We study the self-similar solutions for nonlinear Schrödinger type equations of higher order with nonlinear term |u|αu
by a scaling technique and the contractive mapping method. For some admissible value α, we establish the global well-posedness of the Cauchy problem for nonlinear Schrödinger 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 Schrödinger equations with small data admit a class of self-similar solutions.
1. Introduction
This paper is concerned with the following Cauchy problem for the nonlinear Schrödinger type equation:
iut+(-Δ)mu=μ|u|αu,x∈Rn,t∈R+,u(x,0)=u0(x),x∈Rn,
where μ∈R is a constant, m≥1 is an integer, u=u(t,x) is a complex-valued function defined on R+×Rn(R+≡[0,+∞)), and the initial data u0(x) is a complex-valued function defined in Rn. Pecher and Wahl [1] have established the existence of the classical solution to the Cauchy problem for the higher-order Schrödinger equation (1.1) by making use of Lp-estimates of the associated elliptic equation in conjunction with the compactness method. Recently Sjögren and Sjölin studied the local smoothing effect of the solutions to the Cauchy problem (1.1) by means of the Strichartz estimates in nonhomogeneous spaces ([2, 3]). Moreover, there are some work ([4–6]) which is devoted to the investigation of the global well-posedness and the scattering theory of the problem (1.1). However, little attention is paid to the self-similar solutions of the Cauchy problem (1.1).
Our goal is to prove the existence of the global self-similar solutions to the Cauchy problem (1.1) for some admissible parameter α. From the scaling principle, it is easy to see that if u(t,x) is a solution of the Cauchy problem (1.1), then uλ(t,x)=λ2m/αu(λ2mt,λx) with λ>0 is also a solution of equation in (1.1) with the initial value λ2m/αu0(λx). We thus have the following definition.
Definition 1.1.
u(t,x) is said to be a self-similar solution to the higher-order Schrödinger equation in (1.1) if
u(t,x)=uλ(t,x)=λ2m/αu(λ2mt,λx),∀λ>0.
By Definition 1.1, we know that the self-similar solution to (1.1) is of the form
u(t,x)=t-(1/α)U(xt2m),
where U:Rn→C is called profile of the solution, and the initial value u0 is of the form
u0(x)=Ω(x′)|x|2m/α,
where x′=x/|x| and Ω is defined on the unit sphere Sn of Rn. Therefore the problem (1.1) can be studied through a nonlinear higher-order elliptic equation on U. However, this is usually very complicated. By virtue of this method, Kavian and Weissler [7] have dealt with the radially symmetric solutions of (1.1) in the case m=1,u0(x)=|x|-(2/α).
Another important way of looking for self-similar solutions for the nonlinear Schrödinger equation in (1.1) is to study the small global well-posedness of associated Cauchy problem (1.1) in some suitable function spaces. These global solutions admit a class of self-similar solutions. As a consequence, if u(t,x) is the unique solution of the Cauchy problem (1.1) with the initial data u0 given by (1.4), then u(t,x) is a self-similar solution of the problem.
On the other hand, if u(t,x) is a self-similar solution to the problem (1.1), then the initial function is u0(x)=λ2m/αu0(λx). So u0(x) 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 by S(Rn) and S′(Rn) the Schwartz space and the space of Schwartz distribution functions, respectively. Lr(Rn) denotes the usual Lebesgue space on Rn with the norm ∥·∥r for 1≤r≤∞. For s∈R and 1<r<∞, let Hrs(Rn)=(1-Δ)-(s/2)Lr(Rn), the inhomogeneous Sobolev space in terms of Bessel potentials; let Ḣrs(Rn)=(-Δ)-(s/2)Lr(Rn), the homogeneous Sobolev space in terms of Riesz potentials, and write Hs(Rn)=H2s(Rn) and Ḣs(Rn)=Ḣ2s(Rn). We will omit Rn from spaces and norms. For any interval I⊂R+ (or I=R+) and for any Banach space X, we denote by C(I;X) the space of strongly continuous functions from I to Xand by Lq(I;X) the space of strongly measurable functions from I to X with ∥u(·)∥X∈Lq(I). Finally, let q>0,q′ stands for the dual to q, that is, (1/q)+(1/q′)=1; [a] denotes the largest integer less or equal to a.
