© Hindawi Publishing Corp. ASYMPTOTICS FOR CRITICAL NONCONVECTIVE TYPE EQUATIONS

We study large-time asymptotic behavior of solutions to the Cauchy problem for a model of nonlinear dissipative evolution equation. The linear part is a pseudodifferential operator and the nonlinearity is a cubic pseudodifferential operator defined by means of the inverse Fourier transformation and represented by bilinear and trilinear forms with respect to the direct Fourier transform of the dependent variable. We consider nonconvective type nonlinearity, that is, we suppose that the total mass of the nonlinear term does not vanish. We consider the initial data, which have a nonzero total mass and belong to the weighted Sobolev space with a sufficiently small norm. Then we give the main term of the large-time asymptotics of solutions in the critical case. The time decay rate have an additional logarithmic correction in comparison with the corresponding linear case.


Introduction.
We study large-time asymptotics of solutions to the Cauchy problem for dissipative equations u t + ᏺ(u) + ᏸu = 0, t >0, u(0,x) = u 0 (x), x ∈ R. (1.1) The linear part of (1.1) is a pseudodifferential operator defined by the Fourier transformation and the nonlinearity ᏺ(u) is a cubic pseudodifferential operator of nonconvective type:
Large-time asymptotics of solutions to the Cauchy problems for nonlinear evolution equations was extensively studied.A great success in the study of the large-time asymptotics first of all was achieved by inverse scattering transform method (see [1,3]).In the super critical case, large-time asymptotics of solutions is similar to that for the linearized case (see [2,4,18,20] and the references cited therein).Critical and subcritical dissipative equations with nonconvective type nonlinearities were considered in papers [5,6,7,8,9,10,11,12,13,14,15,16,17,19,22].
In [7,9], the Cauchy problem of the cubic nonlinear heat equation (1.7) was considered and the time decay estimate (1.10) was shown.This result was extended to the case of the porous media equation with critical exponents (see [8]).On the other hand, the asymptotic behavior in the time of the positive solutions to the equation u t − ∆u = −u p , x ∈ R n , t > 0, (1.11) was studied in [5,6,7,9,18,19].In particular, in [7,9], it was shown that if the initial data are not negative and p = 1+2/n, then the solution decays in time as (t log t) −n/2 for any x ∈ R n .It is the log t behavior that is difficult to recognize by standard methods.Gmira and Véron [9] proceeded by making the a priori substitution v(t, x) = (log t) n/2 u(t, x) and estimating the new dependent variable v(t, x).In the present paper, we proceed via a similar change of the solution v(t, x) = e ϕ(t) u(t, x), where the new unknown dependent variable ϕ(t) is determined by the requirement of zero total mass of the nonlinearity (see Section 3).This condition enables us to consider a broader class of equations and to obtain a more detailed asymptotics.
In [12], we obtained the large-time asymptotic behavior of solutions to the Cauchy problem for the nonlinear Schrödinger equation with dissipation in the critical case, that is, when the symbol L(ξ) of the linear pseudodifferential operator ᏸ has the following asymptotic representation L(ξ) ∼ µξ 2 in the origin ξ → 0, where Re µ > 0 and Im µ ≥ 0.
In [13,16], we considered the large-time asymptotics for solutions of the complex Landau-Ginzburg equation in the critical case q = 2/n.The subcritical case q ∈ (2/n−ε, 2/n), where ε is sufficiently small and depends on the size of the data, was studied in [17], where µ, a ∈ C, Reµ > 0, and µ and a satisfy suitable angular conditions.In [14], the asymptotic expansion of small solutions to the Cauchy problem for the complex Landau-Ginzburg equation was considered.
Large-time asymptotics of equations with convective type nonlinearities in the critical case also was studied extensively (see [15] and the literature cited therein).
