Abstract
In this paper, we would consider the dynamical behaviors of the chemical model represented by Satnoianu et al. (2001). Using the Kuratowski measure of noncompactness method, we prove the existence of global attractor for the weak solution semiflow of system. Finally, several numerical experiments confirm the theoretical results.
1. Introduction
Satnoianu et al. [1] considered a chemical model which assumed that a precursor is present in excess. They required further species to be present in the reactor at a concentration similar to that of though this species does not take part in the reaction. Assume that decays at a constant rate to form the substrate via , , where is the initial concentration of the reservoir species . The substrate and autocatalyst subsequently react by following a linear combination of quadratic and cubic steps according to the scheme where , , are the constants and , are the concentrations of and .
This leads to the dimensionless reactor modelwhere , , , are the dimensionless concentrations and time and distance along the reactor, respectively. With the Dirichlet boundary conditionand an initial conditionwhere is the bounded set .
Here, is the radio of diffusion coefficients of substrate and autocatalyst . is a parameter denoting the intensity of the applied electric field. The parameter measures the strength of the quadratic step against the cubic step. For , the nonlinearity in the kinetics is purely quadratic, while for , it is purely cubic.
Satnoianu et al. discussed that diffusion-distributed structures (FDS) are predicted in a wider domain and are more robust than the classical Turing instability patterns. FDS also represent a natural extension of flow-distributed oscillations. Nonlinear bifurcation analysis and numerical simulations in one-dimensional spatial domains show that FDS have much richer solution behavior than Turing structures. Tang and Wang [2] showed a rigorous bifurcation analysis for the model with combination of quadratic and cubic steps by the parameters and .
For the infinite-dimensional dynamical systems about the chemical model You [3] considered the Brusselator equations; he explored a decomposition technique to show the -contraction of the solution semiflow and proved the existence of a global attractor for the solution semiflow of the Brusselator equation and the Hausdorff dimension and the fractal dimension of the global attractors are finite. You [4] studied the reversible Selkov equations with Dirichlet boundary condition on a bounded domain which is a representative cubic-autocatalytic reaction-diffusion system. He obtained the existence of global attractor and the upper semicontinuity of the global attractors in product space when the reverse reaction rate tends to zero. For more details about the infinite-dimension dynamics systems about reaction-diffusion models, we can refer to [5–11].
In this paper we will discuss the long-time behavior of the solutions to (2)–(4) for ; that is,We prove the existence of the global attractor by Kuratowski measure. The paper is organized as follows. In Section 2, we prove the existence the weak solution of the system and absorbing property. In Section 3, we give asymptotic compactness of the semiflow by Kuratowski measure and prove the existence of the global attractor. Finally, in Section 4, we give the simulation of the system.
2. Absorbing Property
Define the product Hilbert spaces as follows:The norm and inner-product of or the component space will be denoted by and , respectively. The norm of will be represented by if .
By the poincaré inequality and the homogenous Dirichlet boundary condition there is a constant such thatWe will take to the norm of the space .
Using the analytic semigroup generation theorem and Lumer-Phillips theorem, it is easy to check that the densely sectorial operatoris the generator of an analytic -semigroup on the Hilbert space.
By the fact that is a continuous embedding, we can verify that the nonlinear mappingis well defined on and the mapping is locally Lipschitz continuous. Let , and then the initial-boundary value problem (5) is formulated into an initial value problem of the abstract evolutionary equation:
A function is called a weak solution to the problem of the parabolic evolutionary equation (10), if the following two conditions are satisfied:(1) is satisfied for a.e. and for any .(2), where denote the space of weakly continuous functions on valued in such that .
By the Galerkin method, analogous to the arguments in [3, 4], we can show the following conclusion about existence of solution.
Lemma 1. For any initial , there exists a unique local weak solution for some of the evolutionary equation (10) such that satisfies
In order to prove the existence of global attractor, we will investigate the absorbing property of the solution semiflow of problem (5).
Lemma 2. Assume , and then, for any initial data , there exists an absorbing set for the solution semiflow of the Cauchy problem of (5):where is a positive constant independent of initial data.
Proof. Taking the inner-product of the first equation of (5) with over , we obtainLet , and since , we have that is,Poincáre inequality and Gronwall’s inequality yieldLet , and since , we haveAdding two equations in (5) to get an equation for the sum , one hasTaking the inner-product (18) with and integrating by parts, we obtain Thanks to (16), we have Let , and then we havethat is,integrating (22) from to , and thenwhere Now we treat the term in (23); multiplying (15) by and integrating from to , we get and, by (16), we havewhere Substituting (26) into (23), we obtainNoticing that , , as , therefore we conclude thatLet , which is independent of any initial data, and then we getCombining (17) and (30), we can deduce that for any initial data the solution satisfiesSo set in (12) is the absorbing set for the solution semiflow to problem (5). Then proof is completed.
3. The Existence of Global Attractor
In this section we will prove that the solution semiflow of (5) has a global attractor. According the existence of criterion in [6, 12], we just need to prove that the semiflow is asymptotic compactness. Firstly, the following proposition is about the uniform Gronwall’s inequality which is a basic tool in the analysis of asymptotics.
Lemma 3. Let and be nonnegative function in . Assume that is absolutely continuous on and the following differential inequality is satisfied: If there is a finite time and some , such that for any , where , , and are positive constants thenfor any .
Lemma 4. There is a constant , such that for any , , where is a positive constant depending only on the absorbing set .
Proof. From (15) we haveSince the absorbing set absorbs itself, there exists such that for any , so we obtainFrom (21) and (37), we getSince , it follows from (37) and (38); we obtain for any The proof is completed.
Recall the definition of the Kuratowski measure of noncompactness for bounded sets in a Banach space [13, 14]:
A semiflow on a complete metric space is called -contracting if, for every bounded subset in , one has
Lemma 5. Let be a semiflow on a Banach space , if the following conditions are satisfied: (1) has a bounded absorbing set in .(2) is -contracting. Then is asymptotically compact and there exists a global attractor in .
Lemma 6 (see [4]). Let be a semiflow on , and there exists a global attractor in for this semiflow, if and only if the following conditions are satisfied:(1)There exists a bounded absorbing set in for this semiflow.(2)For any , there are positive constants and such that where , and is the characteristic function of the subset .
Lemma 7. For any , there exist positive constants and , such that the -component of the solution to problem (5) satisfies
Proof. Since is a bounded absorbing set, there exist and such thatHence we have and hence there exist such thatTaking the inner-product (5) with where is given in (46) and we obtain It follows that By Poincáre inequality, we have By Gronwall’s lemma, we obtain Thus there exist , such that, for any and any , one has Symmetrically, we can prove that there exists such that for any and any one has where Therefore, let , for any and , and there hold Next we can deduce that there exists a positive integer , such that Indeed it follows that Since , where for a sufficiently large integer , (43) holds, with , for any . The proof is completed.
Lemma 8. Let be the bounded absorbing set of solution semiflow to problem (5), and then, for any given , it hold that where is the orthogonal projection from the product Hilbert space onto the first component space associated with the -component.
Proof. Taking the inner-product (5) with over the set , so we obtain and since , we haveIt follows that . By Lemma 4, we get that there exits such that for all Combining results in Lemma 3, (61), and (62), we conclude that there exist , such thatInequality (63) is uniform constant depending on the absorbing set and the given and shows that, for any fixed , is bounded in . Due to the compact embedding , it turns out that, for any fixed , is a precompact set in , so we have , for in the space . The proof is completed.
Lemma 9. For any , there exist positive constants and , such that the -component of the solution semiflow of problem (5) satisfies
Proof. Taking the inner-product (18) with over , we get We obtain Integrating both sides of the above inequality from to , we haveUsing inequality (26), the right hand of (67) can be estimated as follows: for all , , we find that there exist , such thatCombining (43) and (69), we have that there exist positive , such that The proof is completed.
Lemma 10. Let be the bounded absorbing set of solution semiflow to problem (2)–(4), and then, for any given , it holds that where is the orthogonal projection from the product Hilbert space onto the first component space associated with the -component.
Proof. Taking the inner-product (5) with over the set , we have and, thus, it follows thatfor . In view of (62) and, applying Lemma 3, there exist , such thatit shows that is bounded set in for any , so that is a precompact set in . It is in for . The proof is completed.
Theorem 11. If , , there exists a global attractor in for the solution semiflow generated by problem (5).
Proof. By Lemma 2, we have that has a bounded absorbing set in and by Lemmas 8 and 10 we have that the conditions in Lemma 4 are satisfied. So by Lemma 4 there exists a global attractor in for the solution semiflow generated by problem (5). The proof is completed.
4. Simulation
In this section, using the finite difference method to (5), the discrete equation of (5) was obtained as follows:Here is space stepsize, is time stepsize, and is a numerical approximation to at . Then we apply Euler’s Method with the Dirichlet boundary condition to solve the discrete equation (5). In what follows, we always set space stepsize and time stepsize to simulate the problem under different conditions.
Now we set the parameters , , , and let initial data , . The numerical results are shown in Figure 1. These plots indicate that the global attractor in Section 3 is complicated and consist of many period trajectories. In the meanwhile, when we let initial data , the plots can be shown in Figure 2, and it indicates that the solution is convergent to a stationary solution.
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5. Conclusion
In this paper, it can be seen from the numerical results that the global attractor of problem (5) exists for certain and . It includes several types of periodic solutions and quasiperiodic solutions. Besides, the solutions also reduce to stationary solution in some parameter value. All these numerical simulations are consistent with our theoretical conclusions.
Conflict of Interests
The authors declare that there is no conflict of interests.
Acknowledgments
This work is supported by National Natural Science Foundation of China (11272277, 11471129), Innovation Scientists and Technicians Troop Construction Projects of Henan Province (134100510013), Innovative Research Team in University of Henan Province (13IRTSTHN019), and Key Project of the Education Department Henan Province (15A110043).