When m=1, the equation in (1.1) becomes the classical Schrödinger equation
iut-Δu=μ|u|αu,x∈Rn,t∈R+,u(x,0)=u0(x),x∈Rn,
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 Weissler [12, 13] (also Ribaud and Youssfi [14]) have studied the self-similar solutions of the equation in (1.5) with initial value u0(x) as (1.4). Their common ideas are to introduce the new function space Es,p=Es,p(R+×Rn) which consists of all Bochner measurable functions u:(0,∞)→Ḣps(Rn) such that ∥u∥Es,p=supt>0tσ∥u(t,x)∥Ḣps<∞, where 2≤p<∞,0≤s<n/p and σ=σ(s,p)=(1/2)((2/α)-(n/p)+s). They then established the existence of global self-similar solutions in Es,p for the problem (1.5) under the condition that ∥u0∥Es,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 in Section 3 we state the main results and the related propositions. The last section is devoted to the proof of main results.
2. Function Spaces2.1. Lorentz Spaces Lp,q(Rn)Definition 2.1.
Let f*(t),t∈(0,∞), be the nonincreasing rearrangement of a measurable functionf(x),x∈Rn, thenf∈S′(Rn) is said to be inLp,q(Rn)if and only if
∥f∥p,q={∫0∞(t1/pf*(t))qdtt}1/q,
when 1≤p,q<∞, and
∥f∥p,∞=supt≥0t1/pf*(t)<+∞,
when 1≤p<∞, where ∥u∥p,q is the quasinorm of space Lp,q(Rn).
We refer the reader to [15, 16] for the definitions and detailed properties of the nonincreasing rearrangement functions and Lorentz spaces. In fact, Lorentz space Lp,q(Rn) is a generalization of Lebesgue space Lp(Rn). We have Lp,q(Rn)=Lp(Rn) as p=q, and Lp(Rn)⊂Lp,q(Rn)⊂Lp,∞(Rn) as q>p. Meanwhile, a lot of properties of Lebesgue spaces are still valid in Lorentz spaces.
We may prove the following results according to Definition 2.1.
Proposition 2.2.
Suppose that1≤p<∞,1≤q≤∞, then
|∫Rnf(x)g(x)dx|≤C∥f∥p,q·∥g∥p′,q′,∥∫Rnf(·,y)dy∥p,q≤C∫Rn∥f(·,y)∥p,qdy,∥|f|α∥p,q=∥f∥pα,qαα.
The inequalities (2.3) and (2.4) are essentially the Hölder and Minkowski inequality in Lorentz spaces, respectively, and they can be proved by using Definition 2.1. Furthermore, noting that Lp,q(Rn) 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
∥∫Rnf(y)|x-y|n-αdy∥r,q≤C∥f∥p,q.
2.2. Besov Spaces
We first recall briefly the definition of Besov spaces. For detailed properties and embedding theorems, we are referred to [15, 17].
Let φ0∈S(Rn) satisfy φ̂0(ξ)=1 as |ξ|≤1 and φ̂0(ξ)=0 as |ξ|≥2,
φ̂j(ξ)=φ̂0(2-jξ),ψ̂j(ξ)=φ̂0(2-jξ)-φ̂0(2-j+1ξ),j∈Z,
then we have the Littlewood-Paley decomposition
For convenience, we introduce the following notions:
Δjf=ℱ-1ψ̂jℱf=ψj*f,Sjf=ℱ-1φ̂jℱf=φj*f,j∈Z,
where ℱ and ℱ-1 stand for Fourier and inverse Fourier transforms, respectively.
Definition 2.4.
Assume thats∈R,1≤q≤∞, then
Bps,q={f∈S′(Rn)∣∥f∥Bps,q=∥S0f∥p+(∑j=1∞2jsq∥Δjf∥pq)1/q=∥φ0*f∥p+(∑j=1∞2jsq∥ψj*f∥pq)1/q<∞}
is called Besov space, and
Ḃps,q={f∈S′(Rn)∣∥f∥Ḃps,q=(∑j=-∞∞2jsq∥Δjf∥pq)1/q=(∑j∈Z2jsq∥ψj*f∥pq)1/q<∞}
is homogeneous Besov space.
Besides the classical Besov spaces, we also need the so-called generalized Besov spaces.
