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On inextensible ruled surfaces generated via a curve derived from a curve with constant torsion

  • Received: 20 December 2022 Revised: 28 February 2023 Accepted: 01 March 2023 Published: 13 March 2023
  • MSC : 53A05

  • If both the arc length and the intrinsic curvature of a curve or surface are preserved, then the flow of the curve or surface is said to be inextensible. The absence of motion-induced strain energy is the physical characteristic of inextensible curve and surface flows. In this paper, we study inextensible tangential, normal and binormal ruled surfaces generated by a curve with constant torsion, which is also called a Salkowski curve. We investigate whether or not these surfaces are minimal or can be developed. In addition, we prove some theorems which are related to inextensible ruled surfaces within three-dimensional Euclidean space.

    Citation: Nural Yüksel, Burçin Saltık. On inextensible ruled surfaces generated via a curve derived from a curve with constant torsion[J]. AIMS Mathematics, 2023, 8(5): 11312-11324. doi: 10.3934/math.2023573

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  • If both the arc length and the intrinsic curvature of a curve or surface are preserved, then the flow of the curve or surface is said to be inextensible. The absence of motion-induced strain energy is the physical characteristic of inextensible curve and surface flows. In this paper, we study inextensible tangential, normal and binormal ruled surfaces generated by a curve with constant torsion, which is also called a Salkowski curve. We investigate whether or not these surfaces are minimal or can be developed. In addition, we prove some theorems which are related to inextensible ruled surfaces within three-dimensional Euclidean space.



    This paper considers the Cauchy problem for the coupled non-linear Schrödinger system, shortened to CNLS,

    {ituj+Δuj=τ(1knajk|uk|σ)|uj|σ2uj;u|t=0=u0. (CNLS)

    Here and hereafter, for j[1,n], uj is a complex valued function of the variable (t,y)R+×RN, and u:=(u1,,un). The constant 0τC and the coupling parameters satisfy:

    ajk=akj0 and ajj>0for anyj,k[1,n].

    The coupled Schrödinger system (CNLS) models many physical phenomena, such as the propagation in birefringent optical fibers, Kerr-like photorefractive media in optic and Bose-Einstein condensates [1,4,7], see also [2,3,5,6]. The passage of a light beam with two components along an optical fiber produces the decomposition of the ray into two. Then, the model of a scalar Schrödinger equation can be improved by a system of coupled Shrödinger equations [8]. As a consequence, new kinds of solutions appear, first observed by Manakov [9] in a Kerr medium.

    In mathematical point of view, many authors focused their attention on coupled nonlinear equations of Schrödinger type. So, the list of references is necessarily incomplete. The local well-posedness in the energy space was proved in [10,11]. The existence of ground states was investigated in [12,13,14]. The scattering of defocusing global solutions was treated in [15]. The scattering in the repulsive regime with small data was obtained in [16,17]. See also [18,19,20,21] for the inhomogeneous case and [22] for the case of harmonic potential and [23] for the parabolic context.

    To the author's knowledge, the above works treat only the case σ2 in (CNLS). This is to avoid a singularity of the source term for σ<2. The contribution of this paper is to try to fill in this gap in the literature.

    The purpose of this paper is to prove the existence of a local solution to the coupled Schrödinger problem (CNLS) in the case 32<σ<2. Then, one establishes the existence of global solutions which scatter in some Sobolev spaces and the existence of non-global solutions with finite variance.

    The paper is organized as follows: Section 2 contains the main results and some standard estimates needed in the sequel. Section 3 proves the local existence of solutions to (CNLS). Section 4 establishes the scattering of global solutions and the existence of non-global solutions to (CNLS).

    Throughout this paper, we denote the spaces and norms

    Ws,p:=Ws,p(RN),Hs:=Ws,2,Lr:=Lr(RN);r:=Lr,:=2;(u1,,un)p:=(1jnujpp)1p,(u1,,un):=(u1,,un)2.

    Lastly, T>0 denotes the lifespan for an eventual solution to (CNLS).

    In this section, we give the main results and some standard estimates.

    Solutions of (CNLS) formally satisfy the conservation of the following real quantities, respectively the mass and the energy

    M(u):=1jnRN|uj(t,y)|2dy;E(u):=1jn(|uj(t,y)|2+|uj(t,y)|2dy+τσ1knajk|uj(t,y)uk(t,y)|σ)dy.

    Let the non-linear terms

    F:=F(u):=[(1kna1k|uk|σ)|u1|σ2u1,,(1knank|uk|σ)|un|σ2un];Fj:=Fj(u):=(1knajk|uk|σ)|uj|σ2uj;Fj,k:=Fj,k(u):=|uk|σ|uj|σ2uj.

