ ( u ( s ) ) d s , 0 ≤ t < T {:(2.7)u(t)=S(t)u_(0)+int_(0)^(t)S(t-s) tilde(f)(u(s))ds","quad0 <= t < T:}\begin{equation*} u(t)=S(t) u_{0}+\int_{0}^{t} S(t-s) \tilde{f}(u(s)) d s, \quad 0 \leq t<T \tag{2.7} \end{equation*}(2.7)u(t)=S(t)u0+∫0tS(t−s)f~(u(s))ds,0≤t<T
If problem (2.5), (2.6) has a strong solution u u uuu on [ 0 , T ) [ 0 , T ) [0,T)[0, T)[0,T) (in the usual sense, see, e.g., [14, p. 27] or [10, Chap. 4]), then u u uuu is also a mild solution of this problem on [ 0 , T ) [ 0 , T ) [0,T)[0, T)[0,T).
Lemma 1. Assume that D ⊂ R N D ⊂ R N D subR^(N)D \subset \mathbb{R}^{N}D⊂RN is a compact set, condition (1.3) is satisfied, and A A AAA is the infinitesimal generator of a C 0 C 0 C_(0)C_{0}C0-semigroup { S ( t ) : X → X , t ≥ 0 } { S ( t ) : X → X , t ≥ 0 } {S(t):X rarr X,t >= 0}\{S(t): X \rightarrow X, t \geq 0\}{S(t):X→X,t≥0} such that S ( t ) S ( t ) S(t)S(t)S(t) is a compact operator for all t > 0 t > 0 t > 0t>0t>0 and S ( t ) D ~ ⊂ D ~ S ( t ) D ~ ⊂ D ~ S(t) tilde(D)sub tilde(D)S(t) \tilde{D} \subset \tilde{D}S(t)D~⊂D~ for all t ≥ 0 t ≥ 0 t >= 0t \geq 0t≥0. Then, for every u 0 ∈ D ~ u 0 ∈ D ~ u_(0)in tilde(D)u_{0} \in \tilde{D}u0∈D~ there exists a unique mild solution u u uuu of problem (2.5), (2.6) with D ( u ) = [ 0 , ∞ ) D ( u ) = [ 0 , ∞ ) D(u)=[0,oo)D(u)=[0, \infty)D(u)=[0,∞) such that u ( t ) ∈ D ~ u ( t ) ∈ D ~ u(t)in tilde(D)u(t) \in \tilde{D}u(t)∈D~ for all t ≥ 0 t ≥ 0 t >= 0t \geq 0t≥0.
Proof. Since f ~ f ~ tilde(f)\tilde{f}f~ is a Lipschitz operator on X X XXX, it follows that for each u 0 ∈ X u 0 ∈ X u_(0)in Xu_{0} \in Xu0∈X there exists a unique mild solution u ∈ C ( [ 0 , ∞ ) ; X ) u ∈ C ( [ 0 , ∞ ) ; X ) u in C([0,oo);X)u \in C([0, \infty) ; X)u∈C([0,∞);X) to problem (2.5), (2.6) (cf., e.g., [14, Theorem 2.0.28, p. 30]). In order to prove that the trajectory of u u uuu lies in D ~ D ~ tilde(D)\tilde{D}D~, we shall make use of [ 15 [ 15 [15[15[15, Theorem 1.1]. To this purpose, let us show that f ~ f ~ tilde(f)\tilde{f}f~ and D ~ D ~ tilde(D)\tilde{D}D~ satisfy the tangency condition
(2.8) lim h ↓ 0 1 h d X ( S ( h ) v + h f ~ ( v ) , D ~ ) = 0 , ∀ v ∈ D ~ (2.8) lim h ↓ 0   1 h d X ( S ( h ) v + h f ~ ( v ) , D ~ ) = 0 , ∀ v ∈ D ~ {:(2.8)lim_(h darr0)(1)/(h)d_(X)(S(h)v+h tilde(f)(v)"," tilde(D))=0","quad AA v in tilde(D):}\begin{equation*} \lim _{h \downarrow 0} \frac{1}{h} d_{X}(S(h) v+h \tilde{f}(v), \tilde{D})=0, \quad \forall v \in \tilde{D} \tag{2.8} \end{equation*}(2.8)limh↓01hdX(S(h)v+hf~(v),D~)=0,∀v∈D~
where d X ( w , D ~ ) := inf y ∈ D ~ ‖ w − y ‖ X d X ( w , D ~ ) := inf y ∈ D ~   ‖ w − y ‖ X d_(X)(w, tilde(D)):=i n f_(y in tilde(D))||w-y||_(X)d_{X}(w, \tilde{D}):=\inf _{y \in \tilde{D}}\|w-y\|_{X}dX(w,D~):=infy∈D~‖w−y‖X. By Remark 1 and Proposition 1, we have
(2.9) lim h ↓ 0 1 h d X ( v + h f ~ ( v ) , D ~ ) = 0 , ∀ v ∈ D ~ . (2.9) lim h ↓ 0   1 h d X ( v + h f ~ ( v ) , D ~ ) = 0 , ∀ v ∈ D ~ . {:(2.9)lim_(h darr0)(1)/(h)d_(X)(v+h tilde(f)(v)"," tilde(D))=0","quad AA v in tilde(D).:}\begin{equation*} \lim _{h \downarrow 0} \frac{1}{h} d_{X}(v+h \tilde{f}(v), \tilde{D})=0, \quad \forall v \in \tilde{D} . \tag{2.9} \end{equation*}(2.9)limh↓01hdX(v+hf~(v),D~)=0,∀v∈D~.