Definition 2.5.
LetEbe a Banach space, then, fors∈Rand1≤q≤∞, definesḂEs,qas
ḂEs,q={f∈E∣∥f∥ḂEs,q=(∑j∈Z2jsq∥Δjf∥Eq)1/q<∞},
where Δj is the Littlewood-Paley operator on Rn defined as above.
Remark 2.6.
If E is the Lorentz space Lp,r(Rn), then
ḂLp,rs,q={f∈Lp,r∣∥f∥ḂLp,rs,q=(∑j∈Z2jsq∥Δjf∥Lp,rq)1/q<∞}.
This space is useful in the study of self-similar solutions.
Remark 2.7.
Let E=Lq(I;Lr) with I=R+ or I⊂R+ being an interval, then we have
ḂLq(I;Lr)s,p={f∈Lq(I;Lr)∣∥f∥ḂLq(I;Lr)s,p=(∑j∈Z2jsp∥Δjf∥Lq(I;Lr)p)1/p<∞},ḂLq(I;Lr)s,∞={f∈Lq(I;Lr)∣∥f∥ḂLq(I;Lr)s,∞=supj∈Z2js∥Δjf∥Lq(I;Lr)<∞},
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 Ḃps,q and Bps,q:
∥v∥Ḃps,q=∑|α|=N(∫0∞t-qσsup|y|≤t∥Δy2∂αv∥pqdtt)1/q,∥v∥Bps,q=∥v∥p+∥v∥Ḃps,q,
where Δy2v=τyv+τ-yv-2v,τ±yv(·)=v(·±y); ∂α=∂1α1∂2α2⋯∂nαn,∂i=∂/∂xi,i=1,2,…,n:α=(α1,α2,…,αn), and s=N+σ with a nonnegative integer N and 0<σ<2. When s is not an integer, (2.16) is also equivalent to the following norm:
∥v∥Ḃps,q=∑|α|=[s](∫0∞t-q(s-[s])sup|y|≤t∥Δy∂αv∥pqdtt)1/q,
where Δ±yv(·)=τ±yv-v. In the case when q=∞, the above norm should be modified as follows:
∥v∥Ḃps,∞=∑|α|=Nsupt>0sup|y|≤tt-σ∥Δy2∂αv∥p,s∈R,∥v∥Ḃps,∞=∑|α|=[s]supt>0sup|y|≤tt-s+[s]∥Δy∂αv∥p,s∉Z.
3. Main Results
To solve our problems, we may rewrite (1.1) in the equivalent integral equation of the form
u(t)=S(t)u0(x)-iμ∫0tS(t-τ)(|u(τ)|αu(τ))dτ,
where S(t)=ei(-Δ)mt=ℱ-1(ei|ξ|2mtℱ·) is the free group generated by the free equation of Schrödinger type ivt+(-Δ)mv=0.
Definition 3.1.
One calls (q,r)a classical admissible pair with respect to the 2m-order Schrödinger operator if
2q=nm(12-1r),
where 2≤r<∞ for n≤2m; 2≤r≤2n/(n-2m) for n>2m.
To prove Theorem 3.3 we need the following generalized Strichartz estimates which follow directly from the stationary phase method, the Strichartz estimates, and interpolation theorems (see [5, 15, 18] for details).
Proposition 3.2.
Let S(t)=ei(-Δ)mt, 2≤p,l≤∞ and (q,r) satisfy (3.2); then
∥S(t)φ(x)∥p,l≤C|t|-(n/m)((1/2)-(1/p))∥φ(x)∥p′,l,∥S(t)φ(x)∥Lq,2(I;Lr,2)≤C∥φ(x)∥2,∥∫0tS(t-τ)f(x,τ)dτ∥L∞(I;L2)≤C∥f∥Lq′,2(I;Lr′,2),∥∫0tS(t-τ)f(x,τ)dτ∥Lq,2(I;Lr,2)≤C∥f∥Lq′,2(I;Lr′,2).
Moreover, if α>4m/n,2/β=(n/m)((1/2)-(sc/n)-(1/(α+2))), then
∥S(t)φ(x)∥Lβ,∞(I;Lα+2,∞)≤C∥φ(x)∥Ḃ2sc,∞,
where sc=(n/2)-(2m/α).
Our main results state as follows.