    Take for sR,

    y2:=1+|y|2andIγ:=(1Δ)s2.

    Here and hereafter, one defines some real numbers

    2σ<α<min{σN2m,p1}; (2.1)
    max{N4(σ1),N2(σ1α),N2}<m<N2(2σ); (2.2)
    M0>N+m; (2.3)
    M>4E[N2]+5+m, (2.4)

    where E[] refers to the integer part. Take the Banach space

    Y:={u(HN+M0+M)n,uY:=1jn(|α|E[N2]ymαuj+E[N2]<|α|Mymαuj+M<|α|N+M0+Mαuj)<}.

    Let the centered ball of radius R>0, denoted by BT(R):={uCT(Y),uLT(Y)R}, and for ν>0, take

    Bν,T(R):={uBT(R),2inft,x|ymuj(t,y)|ν,j[1,n]}.

    Finally, we define the vector Schrödinger group

    eiΔ(u1,,un):=(eiΔu1,,eiΔun).

    In what follows, we list the results proved in this paper. The first purpose of this note is to prove the existence of local solutions to (CNLS).

    Theorem 2.1. Let N1, 32<σ<2, and u0Y, satisfying

    infxRN|ymu0(x)|>0. (2.5)

    Then, there are T>0 and a unique solution to (CNLS) denoted by uCT(Y). Moreover, the flow is locally continuous.

    Remarks 2.1.

    1)The condition σ>32, which seems to be technical, is necessary in order to have (2.1);

    2) the continuity of the flow follows with standard arguments;

    3) assumption (2.5) avoids in some meaning the singularity of the source term for σ<2;

    4) the proof follows ideas in [24], where the scalar case is treated;

    5) some tools needed in the proof are taken from [25];

    6) the proof is based on a Picard fixed point argument.

    The second result is about the scattering of global solutions to (CNLS) in some Sobolev spaces.

    Theorem 2.2. Let N2 and max{1+1N,32}<σ<2. Take v0Y satisfying (2.5), and u0:=eiκt|x|24v0 for some real number κ1. Take 0s<mN2. Then, there is a unique global solution to (CNLS) in C(R+,Hs)yN2L(R+×RN), which scatters in Hs.

    Remarks 2.2.

    1)The proof is based on the pseudo-conformal transformation [26] and Theorem 2.1;

    2) a similar result was first proved [24] in the scalar case;

    3) a part of the proof is omitted because it follows [25];

    4) the real number κ depends on ν,v0Y,N,m.

    Finally, one proves the existence of non-global solutions to (CNLS).

    Proposition 2.1. Take the assumptions of Theorem 2.1, m>1+1N and τ<0. Then, the solution to (CNLS) is non-global if one of the following statements holds:

    1)p1+2N and E(u0)<0;

    2) E(u0)<d and I(u)<0.

    Remarks 2.3.

    1)The blow-up in the first case follows with classical variance method;

    2) the second case follows like [18,Theorem 2.8];

    3) d denotes the ground state energy d:=inf0u(H1)m{E(u),I(u)=0}, where I(u):=BE(u)(B2)u2 and B:=N(σ1).

    Some intermediate results are listed in what follows.

    In order to investigate the blow-up of solutions, one needs the next variance identity [27].

    Proposition 2.2. Assume that uCT([H1]n) is a solution to (CNLS) for τ=1, and satisfies xuj(t)L2, for any 1jn. Then,

    18(1jnxuj2)=1jnuj2B2σ1j,knajkRN(|ukuj|)σdy.

    The following estimate will be useful [25,Lemma 2.9].

    Lemma 2.1. Let α0. Then, for all real number: t, we have

    yαeitΔucαtα(Iαu+yαu).

    Recall also [25,Proposition 2.10].

    Lemma 2.2. Let α0, k,KN, and s2k+K+3+E[N2]+α. Then, for any tR and any |μ|K,

    yαμeitΔutk|γ|kyαγu+t1+k+α(Isu+k<|γ|2k+K+3+E[N2]yαγu).

    One recalls the nonlinear estimates [25,Proposition 3.1].

    Lemma 2.3. Let α0, uBν,T(R), and r[1,]. Then, for all |μ|N+M0+M, we have

    1)

    yαμ(|u|σ)L((0,T),Lr)Rσyαmσr+1k|μ|ν(2kσ)R2k1(Ryαmσr+E[N2]<|γ||μ|yα+m(1σ)γuL((0,T),Lr)).