It is easily seen that (2.9) implies (2.8). Indeed, for h > 0 h > 0 h > 0h>0h>0 and v , w ∈ D ~ v , w ∈ D ~ v,w in tilde(D)v, w \in \tilde{D}v,w∈D~, we have
(2.10) ‖ S ( h ) v + h f ~ ( v ) − S ( h ) w ‖ X ≤ ‖ S ( h ) [ v + h f ~ ( v ) − w ] ‖ X + h ‖ f ~ ( v ) − S ( h ) f ~ ( v ) ‖ X (2.10) ‖ S ( h ) v + h f ~ ( v ) − S ( h ) w ‖ X ≤ ‖ S ( h ) [ v + h f ~ ( v ) − w ] ‖ X + h ‖ f ~ ( v ) − S ( h ) f ~ ( v ) ‖ X {:(2.10)||S(h)v+h tilde(f)(v)-S(h)w||_(X) <= ||S(h)[v+h tilde(f)(v)-w]||_(X)+h|| tilde(f)(v)-S(h) tilde(f)(v)||_(X):}\begin{equation*} \|S(h) v+h \tilde{f}(v)-S(h) w\|_{X} \leq\|S(h)[v+h \tilde{f}(v)-w]\|_{X}+h\|\tilde{f}(v)-S(h) \tilde{f}(v)\|_{X} \tag{2.10} \end{equation*}(2.10)‖S(h)v+hf~(v)−S(h)w‖X≤‖S(h)[v+hf~(v)−w]‖X+h‖f~(v)−S(h)f~(v)‖X
and, since S ( h ) w ∈ D ~ S ( h ) w ∈ D ~ S(h)w in tilde(D)S(h) w \in \tilde{D}S(h)w∈D~ and ‖ S ( h ) ‖ ≤ M e ω h ‖ S ( h ) ‖ ≤ M e ω h ||S(h)|| <= Me^(omega h)\|S(h)\| \leq M e^{\omega h}‖S(h)‖≤Meωh for some M ≥ 1 M ≥ 1 M >= 1M \geq 1M≥1 and ω ∈ R ω ∈ R omega inR\omega \in \mathbb{R}ω∈R (note that such an estimate holds for every C 0 C 0 C_(0)C_{0}C0-semigroup), we derive from (2.10)
1 h d X ( S ( h ) v + h f ~ ( v ) , D ~ ) ≤ M e ω h 1 h d X ( v + h f ~ ( v ) , D ~ ) + ‖ f ~ ( v ) − S ( h ) f ~ ( v ) ‖ X 1 h d X ( S ( h ) v + h f ~ ( v ) , D ~ ) ≤ M e ω h 1 h d X ( v + h f ~ ( v ) , D ~ ) + ‖ f ~ ( v ) − S ( h ) f ~ ( v ) ‖ X (1)/(h)d_(X)(S(h)v+h tilde(f)(v), tilde(D)) <= Me^(omega h)(1)/(h)d_(X)(v+h tilde(f)(v), tilde(D))+|| tilde(f)(v)-S(h) tilde(f)(v)||_(X)\frac{1}{h} d_{X}(S(h) v+h \tilde{f}(v), \tilde{D}) \leq M e^{\omega h} \frac{1}{h} d_{X}(v+h \tilde{f}(v), \tilde{D})+\|\tilde{f}(v)-S(h) \tilde{f}(v)\|_{X}1hdX(S(h)v+hf~(v),D~)≤Meωh1hdX(v+hf~(v),D~)+‖f~(v)−S(h)f~(v)‖X
which shows that (2.9) implies (2.8), as claimed. Now, since (2.8) holds, it follows by [15, Theorem 1.1] that for each u 0 ∈ D ~ u 0 ∈ D ~ u_(0)in tilde(D)u_{0} \in \tilde{D}u0∈D~, there exists a local mild solution u : [ 0 , a ] → X u : [ 0 , a ] → X u:[0,a]rarr Xu:[0, a] \rightarrow Xu:[0,a]→X to problem (2.5), (2.6) whose trajectory is included into D ~ D ~ tilde(D)\tilde{D}D~. This solution is unique (in fact there exists a unique mild solution defined on [ 0 , ∞ ) [ 0 , ∞ ) [0,oo)[0, \infty)[0,∞), as remarked above) and can be extended uniquely in the future to a maximal interval, say [ 0 , a max ) 0 , a max [0,a_(max))\left[0, a_{\max }\right)[0,amax), such that its trajectory remains in D ~ D ~ tilde(D)\tilde{D}D~. It is easily seen that a max a max a_(max)a_{\max }amax cannot be finite. Indeed, in this case the mild solution could be extended to the closed interval [ 0 , a max ] 0 , a max [0,a_(max)]\left[0, a_{\max }\right][0,amax] with u ( a max ) ∈ D ~ u a max ∈ D ~ u(a_(max))in tilde(D)u\left(a_{\max }\right) \in \tilde{D}u(amax)∈D~, and eventually to a larger interval, so contradicting the maximality of a max a max  a_("max ")a_{\text {max }}amax . Therefore a max = + ∞ a max  = + ∞ a_("max ")=+ooa_{\text {max }}=+\inftyamax =+∞.
Lemma 2. If { T ( t ) : L 2 ( Ω ) → L 2 ( Ω ) ; t ≥ 0 } T ( t ) : L 2 ( Ω ) → L 2 ( Ω ) ; t ≥ 0 {T(t):L^(2)(Omega)rarrL^(2)(Omega);t >= 0}\left\{T(t): L^{2}(\Omega) \rightarrow L^{2}(\Omega) ; t \geq 0\right\}{T(t):L2(Ω)→L2(Ω);t≥0} is the C 0 C 0 C_(0)C_{0}C0-semigroup generated by − α Δ , α > 0 − α Δ , α > 0 -alpha Delta,alpha > 0-\alpha \Delta, \alpha>0−αΔ,α>0, with the homogeneous Neumann boundary condition, and F F FFF is the set { θ ∈ L 2 ( Ω ) ; a ≤ θ ( x ) ≤ a ¯ θ ∈ L 2 ( Ω ) ; a ≤ θ ( x ) ≤ a ¯ {theta inL^(2)(Omega);a <= theta(x) <= ( bar(a)):}\left\{\theta \in L^{2}(\Omega) ; a \leq \theta(x) \leq \bar{a}\right.{θ∈L2(Ω);a≤θ(x)≤a¯, for a.a. x ∈ Ω } , a , a ¯ ∈ R , a < a ¯ x ∈ Ω , a , a ¯ ∈ R , a < a ¯ {:x in Omega},a, bar(a)inR,a < bar(a)\left.x \in \Omega\right\}, a, \bar{a} \in \mathbb{R}, a<\bar{a}x∈Ω},a,a¯∈R,a<a¯, then T ( t ) F ⊂ F T ( t ) F ⊂ F T(t)F sub FT(t) F \subset FT(t)F⊂F, for all t ≥ 0 t ≥ 0 t >= 0t \geq 0t≥0.
Proof. Obviously, for any constant c ∈ R c ∈ R c inRc \in \mathbb{R}c∈R, we have T ( t ) c = c T ( t ) c = c T(t)c=cT(t) c=cT(t)c=c for all t ≥ 0 t ≥ 0 t >= 0t \geq 0t≥0. Using the positivity preserving of the Neumann heat semigroup we can write for θ ∈ F θ ∈ F theta in F\theta \in Fθ∈F and t ≥ 0 t ≥ 0 t >= 0t \geq 0t≥0
0 ≤ T ( t ) ( θ ( ⋅ ) − a ) = T ( t ) θ − a , 0 ≤ T ( t ) ( a ¯ − θ ( ⋅ ) ) = a ¯ − T ( t ) θ a.e. in Ω 0 ≤ T ( t ) ( θ ( ⋅ ) − a ) = T ( t ) θ − a , 0 ≤ T ( t ) ( a ¯ − θ ( ⋅ ) ) = a ¯ − T ( t ) θ  a.e. in  Ω 0 <= T(t)(theta(*)-a)=T(t)theta-a,quad0 <= T(t)( bar(a)-theta(*))= bar(a)-T(t)thetaquad" a.e. in "Omega0 \leq T(t)(\theta(\cdot)-a)=T(t) \theta-a, \quad 0 \leq T(t)(\bar{a}-\theta(\cdot))=\bar{a}-T(t) \theta \quad \text { a.e. in } \Omega0≤T(t)(θ(⋅)−a)=T(t)θ−a,0≤T(t)(a¯−θ(⋅))=a¯−T(t)θ a.e. in Ω
i.e., T ( t ) θ ∈ F T ( t ) θ ∈ F T(t)theta in FT(t) \theta \in FT(t)θ∈F for all t ≥ 0 t ≥ 0 t >= 0t \geq 0t≥0.
Next, we are going to exploit the above information in order to establish the following existence and invariance result for problem (1.6), (1.7), (1.8):
Theorem 2. Assume that f : R N → R N , N ≥ 3 f : R N → R N , N ≥ 3 f:R^(N)rarrR^(N),N >= 3f: \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}, N \geq 3f:RN→RN,N≥3, is locally Lipschitz, D ⊂ R N D ⊂ R N D subR^(N)D \subset \mathbb{R}^{N}D⊂RN defined by (1.4) is positively invariant with respect to Eq. (1.1), and Ω ⊂ R n , n ≤ 3 Ω ⊂ R n , n ≤ 3 Omega subR^(n),n <= 3\Omega \subset \mathbb{R}^{n}, n \leq 3Ω⊂Rn,n≤3, is an open bounded set with smooth boundary ∂ Ω ∂ Ω del Omega\partial \Omega∂Ω. Then, for every u 0 ∈ D ~ u 0 ∈ D ~ u_(0)in tilde(D)u_{0} \in \tilde{D}u0∈D~ (defined by (2.1) above, with the set D D DDD given by (1.4)) there exists a unique strong solution u u uuu to problem (1.6), (1.7), (1.8) (expressed as the Cauchy problem in X ( 2.5 ) , ( 2.6 ) X ( 2.5 ) , ( 2.6 ) X(2.5),(2.6)X(2.5),(2.6)X(2.5),(2.6) ), defined on [ 0 , ∞ ) × Ω [ 0 , ∞ ) × Ω [0,oo)xx Omega[0, \infty) \times \Omega[0,∞)×Ω, with u ( t , x ) ∈ D u ( t , x ) ∈ D u(t,x)in Du(t, x) \in Du(t,x)∈D for all t ≥ 0 t ≥ 0 t >= 0t \geq 0t≥0 and a.a. x ∈ Ω x ∈ Ω x in Omegax \in \Omegax∈Ω.
Proof. Denote S ( t ) = ( T 1 ( t ) , … , T N ( t ) ) : X → X , t ≥ 0 S ( t ) = T 1 ( t ) , … , T N ( t ) : X → X , t ≥ 0 S(t)=(T_(1)(t),dots,T_(N)(t)):X rarr X,t >= 0S(t)=\left(T_{1}(t), \ldots, T_{N}(t)\right): X \rightarrow X, t \geq 0S(t)=(T1(t),…,TN(t)):X→X,t≥0, where { T i ( t ) : L 2 ( Ω ) → L 2 ( Ω ) ; t ≥ 0 } T i ( t ) : L 2 ( Ω ) → L 2 ( Ω ) ; t ≥ 0 {T_(i)(t):L^(2)(Omega)rarrL^(2)(Omega);t >= 0}\left\{T_{i}(t): L^{2}(\Omega) \rightarrow L^{2}(\Omega) ; t \geq 0\right\}{Ti(t):L2(Ω)→L2(Ω);t≥0} is the C 0 C 0 C_(0)C_{0}C0-semigroup generated by − α i Δ , α i > 0 , i = 1 , N ― − α i Δ , α i > 0 , i = 1 , N ¯ -alpha_(i)Delta,alpha_(i) > 0,i= bar(1,N)-\alpha_{i} \Delta, \alpha_{i}>0, i=\overline{1, N}−αiΔ,αi>0,i=1,N―, with the homogeneous Neumann boundary condition. By Lemma 2, we have S ( t ) D ~ ⊂ D ~ S ( t ) D ~ ⊂ D ~ S(t) tilde(D)sub tilde(D)S(t) \tilde{D} \subset \tilde{D}S(t)D~⊂D~, for all t ≥ 0 t ≥ 0 t >= 0t \geq 0t≥0. On the other hand, as it is well-known, T i ( t ) T i ( t ) T_(i)(t)T_{i}(t)Ti(t) is a compact operator for all i = 1 , N ― , t > 0 i = 1 , N ¯ , t > 0 i= bar(1,N),t > 0i=\overline{1, N}, t>0i=1,N―,t>0, hence S ( t ) : X → X S ( t ) : X → X S(t):X rarr XS(t): X \rightarrow XS(t):X→X is also compact for all t > 0 t > 0 t > 0t>0t>0. Therefore, for any u 0 ∈ D ~ u 0 ∈ D ~ u_(0)in tilde(D)u_{0} \in \tilde{D}u0∈D~, Lemma 1 insures the existence of a unique mild solution u u uuu to problem (1.6), (1.7), (1.8) with values in D D DDD. In fact u u uuu is a strong solution with t 1 / 2 u ( t , ⋅ ) ∈ L 2 ( Ω ) N t 1 / 2 u ( t , ⋅ ) ∈ L 2 ( Ω ) N t^(1//2)u(t,*)inL^(2)(Omega)^(N)t^{1 / 2} u(t, \cdot) \in L^{2}(\Omega)^{N}t1/2u(t,⋅)∈L2(Ω)N (cf. [16, Prop. 3.12, p. 106] and [17, Prop. 2.9, p. 63]).
Remark 3. Obviously, any periodic solution p = p ( t ) p = p ( t ) p=p(t)p=p(t)p=p(t) of Eq. (1.1) defining an attractive closed orbit (attractor) is also a solution of system (1.6) with the same f = ( f 1 , … , f N ) f = f 1 , … , f N f=(f_(1),dots,f_(N))f=\left(f_{1}, \ldots, f_{N}\right)f=(f1,…,fN) (since p p ppp is independent of x x xxx ). We expect that, for t t ttt large, the orbit of the solution u ( t , ⋅ ) u ( t , ⋅ ) u(t,*)u(t, \cdot)u(t,⋅) of the Hanusse system (1.6), (1.7), (1.8) starting from any u 0 ∈ D ~ u 0 ∈ D ~ u_(0)in tilde(D)u_{0} \in \tilde{D}u0∈D~ gets closer and closer (with respect to ‖ ⋅ ‖ X ‖ ⋅ ‖ X ||*||_(X)\|\cdot\|_{X}‖⋅‖X ) to an attractive orbit defined by a periodic solution p = p ( t ) p = p ( t ) p=p(t)p=p(t)p=p(t) of Eq. (1.1) with the same f = ( f 1 , … , f N ) f = f 1 , … , f N f=(f_(1),dots,f_(N))f=\left(f_{1}, \ldots, f_{N}\right)f=(f1,…,fN); in other words, there are no attractors of system (1.6), other than those corresponding to Eq. (1.1) with the same f = ( f 1 , … , f N ) f = f 1 , … , f N f=(f_(1),dots,f_(N))f=\left(f_{1}, \ldots, f_{N}\right)f=(f1,…,fN). We are going to show by numerical simulations that this assertion holds true (see Section 3 where specific comments are included).
Remark 4. Since the reaction terms f i f i f_(i)f_{i}fi are polynomial functions and all the inequalities in (1.5) are strict, it follows that these inequalities remain valid for small variations of the a i a i a_(i)a_{i}ai 's, so the invariant sets D D DDD are stable to small perturbations.
Note that the content of this section is partially based on known material which is here adapted to our specific framework. For related material, we refer the reader to [18,19], and the references therein.

3. Numerical simulations

Previous results in [3] have shown the existence of attractors for the Hanusse-type ODE models with certain coefficients of the polynomial reaction terms. We perform numerical simulations for both the ODE and PDE models, noticing that the attractors of the ODE systems are also attractors, in fact the only attractors, for the corresponding PDE systems. The initial values for each component of every system have been randomly generated in a subset of the positive orthant following a uniform distribution. Small perturbations of the coefficients in the polynomial expressions and of the α i α i alpha_(i)\alpha_{i}αi 's in the PDE systems have been considered, leading to the same limit cycle behavior. Multiple simulations have given the same results in terms of attractors, although we have not followed a statistical approach. We shall give results only for certain coefficients and for certain domains, with some insight into the solutions' transient behavior.

3.1. ODE models

We first consider the ODE systems (Hanusse-type models) previously analyzed in [3]:
\begin{align*} \frac{d}{d t}\left(\begin{array}{l} u_{1} \\ u_{2} \\ u_{3} \end{array}\right) & =\left(\begin{array}{c} u_{1}-0.1 u_{1}^{2}-u_{1} u_{2}+0.1 u_{2}^{2} \\ u_{1} u_{2}-0.1 u_{2}^{2}-u_{2} u_{3}+0.1 u_{3}^{2} \\ u_{2} u_{3}-0.1 u_{3}^{2}-u_{1} u_{3}+0.005 u_{1} \end{array}\right) \tag{$H\cdotDE$}\\ \frac{d}{d t}\left(\begin{array}{l} u_{1} \\ u_{2} \\ u_{3} \\ u_{4} \end{array}\right) & =\left(\begin{array}{c} u_{1}-0.1 u_{1}^{2}-u_{1} u_{2}+0.1 u_{2}^{2} \\ u_{1} u_{2}-0.1 u_{2}^{2}-u_{2} u_{3}+0.1 u_{3}^{2} \\ u_{2} u_{3}-0.1 u_{3}^{2}-u_{3} u_{4}+0.1 u_{4}^{2}-u_{1} u_{3}+0.005 u_{1} \\ u_{3} u_{4}-0.1 u_{4}^{2}-u_{2} u_{4}+0.1 u_{2} \end{array}\right) \tag{1}\\ \frac{d}{d t}\left(\begin{array}{l} u_{1} \\ u_{2} \\ u_{3} \\ u_{4} \\ u_{5} \end{array}\right) & =\left(\begin{array}{c} u_{1}-0.1 u_{1}^{2}-u_{1} u_{2}+0.1 u_{2}^{2} \\ u_{1} u_{2}-0.1 u_{2}^{2}-u_{2} u_{3}+0.1 u_{3}^{2} \\ u_{2} u_{3}-0.1 u_{3}^{2}-u_{3} u_{4}+0.1 u_{4}^{2}-u_{1} u_{3}+0.005 u_{1} \\ u_{3} u_{4}-0.2 u_{4}^{2}-u_{2} u_{4}+0.1 u_{5}^{2}-u_{4} u_{5}+0.1 u_{2}+u_{4} \\ u_{4} u_{5}-0.1 u_{5}^{2}-u_{3} u_{5}+0.1 u_{3} \end{array}\right) \tag{2}\\ \frac{d}{d t}\left(\begin{array}{l} u_{1} \\ u_{2} \\ u_{3} \\ u_{4} \\ u_{5} \\ u_{6} \end{array}\right) & =\left(\begin{array}{c} u_{1}-0.1 u_{1}^{2}-u_{1} u_{2}+0.1 u_{2}^{2} \\ u_{1} u_{2}-0.1 u_{2}^{2}-u_{2} u_{3}+0.1 u_{3}^{2} \\ u_{2} u_{3}-0.1 u_{3}^{2}-u_{3} u_{4}+0.1 u_{4}^{2}-u_{1} u_{3}+0.005 u_{1} \\ u_{3} u_{4}-0.2 u_{4}^{2}-u_{2} u_{4}+0.1 u_{5}^{2}-u_{4} u_{5}+0.1 u_{2}+u_{4} \\ u_{4} u_{5}-0.2 u_{5}^{2}-u_{3} u_{5}+0.1 u_{6}^{2}-u_{5} u_{6}+0.1 u_{3}+u_{5} \\ u_{5} u_{6}-0.1 u_{6}^{2}-u_{4} u_{6}+0.1 u_{4} \end{array}\right) \tag{3} \end{align*}Undefined control sequence \cdotDE
As already noticed in [3], numerical simulations indicate the existence of limit cycles for trajectories starting from the positively invariant set D D DDD. Fig. 1 gives the projections of the limit cycles on various phase planes when starting for each component with random initial data uniformly distributed in the interval [ 0.1 , 100 ] [ 0.1 , 100 ] [0.1,100][0.1,100][0.1,100]. We have integrated on the time interval [ 0 , 10 5 ] 0 , 10 5 [0,10^(5)]\left[0,10^{5}\right][0,105], using the Runge-Kutta solver ode45 in MATLAB with reltol = 10 − 6 = 10 − 6 =10^(-6)=10^{-6}=10−6 and abstol = 10 − 9 = 10 − 9 =10^(-9)=10^{-9}=10−9.

