Attractors of semigroups associated with nonlinear systems for diffusive phase separation

We consider a model for diffusive phase transitions,for instance, the component separation in a binary mixture. Our model is described by two functions,the absolutete temperature θ := θ(t, x) and the order pa- rameter w := w(t, x), which are governed by a system of two nonlinear parabolic PDEs. The order parameter w is constrained to have double ob- stacles σ∗ ≤ w ≤ σ ∗ (i.e., σ∗ and σ ∗ are the threshold values of w). The objective of this paper is to discuss the semigroup {S(t)} associated with the phase separation model,and construct its global attractor.

In the one-dimensional case, i.e.N = 1, the existence and uniqueness of a global solution of (PSC) was proved in [10], and in [14] for the case without constraint (1.3).In the higher dimensional case (N = 2 or 3), any uniqueness result has not been noticed in the general setting; for a model in which the mass balance equation includes a viscosity term −µ∆w t , the uniqueness was obtained in [10].Recently, in [6] a uniqueness result was established in a very wide space of distributions under the additional assumption that (1.7) λ is convex on D(β) and D(ρ) ⊂ (−∞, 0].So far as the large time behaviour of solutions is concerned, we have noticed a few papers (e.g.[5,9,10,14]) including some results about the ω-limit set of each single solution as time t goes to +∞, but no results, except [12], on attractors so far for non-isothermal phase separation models; in [12] the regular case of ρ was treated, so this result is not applicable to (PSC) including a singular function ρ.
In this paper, assuming (1.7), we shall give a new existence result for problem (PSC) with initial data [u o , w o ] in a larger class than that in [10].Also, based on our existence result, we shall consider a semigroup {S(t)} t≥0 consisting of operators S(t) which assign to each initial data [u o , w o ] the element [u(t), w(t)], {u, w} being the solution.Moreover we shall construct the global attractor for {S(t)} in the product space L 2 (Ω) × H 1 (Ω).Unfortunately, the mapping t → S(t)[u o , w o ] lacks the continuity at t = 0 in L 2 (Ω)×H 1 (Ω) for bad initial data [u o , w o ], which comes from the singularity of ρ(u).Therefore the general theory on attractors (cf.[4,17]) cannot be directly applied to our case.However, the construction of the global attractor will be done by introducing a Lyapunov-like functional and by appropriately modified versions of some results in [4,17].Especially, the term νρ(u) with positive ν in (1.1) is very important in order to find an absorbing set.Notation.In general, for a (real) Banach space W we denote by | • | W the norm and by W * its dual space endowed with the dual norm.For any compact time interval [t o , t 1 ] we denote by C w ([t o , t 1 ]; W ) the space of all weakly continuous functions from [t o , t 1 ] into W , and mean by " as n → +∞, where •, • W * ,W stands for the duality pairing between W * and W .
For two real valued functions u, v we define Throughout this paper, let Ω be a bounded domain in R N (1 ≤ N ≤ 3) with smooth boundary Γ := ∂Ω, and for simplicity fix some notation as follows: •, • : duality pairing between V * and V. and clearly we have standard relations in which all the injections are compact and densely defined.Associated with the above norms, the duality mappings    3) In terms of the duality mapping We now introduce some functions and spaces in order to formulate an existence-uniqueness result.Let u ∞ be the unique solution of (2.7)

7) has one and only one solution u
In case of ν = 0 we suppose (2.8) holds.
Next we define a functional J(•, •) on the set (2.9) where ε o is a (small) positive number determined later and for all r ∈ R and a.e.x ∈ Ω.With the functional J 1 and a number m o with Also, for a number m o with σ * ≤ m o ≤ σ * , we put (2.13) ).First we recall the following theorem which guarantees the uniqueness of the solution of (PSC).

Theorem 2.1. ([5; Theorem 2.1]) Assume that (H1)-(H5) hold, and let f ∈
, and {u i , w i } be any solution of (P SC) i := ) is a positive constant dependent only on κ, n o and the Lipschitz constants of λ and g.Hypothesis (2.1) of (H3) is essential for the proof of inequality (2.15).An existence result is stated as follows.