    2)

    yαμ(|u|σ2u)L((0,T),Lr)0k|μ|ν(2k+2σ)R2k(Ryαm(σ1)r+E[N2]<|γ||μ|yα+m(2σ)γuL((0,T),Lr)).

    Finally, let us give some nonlinear estimates [25,Proposition 3.2].

    Lemma 2.4. Let α0, u,vBν,T(R), and r[1,]. Then, for all |μ|N+M0+M, we have

    1)

    yαμ(|u|σ|v|σ)L((0,T),Lr)(Rσ1+ν2(3σ)R5σ)yαmσruvLT(Y)+1k|μ|ν2(2kσ)R4kσ2(Ryαmσr+E[N2]<|γ||μ|yα+m(1σ)γ(uv)L((0,T),Lr))+1k|μ|ν(2kσ)(R2k1[uvLT(Y)yαmσr+E[N2]<|γ||μ|yα+m(1σ)γ(uv)L((0,T),Lr)]+R2(k1)uvLT(Y)[E[N2]<|γ||μ|yα+m(1σ)γuL((0,T),Lr)+yα+m(1σ)γvL((0,T),Lr)]);

    2)

    yαμ(|u|σ2u|v|σ2v)LT(L2)ν2(3σ)R4σyαmσruvLT(Y)+1k|μ|ν2(2k+2σ)R4kσ+1(Ryαm(σ1)r+E[N2]<|γ||μ|yα+m(2σ)γ(uv)L((0,T),Lr))+1k|μ|ν(2(1+k)σ)(R2k[uvLT(Y)yαm(σ1)r+E[N2]<|γ||μ|yα+m(2σ)γ(uv)L((0,T),Lr)]+R2k1uvLT(Y)[E[N2]<|γ||μ|yα+m(2σ)γuL((0,T),Lr)+yα+m(2σ)γvL((0,T),Lr)]).

    This section is devoted to Theorem 2.1, which deals with the existence of a unique solution of (CNLS) in CT(Y). Take the function

    f(u):=eiΔu0iτ0ei(s)ΔFds:=(f1(u),,fm(u)),

    where u0:=(u0,1,,u0,n), and

    fj(u):=eiΔu0,jiτ0ei(s)ΔFjds.

    Let us prove that f(Bν,T(R))Bν,T(R). By Lemma 2.2, for the choice

    k:=E[N2],s=N+M0+M:=K+2k+3+E[N2]+M0N,

    and for |α|E[N2], we have

    ymαfj(u(t,y))LT(L)TE[N2]+1+m(|γ|E[N2]ymγu0+IN+M0+Mu0+E[N2]<|γ|Mymγu0)+TTE[N2]+1+m(|γ|E[N2]ymγFLT(L)+IN+M0+MFLT(L2)+E[N2]<|γ|MymγFLT(L2)).

    Because p1, by Lemma 2.3, for |γ1|E[N2], we have

    ym(2σ)γ1(|uk|σ)LT(L)Rσy2m(σ1)+1lE[N2]ν(2lσ)R2l1(Ry2m(σ1)+E[N2]<|γ||γ1|ym(32σ)γukLT(L))Rσ+1lE[N2]ν(2lσ)R2l.

    Also, by Lemma 2.3, we get

    ym(σ1)γ2(|uj|σ2uj)LT(L)0lE[N2]ν(2l+2σ)R1+2l.

    Thus, with the Leibniz rule, if |γ|E[N2], we get

    ymγFLT(L)nj,k=1γ=γ1+γ2ym(2σ)γ1(|uk|p)LT(L)ym(σ1)γ2(|uj|σ2uj)LT(L)(Rσ+1lE[N2]ν(2lσ)R2l)(0lE[N2]ν(2l+2σ)R1+2l). (3.1)

    Now, with the Leibniz rule via Lemma 2.3, for E[N2]|γ|M, 12=1r1+1r2, and αR,

    ymγFLT(L2)nj,k=1γ=γ1+γ2,|γ2|Nymαγ1(|uk|p)L((0,T),Lr1)ym(1α)γ2(|uj|σ2uj)L((0,T),Lr2)+nj,k=1γ=γ1+γ2,|γ2|Nymαγ1(|uk|p)L((0,T),Lr1)ym(1α)γ2(|uj|σ2uj)L((0,T),Lr2):=(I|γ2|N)+(I|γ2|N).