3.2. PDE models

Let us start with the case of one space dimension PDE systems by taking Ω = [ a , b ] Ω = [ a , b ] Omega=[a,b]\Omega=[a, b]Ω=[a,b]. With diffusion taken into account the Hanusse-type models become
\frac{\partial}{\partial t}\left(\begin{array}{l} u_{1} \tag{$H\cdotPDE$}\\ u_{2} \\ u_{3} \end{array}\right)=\left(\begin{array}{c} \alpha_{1} \frac{\partial^{2} u_{1}}{\partial x^{2}} \\ \alpha_{2} \frac{\partial^{2} u_{2}}{\partial x^{2}} \\ \alpha_{3} \frac{\partial^{2} u_{3}}{\partial x^{2}} \end{array}\right)+\left(\begin{array}{c} u_{1}-0.1 u_{1}^{2}-u_{1} u_{2}+0.1 u_{2}^{2} \\ u_{1} u_{2}-0.1 u_{2}^{2}-u_{2} u_{3}+0.1 u_{3}^{2} \\ u_{2} u_{3}-0.1 u_{3}^{2}-u_{1} u_{3}+0.005 u_{1} \end{array}\right)Undefined control sequence \cdotPDE
Fig. 1. Limit cycles for the ODE systems.
∂ ∂ t ( u 1 u 2 u 3 u 4 ) = ( α 1 ∂ 2 u 1 ∂ x 2 α 2 ∂ 2 u 2 ∂ x 2 α 3 ∂ 2 u 3 ∂ x 2 α 4 ∂ 2 u 4 ∂ x 2 ) + ( u 1 − 0.1 u 1 2 − u 1 u 2 + 0.1 u 2 2 u 1 u 2 − 0.1 u 2 2 − u 2 u 3 + 0.1 u 3 2 u 2 u 3 − 0.1 u 3 2 − u 3 u 4 + 0.1 u 4 2 − u 1 u 3 + 0.005 u 1 u 3 u 4 − 0.1 u 4 2 − u 2 u 4 + 0.1 u 2 ) ∂ ∂ t u 1 u 2 u 3 u 4 = α 1 ∂ 2 u 1 ∂ x 2 α 2 ∂ 2 u 2 ∂ x 2 α 3 ∂ 2 u 3 ∂ x 2 α 4 ∂ 2 u 4 ∂ x 2 + u 1 − 0.1 u 1 2 − u 1 u 2 + 0.1 u 2 2 u 1 u 2 − 0.1 u 2 2 − u 2 u 3 + 0.1 u 3 2 u 2 u 3 − 0.1 u 3 2 − u 3 u 4 + 0.1 u 4 2 − u 1 u 3 + 0.005 u 1 u 3 u 4 − 0.1 u 4 2 − u 2 u 4 + 0.1 u 2 (del)/(del t)([u_(1)],[u_(2)],[u_(3)],[u_(4)])=([alpha_(1)(del^(2)u_(1))/(delx^(2))],[alpha_(2)(del^(2)u_(2))/(delx^(2))],[alpha_(3)(del^(2)u_(3))/(delx^(2))],[alpha_(4)(del^(2)u_(4))/(delx^(2))])+([u_(1)-0.1u_(1)^(2)-u_(1)u_(2)+0.1u_(2)^(2)],[u_(1)u_(2)-0.1u_(2)^(2)-u_(2)u_(3)+0.1u_(3)^(2)],[u_(2)u_(3)-0.1u_(3)^(2)-u_(3)u_(4)+0.1u_(4)^(2)-u_(1)u_(3)+0.005u_(1)],[u_(3)u_(4)-0.1u_(4)^(2)-u_(2)u_(4)+0.1u_(2)])\frac{\partial}{\partial t}\left(\begin{array}{l}u_{1} \\ u_{2} \\ u_{3} \\ u_{4}\end{array}\right)=\left(\begin{array}{c}\alpha_{1} \frac{\partial^{2} u_{1}}{\partial x^{2}} \\ \alpha_{2} \frac{\partial^{2} u_{2}}{\partial x^{2}} \\ \alpha_{3} \frac{\partial^{2} u_{3}}{\partial x^{2}} \\ \alpha_{4} \frac{\partial^{2} u_{4}}{\partial x^{2}}\end{array}\right)+\left(\begin{array}{c}u_{1}-0.1 u_{1}^{2}-u_{1} u_{2}+0.1 u_{2}^{2} \\ u_{1} u_{2}-0.1 u_{2}^{2}-u_{2} u_{3}+0.1 u_{3}^{2} \\ u_{2} u_{3}-0.1 u_{3}^{2}-u_{3} u_{4}+0.1 u_{4}^{2}-u_{1} u_{3}+0.005 u_{1} \\ u_{3} u_{4}-0.1 u_{4}^{2}-u_{2} u_{4}+0.1 u_{2}\end{array}\right)∂∂t(u1u2u3u4)=(α1∂2u1∂x2α2∂2u2∂x2α3∂2u3∂x2α4∂2u4∂x2)+(u1−0.1u12−u1u2+0.1u22u1u2−0.1u22−u2u3+0.1u32u2u3−0.1u32−u3u4+0.1u42−u1u3+0.005u1u3u4−0.1u42−u2u4+0.1u2),
( E 1 ⋅ E 1 ⋅ E_(1)*E_{1} \cdotE1⋅ PDE )
∂ ∂ t ( u 1 u 2 u 3 u 4 u 5 ) = ( α 1 ∂ 2 u 1 ∂ x 2 α 2 ∂ 2 u 2 ∂ x 2 α 3 ∂ 2 u 3 ∂ x 2 α 4 ∂ 2 u 4 ∂ x 2 α 5 ∂ 2 u 5 ∂ x 2 ) + ( u 1 − 0.1 u 1 2 − u 1 u 2 + 0.1 u 2 2 u 1 u 2 − 0.1 u 2 2 − u 2 u 3 + 0.1 u 3 2 u 2 u 3 − 0.1 u 3 2 − u 3 u 4 + 0.1 u 4 2 − u 1 u 3 + 0.005 u 1 u 3 u 4 − 0.2 u 4 2 − u 2 u 4 + 0.1 u 5 2 − u 4 u 5 + 0.1 u 2 + u 4 u 4 u 5 − 0.1 u 5 2 − u 3 u 5 + 0.1 u 3 ) ∂ ∂ t u 1 u 2 u 3 u 4 u 5 = α 1 ∂ 2 u 1 ∂ x 2 α 2 ∂ 2 u 2 ∂ x 2 α 3 ∂ 2 u 3 ∂ x 2 α 4 ∂ 2 u 4 ∂ x 2 α 5 ∂ 2 u 5 ∂ x 2 + u 1 − 0.1 u 1 2 − u 1 u 2 + 0.1 u 2 2 u 1 u 2 − 0.1 u 2 2 − u 2 u 3 + 0.1 u 3 2 u 2 u 3 − 0.1 u 3 2 − u 3 u 4 + 0.1 u 4 2 − u 1 