Theorem 3.3.
(i) Let β=2mα(α+2)/(4m-(n-2m)α),4m/n<α<∞forn≤2m; 4m/n<α<4m/(n-2m) for n>2m. There exists an ε>0 such that if u0∈Ḃ2sc,∞ with ∥u0∥Ḃ2sc,∞≤ε, then the Cauchy problem (1.1) (or (3.1)) has a unique global solution u(t,x) with
u(t,x)∈L∞(R+;Ḃ2sc,∞)∩Lβ,∞(R+;Lα+2,∞),n≤2mu(t,x)∈L∞(R+;Ḃ2sc,∞)∩ḂL2(R+;L2n/(n-2m),2)sc,∞∩Lβ,∞(R+;Lα+2,∞),n>2m.
(ii)Letα∈2N, n>2m, and α≥4m/(n-2m). There exists an ε>0 such that if u0∈Ḃ2sc,∞ with ∥u0∥Ḃ2sc,∞≤ε, then (1.1) has a unique global solution
u(t,x)∈L∞(R+;Ḃ2sc,∞)∩ḂL2(R+;L2n/(n-2m),2)sc,∞.
(iii)Letα∉2N, and let the condition (a) 2m<n<42m for 1≤m<8,α≥4m/(n-2m); or (b) n>2m for m≥8, α∈[4m/(n-2m),α-)∪(α+,∞) be satisfied, where α- and α+ are two positive roots of equation 2x2-nx+4m=0 and α-<α+. There exists an ε>0 such that if u0∈Ḃ2sc,∞ with ∥u0∥Ḃ2sc,∞≤ε, then the problem (1.1) has a unique global solution:
u(t,x)∈L∞(R+;Ḃ2sc,∞)∩ḂL2(R+;L2n/(n-2m),2)sc,∞.
Corollary 3.4 (see [19]).
Let u0(x)=ε0|x|-(2m/α), where ε0 is a positive constant, α satisfies the assumptions in Theorem 3.3; then there exists a unique global self-similar solution for the Cauchy problem (1.1) with the initial value u0(x).
Theorem 3.5.
Let u0(x)∈Ḣsc satisfy the conditions of Theorem 3.3; then the global solution u(t,x) obtained in Theorem 3.3 satisfies u(t,x)∈C(R+;Ḣsc).
4. The Proof of Main Results
To prove the main results, we need the following lemmas.
Lemma 4.1 (see [20]).
Let δp=n·max(0,(1/p)-1) and m∈N with m≥2. Suppose that
mink=1,2,…,m∑k≠j1rk<1,1p=1pj+∑k≠j1rk,j=1,2,…,m.
If s>δp, then there exists a constant C>0 such that
∥∏i=1mfi∥Ḃps,q≤C∑j=1m(∥fj∥Ḃpjs,q)∏k≠j∥fk∥Lrk
for all (f1,f2,…,fm)∈∏j=1m(Ḃpj,qs∩Lrj).
Lemma 4.2.
Let F=Lβ,∞(R+;Lα+2,∞), where β=2mα(α+2)/(4m-(n-2m)α),0<α<∞ for n≤2m; 0<α<4m/(n-2m) for n>2m, then
∥∫0tS(t-τ)(|u(τ)|αu(τ))dτ∥F≤C∥u∥Fα+1.
Proof.
By (2.4) in Proposition 2.2, we have
∥∫0tS(t-τ)(|u(τ)|αu(τ))dτ∥F≤C∥∫0t∥S(t-τ)(|u(τ)|αu(τ))∥Lα+2,∞dt∥Lβ,∞.
We get from (3.3) in Proposition 3.2∥S(t-τ)(|u(τ)|αu(τ))∥Lα+2,∞≤C|t-τ|-nα/(2m(α+2))∥|u(τ)|αu(τ)∥L(α+2)/(α+1),∞≤C|t-τ|-nα/(2m(α+2))∥u(τ)∥Lα+2,∞α+1.
Therefore, we obtain from Proposition 2.3 and (2.5)
∥∫0tS(t-τ)(|u(τ)|αu(τ))dτ∥F≤C∥∫0tC|t-τ|-nα/(2m(α+2))∥u(τ)∥Lα+2,∞α+1dτ∥Lβ,∞≤C∥∥u∥Lα+2,∞α+1∥L2mα(α+2)/[4m-(n-2m)α](α+1),∞≤C∥u∥Fα+1.
