Volume 2009, Article ID 298980,15pages doi:10.1155/2009/298980
Research Article
Self-Similar Solutions for Nonlinear Schr ¨odinger Equations
Yaojun Ye
Department of Mathematics and Information Science, Zhejiang University of Science and Technology, Hangzhou 310023, China
Correspondence should be addressed to Yaojun Ye,[email protected] Received 19 March 2009; Accepted 22 August 2009
Recommended by Ben T. Nohara
We study the self-similar solutions for nonlinear Schr ¨odinger type equations of higher order with nonlinear term|u|αuby a scaling technique and the contractive mapping method. For some admissible valueα, we establish the global well-posedness of the Cauchy problem for nonlinear Schr ¨odinger equations of higher order in some nonstandard function spaces which contain many homogeneous functions. we do this by establishing some nonlinear estimates in the Lorentz spaces or Besov spaces. These new global solutions to nonlinear Schr ¨odinger equations with small data admit a class of self-similar solutions.
Copyrightq2009 Yaojun Ye. 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.
1. Introduction
This paper is concerned with the following Cauchy problem for the nonlinear Schr ¨odinger type equation:
iut −Δmuμ|u|αu, x∈Rn, t∈R,
ux,0 u0x, x∈Rn, 1.1
whereμ∈Ris a constant,m≥1 is an integer,uut, xis a complex-valued function defined onR×Rn R≡0,∞, and the initial datau0xis a complex-valued function defined in Rn. Pecher and Wahl1have established the existence of the classical solution to the Cauchy problem for the higher-order Schr ¨odinger equation1.1by making use ofLp-estimates of the associated elliptic equation in conjunction with the compactness method. Recently Sj ¨ogren and Sj ¨olin studied the local smoothing effect of the solutions to the Cauchy problem 1.1 by means of the Strichartz estimates in nonhomogeneous spaces2,3. Moreover, there are
some work4–6which is devoted to the investigation of the global well-posedness and the scattering theory of the problem1.1. However, little attention is paid to the self-similar solutions of the Cauchy problem1.1.
Our goal is to prove the existence of the global self-similar solutions to the Cauchy problem1.1for some admissible parameterα. From the scaling principle, it is easy to see that ifut, xis a solution of the Cauchy problem1.1, thenuλt, x λ2m/αuλ2mt, λxwith λ >0 is also a solution of equation in1.1with the initial valueλ2m/αu0λx. We thus have the following definition.
Definition 1.1. ut, x is said to be a self-similar solution to the higher-order Schr ¨odinger equation in1.1if
ut, x uλt, x λ2m/αu
λ2mt, λx
, ∀λ >0. 1.2
ByDefinition 1.1, we know that the self-similar solution to1.1is of the form ut, x t−1/αU
x
2m√ t
, 1.3
whereU: Rn → Cis called profile of the solution, and the initial valueu0is of the form
u0x Ωx
|x|2m/α, 1.4
wherex x/|x|andΩis defined on the unit sphereSnofRn. Therefore the problem1.1can be studied through a nonlinear higher-order elliptic equation onU. However, this is usually very complicated. By virtue of this method, Kavian and Weissler 7 have dealt with the radially symmetric solutions of1.1in the casem1, u0x |x|−2/α.
Another important way of looking for self-similar solutions for the nonlinear Schr ¨odinger equation in 1.1 is to study the small global well-posedness of associated Cauchy problem1.1in some suitable function spaces. These global solutions admit a class of self-similar solutions. As a consequence, if ut, xis the unique solution of the Cauchy problem1.1with the initial datau0given by1.4, thenut, xis a self-similar solution of the problem.
On the other hand, if ut, x is a self-similar solution to the problem 1.1, then the initial function is u0x λ2m/αu0λx. So u0x is homogeneous of degree −2m/α.
In general, such homogeneous functions do not belong to the usual Lebesgue spaces and Sobolev spaces.
To do our work, several definitions and notations are required. Denote bySRnand SRn the Schwartz space and the space of Schwartz distribution functions, respectively.
LrRn denotes the usual Lebesgue space on Rn with the norm · r for 1 ≤ r ≤ ∞. For s∈Rand 1< r <∞, letHrsRn 1−Δ−s/2LrRn, the inhomogeneous Sobolev space in terms of Bessel potentials; let ˙HrsRn −Δ−s/2LrRn, the homogeneous Sobolev space in terms of Riesz potentials, and writeHsRn H2sRnand ˙HsRn H˙2sRn. We will omitRn from spaces and norms. For any intervalI ⊂R orIRand for any Banach spaceX, we denote byCI;Xthe space of strongly continuous functions fromItoX and byLqI;Xthe
space of strongly measurable functions fromItoXwithu·X∈LqI. Finally, letq >0, q stands for the dual toq, that is,1/q 1/q 1;adenotes the largest integer less or equal toa.
Whenm1, the equation in1.1becomes the classical Schr ¨odinger equation
iut−Δuμ|u|αu, x∈Rn, t∈R,
ux,0 u0x, x∈Rn, 1.5
which describes many physical phenomena, and the well-posedness as well as the scattering theory for the Cauchy problem 1.5 has been extensively studied by many authors 8–
11. Cazenave and Weissler12,13 also Ribaud and Youssfi14have studied the self- similar solutions of the equation in1.5with initial valueu0xas1.4. Their common ideas are to introduce the new function spaceEs,p Es,pR×Rnwhich consists of all Bochner measurable functionsu:0,∞ → H˙psRnsuch thatuEs,p supt>0tσut, xH˙ps<∞, where 2≤p <∞,0≤s < n/pandσσs, p 1/22/α−n/p s. They then established the existence of global self-similar solutions inEs,pfor the problem1.5under the condition that u0Es,p < ε.
This paper is organized as follows. In the next section, we will recall the definition and basic properties of function spaces that we require. Then inSection 3we state the main results and the related propositions. The last section is devoted to the proof of main results.
2. Function Spaces
2.1. Lorentz SpacesLp,qRn
Definition 2.1. Let f∗t, t ∈ 0,∞, be the nonincreasing rearrangement of a measurable functionfx, x∈Rn, thenf∈SRn is said to be inLp,qRnif and only if
f
p,q ∞
0
t1/pf∗tqdt t
1/q
, 2.1
when 1≤p, q <∞, and
fp,∞sup
t≥0t1/pf∗t<∞, 2.2
when 1≤p <∞, whereup,qis the quasinorm of spaceLp,qRn.
We refer the reader to 15, 16 for the definitions and detailed properties of the nonincreasing rearrangement functions and Lorentz spaces. In fact, Lorentz spaceLp,qRn is a generalization of Lebesgue space LpRn. We have Lp,qRn LpRn as p q, and LpRn ⊂ Lp,qRn ⊂ Lp,∞Rnasq > p. Meanwhile, a lot of properties of Lebesgue spaces are still valid in Lorentz spaces.
We may prove the following results according toDefinition 2.1.
Proposition 2.2. Suppose that 1≤p <∞, 1≤q≤ ∞, then
Rn
fxgxdx
≤Cf
p,q·g
p,q, 2.3
Rn
f·, ydy p,q
≤C
Rn
f·, y
p,qdy, 2.4
fα
p,qfα
pα,qα. 2.5
The inequalities2.3and2.4are essentially the H ¨older and Minkowski inequality in Lorentz spaces, respectively, and they can be proved by usingDefinition 2.1. Furthermore, noting that Lp,qRn is a real interpolation of Lebesgue space, we immediately obtain the following proposition.
Proposition 2.3. Let 0< α < n, 1≤p < r <∞, 1≤q≤ ∞and 1/r 1/p−α/n, then
Rn
xfy−yn−αdy
r,q
≤Cf
p,q. 2.6
2.2. Besov Spaces
We first recall briefly the definition of Besov spaces. For detailed properties and embedding theorems, we are referred to15,17.
Letϕ0∈SRnsatisfyϕ0ξ 1 as|ξ| ≤1 andϕ0ξ 0 as|ξ| ≥2,
ϕjξ ϕ0
2−jξ
, ψjξ ϕ0
2−jξ
−ϕ0
2−j1ξ
, j ∈Z, 2.7
then we have the Littlewood-Paley decomposition
ϕ0ξ ∞
j0
ψjξ 1, ξ∈Rn,
j∈Z
ψjξ 1, ξ∈Rn\ {0},
jlim→∞ϕjξ 1, ξ∈Rn.
2.8
For convenience, we introduce the following notions:
ΔjfF−1ψjFf ψj∗f, SjfF−1ϕjFf ϕj∗f, j ∈Z, 2.9
whereFandF−1stand for Fourier and inverse Fourier transforms, respectively.
Definition 2.4. Assume thats∈R, 1≤q≤ ∞, then
Bs,qp
⎧⎪
⎨
⎪⎩f ∈SRn|f
Bps,q S0f
p
⎛
⎝∞
j1
2jsqΔjfq
p
⎞
⎠
1/q
ϕ0∗f
p
⎛
⎝∞
j1
2jsqψj∗fq
p
⎞
⎠
1/q
<∞
⎫⎪
⎬
⎪⎭
2.10
is called Besov space, and
B˙s,qp
⎧⎪
⎨
⎪⎩f∈SRn|f˙
Bps,q
⎛
⎝∞
j−∞
2jsqΔjfq
p
⎞
⎠
1/q
⎛
⎝
j∈Z
2jsqψj∗fq
p
⎞
⎠
1/q
<∞
⎫⎪
⎬
⎪⎭ 2.11
is homogeneous Besov space.
In particular, we have
B˙ps,∞
f∈SRn|f˙
Bs,∞p sup
j∈Z 2jsΔjf
psup
j∈Z 2jsψj∗f
p<∞
. 2.12
Besides the classical Besov spaces, we also need the so-called generalized Besov spaces.
Definition 2.5. LetEbe a Banach space, then, fors∈R and 1≤q≤ ∞, defines ˙Bs,qE as
B˙s,qE
⎧⎪
⎨
⎪⎩f∈E|f˙
BEs,q
⎛
⎝
j∈Z
2jsqΔjfq
E
⎞
⎠
1/q
<∞
⎫⎪
⎬
⎪⎭, 2.13
whereΔjis the Littlewood-Paley operator onRndefined as above.
Remark 2.6. IfEis the Lorentz spaceLp,rRn, then
B˙s,qLp,r
⎧⎪
⎨
⎪⎩f ∈Lp,r|f˙
Bs,qLp,r
⎛
⎝
j∈Z
2jsqΔjfq
Lp,r
⎞
⎠
1/q
<∞
⎫⎪
⎬
⎪⎭. 2.14
This space is useful in the study of self-similar solutions.
Remark 2.7. LetELqI;LrwithI RorI⊂Rbeing an interval, then we have
B˙s,pLqI;Lr
⎧⎪
⎨
⎪⎩f ∈LqI;Lr|f˙
Bs,pLqI;Lr
⎛
⎝
j∈Z
2jspΔjfp
LqI;Lr
⎞
⎠
1/p
<∞
⎫⎪
⎬
⎪⎭,
B˙s,∞LqI;Lr
f ∈LqI;Lr|f˙
BLqs,∞I;Lr sup
j∈Z 2jsΔjf
LqI;Lr<∞
,
2.15
where 1≤q≤ ∞, 1≤r≤ ∞, 1≤p <∞.
Remark 2.8. In addition to the Besov spaces norm in Definition 2.4, we usually use the following equivalent norms for the Besov spaces ˙Bs,qp andBps,q:
vB˙s,qp
|α|N
⎛
⎝∞
0
t−qσsup
|y|≤t
Δ2y∂αvq
p
dt t
⎞
⎠
1/q
, vBs,qp vpvB˙s,qp ,
2.16
whereΔ2yvτyvτ−yv−2v, τ±yv· v· ±y;∂α∂α11∂α22· · ·∂αnn, ∂i∂/∂xi, i1,2, . . . , n: α α1, α2, . . . , αn,andsNσwith a nonnegative integerNand 0< σ <2. Whensis not an integer,2.16is also equivalent to the following norm:
vB˙ps,q
|α|s
∞
0
t−qs−ssup
|y|≤t
Δy∂αvq
p
dt t
1/q
, 2.17
whereΔ±yv· τ±yv−v. In the case whenq ∞, the above norm should be modified as follows:
vB˙ps,∞
|α|N
sup
t>0
sup
|y|≤tt−σΔ2y∂αv
p, s∈R, vB˙ps,∞
|α|s
sup
t>0
sup
|y|≤t
t−ssΔy∂αv
p, s /∈Z.
2.18
3. Main Results
To solve our problems, we may rewrite1.1in the equivalent integral equation of the form
ut Stu0x−iμ t
0
St−τ
|uτ|αuτ
dτ, 3.1
where St ei−Δmt F−1ei|ξ|2mtF· is the free group generated by the free equation of Schr ¨odinger typeivt −Δmv0.
Definition 3.1. One calls q, ra classical admissible pair with respect to the 2m-order Schr ¨odinger operator if
2 q n
m 1
2 −1 r
, 3.2
where 2≤r <∞forn≤2m; 2≤r≤2n/n−2mforn >2m.
To proveTheorem 3.3we need the following generalized Strichartz estimates which follow directly from the stationary phase method, the Strichartz estimates, and interpolation theoremssee5,15,18for details.
Proposition 3.2. LetSt ei−Δmt, 2≤p, l≤ ∞andq, rsatisfy3.2; then Stϕx
p,l≤C|t|−n/m1/2−1/pϕx
p,l, 3.3
Stϕx
Lq,2I;Lr,2≤Cϕx
2, 3.4
t
0
St−τfx, τdτ
L∞I;L2≤Cf
Lq,2I;Lr,2, 3.5
t
0
St−τfx, τdτ
Lq,2I;Lr,2≤Cf
Lq,2I;Lr,2. 3.6
Moreover, ifα >4m/n,2/β n/m1/2−sc/n−1/α2, then Stϕx
Lβ,∞I;Lα2,∞≤Cϕx˙
B2sc,∞, 3.7
wheresc n/2−2m/α.
Our main results state as follows.
Theorem 3.3. iLetβ 2mαα2/4m−n−2mα,4m/n < α <∞forn≤ 2m; 4m/n <
α <4m/n−2mforn >2m. There exists anε >0 such that ifu0 ∈B˙s2c,∞withu0B˙sc,∞2 ≤ε, then the Cauchy problem1.1(or3.1) has a unique global solutionut, xwith
ut, x∈L∞
R; ˙Bs2c,∞
∩Lβ,∞
R;Lα2,∞
, n≤2m ut, x∈L∞
R; ˙B2sc,∞
∩B˙sLc2,∞R;L2n/n−2m,2∩Lβ,∞
R;Lα2,∞
, n >2m.
3.8
iiLet α∈2N,n >2m,andα≥4m/n−2m. There exists anε >0 such that ifu0∈B˙s2c,∞
withu0B˙sc,∞2 ≤ε, then1.1has a unique global solution ut, x∈L∞
R; ˙Bs2c,∞
∩B˙sLc2,∞R;L2n/n−2m,2. 3.9
iiiLet α /∈2N, and let the condition (a) 2m < n <4√
2mfor 1≤m <8, α≥4m/n−2m;
or (b)n > 2mform≥ 8,α∈ 4m/n−2m, α−∪α,∞be satisfied, whereα−andα are two positive roots of equation 2x2−nx4m0 andα−< α. There exists anε >0 such that ifu0∈B˙s2c,∞
withu0B˙sc,∞2 ≤ε, then the problem1.1has a unique global solution:
ut, x∈L∞
R; ˙Bs2c,∞
∩B˙sc,∞
L2R;L2n/n−2m,2. 3.10
Corollary 3.4see19. Letu0x ε0|x|−2m/α, whereε0 is a positive constant,αsatisfies the assumptions inTheorem 3.3; then there exists a unique global self-similar solution for the Cauchy problem1.1with the initial valueu0x.
Theorem 3.5. Letu0x∈H˙sc satisfy the conditions ofTheorem 3.3; then the global solutionut, x obtained inTheorem 3.3satisfiesut, x∈CR; ˙Hsc.
4. The Proof of Main Results
To prove the main results, we need the following lemmas.
Lemma 4.1see20. Letδpn·max0,1/p−1andm∈Nwithm≥2. Suppose that
k1,2,...,mmin
k /j
1
rk <1, 1 p 1
pj
k /j
1
rk, j1,2, . . . , m. 4.1
Ifs > δp, then there exists a constantC >0 such that
m
i1
fi
˙
Bs,qp
≤C m
j1
fj˙
Bs,qpj
k /j
fk
Lrk 4.2
for allf1, f2, . . . , fm∈m
j1B˙psj,q∩Lrj.
Lemma 4.2. LetFLβ,∞ R;Lα2,∞, whereβ2mαα2/4m−n−2mα, 0< α <∞for n≤2m; 0< α <4m/n−2mforn >2m, then
t
0
St−τ
|uτ|αuτ dτ
F
≤Cuα1F . 4.3
Proof. By2.4inProposition 2.2, we have
t
0
St−τ
|uτ|αuτ dτ
F
≤C
t
0
St−τ
|uτ|αuτ
Lα2,∞dt
Lβ,∞
. 4.4
We get from3.3inProposition 3.2 St−τ
|uτ|αuτ
Lα2,∞≤C|t−τ|−nα/2mα2|uτ|αuτ
Lα2/α1,∞
≤C|t−τ|−nα/2mα2uτα1Lα2,∞.
4.5
Therefore, we obtain fromProposition 2.3and2.5
t
0
St−τ
|uτ|αuτ dτ
F
≤C
t
0
C|t−τ|−nα/2mα2uτα1Lα2,∞dτ
Lβ,∞
≤Cuα1Lα2,∞
L2mαα2/4m−n−2mαα1,∞≤Cuα1F .
4.6
Lemma 4.3see21. Suppose thatEB˙sLc4mα2/nα,∞ R;Lα2,2;F Lβ,∞R;Lα2,∞, then one has |u|αu˙
Bsc,∞
L4mα2/8m−n−4mα,2R;Lα2/α1,2 ≤CuαFuE 4.7
forn≤2m.
Lemma 4.4see22. Letfu |u|αu, sc n/2−2m/αand 1≤sc< α, then fu˙
Bsc,∞
L2R;L2n/n2m,2≤CuB˙sc,∞
L2R;L2n/n2m,2uαL∞R; ˙Bsc,∞2 , 4.8 fu˙
Bsc,∞
L2R;Ll,2≤CuB˙sc,∞
L2R;L2n/n2m,2uα−1L∞R; ˙Bsc,∞2 , 4.9 wherel2nα/n2mα−4m.
4.1. The Proof ofTheorem 3.3
We first provei. Defining the following map by3.1,
Φut Stu0x−iμ t
0
St−τ
|uτ|αuτ
dτ. 4.10
Forn≤2m, we have fromLemma 4.2and3.7inProposition 3.2, ΦuF ≤C
u0B˙2sc,∞uα1F
, 4.11
Φu−ΦvF ≤C
uαFvαF
u−vF. 4.12
LetFε{u|u∈F,uF ≤2Cε} ⊂Fand choose
ε≤ 1 2C
11/α
, 4.13
then we get by4.11and4.12
ΦuFε ≤2Cε Φu−ΦvFε ≤ 1
2u−vFε 4.14
for allu, v∈Fε.
This implies thatΦis a contraction map fromFε intoFε. Thus, there exists a unique solutionu∈Fof1.1withuF ≤2Cε.
LetEB˙sΠc,∞, whereΠ L4mα2/nα,2R;Lα2,2. Then we derive from3.4and3.6
uE≤ Stu0Eμ
t
0
St−τ|u|αudτ
E
≤C
sup
j
2jscΔjStu0
Πsup
j
2jsc
t
0
St−τΔj
|u|αu dτ
Π
≤C
u0B˙sc,∞2 |u|αu˙
Bsc,∞Π
4.15
foru ∈ Fε, whereΠ L4mα2/8m−n−4mα,2R;Lα2/α1,2. As a consequence, we get by Lemma 4.3that
uE≤C
u0B˙2sc,∞uEuαF
. 4.16
It follows that from4.13,CuαF ≤1/2. So,4.16implies that
uE≤2Cu0B˙2sc,∞ <∞. 4.17
Taking theL∞R; ˙B2sc,∞norm in both sides of3.1, we obtain from the definition of generalized Besov spaces,Lemma 4.3and3.4and3.5
uL∞R; ˙Bsc,∞
2 ≤ Stu0L∞R; ˙Bsc,∞2
μ
t
0
St−τ|u|αuτdτ
L∞R; ˙Bsc,∞2
≤C
u0B˙2sc,∞|u|αu˙
BΠsc,∞
≤C
u0B˙2sc,∞uEuαF
<∞,
4.18
which impliesu∈L∞R; ˙B2sc,∞. Therefore, in the case ofn≤2m, we have
ut, x∈L∞
R; ˙B2sc,∞
∩Lβ,∞
R;Lα2,∞
. 4.19
Forn >2m, letG L∞R; ˙B2sc,∞,H B˙Lsc2,∞R;L2n/n−2m,2 andX F∩G∩H, then we obtain from the assumption ini0 < sc < m. In the case of 0 < sc < 1, according to the equivalent norm of Besov spaces and H ¨older inequality it follows that
|u|αuB˙sc,∞
L2R;L2n/n2m,2
sup
|y|≤ττ−scΔy|u|αu
L2R;L2n/n−2m,2
≤Csup
|y|≤ττ−scΔyu
L2R;L2n/n−2m,2τyuα|u|α
L∞R;Ln/2m,∞
≤Csup
|y|≤τ
τ−scΔyu
L2R;L2n/n−2m,2uαL∞R;Lnα/2m,∞.
4.20
Using the Sobolev embedding theorem ˙Bs2c,∞→Lnα/2m,∞, we get that |u|αu˙
Bsc,∞
L2R;L2n/n2m≤CuH· uαG. 4.21
Consequently, fromRemark 2.7,3.4,3.5, and4.21, it follows that
ΦuH≤ Stu0Hμ
t
0
St−τ|u|αudτ
H
≤C
u0B˙2sc,∞|u|αu˙
Bsc,∞
≤C
u0B˙2sc,∞uH· uαG
≤C
u0B˙2sc,∞uα1X .
4.22
Similarly
Φu−ΦvH≤μ
t
0
St−τ|u|αu− |v|αvdτ
H
≤C
uαGvαG
u−vH
≤C
uαXvαX
u−vX.
4.23
By using3.4,3.5, and4.21, and arguing similarly as in deriving4.18one obtain that
ΦuG≤ Stu0Gμ
t
0
St−τ|u|αudτ
G
≤C
u0B˙sc,∞2 |u|αu
L2R;L2n/n2m
≤C
u0B˙sc,∞2 uH· uαG
≤C
u0B˙sc,∞2 uα1X ,
4.24
Φu−ΦvG≤C
uαXvαX
u−vX. 4.25
From4.11and4.12, it follows that
ΦuF ≤C
u0B˙sc,∞2 uα1X
4.26 Φu−ΦvF ≤C
uαXvαX
u−vX. 4.27
Thus, by4.22–4.27we have
ΦuX≤C
u0B˙sc,∞2 uα1X
, 4.28
Φu−ΦvX≤C
uαXvαX
u−vX. 4.29
LettingXε{u|u∈X,uX≤2Cε}, and choosingε≤1/2Cα11/α, then4.28and 4.29imply thatΦis a contraction map fromXεintoXε. By the Banach contraction mapping principle we conclude that there is a unique solutionut, x∈Xε⊂Xsuch that
ut, x∈L∞
R; ˙Bs2c,∞
∩Lβ,∞
R;Lα2,∞
∩B˙sLc2,∞R;L2n/n−2m,2. 4.30
In the case of 1< sc< m, the proof above can see that ofiiibelow.
For a proof ofiisee18.
We now proveiii. Note thatsc n/2−2m/α≥m >1 andsc n/2−2m/α≤α under the assumption iniii.
LetY L∞R; ˙B2sc,∞∩B˙Lsc2,∞R;L2n/n−2m,2G∩H, then by using4.8inLemma 4.4and arguing similarly as in deriving4.24we have
ΦuH≤C
u0B˙sc,∞2 uα1Y
. 4.31
On the other hand, sincefu−fv 10u−v·fuθv−udθ, wherefu |u|αu, it follows fromProposition 3.2,Lemma 4.1, and4.9inLemma 4.4that
Φu−ΦvHμ
t
0
St−τ
fu−fv dτ
H
≤Cfu−fv˙
Bsc,∞
L2R;L2n/n2m,2
C u−v
1
0
fuθv−udθ ˙
Bsc,∞
L2R;L2n/n2m,2
≤Cv−uB˙sc,∞
L2R;L2n/n−2m,2·
1
0
fuθv−udθ
L∞R;Ln/2m
u−vL∞R;Lnα/2m·
1
0
fuθv−udθ ˙
Bsc,∞
L2R,Ll,2
,
4.32
wherel2nα/n2mα−4m.
Becausefuθu−v α1|1θu−θv|α, So we derive from 4.9and the Sobolev embedding theoremL∞R; ˙B2sc,∞→L∞R;Lnα/2mthat
Φu−ΦvH≤ u−vH·|u|α|v|α
L∞R;Ln/2mCu−vH· u−vαG
≤Cu−vH
uαGvαG
≤Cu−vY
uαY vαY .
4.33
By arguing similarly as in deriving4.24and4.25we get ΦuG≤C
u0B˙sc,∞2 uα1Y
, 4.34
Φu−ΦvG≤C
uαY vαY
u−vY. 4.35
it follows from4.31–4.35that
ΦuY ≤C
u0B˙sc,∞2 uα1Y
4.36 Φu−ΦvY ≤C
uαY vαY
u−vY. 4.37
LetYM{u|u∈Y, uY ≤M} withM2Cu0B˙2sc,∞ and chooseε <1/2Cα1/α, then 4.36and4.37 imply thatΦis a contraction map fromYM intoYM. By the Banach contraction mapping principle we obtain that there is a unique solutionut, x ∈ YM ⊂ Y such that
ut, x∈L∞
R; ˙Bs2c,∞
∩B˙sLc2,∞R;L2n/n−2m,2. 4.38
This complete the proof ofTheorem 3.3.
4.2. The Proof ofTheorem 3.5
Without loss of generality we only consider the casen > 2m. FromTheorem 3.3it follows that the Cauchy problem1.1has a unique solutionut, x∈L∞R; ˙Bs2c,∞∩B˙sLc2,∞R;L2n/n−2m,2 provided thatu0B˙sc,∞2 is suitably small. If, in addition,u0 ∈ H˙sc B˙s2c,2, then we have by lettingI L∞R; ˙B2sc,2∩B˙sLc2,2R;L2n/n−2m,2that
uI ≤C
u0H˙sc
t
0
St−τ
|u|αu dτ
I
≤C
u0H˙sc |u|αu˙
Bsc,2
L2R;L2n/n−2m,2
≤C
u0H˙scuαL∞R;Lnα/2m,∞uB˙L2R;L2n/n−2m,2
≤C
u0H˙scuαL∞R; ˙B2sc,∞uI .
4.39
FromTheorem 3.3it follows that the problem1.1–1.4has a unique solutionut, x such thatuαL∞R; ˙Bsc,∞2 ≤1/2 provided thatu0B˙sc,∞2 ≤δwith enough smallδ. Then we have that from4.39
uI ≤2Cu0H˙sc <∞. 4.40
The continuity with respect totofut, xis obvious; sout, x∈CR; ˙Hsc. The proof ofTheorem 3.5is thus completed.
Acknowledgment
This research was supported by the Natural Science Foundation of Henan Province Education Commissionno. 200711013, The Research Foundation of Zhejiang University of Science and Technologyno. 200806, and the Middle-aged and Young Leader in Zhejiang University of Science and Technology2008–2010. The Science and Research Project of Zhejiang Province Education Commissionno. Y200803804.
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