u 3 + 0.005 u 1 u 3 u 4 − 0.2 u 4 2 − u 2 u 4 + 0.1 u 5 2 − u 4 u 5 + 0.1 u 2 + u 4 u 4 u 5 − 0.1 u 5 2 − u 3 u 5 + 0.1 u 3 (del)/(del t)([u_(1)],[u_(2)],[u_(3)],[u_(4)],[u_(5)])=([alpha_(1)(del^(2)u_(1))/(delx^(2))],[alpha_(2)(del^(2)u_(2))/(delx^(2))],[alpha_(3)(del^(2)u_(3))/(delx^(2))],[alpha_(4)(del^(2)u_(4))/(delx^(2))],[alpha_(5)(del^(2)u_(5))/(delx^(2))])+([u_(1)-0.1u_(1)^(2)-u_(1)u_(2)+0.1u_(2)^(2)],[u_(1)u_(2)-0.1u_(2)^(2)-u_(2)u_(3)+0.1u_(3)^(2)],[u_(2)u_(3)-0.1u_(3)^(2)-u_(3)u_(4)+0.1u_(4)^(2)-u_(1)u_(3)+0.005u_(1)],[u_(3)u_(4)-0.2u_(4)^(2)-u_(2)u_(4)+0.1u_(5)^(2)-u_(4)u_(5)+0.1u_(2)+u_(4)],[u_(4)u_(5)-0.1u_(5)^(2)-u_(3)u_(5)+0.1u_(3)])\frac{\partial}{\partial t}\left(\begin{array}{l}u_{1} \\ u_{2} \\ u_{3} \\ u_{4} \\ u_{5}\end{array}\right)=\left(\begin{array}{c}\alpha_{1} \frac{\partial^{2} u_{1}}{\partial x^{2}} \\ \alpha_{2} \frac{\partial^{2} u_{2}}{\partial x^{2}} \\ \alpha_{3} \frac{\partial^{2} u_{3}}{\partial x^{2}} \\ \alpha_{4} \frac{\partial^{2} u_{4}}{\partial x^{2}} \\ \alpha_{5} \frac{\partial^{2} u_{5}}{\partial x^{2}}\end{array}\right)+\left(\begin{array}{c}u_{1}-0.1 u_{1}^{2}-u_{1} u_{2}+0.1 u_{2}^{2} \\ u_{1} u_{2}-0.1 u_{2}^{2}-u_{2} u_{3}+0.1 u_{3}^{2} \\ u_{2} u_{3}-0.1 u_{3}^{2}-u_{3} u_{4}+0.1 u_{4}^{2}-u_{1} u_{3}+0.005 u_{1} \\ u_{3} u_{4}-0.2 u_{4}^{2}-u_{2} u_{4}+0.1 u_{5}^{2}-u_{4} u_{5}+0.1 u_{2}+u_{4} \\ u_{4} u_{5}-0.1 u_{5}^{2}-u_{3} u_{5}+0.1 u_{3}\end{array}\right)∂∂t(u1u2u3u4u5)=(α1∂2u1∂x2α2∂2u2∂x2α3∂2u3∂x2α4∂2u4∂x2α5∂2u5∂x2)+(u1−0.1u12−u1u2+0.1u22u1u2−0.1u22−u2u3+0.1u32u2u3−0.1u32−u3u4+0.1u42−u1u3+0.005u1u3u4−0.2u42−u2u4+0.1u52−u4u5+0.1u2+u4u4u5−0.1u52−u3u5+0.1u3),
( E 2 ⋅ E 2 ⋅ E_(2)*E_{2} \cdotE2⋅ PDE)
Fig. 2. Attractors for the 1 d PDE systems.
Fig. 3. Surf plots for the 1d PDE systems, Ω = [ − 1 , 1 ] Ω = [ − 1 , 1 ] Omega=[-1,1]\Omega=[-1,1]Ω=[−1,1].
(3) ∂ ∂ t ( u 1 u 2 u 3 u 4 u 5 u 6 ) = ( α 1 ∂ 2 u 1 ∂ x 2 α 2 ∂ 2 u 2 ∂ x 2 α 3 ∂ 2 u 3 ∂ x 2 α 4 ∂ 2 u 4 ∂ x 2 α 5 ∂ 2 u 5 ∂ x 2 α 6 ∂ 2 u 6 ∂ x 2 ) + ( u 1 − 0.1 u 1 2 − u 1 u 2 + 0.1 u 2 2 u 1 u 2 − 0.1 u 2 2 − u 2 u 3 + 0.1 u 3 2 u 2 u 3 − 0.1 u 3 2 − u 3 u 4 + 0.1 u 4 2 − u 1 u 3 + 0.005 u 1 u 3 u 4 − 0.2 u 4 2 − u 2 u 4 + 0.1 u 5 2 − u 4 u 5 + 0.1 u 2 + u 4 u 4 u 5 − 0.2 u 5 2 − u 3 u 5 + 0.1 u 6 2 − u 5 u 6 + 0.1 u 3 + u 5 u 5 u 6 − 0.1 u 6 2 − u 4 u 6 + 0.1 u 4 ) . (3) ∂ ∂ t u 1 u 2 u 3 u 4 u 5 u 6 = α 1 ∂ 2 u 1 ∂ x 2 α 2 ∂ 2 u 2 ∂ x 2 α 3 ∂ 2 u 3 ∂ x 2 α 4 ∂ 2 u 4 ∂ x 2 α 5 ∂ 2 u 5 ∂ x 2 α 6 ∂ 2 u 6 ∂ x 2 + u 1 − 0.1 u 1 2 − u 1 u 2 + 0.1 u 2 2 u 1 u 2 − 0.1 u 2 2 − u 2 u 3 + 0.1 u 3 2 u 2 u 3 − 0.1 u 3 2 − u 3 u 4 + 0.1 u 4 2 − u 1 u 3 + 0.005 u 1 u 3 u 4 − 0.2 u 4 2 − u 2 u 4 + 0.1 u 5 2 − u 4 u 5 + 0.1 u 2 + u 4 u 4 u 5 − 0.2 u 5 2 − u 3 u 5 + 0.1 u 6 2 − u 5 u 6 + 0.1 u 3 + u 5 u 5 u 6 − 0.1 u 6 2 − u 4 u 6 + 0.1 u 4 . {:(3)(del)/(del t)([u_(1)],[u_(2)],[u_(3)],[u_(4)],[u_(5)],[u_(6)])=([alpha_(1)(del^(2)u_(1))/(delx^(2))],[alpha_(2)(del^(2)u_(2))/(delx^(2))],[alpha_(3)(del^(2)u_(3))/(delx^(2))],[alpha_(4)(del^(2)u_(4))/(delx^(2))],[alpha_(5)(del^(2)u_(5))/(delx^(2))],[alpha_(6)(del^(2)u_(6))/(delx^(2))])+([u_(1)-0.1u_(1)^(2)-u_(1)u_(2)+0.1u_(2)^(2)],[u_(1)u_(2)-0.1u_(2)^(2)-u_(2)u_(3)+0.1u_(3)^(2)],[u_(2)u_(3)-0.1u_(3)^(2)-u_(3)u_(4)+0.1u_(4)^(2)-u_(1)u_(3)+0.005u_(1)],[u_(3)u_(4)-0.2u_(4)^(2)-u_(2)u_(4)+0.1u_(5)^(2)-u_(4)u_(5)+0.1u_(2)+u_(4)],[u_(4)u_(5)-0.2u_(5)^(2)-u_(3)u_(5)+0.1u_(6)^(2)-u_(5)u_(6)+0.1u_(3)+u_(5)],[u_(5)u_(6)-0.1u_(6)^(2)-u_(4)u_(6)+0.1u_(4)]).:}\frac{\partial}{\partial