Lemma 4.3 (see [21]).
Suppose that E=ḂL4m(α+2)/nα(R+;Lα+2,2)sc,∞; F=Lβ,∞(R+;Lα+2,∞), then one has
∥|u|αu∥ḂL4m(α+2)/(8m-(n-4m)α),2(R+;L(α+2)/(α+1),2)sc,∞≤C∥u∥Fα∥u∥E
for n≤2m.
Lemma 4.4 (see [22]).
Let f(u)=|u|αu,sc=(n/2)-(2m/α) and 1≤sc<α, then
∥f(u)∥ḂL2(R+;L2n/(n+2m),2)sc,∞≤C∥u∥ḂL2(R+;L2n/(n+2m),2)sc,∞∥u∥L∞(R+;Ḃ2sc,∞)α,∥f′(u)∥ḂL2(R+;Ll,2)sc,∞≤C∥u∥ḂL2(R+;L2n/(n+2m),2)sc,∞∥u∥L∞(R+;Ḃ2sc,∞)α-1,
wherel=2nα/((n+2m)α-4m).
4.1. The Proof of Theorem 3.3
We first prove (i). Defining the following map by (3.1),
Φ(u)(t)=S(t)u0(x)-iμ∫0tS(t-τ)(|u(τ)|αu(τ))dτ.
For n≤2m, we have from Lemma 4.2 and (3.7) in Proposition 3.2,
∥Φ(u)∥Fε≤2Cε∥Φ(u)-Φ(v)∥Fε≤12∥u-v∥Fε
for all u,v∈Fε.
This implies that Φ is a contraction map from Fε into Fε. Thus, there exists a unique solution u∈F of (1.1) with ∥u∥F≤2Cε.
Let E=ḂΠsc,∞, where Π=L4m(α+2)/nα,2(R+;Lα+2,2). Then we derive from (3.4) and (3.6)
∥u∥E≤∥S(t)u0∥E+|μ|∥∫0tS(t-τ)(|u|αu)dτ∥E≤C(supj2jsc∥ΔjS(t)u0∥Π+supj2jsc∥∫0tS(t-τ)Δj(|u|αu)dτ∥Π)≤C(∥u0∥Ḃ2sc,∞+∥|u|αu∥ḂΠ′sc,∞)
for u∈Fε, where Π′=L4m(α+2)/(8m-(n-4m)α),2(R+;L(α+2)/(α+1),2). As a consequence, we get by Lemma 4.3 that
∥u∥E≤C(∥u0∥Ḃ2sc,∞+∥u∥E∥u∥Fα).
It follows that from (4.13), C∥u∥Fα≤1/2. So, (4.16) implies that
∥u∥E≤2C∥u0∥Ḃ2sc,∞<∞.
Taking the L∞(R+;Ḃ2sc,∞) norm in both sides of (3.1), we obtain from the definition of generalized Besov spaces, Lemma 4.3 and (3.4) and (3.5)
∥u∥L∞(R+;Ḃ2sc,∞)≤∥S(t)u0∥L∞(R+;Ḃ2sc,∞)+|μ|∥∫0tS(t-τ)(|u|αu)(τ)dτ∥L∞(R+;Ḃ2sc,∞)≤C(∥u0∥Ḃ2sc,∞+∥|u|αu∥ḂΠ′sc,∞)≤C(∥u0∥Ḃ2sc,∞+∥u∥E∥u∥Fα)<∞,
which implies u∈L∞(R+;Ḃ2sc,∞). Therefore, in the case of n≤2m, we have
u(t,x)∈L∞(R+;Ḃ2sc,∞)∩Lβ,∞(R+;Lα+2,∞).
For n>2m, let G=L∞(R+;Ḃ2sc,∞), H=ḂL2(R+;L2n/(n-2m),2)sc,∞ and X=F∩G∩H, then we obtain from the assumption in (i) 0<sc<m. In the case of 0<sc<1, according to the equivalent norm of Besov spaces and Hölder inequality it follows that
Letting Xε={u∣u∈X,∥u∥X≤2Cε}, and choosing ε≤(1/(2C)α+1)1/α, then (4.28) and (4.29) imply that Φ is a contraction map from Xε into Xε. By the Banach contraction mapping principle we conclude that there is a unique solution u(t,x)∈Xε⊂X such that
u(t,x)∈L∞(R+;Ḃ2sc,∞)∩Lβ,∞(R+;Lα+2,∞)∩ḂL2(R+;L2n/(n-2m),2)sc,∞.