The aim of the present paper is to obtain the large-time asymptotic behavior of solutions to the Cauchy problem for the nonlinear evolution equations (1.1) in the critical case.Note that the method of [7,9] does not work for (1.1).In the present paper, we generalize the approach developed in our previous papers [10,12,13,14,16,17], where we considered mainly the case of Laplacian ᏸ = −∆, the power nonlinearity ᏺ(u) = |u| q 1 u, 0 < q 1 ≤ 2/n or nonlinearity of convective type ᏺ(u) = ∂ x 1 |u| q 2 +1 , 0 < q 2 ≤ 1/n, and applied the L p -estimates of the Green operator e t∆ to show the positivity of the value ᏺ(u 1 )dx, where u 1 is the first approximation of the solution and (ᏺ(u)−ᏺ(u 1 ))dx is the remainder term in our function space.In comparison with our previous works [12,13,16], we are working here in the Lebesgue spaces for the Fourier transform of the solution in order to treat the case of nonlocal nonlinearities of nonconvective type involving derivatives of unknown function and to be able to show that (ᏺ(u)−ᏺ(u 1 ))dx is the remainder term in our function space defined below.To obtain the estimates of the remainder terms of the large-time asymptotic formulas, we have to assume that the initial data satisfy some decay condition at infinity.Suppose that the linear operator ᏸ satisfies the dissipation condition which in terms of the symbol L(ξ) has the form Re L(ξ) ≥ µ{ξ} δ ξ ν (1.14) for all ξ ∈ R, where µ > 0, ν ≥ 0, and δ > 0. Also we suppose that the symbol is a smooth L(ξ) ∈ C 1 (R) and has the estimate for all ξ ∈ R \{0}, l = 0, 1.
To find the asymptotic formulas for the solution, we assume that the symbol L(ξ) has the following asymptotic representation in the origin: We suppose that the symbols of the nonlinear operator ᏺ are such that for all ξ, y ∈ R, t > 0, and l = 0, 1, and for all ξ, y, z ∈ R, t > 0, and l = 0, 1, where α ≥ 0, β ≥ 0, σ = 0 if ν = 0, and σ ∈ [0,ν) Here and below, we denote {ξ} = |ξ|/ ξ , ξ = 1 + ξ 2 .We consider the case of nonlinearity of the nonconvective type, that is, we suppose that a(t, 0,y) ≠ 0 or b(t, 0,y,z) ≠ 0. ( The critical case with respect to the large-time asymptotic behavior of solutions means that We assume that the symbols of the nonlinearity have the asymptotics for all y,z ∈ R and t > 0, where γ ∈ (0, 1), a 0 (y) is homogeneous of order α, and b 0 (y, z) is homogeneous of order β.For example, the equation where ).Also we suppose that the total mass of the initial data is not zero: a 0 (y)e −L 0 (−y)−L 0 (y) dy where ω = 0 if a 0 ≠ 0 and ω = 1 if a 0 ≡ 0. To obtain asymptotics of solutions in the critical case, we assume below that > 0. The condition > 0 implies the restriction on the nonlinearity and yields the positivity of the value ᏺ(u 1 )dx > 0, where u 1 is the first approximation of the solution.We easily see that, for example, the equation (1.27) Define W = A 0,∞ ∩ B 0,1 ∩ D 0,0 , where and define the usual Lebesgue space L p = {ϕ ∈ ; ϕ L p < ∞}, where and we define the majorant of the fractional derivative of order γ ∈ (0, 1) as (1.31) In the present paper we prove the following result.
Remark 1.2.The conditions of the theorem on the initial data u 0 can also be expressed in terms of the usual weighted Sobolev spaces However, the conditions on the initial data u 0 are described more precisely in the norm W.
To help the reader to assimilate the assumptions regarding a pair of pseudodifferential operators and the initial data, we state the following list.

Assumptions on ᏸ(
Assumptions on ᏺ.We assume the following: Assumptions on the initial data u 0 (x).We assume the following: and some ε > 0 sufficiently small.
is true for large t.
(2) As another example, we consider the cubic nonlinear heat equation with convection (1.41) It represents an interaction of the quadratic convective nonlinearity with cubic nonconvective nonlinear term; both are critical.We have The conditions (1.18) and (1.22) are fulfilled with σ = α = 1, β = 0, and δ = ν = 2. Then for small initial data u 0 such that θ > 0 and the norm u 0 H ρ,0 + u 0 H 0,ρ ≤ ε, ρ > 1/2, the asymptotics (1.40) is true for large t.Thus the cubic nonconvective nonlinearity dominates under convective term and defines the main term of the large-time asymptotic behavior of solutions.
We organize the rest of the paper as follows.Some preliminary estimates of the Green operator solving the linearized Cauchy problem corresponding to (1.1) are obtained in Section 2. The proof of Theorem 1.1 will be given in Section 3.