Theorem 2.2. Assume that (H1)-(H5) hold as well as (2.8), and let m o be any number with σ
and there are constants Then, for each (2.18) Remark 2.2.From (2.18) and (2.20) of Theorem 2.2 we further derive an estimate of the form for all z ∈ H 1 (Ω) and a.e.t ∈ [0, T ] and u(0) = u o .

Approximate Problems and Estimates for Their Solutions
The solution of (PSC) will be constructed as a limit of solutions {u µεη , w µεη } of approximate problems (PSC) µεη , defined below, as µ, ε, η → 0; parameters ε, η concern with approximation ρ εη of function ρ, while parameter µ concerns with the coefficient of viscosity term, i.e. −µ∆w t , added in the mass balance equation.
The main idea for approximation is found in [7,10,15], and uniform estimates for approximate solutions with respect to parameters are quite similar to those in the above cited papers.Therefore, we mention very briefly some estimates for approximate solutions.In the rest of this section, we make all the assumptions of Theorem 2.2 as well as (2.4).

Estimate (I)
By regularity (3.9) we can compute rigorously

Convergence of Approximate Solutions and Proof of the Existence Result
The solution of (PSC) is constructed in two steps of limiting process as η → 0 and ε, µ → 0.
In the first step, parameters µ and ε are fixed, and parameter η goes to 0. For each ε ∈ (0, 1], we write J 1ε for J 1ε0 and u ∞ ε for u ∞ ε0 . where Here R 1 (•) and R 3 (•) are the same ones as in Estimates (II), (IV ).

Proof of Theorem 2.2. First assume that [u
In this case we can use the estimates (4.11) and (4.12) in addition to (4.1) -(4.3).Therefore, for suitable sequences {µ n } and {ε n } with µ n ↓ 0 and ε n ↓ 0 (as n → +∞) and some functions u ∈ L ∞ (0, T ; V ), ρ ∈ L ∞ (0, T ; H), w ∈ C w ([0, T ]; H 2 (Ω)) and ξ ∈ L ∞ (0, T ; H), the solution {u n , w n } of (PSC) n := (PSC) µnεn converges to the couple {u, w} in the sense that (4.13) ) ) In this paper, we say that a set A is a global attractor for the semigroup {S(t)}, if it has properties (i), (ii) and (iii) of Theorem 5.2.The key for the proof of Theorem 5.2 is to find an absorbing set B o for the semigroup {S(t)}.
To do so we prove a lemma under the same assumptions of Theorem 5.2.
Next, by contradiction we show the connectedness of A. Assume that A is not connected in H × V .Then there would exist two compact sets A 1 and Without loss of generality we may assume that  This implies by (b) of Theorem 5.1 that S(t 1 n )L n (τ ), τ ∈ [0, 1], is a continuous curve combining S(t 1 n )X 1 n and S(t 1 n ) X2 n in H × V .Now, take εneighborhoods U i ε of A i for i = 1, 2, for a sufficiently small number ε > 0 so

Journal of Applied Mathematics and Decision Sciences
Special Issue on Intelligent Computational Methods for Financial Engineering

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning .4) ∂u ∂n + n o u = h(x) on Σ := (0, +∞) × Γ, + ξ + g(w) − λ (w)u} = 0 on Σ, (1.6) u(0, •) = u o , w(0, •) = w o in Ω.
As easily understood from the above definition, since σ * ≤ w ≤ σ * for any solution {u, w} of (PSC), the behaviour of g, λ on the outside of [σ * , σ * ] gives no influence to the solution and we may assume without loss of generality that (2.4) the support of g is compact in R and λ is linear on the outside of [σ * , σ * ].
a.e.t ≥ 0, for all global solutions {u, w} with initial data [u o , w o ] ∈ D(m o ).
3, we see that A ⊂ D o (m o ) ∩ B o , and in (5.13) the sequences {t n } and {[z n , v n ]} can be chosen so as to satisfy further that for some [z, ṽ] ∈ D o (m o ) 2, and choose by (5.13) and (5.14) sequences {t i n } with t i n ↑ +∞ and X i n

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation