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Abstract and Applied Analysis
Volumeย 2012ย (2012), Article IDย 627813, 12 pages
http://dx.doi.org/10.1155/2012/627813
Research Article

Long-Time Decay to the Global Solution of the 2D Dissipative Quasigeostrophic Equation

Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia

Received 10 January 2012; Accepted 8 March 2012

Academic Editor: Muhammad Aslamย Noor

Copyright ยฉ 2012 Jamel Benameur and Mongi Blel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the behavior at infinity in time of any global solution ๐œƒโˆˆ๐’ž(โ„+,ฬ‡๐ป2โˆ’2๐›ผ(โ„2)) of the surface quasigeostrophic equation with subcritical exponent 2/3โ‰ค๐›ผโ‰ค1. We prove that lim๐‘กโ†’โˆžโ€–๐œƒ(๐‘ก)โ€–ฬ‡๐ป2โˆ’2๐›ผ=0. Moreover, we prove also the nonhomogeneous version of the previous result, and we prove that if ๐œƒโˆˆ๐’ž(โ„+,ฬ‡๐ป2โˆ’2๐›ผ(โ„2)) is a global solution, then lim๐‘กโ†’โˆžโ€–๐œƒ(๐‘ก)โ€–๐ป2โˆ’2๐›ผ=0.

1. Introduction

We consider the 2๐ท dissipative quasi-geostrophic equation with subcritical exponent 1/2<๐›ผโ‰ค1, ๐œ•๐‘ก๐œƒ+(โˆ’ฮ”)๐›ผ๐œƒ+(๐‘ขโ‹…โˆ‡)๐œƒ=0inโ„+ร—โ„2,๐œƒ(0,๐‘ฅ)=๐œƒ0(๐‘ฅ)inโ„2,(๐’ฎ๐›ผ) where ๐‘ฅโˆˆโ„2, ๐‘ก>0, ๐œƒ=๐œƒ(๐‘ฅ,๐‘ก) is the unknown potential temperature, and ๐‘ข=(๐‘ข1,๐‘ข2) is the divergence free velocity which is determined by the Riesz transformation of ๐œƒ in the following way: ๐‘ข1=โˆ’โ„›2๐œƒ=โˆ’๐œ•2(โˆ’ฮ”)โˆ’1/2๐‘ข๐œƒ,2=โ„›1๐œƒ=๐œ•1(โˆ’ฮ”)โˆ’1/2๐œƒ.(1.1) This equation is a two-dimensional model of the 3๐ท incompressible Euler equations, and if ๐›ผ=1, the equation (๐’ฎ1) is the 2๐ท Navier-Stokes equation. We refer the reader to [1] where the authors explain the physical origin and the signification of the parameters of this equation.

The critical homogeneous Sobolev space of the system (๐’ฎ๐›ผ) is ฬ‡๐ป2โˆ’2๐›ผ(โ„2), and we have โ€–โ€–๐œ†2๐›ผโˆ’1๐‘“โ€–โ€–(๐œ†.)ฬ‡๐ป2โˆ’2๐›ผ=โ€–๐‘“โ€–ฬ‡๐ป2โˆ’2๐›ผ,โˆ€๐œ†>0.(1.2)

The local well-posedness of (๐’ฎ๐›ผ) with ฬ‡๐ป2โˆ’2๐›ผ(โ„2) data is established by [2] and [3] separately if ๐›ผโˆˆ(0,1/2]. In [4], Dong and Du study the critical case ๐›ผ=1/2 in the critical space ฬ‡๐ป1(โ„2). They prove the global existence if the initial condition is in the critical space ๐ป1(โ„2).

The global existence when ๐›ผโˆˆ(1/2,1] is an open problem. We have only the local existence. In this case [5], Niche and Schonbek prove that if the initial data ๐œƒ0 is in ๐ฟ2(โ„2), then the ๐ฟ2 norm of the solution tends to zero but with no uniform rate, that is, there are solutions with arbitrary slow decay. If ๐œƒ0โˆˆ๐ฟ๐‘(โ„2), with 1โ‰ค๐‘โ‰ค2, they obtain a uniform decay rate in ๐ฟ2. They consider also the solution in other ๐ฟ๐‘ž spaces. For the proof of their results, they use the kernel ๐‘ƒ๐›ผ(๐‘ก,๐‘ฅ) associated to the operator ๐œ•๐‘ก+(โˆ’ฮ”)๐›ผ, and they use the Littlewood-Paley decomposition. Our main result is the following.

Theorem 1.1. Assume that 2/3โ‰ค๐›ผโ‰ค1.(i)If ๐œƒโˆˆ๐’ž(โ„+,ฬ‡๐ป2โˆ’2๐›ผ(โ„2)) is a global solution of (๐’ฎ๐›ผ), then lim๐‘กโ†’โˆžโ€–๐œƒ(๐‘ก)โ€–ฬ‡๐ป2โˆ’2๐›ผ=0.(1.3)(ii)If ๐œƒโˆˆ๐’ž(โ„+,๐ป2โˆ’2๐›ผ(โ„2)) is a global solution of (๐’ฎ๐›ผ), then lim๐‘กโ†’โˆžโ€–๐œƒ(๐‘ก)โ€–๐ป2โˆ’2๐›ผ=0.(1.4)

2. Notations and Preliminary Results

2.1. Notations and Technical Lemmas

In this short section, we collect some notations and definitions that will be used later, and we give some technical lemmas.(i)The Fourier transformation in โ„2 is normalized as โ„ฑ(๐‘“)(๐œ‰)=โˆง๐‘“๎€œ(๐œ‰)=โ„2๎€ท๐œ‰exp(โˆ’๐‘–๐‘ฅโ‹…๐œ‰)๐‘“(๐‘ฅ)๐‘‘๐‘ฅ,๐œ‰=1,๐œ‰2๎€ธโˆˆโ„2.(2.1)(ii)The inverse Fourier formula is โ„ฑโˆ’1(๐‘”)(๐‘ฅ)=(2๐œ‹)โˆ’2๎€œโ„2๎€ท๐‘ฅexp(๐‘–๐œ‰โ‹…๐‘ฅ)๐‘“(๐œ‰)๐‘‘๐œ‰,๐‘ฅ=1,๐‘ฅ2๎€ธโˆˆโ„2.(2.2)(iii)For ๐‘ โˆˆโ„, ๐ป๐‘ (โ„2) denotes the usual nonhomogeneous Sobolev space on โ„2 and โŸจโ‹…,โ‹…โŸฉ๐ป๐‘ (โ„2) its scalar product.(iv)For ๐‘ โˆˆโ„, ฬ‡๐ป๐‘ (โ„2) denotes the usual homogeneous Sobolev space on โ„2 and โŸจโ‹…,โ‹…โŸฉฬ‡๐ป๐‘ (โ„2) its scalar product.(v)For ๐‘ ,๐‘ ๎…žโˆˆโ„ and ๐‘กโˆˆ[0,1],โ€–๐‘“โ€–๐ปโ€ฒ๐‘ก๐‘ +(1โˆ’๐‘ก)๐‘ โ‰คโ€–๐‘“โ€–๐‘ก๐ป๐‘ โ€–๐‘“โ€–๐ป1โˆ’๐‘ก๐‘ โ€ฒ,(2.3)โ€–๐‘“โ€–ฬ‡๐ปโ€ฒ๐‘ก๐‘ +(1โˆ’๐‘ก)๐‘ โ‰คโ€–๐‘“โ€–๐‘กฬ‡๐ป๐‘ โ€–๐‘“โ€–ฬ‡๐ป1โˆ’๐‘ก๐‘ โ€ฒ.(2.4) These two inequalities are called the interpolation inequalities, respectively, in the homogeneous and nonhomogeneous Sobolev spaces.(i)For any Banach space (๐ต,โ€–โ‹…โ€–), any real number 1โ‰ค๐‘โ‰คโˆž, and any time ๐‘‡>0, we denote by ๐ฟ๐‘๐‘‡(๐ต) the space of measurable functions ๐‘กโˆˆ[0,๐‘‡]โ†ฆ๐‘“(๐‘ก)โˆˆ๐ต such that (๐‘กโ†ฆโ€–๐‘“(๐‘ก)โ€–)โˆˆ๐ฟ๐‘([0,๐‘‡]).(ii)If ๐‘“=(๐‘“1,๐‘“2) and ๐‘”=(๐‘”1,๐‘”2) are two vector fields, we set๎€ท๐‘”๐‘“โŠ—๐‘”โˆถ=1๐‘“,๐‘”2๐‘“๎€ธ,๎€ท๎€ท๐‘”div(๐‘“โŠ—๐‘”)โˆถ=div1๐‘“๎€ธ๎€ท๐‘”,div2๐‘“.๎€ธ๎€ธ(2.5) We recall a fundamental lemma concerning some product laws in homogeneous Sobolev spaces.

Lemma 2.1 (see [6]). Let ๐‘ 1, ๐‘ 2 be two real numbers such that ๐‘ 1<1,๐‘ 1+๐‘ 2>0.(2.6) There exists a constant ๐ถโˆถ=๐ถ(๐‘ 1,๐‘ 2), such that for all ฬ‡๐ป๐‘“,๐‘”โˆˆ๐‘ 1(โ„2ฬ‡๐ป)โˆฉ๐‘ 2โ„2), โ€–๐‘“๐‘”โ€–ฬ‡๐ป๐‘ 12+๐‘ โˆ’1(โ„2)๎€ทโ‰ค๐ถโ€–๐‘“โ€–ฬ‡๐ป๐‘ 1(โ„2)โ€–๐‘”โ€–ฬ‡๐ป๐‘ 2+โ€–๐‘“โ€–ฬ‡๐ป๐‘ 2โ€–๐‘”โ€–ฬ‡๐ป๐‘ 1๎€ธ.(2.7) If ๐‘ 1,๐‘ 2<1 and ๐‘ 1+๐‘ 2>0, there exists a constant ๐‘=๐‘(๐‘ 1,๐‘ 2) such that for all ฬ‡๐ป๐‘“โˆˆ๐‘ 1(โ„2) and ฬ‡๐ป๐‘”โˆˆ๐‘ 2โ„2), โ€–๐‘“๐‘”โ€–ฬ‡๐ป๐‘ 12+๐‘ โˆ’1(โ„2)โ‰ค๐‘โ€–๐‘“โ€–ฬ‡๐ป๐‘ 1โ€–๐‘”โ€–ฬ‡๐ป๐‘ 2.(2.8)

For the proof of the main result, we need the following lemma.

Lemma 2.2. With the same conditions of Theorem 1.1, for all ๐œŽโ‰ฅ0, ๎€œโ„2||๐œ‰||2๐œŽ||||โ„ฑ((๐‘ขโ‹…โˆ‡)๐œƒ)โ„ฑ(๐‘ค)๐‘‘๐œ‰โ‰ค๐ถโ€–๐œƒโ€–ฬ‡๐ป2โˆ’2๐›ผโ€–๐œƒโ€–ฬ‡๐ป๐œŽ+๐›ผโ€–๐‘คโ€–ฬ‡๐ป๐œŽ+๐›ผ.(2.9)

Remark 2.3. (i) In the case where ๐œŽ=0, the formula (2.9) gives ๎€œโ„2||||โ„ฑ((๐‘ขโ‹…โˆ‡)๐œƒ)โ„ฑ(๐‘ค)๐‘‘๐œ‰โ‰ค๐ถโ€–๐œƒโ€–ฬ‡๐ป2โˆ’2๐›ผโ€–๐œƒโ€–ฬ‡๐ป๐›ผโ€–๐‘คโ€–ฬ‡๐ป๐›ผ.(2.10)
In the case where ๐œŽ=2โˆ’2๐›ผ, the formula (2.9) gives ๎€œโ„2||๐œ‰||2(2โˆ’2๐›ผ)||||โ„ฑ((๐‘ขโ‹…โˆ‡)๐œƒ)โ„ฑ(๐‘ค)๐‘‘๐œ‰โ‰ค๐ถโ€–๐œƒโ€–ฬ‡๐ป2โˆ’2๐›ผโ€–๐œƒโ€–ฬ‡๐ป2โˆ’๐›ผโ€–๐‘คโ€–ฬ‡๐ป2โˆ’๐›ผ.(2.11)

Proof of Lemma 2.2. From the Cauchy-Schwarz inequality, we have ๎€œโ„2||๐œ‰||2๐œŽ||||๎€œโ„ฑ((๐‘ขโ‹…โˆ‡)๐œƒ)โ„ฑ(๐‘ค)๐‘‘๐œ‰โ‰คโ„2||๐œ‰||๐œŽโˆ’๐›ผ||||||๐œ‰||โ„ฑ((๐‘ขโ‹…โˆ‡)๐œƒ)๐œŽ+๐›ผ||||โ‰ค๎‚ต๎€œโ„ฑ(๐‘ค)(๐œ‰)๐‘‘๐œ‰โ„2||๐œ‰||2(๐œŽโˆ’๐›ผ)||||โ„ฑ((๐‘ขโ‹…โˆ‡)๐œƒ)2๎‚ถ๐‘‘๐œ‰1/2โ€–๐‘คโ€–ฬ‡๐ป๐œŽ+๐›ผ.(2.12) Using the weak derivatives properties, the product laws (Lemma 2.1), with ๐‘ 1+๐‘ 2=๐œŽโˆ’๐›ผ+2>0, ๐‘ 1=2โˆ’2๐›ผ<1, and ๐‘ 2=๐œŽ+๐›ผ, we can dominate the nonlinear part as follows: ๎€œโ„2||๐œ‰||2(๐œŽโˆ’๐›ผ)||||โ„ฑ((๐‘ขโ‹…โˆ‡)๐œƒ)2๎€œ๐‘‘๐œ‰โ‰คโ„2||๐œ‰||2(๐œŽโˆ’๐›ผ+1)๎€ท||||๎€ธโ„ฑ(๐œƒ)โˆ—|โ„ฑ(๐œƒ)|2๐‘‘๐œ‰โ‰ค๐ถโ€–๐œƒโ€–2ฬ‡๐ป2โˆ’2๐›ผโ€–๐œƒโ€–2ฬ‡๐ป๐œŽ+๐›ผ.(2.13)

2.2. Existence Theorem

In [7], Wu proves an existence and uniqueness theorem of (๐’ฎ๐›ผ) in the well-known Besov spaces ฬ‡๐ต๐‘Ÿ๐‘,๐‘ž. We recall this theorem in the special case, where ๐‘=๐‘ž=2.

Theorem 2.4. Assume that ๐›ผโˆˆ(0,1] and ๐œƒ0โˆˆฬ‡๐ป2โˆ’2๐›ผ(โ„2), then there exists a constant ๐‘๐›ผ>0 such that if โ€–โ€–๐œƒ0โ€–โ€–ฬ‡๐ป2โˆ’2๐›ผ<๐‘๐›ผ,(2.14) then the initial value problem (๐’ฎ๐›ผ) has a unique solution in ๐’ž๐‘(โ„+,ฬ‡๐ป2โˆ’2๐›ผ(โ„2))โˆฉ๐ฟ2(โ„+,ฬ‡๐ป2โˆ’๐›ผ(โ„2)). Moreover, โ€–๐œƒ(๐‘ก)โ€–2ฬ‡๐ป2โˆ’2๐›ผ+๎€œ๐‘ก0โ€–๐œƒ(๐œ)โ€–2ฬ‡๐ป2โˆ’๐›ผ๐‘‘๐œโ‰ค๐‘๎…ž๐›ผ,โˆ€๐‘กโ‰ฅ0,(2.15) where ๐’ž๐‘(โ„+,ฬ‡๐ป2โˆ’2๐›ผ(โ„2)) is the space of continuous and bounded functions from โ„+ to ฬ‡๐ป2โˆ’2๐›ผ(โ„2).

In use of the fact that ฬ‡๐ป2โˆ’2๐›ผ(โ„2) is a Hilbert space, one deduces the following.

Corollary 2.5. Assume that ๐›ผโˆˆ(1/2,1] and ๐œƒ0โˆˆฬ‡๐ป2โˆ’2๐›ผ(โ„2), then there exists a constant ๐‘๐›ผ>0 such that if โ€–โ€–๐œƒ0โ€–โ€–ฬ‡๐ป2โˆ’2๐›ผ<๐‘๐›ผ,(2.16) then the initial value problem (๐’ฎ๐›ผ) has a unique solution in ๐’ž๐‘(โ„+,ฬ‡๐ป2โˆ’2๐›ผ(โ„2))โˆฉ๐ฟ2(โ„+,ฬ‡๐ป2โˆ’๐›ผ(โ„2)). Moreover, โ€–๐œƒ(๐‘ก)โ€–2ฬ‡๐ป2โˆ’2๐›ผ+๎€œ๐‘ก0โ€–๐œƒ(๐œ)โ€–2ฬ‡๐ป2โˆ’๐›ผโ€–โ€–๐œƒ๐‘‘๐œโ‰ค0โ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผ,โˆ€๐‘กโ‰ฅ0.(2.17)

Proof . Taking the scalar product in ฬ‡๐ป2โˆ’2๐›ผ(โ„2), we get 12๐œ•๐‘กโ€–๐œƒโ€–2ฬ‡๐ป2โˆ’2๐›ผ+โ€–๐œƒโ€–2ฬ‡๐ป2โˆ’๐›ผโ‰ค||โŸจ(๐‘ขโ‹…โˆ‡)๐œƒ,๐œƒโŸฉฬ‡๐ป2โˆ’2๐›ผ||โ‰ค||โŸจdiv(๐œƒ๐‘ข),๐œƒโŸฉฬ‡๐ป2โˆ’2๐›ผ||โ‰คโ€–div(๐œƒ๐‘ข)โ€–ฬ‡๐ป2โˆ’3๐›ผโ€–๐œƒโ€–ฬ‡๐ป2โˆ’๐›ผโ‰คโ€–๐œƒ๐‘ขโ€–ฬ‡๐ป3โˆ’3๐›ผโ€–๐œƒโ€–ฬ‡๐ป2โˆ’๐›ผ.(2.18)
Using Lemma 2.1 with ๐‘ 1=2โˆ’2๐›ผ<1 and ๐‘ 2=2โˆ’๐›ผ, we obtain 12๐œ•๐‘กโ€–๐œƒโ€–2ฬ‡๐ป2โˆ’2๐›ผ+โ€–๐œƒโ€–2ฬ‡๐ป2โˆ’๐›ผโ‰ค๐ถ๐›ผโ€–๐œƒโ€–ฬ‡๐ป2โˆ’2๐›ผโ€–๐œƒโ€–2ฬ‡๐ป2โˆ’๐›ผ,๎‚ต๐ถ๐›ผ=12๐‘๐›ผ๎‚ถ.(2.19) Then the quadratic term can be absorbed, 12๐œ•๐‘กโ€–๐œƒโ€–2ฬ‡๐ป2โˆ’2๐›ผ+โ€–๐œƒโ€–2ฬ‡๐ป2โˆ’๐›ผโ‰ค0.(2.20) Taking the integral on the interval [0,๐‘ก], we obtain โ€–๐œƒ(๐‘ก)โ€–2ฬ‡๐ป2โˆ’2๐›ผ+๎€œ๐‘ก0โ€–๐œƒ(๐œ)โ€–2ฬ‡๐ป2โˆ’๐›ผโ€–โ€–๐œƒ๐‘‘๐œโ‰ค0โ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผ,โˆ€๐‘กโ‰ฅ0.(2.21)

3. Proof of the Main Theorem

The proof of the first part will be in two steps.

First Step (Small Initial Data)
In this case, we suppose that โ€–โ€–๐œƒ0โ€–โ€–ฬ‡๐ป2โˆ’2๐›ผ<๐‘๐›ผ,(3.1) with ๐‘๐›ผ a sufficient small number. Then from Corollary 2.5, ๐œƒโˆˆ๐’ž๐‘๎€ทโ„+,ฬ‡๐ป2โˆ’2๐›ผ๎€ทโ„2๎€ธ๎€ธโˆฉ๐ฟ2๎€ทโ„+,ฬ‡๐ป2โˆ’๐›ผ๎€ทโ„2,๎€ธ๎€ธ(3.2)โ€–๐œƒโ€–2ฬ‡๐ป2โˆ’2๐›ผ+๎€œ๐‘ก0โ€–๐œƒโ€–2ฬ‡๐ป2โˆ’๐›ผโ‰คโ€–โ€–๐œƒ0โ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผ,โˆ€๐‘กโ‰ฅ0.(3.3)
For a strictly positive real number ๐›ฟ and a given distribution ๐‘“, we define the operators ๐ด๐›ฟ(๐ท) and ๐ต๐›ฟ(๐ท), respectively, by the following: ๐ด๐›ฟ(๐ท)๐‘“โˆถ=๐œ’๐ต(0,๐›ฟ)๎€ท||๐ท||๎€ธ๐‘“=โ„ฑโˆ’1๎€ท๐œ’๐ต(0,๐›ฟ)๎€ธ,๐ตโ„ฑ(๐‘“)๐›ฟ๎€ท(๐ท)๐‘“โˆถ=1โˆ’๐ด๐›ฟ๎€ธ(๐ท)๐‘“=โ„ฑโˆ’1๎€ท๎€ท1โˆ’๐œ’๐ต(0,๐›ฟ)๎€ธ๎€ธ.โ„ฑ(๐‘“)(3.4) We define ๐‘ค๐›ฟ=๐ด๐›ฟ(๐ท)๐œƒ and ๐‘ฃ๐›ฟ=๐ต๐›ฟ(๐ท)๐œƒ; โ„ฑ(๐œƒ)=โ„ฑ(๐‘ค๐›ฟ)+โ„ฑ(๐‘ฃ๐›ฟ). Then, ๐œ•๐‘ก๐‘ค๐›ฟ+(โˆ’ฮ”)๐›ผ๐‘ค๐›ฟ+๐ด๐›ฟ๐œ•(๐ท)(๐‘ขโ‹…โˆ‡๐œƒ)=0,๐‘กโ€–โ€–๐‘ค๐›ฟโ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผโ€–โ€–๐‘ค+2๐›ฟโ€–โ€–2ฬ‡๐ป2โˆ’๐›ผโ‰ค๐ถโ€–๐œƒโ€–ฬ‡๐ป2โˆ’2๐›ผโ‹…โ€–๐œƒโ€–ฬ‡๐ป2โˆ’๐›ผโ‹…โ€–โ€–๐‘ค๐›ฟโ€–โ€–ฬ‡๐ป2โˆ’๐›ผ.(3.5) We deduce that โ€–โ€–๐‘ค๐›ฟโ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผโ‰คโ€–โ€–๐‘ค๐›ฟโ€–โ€–(0)2ฬ‡๐ป2โˆ’2๐›ผ+๐ถโ€–๐œƒ(0)โ€–ฬ‡๐ป2โˆ’2๐›ผ๎€œโˆž0โ€–๐œƒโ€–ฬ‡๐ป2โˆ’๐›ผโ€–โ€–๐‘ค๐›ฟโ€–โ€–ฬ‡๐ป2โˆ’๐›ผ๐‘‘๐œ.(3.6) Since โ€–๐‘ค๐›ฟโ€–ฬ‡๐ป2โˆ’๐›ผโ‰คโ€–๐œƒโ€–ฬ‡๐ป2โˆ’๐›ผ, then from the dominate convergence theorem and (3.3), we have lim๐›ฟโ†’0Sup๐‘กโ‰ฅ0โ€–โ€–๐‘ค๐›ฟโ€–โ€–ฬ‡๐ป2โˆ’2๐›ผ=0.(3.7) The function ๐‘ฃ๐›ฟ satisfies ๐œ•๐‘ก๐‘ฃ๐›ฟ+(โˆ’ฮ”)๐›ผ๐‘ฃ๐›ฟ+๐ต๐›ฟ๐œ•(๐ท)(๐‘ขโ‹…โˆ‡๐œƒ)=0,๐‘ก||โ„ฑ๎€ท๐‘ฃ๐›ฟ๎€ธ||2||๐œ‰||+22๐›ผ||โ„ฑ๎€ท๐‘ฃ๐›ฟ๎€ธ||2โ‰ค||๎€ท๐‘ฃโ„ฑ(๐‘ขโ‹…โˆ‡๐œƒ)โ„ฑ๐›ฟ๎€ธ||.(3.8) Multiplying this equation by |๐œ‰|2(2โˆ’2๐›ผ)๐‘’2๐‘ก|๐œ‰|2๐›ผ, we deduce that โ€–โ€–๐‘ฃ๐›ฟโ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผโ‰ค๎€œ|๐œ‰|>๐›ฟ||๐œ‰||2(2โˆ’2๐›ผ)๐‘’โˆ’2๐‘ก|๐œ‰|2๐›ผ||โ„ฑ๎€ท๐‘ฃ0๐›ฟ๎€ธ||2+๎€œ๐‘ก0๎€œ|๐œ‰|>๐›ฟ||๐œ‰||2(2โˆ’2๐›ผ)๐‘’โˆ’2(๐‘กโˆ’๐œ)|๐œ‰|2๐›ผ||โ„ฑ๎€ท๐‘ฃ(๐‘ขโ‹…โˆ‡๐œƒ)โ„ฑ๐›ฟ๎€ธ||๐‘‘๐œ‰๐‘‘๐œโ‰ค๐‘’โˆ’2๐‘ก๐›ฟ2๐›ผโ€–โ€–๐‘ฃ0๐›ฟโ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผ๎€œ+๐ถ๐‘ก0๐‘’โˆ’2(๐‘กโˆ’๐œ)๐›ฟ2๐›ผ๎€œ๐œ‰||๐œ‰||2(2โˆ’2๐›ผ)||๎€ท๐‘ฃโ„ฑ(๐‘ขโ‹…โˆ‡๐œƒ)โ„ฑ๐›ฟ๎€ธ||๐‘‘๐œ‰๐‘‘๐œ.(3.9) Using Remark 2.3 and (3.3), we get โ€–โ€–๐‘ฃ๐›ฟโ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผโ‰ค๐‘’โˆ’2๐‘ก๐›ฟ2๐›ผโ€–โ€–๐‘ฃ0๐›ฟโ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผโ€–โ€–๐œƒ+๐ถ0โ€–โ€–ฬ‡๐ป2โˆ’2๐›ผ๎€œ๐‘ก0๐‘’โˆ’2(๐‘กโˆ’๐œ)๐›ฟ2๐›ผโ€–๐œƒโ€–2ฬ‡๐ป2โˆ’๐›ผ๐‘‘๐œ.(3.10) We set ๐น๐›ฟ(๐‘ก)=๐‘’โˆ’2๐‘ก๐›ฟ2๐›ผโ€–โ€–๐‘ฃ0๐›ฟโ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผโ€–โ€–๐œƒ+๐ถ0โ€–โ€–ฬ‡๐ป2โˆ’2๐›ผ๎€œ๐‘ก0๐‘’โˆ’2(๐‘กโˆ’๐œ)๐›ฟ2๐›ผโ€–๐œƒโ€–2ฬ‡๐ป2โˆ’๐›ผ๎€œ๐‘‘๐œ,0+โˆž๐‘’โˆ’2๐‘ก๐›ฟ2๐›ผโ€–โ€–๐‘ฃ0๐›ฟโ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผโ€–โ€–๐‘ฃ๐‘‘๐‘ก=0๐›ฟโ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผ2๐›ฟ2๐›ผโ‰คโ€–โ€–๐œƒ0โ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผ2๐›ฟ2๐›ผ,๎€œ0+โˆž๎€œ๐‘ก0๐‘’โˆ’2(๐‘กโˆ’๐œ)๐›ฟ2๐›ผโ€–๐œƒโ€–2ฬ‡๐ป2โˆ’๐›ผ๎€œ๐‘‘๐œ๐‘‘๐‘ก=0+โˆž๎‚ต๎€œ๐œ+โˆž๐‘’โˆ’2(๐‘กโˆ’๐œ)๐›ฟ2๐›ผ๎‚ถ๐‘‘๐‘กโ€–๐œƒโ€–2ฬ‡๐ป2โˆ’๐›ผ=1๐‘‘๐œ2๐›ฟ2๐›ผ๎€œ0+โˆžโ€–๐œƒโ€–2ฬ‡๐ป2โˆ’๐›ผโ€–โ€–๐œƒ๐‘‘๐‘กโ‰ค0โ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผ4๐›ฟ2๐›ผ.(3.11) Then, ๎€œ0+โˆž๐น๐›ฟโ€–โ€–๐œƒ(๐‘ก)๐‘‘๐‘กโ‰ค0โ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผ๐›ฟ2๐›ผ.(3.12) Let ๐œ€>0, from (3.7), there exists ๐›ฟ0>0 such that โ€–โ€–๐‘ค๐›ฟ0โ€–โ€–ฬ‡๐ป2โˆ’2๐›ผโ‰ค๐œ€2,โˆ€๐‘กโ‰ฅ0.(3.13) Let ๐ธ๐›ฟ0={๐‘กโ‰ฅ0;โ€–๐‘ฃ๐›ฟ0โ€–ฬ‡๐ป2โˆ’2๐›ผ>๐œ€/2}, then ๎€œ0+โˆžโ€–โ€–๐‘ฃ๐›ฟ0โ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผ๎€œ๐‘‘๐‘กโ‰ฅ๐ธ๐›ฟ0โ€–โ€–๐‘ฃ๐›ฟ0โ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผ๎‚€๐œ€๐‘‘๐‘กโ‰ฅ2๎‚2๐œ†1๎€ท๐ธ๐›ฟ0๎€ธ,(3.14) where ๐œ†1(๐ธ๐›ฟ0) is the Lebesgue measure of ๐ธ๐›ฟ0. If ๐‘‡๐œ€=๎‚€2๐œ€๎‚2๎€œ0+โˆžโ€–โ€–๐‘ฃ๐›ฟ0โ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผ๐‘‘๐‘ก,(3.15) then ๐œ†1(๐ธ๐›ฟ0)โ‰ค๐‘‡๐œ€. For ๐œ‚>0, there exists ๐‘ก0โˆˆ[0,๐‘‡๐œ€+๐œ‚] such that ๐‘ก0โˆ‰๐ธ๐›ฟ0, and it results that โ€–โ€–๐‘ฃ๐›ฟ0๎€ท๐‘ก0๎€ธโ€–โ€–ฬ‡๐ป2โˆ’2๐›ผโ‰ค๐œ€2.(3.16) Equation (3.13) and (3.16) give that โ€–โ€–๐œƒ๎€ท๐‘ก0๎€ธโ€–โ€–ฬ‡๐ป2โˆ’2๐›ผโ‰ค๐œ€.(3.17) Thus, lim๐‘กโ†’+โˆžโ€–๐œƒ(๐‘ก)โ€–ฬ‡๐ป2โˆ’2๐›ผ=0, and this finishes the proof in this case.

Second Step (Large Initial Data)
To prove the result for any initial data, it suffices to prove the existence of some ๐‘ก0โ‰ฅ0 such that โ€–โ€–๐œƒ๎€ท๐‘ก0๎€ธโ€–โ€–ฬ‡๐ป2โˆ’2๐›ผ<๐‘๐›ผ.(3.18) Let ๐œƒ0=๐‘Ž0+๐‘Ÿ0, with ๐‘Ž0โˆถ=โ„ฑโˆ’1๎€ท๐Ÿ{1/๐‘<|๐œ‰|<๐‘}โ„ฑ๎€ท๐œƒ0,๐‘Ÿ๎€ธ๎€ธ0โˆถ=๐œƒ0โˆ’๐‘Ž0,โ€–โ€–๐‘Ÿ0โ€–โ€–ฬ‡๐ป2โˆ’2๐›ผ<๐‘๐›ผ.(3.19)
Now, consider the following system: ๐œ•๐‘ก๐‘Ÿ+(โˆ’ฮ”)๐›ผ๐‘Ÿ+(๐‘…โ‹…โˆ‡)๐‘Ÿ=0inโ„+ร—โ„2,๐‘Ÿ(0)=๐‘Ÿ0inโ„2,๐‘…=โˆ‡โŸ‚ฮ”โˆ’1/2๐‘Ÿ.(3.20) By Corollary 2.5, there is a unique solution ๐‘Ÿโˆˆ๐’ž๐‘(โ„+,ฬ‡๐ป2โˆ’2๐›ผ(โ„2))โˆฉ๐ฟ2(โ„+,ฬ‡๐ป2โˆ’๐›ผ(โ„2)) such that (โ€–๐‘Ÿ๐‘ก)โ€–2ฬ‡๐ป2โˆ’2๐›ผ+๎€œ๐‘ก0(โ€–๐‘Ÿ๐œ)โ€–2ฬ‡๐ป2โˆ’๐›ผโ€–โ€–๐‘Ÿ๐‘‘๐œโ‰ค0โ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผ.(3.21) Let ๐‘Žโˆถ=๐œƒโˆ’๐‘Ÿโˆˆ๐’ž(โ„+,ฬ‡๐ป2โˆ’2๐›ผ(โ„2)), then ๐‘Ž is a solution of the following system: ๐œ•๐‘ก๐‘Ž+(โˆ’ฮ”)๐›ผ๐‘Ž+(๐ดโ‹…โˆ‡)๐‘Ž+(๐ดโ‹…โˆ‡)๐‘Ÿ+(๐‘…โ‹…โˆ‡)๐‘Ž=0inโ„+ร—โ„2,๐‘Ž(0)=๐‘Ž0inโ„2,๐ด=โˆ‡โŸ‚ฮ”โˆ’1/2๐‘Ž.(S1) Taking a scalar product in ๐ฟ2(โ„2), we obtain ๐œ•๐‘กโ€–โ€–๐‘Ž(๐‘ก)2๐ฟ2โ€–+2โ€–๐‘Ž(๐‘ก)2ฬ‡๐ป๐›ผ||||๎€œโ‰ค2โ„2||||||||๎€œ(๐ดโ‹…โˆ‡)๐‘Ÿ๐‘Žโ‰ค2โ„2||||div(๐‘Ÿ๐ด)๐‘Žโ‰ค2โ€–๐‘Ÿ๐ดโ€–ฬ‡๐ป1โˆ’๐›ผโ€–๐‘Žโ€–ฬ‡๐ป๐›ผ.(3.22) Using the product law in Lemma 2.1, with ๐‘ 1=2โˆ’2๐›ผ<1 and ๐‘ 2=๐›ผ<1, ||โŸจ(๐ดโ‹…โˆ‡)๐‘Ÿ,๐‘ŽโŸฉ๐ฟ2(โ„2)||โ‰ค๐ถ(๐›ผ)โ€–๐‘Ÿโ€–ฬ‡๐ป2โˆ’2๐›ผโ€–๐ดโ€–ฬ‡๐ป๐›ผโ€–๐‘Žโ€–ฬ‡๐ป๐›ผโ‰ค๐ถ(๐›ผ)โ€–๐‘Ÿโ€–ฬ‡๐ป2โˆ’2๐›ผโ€–๐‘Žโ€–2ฬ‡๐ป๐›ผโ‰คโ€–๐‘Žโ€–2ฬ‡๐ป๐›ผ,(3.23) then, for all ๐‘กโ‰ฅ0, ๐œ•๐‘กโ€–โ€–๐‘Ž(๐‘ก)2๐ฟ2โ€–+โ€–๐‘Ž(๐‘ก)2ฬ‡๐ป๐›ผ(โ‰ค0,โ€–๐‘Ž๐‘ก)โ€–2๐ฟ2+๎€œ๐‘ก0(โ€–๐‘Ž๐œ)โ€–2ฬ‡๐ป๐›ผโ€–โ€–๐‘Ž๐‘‘๐œโ‰ค0โ€–โ€–2๐ฟ2,(3.24) then 2โˆ’2๐›ผ=๐œ†ร—0+(1โˆ’๐œ†)๐›ผ, with ๐œ†โˆถ=3โˆ’(2/๐›ผ)โˆˆ[0,1], โ€–โ€–๐‘Ž(๐‘ก)ฬ‡๐ป2โˆ’2๐›ผโ€–โ‰คโ€–๐‘Ž(๐‘ก)๐ฟ3โˆ’2/๐›ผ2โ€–โ€–๐‘Ž(๐‘ก)ฬ‡๐ป2/๐›ผโˆ’2๐›ผโ‰คโ€–โ€–๐‘Ž0โ€–โ€–๐ฟ3โˆ’2/๐›ผ2โ€–๐‘Ž(๐‘ก)โ€–ฬ‡๐ป2/๐›ผโˆ’2๐›ผ.(3.25) Then, ๎€œโˆž0โ€–๐‘Ž(๐‘ก)โ€–ฬ‡๐ป๐›ผ/(1โˆ’๐›ผ)2โˆ’2๐›ผโ€–โ€–๐‘Ž๐‘‘๐‘กโ‰ค0โ€–โ€–๐ฟ1/(1โˆ’๐›ผ)2.(3.26) Now define the set ๐‘†๐œ€๎€ฝโ€–โˆถ=๐‘กโ‰ฅ0;โ€–๐‘Ž(๐‘ก)ฬ‡๐ป2โˆ’2๐›ผ๎€พ>๐œ€(3.27) as a measurable with respect to the Lebesgue measure. We have ๐œ€๐›ผ/(1โˆ’๐›ผ)๐œ†1๎€ท๐‘†๐œ€๎€ธโ‰ค๎€œ๐‘†๐œ€(โ€–๐‘Ž๐‘ก)โ€–ฬ‡๐ป๐›ผ/(1โˆ’๐›ผ)2โˆ’2๐›ผโ€–โ€–๐‘Ž๐‘‘๐‘กโ‰ค0โ€–โ€–๐ฟ1/(1โˆ’๐›ผ)2.(3.28) So ๐œ†1(๐‘†๐œ€)<โˆž and ๐œ†1(๐‘†๐œ€)โ‰ค๐œ€๐›ผ/(1โˆ’๐›ผ)โ€–๐‘Ž0โ€–๐ฟ1/(1โˆ’๐›ผ)2, then there is ๐‘ก0โˆˆ๎€บ0,๐œ†1๎€ท๐‘†๐œ€๎€ธ๎€ป+1โงต๐‘†๐œ€.(3.29) Then, โ€–โ€–๐‘Ž๎€ท๐‘ก0๎€ธโ€–โ€–ฬ‡๐ป2โˆ’2๐›ผ<๐œ€,(3.30) and then โ€–โ€–๐œƒ๎€ท๐‘ก0๎€ธโ€–โ€–ฬ‡๐ป2โˆ’2๐›ผโ‰คโ€–โ€–๐‘Ÿ๎€ท๐‘ก0๎€ธโ€–โ€–ฬ‡๐ป2โˆ’2๐›ผ+โ€–โ€–๐‘Ž๎€ท๐‘ก0๎€ธโ€–โ€–ฬ‡๐ป2โˆ’2๐›ผ<๐œ€2+๐œ€2=๐œ€.(3.31)

Applying the conclusion of Theorem 1.1 for (๐’ฎ๐›ผ) system starting at ๐œƒ(๐‘ก0), we can deduce the desired result.

In the nonhomogeneous case, we suppose that ๐œƒโˆˆ๐’ž(โ„+,๐ป2โˆ’2๐›ผ), then lim๐‘กโ†’โˆžโ€–๐œƒ(๐‘ก)โ€–ฬ‡๐ป2โˆ’2๐›ผ=0.(3.32)

We can suppose that โ€–๐œƒโ€–ฬ‡๐ป2โˆ’2๐›ผ<๐‘๐›ผ, and for all ๐‘กโ‰ฅ0, (โ€–๐œƒ๐‘ก)โ€–2ฬ‡๐ป2โˆ’2๐›ผ+๎€œ๐‘ก0(โ€–๐œƒ๐œ)โ€–2ฬ‡๐ป2โˆ’๐›ผโ€–โ€–๐œƒ๐‘‘๐œโ‰ค0โ€–โ€–2ฬ‡๐ป2โˆ’2๐›ผ.(3.33)

Thus, it suffices to prove that lim๐‘กโ†’โˆžโ€–๐œƒ(๐‘ก)โ€–๐ฟ2=0.(3.34)

Let ๐›ฟ>0, then we recall the operators ๐ด๐›ฟ(๐ท)๐œƒ=โ„ฑโˆ’1๎€ท๐œ’๐ต(0,๐›ฟ)๎€ธ,๐ตโ„ฑ(๐œƒ)๐›ฟ(๐ท)๐œƒ=โ„ฑโˆ’1๎€ท๎€ท1โˆ’๐œ’๐ต(0,๐›ฟ)๎€ธ๎€ธ.โ„ฑ(๐œƒ)(3.35) We define ๐‘ค๐›ฟ=๐ด๐›ฟ(๐ท)(๐œƒ) and ๐‘ฃ๐›ฟ=๐ต๐›ฟ(๐ท)(๐œƒ). Then, ๐œ•๐‘ก๐‘ค๐›ฟ+(โˆ’ฮ”)๐›ผ๐‘ค๐›ฟ+๐ด๐›ฟ๎€ท๐‘ข(๐ท)๐œƒ๎€ธโ‹…โˆ‡๐œƒ=0,(3.36) and from Lemma 2.2, ๐œ•๐‘กโ€–โ€–๐‘ค๐›ฟโ€–โ€–2๐ฟ2โ€–โ€–๐‘ค+2๐›ฟโ€–โ€–2ฬ‡๐ป๐›ผโ‰ค๐ถโ€–๐œƒโ€–ฬ‡๐ป2โˆ’2๐›ผ.โ€–๐œƒโ€–ฬ‡๐ป๐›ผโ‹…โ€–โ€–๐‘ค๐›ฟโ€–โ€–ฬ‡๐ป๐›ผ.(3.37) We deduce that โ€–โ€–๐‘ค๐›ฟโ€–โ€–2๐ฟ2โ‰คโ€–โ€–๐‘ค๐›ฟโ€–โ€–(0)2๐ฟ2โ€–โ€–๐œƒ+๐ถ0โ€–โ€–ฬ‡๐ป2โˆ’2๐›ผ๎€œ0+โˆžโ€–๐œƒโ€–ฬ‡๐ป๐›ผโ€–โ€–๐‘ค๐›ฟโ€–โ€–ฬ‡๐ป๐›ผ๐‘‘๐œ.(3.38) Then from the dominate convergence theorem and the following ๐ฟ2energy estimate โ€–๐œƒโ€–2๐ฟ2๎€œ+2๐‘ก0โ€–๐œƒโ€–2ฬ‡๐ป๐›ผโ€–โ€–๐œƒ๐‘‘๐œโ‰ค0โ€–โ€–2๐ฟ2,(3.39) we deduce that lim๐›ฟโ†’0Sup๐‘กโ‰ฅ0โ€–โ€–๐‘ค๐›ฟโ€–โ€–๐ฟ2๐œ•=0,๐‘ก๐‘ฃ๐›ฟ+(โˆ’ฮ”)๐›ผ๐‘ฃ๐›ฟ+๐ต๐›ฟ๎€ท๐‘ข(๐ท)๐œƒ๎€ธ๐œ•โ‹…โˆ‡๐œƒ=0,(3.40)๐‘ก||โ„ฑ๎€ท๐‘ฃ๐›ฟ๎€ธ||2||๐œ‰||+22๐›ผ||โ„ฑ๎€ท๐‘ฃ๐›ฟ๎€ธ||2โ‰ค||๎€ท๐‘ฃโ„ฑ(๐‘ขโ‹…โˆ‡๐œƒ)โ„ฑ๐›ฟ๎€ธ||.(3.41)

Multiplying this equation by ๐‘’2๐‘ก|๐œ‰|2๐›ผ, we haveโ€–โ€–๐‘ฃ๐›ฟโ€–โ€–2๐ฟ2โ‰ค๐‘’โˆ’2๐‘ก๐›ฟ2๐›ผโ€–โ€–๐‘ฃ0๐›ฟโ€–โ€–2๐ฟ2๎€œ+๐ถ๐‘ก0๐‘’โˆ’2(๐‘กโˆ’๐œ)๐›ฟ2๐›ผ||||๎ƒก๐‘ขโ‹…โˆ‡๐œƒ๐‘ฃ๐›ฟ๎ƒข||||2๐ฟ2๎€ทโ„2๎€ธ๐‘‘๐œโ‰ค๐‘’โˆ’2๐‘ก๐›ฟ2๐›ผโ€–โ€–๐‘ฃ0๐›ฟโ€–โ€–2๐ฟ2โ€–โ€–๐œƒ+๐ถ0โ€–โ€–ฬ‡๐ป2โˆ’2๐›ผ๎€œ๐‘ก0๐‘’โˆ’2(๐‘กโˆ’๐œ)๐›ฟ2๐›ผโ€–๐œƒโ€–2ฬ‡๐ป๐›ผ๐‘‘๐œ.(3.42) We set ๐น๐›ฟ(๐‘ก)=๐‘’โˆ’2๐‘ก๐›ฟ2๐›ผโ€–โ€–๐‘ฃ0๐›ฟโ€–โ€–2๐ฟ2โ€–โ€–๐œƒ+๐ถ0โ€–โ€–ฬ‡๐ป2โˆ’2๐›ผ๎€œ๐‘ก0๐‘’โˆ’2(๐‘กโˆ’๐œ)๐›ฟ2๐›ผโ€–๐œƒโ€–2ฬ‡๐ป๐›ผ๎€œ๐‘‘๐œ,0+โˆž๐‘’โˆ’2๐‘ก๐›ฟ2๐›ผโ€–โ€–๐‘ฃ0๐›ฟโ€–โ€–2๐ฟ2โ€–โ€–๐œƒ๐‘‘๐‘ก=0โ€–โ€–2๐ฟ22๐›ฟ2๐›ผ,๎€œ0+โˆž๎€œ๐‘ก0๐‘’โˆ’2(๐‘กโˆ’๐œ)๐›ฟ2๐›ผโ€–๐œƒโ€–2ฬ‡๐ป๐›ผ๎€œ๐‘‘๐œ๐‘‘๐‘ก=0+โˆž๎‚ต๎€œ๐œ+โˆž๐‘’โˆ’2(๐‘กโˆ’๐œ)๐›ฟ2๐›ผ๎‚ถ๐‘‘๐‘กโ€–๐œƒโ€–2ฬ‡๐ป๐›ผ=1๐‘‘๐œ2๐›ฟ2๐›ผ๎€œ0+โˆžโ€–๐œƒโ€–2ฬ‡๐ป๐›ผโ€–โ€–๐œƒ๐‘‘๐‘กโ‰ค0โ€–โ€–2๐ฟ22๐›ฟ2๐›ผ.(3.43) Then,๎€œ0+โˆž๐น๐›ฟโ€–โ€–๐œƒ(๐‘ก)๐‘‘๐‘กโ‰ค0โ€–โ€–2๐ฟ2๐›ฟ2๐›ผ.(3.44) Let ๐œ€>0, from (3.40), then there exists ๐›ฟ0>0 such that โ€–โ€–๐‘ค๐›ฟ0โ€–โ€–๐ฟ2โ‰ค๐œ€2,โˆ€๐‘กโ‰ฅ0.(3.45) Let ๐ธ๐›ฟ0={๐‘กโ‰ฅ0;โ€–๐‘ฃ๐›ฟ0โ€–๐ฟ2>๐œ€/2}, then ๎€œ0+โˆžโ€–โ€–๐‘ฃ๐›ฟ0โ€–โ€–2๐ฟ2๎€œ๐‘‘๐‘กโ‰ฅ๐ธ๐›ฟ0โ€–โ€–๐‘ฃ๐›ฟ0โ€–โ€–2๐ฟ2๎‚€๐œ€๐‘‘๐‘กโ‰ฅ2๎‚2๐œ†1๎€ท๐ธ๐›ฟ0๎€ธ,(3.46) where ๐œ†1(๐ธ๐›ฟ0) is the Lebesgue measure of ๐ธ๐›ฟ0. If ๐‘‡๐œ€=๎‚€2๐œ€๎‚2๎€œ0+โˆžโ€–โ€–๐‘ฃ๐›ฟ0โ€–โ€–2๐ฟ2๐‘‘๐‘ก,(3.47) then ๐œ†1(๐ธ๐›ฟ0)โ‰ค๐‘‡๐œ€. For ๐œ‚>0, there exists ๐‘ก0โˆˆ[0,๐‘‡๐œ€+๐œ‚] such that ๐‘ก0โˆ‰๐ธ๐›ฟ0, then โ€–โ€–๐‘ฃ๐›ฟ0๎€ท๐‘ก0๎€ธโ€–โ€–๐ฟ2โ‰ค๐œ€2.(3.48) The equations (3.45) and (3.48) give that โ€–โ€–๐œƒ๎€ท๐‘ก0๎€ธโ€–โ€–๐ฟ2<๐œ€.(3.49) Thus, lim๐‘กโ†’+โˆžโ€–๐œƒ(๐‘ก)โ€–๐ฟ2=0, and this finishes the proof.

Acknowledgments

This research is supported by NPST Program of King Saud University, project number 10-MAT1293-02. The authors thank the referee for his/her careful reading of the paper and corrections.

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