In the case of 1<sc<m, the proof above can see that of (iii) below.
For a proof of (ii) see [18].
We now prove (iii). Note that sc=(n/2)-(2m/α)≥m>1 and sc=(n/2)-(2m/α)≤α under the assumption in (iii).
Let Y=L∞(R+;Ḃ2sc,∞)∩ḂL2(R+;L2n/(n-2m),2)sc,∞=G∩H, then by using (4.8) in Lemma 4.4 and arguing similarly as in deriving (4.24) we have
∥Φ(u)∥H≤C(∥u0∥Ḃ2sc,∞+∥u∥Yα+1).
On the other hand, since f(u)-f(v)=∫01(u-v)·f′(u+θ(v-u))dθ, where f(u)=|u|αu, it follows from Proposition 3.2, Lemma 4.1, and (4.9) in Lemma 4.4 that
∥Φ(u)-Φ(v)∥H=|μ|∥∫0tS(t-τ)(f(u)-f(v))dτ∥H≤C∥f(u)-f(v)∥ḂL2(R+;L2n/(n+2m),2)sc,∞=C∥(u-v)∫01f′(u+θ(v-u))dθ∥ḂL2(R+;L(2n/(n+2m)),2)sc,∞≤C∥v-u∥ḂL2(R+;L2n/(n-2m),2)sc,∞·∥∫01f′(u+θ(v-u))dθ∥L∞(R+;Ln/2m)+∥u-v∥L∞(R+;Lnα/2m)·∥∫01f′(u+θ(v-u))dθ∥ḂL2(R+,Ll,2)sc,∞,
where l=2nα/((n+2m)α-4m).
Because f′(u+θ(u-v))=(α+1)|(1+θ)u-θv|α, So we derive from (4.9) and the Sobolev embedding theorem L∞(R+;Ḃ2sc,∞)↪L∞(R+;Lnα/2m) that
Let YM={u∣u∈Y,∥u∥Y≤M} with M=2C∥u0∥Ḃ2sc,∞ and choose ε<(1/2C)(α+1)/α, then (4.36) and (4.37) imply that Φ is a contraction map from YM into YM. By the Banach contraction mapping principle we obtain that there is a unique solution u(t,x)∈YM⊂Y such that
u(t,x)∈L∞(R+;Ḃ2sc,∞)∩ḂL2(R+;L2n/(n-2m),2)sc,∞.
This complete the proof of Theorem 3.3.
4.2. The Proof of Theorem 3.5
Without loss of generality we only consider the case n>2m. From Theorem 3.3 it follows that the Cauchy problem (1.1) has a unique solution u(t,x)∈L∞(R+;Ḃ2sc,∞)∩ḂL2(R+;L2n/(n-2m),2)sc,∞ provided that ∥u0∥Ḃ2sc,∞ is suitably small. If, in addition, u0∈Ḣsc=Ḃ2sc,2, then we have by letting I=L∞(R+;Ḃ2sc,2)∩ḂL2(R+;L2n/(n-2m),2)sc,2 that
From Theorem 3.3 it follows that the problem (1.1)–(1.4) has a unique solution u(t,x) such that ∥u∥L∞(R+;Ḃ2sc,∞)α≤1/2 provided that ∥u0∥Ḃ2sc,∞≤δ with enough small δ. Then we have that from (4.39)
∥u∥I≤2C∥u0∥Ḣsc<∞.
The continuity with respect to t of u(t,x) is obvious; so u(t,x)∈C(R+;Ḣsc).
The proof of Theorem 3.5 is thus completed.
Acknowledgment
This research was supported by the Natural Science Foundation of Henan Province Education Commission (no. 200711013), The Research Foundation of Zhejiang University of Science and Technology (no. 200806), and the Middle-aged and Young Leader in Zhejiang University of Science and Technology (2008–2010). The Science and Research Project of Zhejiang Province Education Commission (no. Y200803804).
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