Preliminary lemmas. The solution of the linear Cauchy problem
can be written by the Duhamel formula where the Green operator Ᏻ is given by We first collect some preliminary estimates for the Green operator Ᏻ(t) in the norms where ρ, s ∈ R and γ ∈ (0, 1).The norm A ρ,p is responsible for the large-time asymptotic properties of solutions and the norm B s,p describes the regularity of solutions.
Lemma 2.2.Let the linear operator ᏸ satisfy conditions (1.14) and (1.16), then the estimates are valid for all t > 0, where Proof.By virtue of the dissipation condition (1.14) and asymptotics (1.16), we have (2.33) for all t > 0 and |ξ| ≤ 1, whence (2.34) Thus, the first estimate is true.We prove the second estimate by (2.9) and (2.17); we get (2.35) Therefore, the second estimate of the lemma is valid.Hence, lemma 2.2 is proved.
In the next lemma, we estimate the Green operator Ᏻ(t) in our basic norms • X and • Y which depend on the order of L(ξ) (see conditions (1.14) with some κ > 0.
Lemma 2.3.Let the function f (t,x) have a zero mean value f (t,0) = 0. Then the inequality is valid, provided that the right-hand side is finite.
Now we estimate the nonlinearity ᏺ(u) in the norms A 0,p , B 0,p , and D 0,0 .
Lemma 2.4.Let the nonlinear operator ᏺ satisfy conditions (1.17) and (1.18).Then the inequalities are valid for 1 ≤ p ≤ ∞, provided that the right-hand sides are bounded.
Then the inequalities if a 0 ≠ 0 and if a 0 ≡ 0 are valid for all t > 0.

Proof of Theorem 1.1.
For the local existence of classical solutions for the Cauchy problem (1.1), we refer to [2].We make a change of dependent variable u(t, x) = e −ϕ(t) v(t, x), then we get from (1.1) where Now, we require that the real-valued function ϕ(t) satisfies the following condition expressed in terms of the Fourier transform: whence, via (3.1), we get for all t > 0. Therefore, If we choose the initial condition ϕ(0)= 0, we have and we obtain the following system: Multiplying the second equation of system (3.7) by the factor e ϕ(t) then integrating with respect to time t > 0 and making a change of the dependent variables v = Ᏻ(t)u 0 + r and e ϕ(t) = h 1 (t), we get the system of integral equations where In the case a 0 ≡ 0, we denote e 2ϕ(t) = h 2 (t) and obtain the following system of integral equations: where in the case a 0 ≠ 0 and in the case a 0 ≡ 0. prove that (Ꮽ j , Ꮾ j ) is the contraction mapping in the set where here γ ∈ (0, min(1,δ)) is such that γ < α if α > 0 and γ < β if β > 0. First we prove that the mapping (Ꮽ j , Ꮾ j ) transforms the set X into itself.When (r , h j ) ∈ X, we get by Lemma 2.4 (3.17) Hence, for all t > 0. Thus, (Ꮽ j , Ꮾ j ) transforms the set X into itself.
In the same manner, we consider the differences Ꮽ j (r , h j )−Ꮽ j ( r , h j ) and Ꮾ j (r , h j )− Ꮾ j ( r , h j ) to see that the transformation (Ꮽ j , Ꮾ j ) is the contraction mapping.Therefore, there exists a unique solution (r , h j ) of the system of integral equations (3.8) in the space X.From Lemma 2.5, we see that h j (t) = log t + O(log log t) (3.21) for t → ∞.Therefore, via formulas u(t, x) = e −ϕ(t) v(t, x) = e −ϕ(t) (Ᏻv 0 + r ), we obtain the asymptotic formula of the theorem.Theorem 1.1 is proved.

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

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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

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