t}\left(\begin{array}{l} u_{1} \tag{3}\\ u_{2} \\ u_{3} \\ u_{4} \\ u_{5} \\ u_{6} \end{array}\right)=\left(\begin{array}{c} \alpha_{1} \frac{\partial^{2} u_{1}}{\partial x^{2}} \\ \alpha_{2} \frac{\partial^{2} u_{2}}{\partial x^{2}} \\ \alpha_{3} \frac{\partial^{2} u_{3}}{\partial x^{2}} \\ \alpha_{4} \frac{\partial^{2} u_{4}}{\partial x^{2}} \\ \alpha_{5} \frac{\partial^{2} u_{5}}{\partial x^{2}} \\ \alpha_{6} \frac{\partial^{2} u_{6}}{\partial x^{2}} \end{array}\right)+\left(\begin{array}{c} u_{1}-0.1 u_{1}^{2}-u_{1} u_{2}+0.1 u_{2}^{2} \\ u_{1} u_{2}-0.1 u_{2}^{2}-u_{2} u_{3}+0.1 u_{3}^{2} \\ u_{2} u_{3}-0.1 u_{3}^{2}-u_{3} u_{4}+0.1 u_{4}^{2}-u_{1} u_{3}+0.005 u_{1} \\ u_{3} u_{4}-0.2 u_{4}^{2}-u_{2} u_{4}+0.1 u_{5}^{2}-u_{4} u_{5}+0.1 u_{2}+u_{4} \\ u_{4} u_{5}-0.2 u_{5}^{2}-u_{3} u_{5}+0.1 u_{6}^{2}-u_{5} u_{6}+0.1 u_{3}+u_{5} \\ u_{5} u_{6}-0.1 u_{6}^{2}-u_{4} u_{6}+0.1 u_{4} \end{array}\right) .(3)∂∂t(u1u2u3u4u5u6)=(α1∂2u1∂x2α2∂2u2∂x2α3∂2u3∂x2α4∂2u4∂x2α5∂2u5∂x2α6∂2u6∂x2)+(u1−0.1u12−u1u2+0.1u22u1u2−0.1u22−u2u3+0.1u32u2u3−0.1u32−u3u4+0.1u42−u1u3+0.005u1u3u4−0.2u42−u2u4+0.1u52−u4u5+0.1u2+u4u4u5−0.2u52−u3u5+0.1u62−u5u6+0.1u3+u5u5u6−0.1u62−u4u6+0.1u4).
To each ( N N NNN dimensional) system we attach the homogeneous Neumann boundary conditions ∂ u i ∂ x ( a ) = ∂ u i ∂ x ( b ) = 0 , i = 1 , N ― ∂ u i ∂ x ( a ) = ∂ u i ∂ x ( b ) = 0 , i = 1 , N ¯ (delu_(i))/(del x)(a)=(delu_(i))/(del x)(b)=0,i= bar(1,N)\frac{\partial u_{i}}{\partial x}(a)=\frac{\partial u_{i}}{\partial x}(b)=0, i= \overline{1, N}∂ui∂x(a)=∂ui∂x(b)=0,i=1,N― and the initial conditions u i ( x , 0 ) = u i 0 ( x ) , i = 1 , N ― u i ( x , 0 ) = u i 0 ( x ) , i = 1 , N ¯ u_(i)(x,0)=u_(i)^(0)(x),i= bar(1,N)u_{i}(x, 0)=u_{i}^{0}(x), i=\overline{1, N}ui(x,0)=ui0(x),i=1,N―.
Fig. 2 shows for Ω = [ − 1 , 1 ] Ω = [ − 1 , 1 ] Omega=[-1,1]\Omega=[-1,1]Ω=[−1,1] and α i = 1 , i = 1 , 6 ― α i = 1 , i = 1 , 6 ¯ alpha_(i)=1,i= bar(1,6)\alpha_{i}=1, i=\overline{1,6}αi=1,i=1,6―, two phase plane trajectories on the time intervals ( 0 , t 1 ] , ( t 1 , t 2 ] 0 , t 1 , t 1 , t 2 (0,t_(1)],(t_(1),t_(2)]\left(0, t_{1}\right],\left(t_{1}, t_{2}\right](0,t1],(t1,t2] and ( t 2 , t 3 ] t 2 , t 3 (t_(2),t_(3)]\left(t_{2}, t_{3}\right](t2,t3], where t 1 = 300 , t 2 = 2 t 1 , t 3 = 3 t 1 t 1 = 300 , t 2 = 2 t 1 , t 3 = 3 t 1 t_(1)=300,t_(2)=2t_(1),t_(3)=3t_(1)t_{1}=300, t_{2}=2 t_{1}, t_{3}=3 t_{1}t1=300,t2=2t1,t3=3t1. The systems have been solved on the time interval [ 0 , 10 3 ] 0 , 10 3 [0,10^(3)]\left[0,10^{3}\right][0,103] by the parabolic-elliptic solver pdepe from MATLAB (which uses ode15s for integrating in time after spatial discretization) setting reltol = 10 − 6 = 10 − 6 =10^(-6)=10^{-6}=10−6 and abstol = 10 − 9 = 10 − 9 =10^(-9)=10^{-9}=10−9. For the initial data, uniformly distributed random values in the interval [ 0.1 , 10 ] [ 0.1 , 10 ] [0.1,10][0.1,10][0.1,10] have been generated on the space grid for each component.
Fig. 4. Attractors for the 2d PDE systems in the unit disk.
Fig. 5. Attractors for the 3d PDE systems in the unit ball.
Fig. 3 gives the surf plots for two components of the vector solutions of ( E 1 ⋅ P D E E 1 ⋅ P D E E_(1)*PDEE_{1} \cdot P D EE1⋅PDE ) and ( E 2 ⋅ P D E E 2 ⋅ P D E E_(2)*PDEE_{2} \cdot P D EE2⋅PDE ), obtained for the time interval [ 0 , 10 4 0 , 10 4 0,10^(4)0,10^{4}0,104 ] with timestep 100 and initial conditions for each component uniformly generated in [ 0.1 , 2 0.1 , 2 0.1,20.1,20.1,2 ] - periodicity stands out.
The 2d PDE system corresponding to the first Hanusse model is
∂ ∂ t ( u 1 u 2 u 3 ) = ( α 1 Δ u 1 α 2 Δ u 2 α 3 Δ u 3 ) + ( u 1 − 0.1 u 1 2 − u 1 u 2 + 0.1 u 2 2 u 1 u 2 − 0.1 u 2 2 − u 2 u 3 + 0.1 u 3 2 u 2 u 3 − 0.1 u 3 2 − u 1 u 3 + 0.005 u 1 ) . ∂ ∂ t u 1 u 2 u 3 = α 1 Δ u 1 α 2 Δ u 2 α 3 Δ u 3 + u 1 − 0.1 u 1 2 − u 1 u 2 + 0.1 u 2 2 u 1 u 2 − 0.1 u 2 2 − u 2 u 3 + 0.1 u 3 2 u 2 u 3 − 0.1 u 3 2 − u 1 u 3 + 0.005 u 1 . (del)/(del t)([u_(1)],[u_(2)],[u_(3)])=([alpha_(1)Deltau_(1)],[alpha_(2)Deltau_(2)],[alpha_(3)Deltau_(3)])+([u_(1)-0.1u_(1)^(2)-u_(1)u_(2)+0.1u_(2)^(2)],[u_(1)u_(2)-0.1u_(2)^(2)-u_(2)u_(3)+0.1u_(3)^(2)],[u_(2)u_(3)-0.1u_(3)^(2)-u_(1)u_(3)+0.005u_(1)]).\frac{\partial}{\partial t}\left(\begin{array}{l} u_{1} \\ u_{2} \\ u_{3} \end{array}\right)=\left(\begin{array}{l} \alpha_{1} \Delta u_{1} \\ \alpha_{2} \Delta u_{2} \\ \alpha_{3} \Delta u_{3} \end{array}\right)+\left(\begin{array}{c} u_{1}-0.1 u_{1}^{2}-u_{1} u_{2}+0.1 u_{2}^{2} \\ u_{1} u_{2}-0.1 u_{2}^{2}-u_{2} u_{3}+0.1 u_{3}^{2} \\ u_{2} u_{3}-0.1 u_{3}^{2}-u_{1} u_{3}+0.005 u_{1} \end{array}\right) .∂∂t(u1u2u3)=(α1Δu1α2Δu2α3Δu3)+(u1−0.1u12−u1u2+0.1u22u1u2−0.1u22−u2u3+0.1u32u2u3−0.1u32−u1u3+0.005u1).
( H ⋅ P D E − 2 d H ⋅ P D E − 2 d H*PDE-2dH \cdot P D E-2 dH⋅PDE−2d )
We denote by ( E 1 ⋅ P D E − 2 d ) , ( E 2 ⋅ P D E − 2 d ) E 1 ⋅ P D E − 2 d , E 2 ⋅ P D E − 2 d (E_(1)*PDE-2d),(E_(2)*PDE-2d)\left(E_{1} \cdot P D E-2 d\right),\left(E_{2} \cdot P D E-2 d\right)(E1⋅PDE−2d),(E2⋅PDE−2d), respectively ( E 3 ⋅ P D E − 2 d ) E 3 ⋅ P D E − 2 d (E_(3)*PDE-2d)\left(E_{3} \cdot P D E-2 d\right)(E3⋅PDE−2d) the other three corresponding systems. The polynomial expressions in the systems' RHS are the same with those of the ODE systems. To each system we attach the Neumann boundary conditions (1.7) and the initial conditions (1.8).
As a numerical example we now consider the case of the unit disk Ω = D ( 0 , 1 ) Ω = D ( 0 , 1 ) Omega=D(0,1)\Omega=D(0,1)Ω=D(0,1). When solving the systems we have employed the MATLAB PDE Toolbox using linear finite elements. For a randomly chosen mesh point in the triangulation of the spacial domain we plot phase plane trajectories for the solutions' components. Fig. 4 shows the results when solving on the time interval [ 0 , 10 3 ] 0 , 10 3 [0,10^(3)]\left[0,10^{3}\right][0,103] and setting reltol = 10 − 6 = 10 − 6 =10^(-6)=10^{-6}=10−6 and abstol = 10 − 9 − = 10 − 9 − =10^(-9)-=10^{-9}-=10−9− the same attractors arise.
The 3d PDE systems are the same as in the 2d case - to make a distinction we denote them by ( H ⋅ P D E − 3 d ) , ( E 1 ⋅ P D E − 3 d ) ( H ⋅ P D E − 3 d ) , E 1 ⋅ P D E − 3 d ) (H*PDE-3d),(E_(1)*PDE-:}3d)(H \cdot P D E-3 d),\left(E_{1} \cdot P D E-\right. 3 d)(H⋅PDE−3d),(E1⋅PDE−3d), ( E 2 ⋅ E 2 ⋅ E_(2)*E_{2} \cdotE2⋅ PDE − 3 d − 3 d -3d-3 d−3d ), respectively ( E 3 ⋅ E 3 ⋅ E_(3)*E_{3} \cdotE3⋅ PDE − 3 d − 3 d -3d-3 d−3d ). The numerical results are also obtained using the MATLAB PDE Toolbox with linear finite elements. We have considered Ω Ω Omega\OmegaΩ to be the unit ball in R 3 R 3 R^(3)\mathbb{R}^{3}R3. Fig. 5 shows the results obtained in this case for the time interval [ 0 , 10 4 ] 0 , 10 4 [0,10^(4)]\left[0,10^{4}\right][0,104] with reltol = 10 − 6 = 10 − 6 =10^(-6)=10^{-6}=10−6 and abstol = 10 − 9 = 10 − 9 =10^(-9)=10^{-9}=10−9. The same results have been obtained for considering Ω Ω Omega\OmegaΩ a torus.

Acknowledgments

Many thanks are due to the reviewers for their useful remarks and suggestions.

References

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    • Corresponding author at: T. Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, Cluj-Napoca, Romania.
    E-mail addresses: morosanug@ceu.edu (G. Moroşanu), mihai.nechita@ictp.acad.ro (M. Nechita).
2017

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