    Now, (2.2) enables to take 2σ<α<σN2m, r1=2, and r2=. This implies that ym(ασ)L2. Thus, we have

    (I|γ2|N)nj,k=1γ=γ1+γ2[Rσym(ασ)+1l|γ1|ν(2lσ)R2l1(Rym(ασ)+E[N2]<|μ||γ1|ym(α+1σ)μukLT(L2))][0l|γ2|ν(2l+2σ)R2l(Rym(2ασ)+E[N2]<|μ||γ2|ym(3ασ)μujLT(L)]nj,k=1γ=γ1+γ2[Rσ+1l|γ1|ν(2lσ)R2l1(R+E[N2]<|μ||γ1|ymμukLT(L2))][0l|γ2|ν(2l+2σ)R2l(R+E[N2]<|μ||γ2|ymμujLT(L))]nj,k=1γ=γ1+γ2[Rσ+1l|γ1|ν(2lσ)R2l][0l|γ2|ν(2l+2σ)R2l(R+E[N2]<|μ||γ2|+E[N2]+1ymμujLT(L2))].

    Assuming that |γ2|N, MN+E[N2]+1 gives

    (I|γ2|N)nj,k=1γ=γ1+γ2[Rσ+1l|γ1|ν(2lσ)R2l][0l|γ2|ν(2l+2σ)R2l(R+E[N2]<|μ||γ2|+E[N2]+1ymμujLT(L2))]nj,k=1γ=γ1+γ2[Rσ+1l|γ1|ν(2lσ)R2l][1l|γ2|ν(2l+1σ)R2l(R+E[N2]<|μ|MymμujLT(L2))]nj,k=1γ=γ1+γ2[Rσ+1lMν(2lσ)R2l][0lNν(2l+2σ)R1+2l]. (3.2)

    Now, we assume that |γ2|N, thus |γ1|MNME[N2]1. Since by (2.2) we have m>N2(σ1α) and ym(1+ασ)<. So, with Lemma 2.3, we get

    (I|γ2|N)nj,k=1γ=γ1+γ2[Rσym(ασ)+1l|γ1|ν(2lσ)R2l1(Rym(ασ)+E[N2]<|μ||γ1|ym(α+1σ)μukLT(L))][0l|γ2|ν(2l+2σ)R2l(Rym(1+ασ)+E[N2]<|μ||γ2|ym(3ασ)μujLT(L2))]nj,k=1γ=γ1+γ2[Rσ+1l|γ1|ν(2lσ)R2l1(R+E[N2]<|μ||γ1|ym(α+1σ)μukLT(L))][0l|γ2|ν(2l+2σ)R2l(R+E[N2]<|μ||γ2|ym(3ασ)μujLT(L2))].

    So,

    (I|γ2|N)nj,k=1γ=γ1+γ2[Rσ+1l|γ1|ν(2lσ)R2l(R+1+2E[N2]<|μ|MμukLT(L2))][0l|γ2|ν(2l+2σ)R1+2l]nj,k=1γ=γ1+γ2[Rσ+1lMν(2lσ)R1+2l][0lMν(2l+2σ)R1+2l].

    It follows that

    ymγFLT(L2)nj,k=1γ=γ1+γ2[Rσ+1lMν(2lσ)R1+2l][0lMν(2l+2σ)R1+2l]. (3.3)

    We break down the next term as follows:

    γFLT(L2)nj,k=1γ=γ1+γ2,|γ1|<Nym(2σ)γ1(|uk|p)LT(L)ym(2σ)γ2(|uj|σ2uj)LT(L2)+nj,k=1γ=γ1+γ2,N|γ1|Mym(2σ)γ1(|uk|p)LT(L2)ym(2σ)γ2(|uj|σ2uj)LT(L)+nj,k=1γ=γ1+γ2,M<|γ1|N+M0+Mγ1(|uk|p)LT(L2)γ2(|uj|σ2uj)LT(L):=(A|γ1|<N)+(AN|γ1|M)+(AM<|γ1|N+M0+M). (3.4)

    Now, with the Leibniz rule and Lemma 2.3, for |γ|N+M0+M, we have

    (A|γ1|<N)nj,k=1γ=γ1+γ2ym(2σ)γ1(|uk|p)LT(L)ym(2σ)γ2(|uj|σ2uj)LT(L2)nj,k=1γ=γ1+γ2[Rσy2m(1σ)+1l|γ1|ν(2lσ)R2l1(Ry2m(1σ)+E[N2]<|μ||γ1|ym(32σ)μukLT(L))][0l|γ2|ν(2l+2σ)R2l(Rym+E[N2]<|μ||γ2|μujLT(L2)]. (3.5)

    (2.2) gives ymL2, and

    (A|γ1|<N)nj,k=1γ=γ1+γ2[Rσ+1l|γ1|ν(2lσ)R2l1(R+E[N2]<|μ||γ1|ym(32σ)μukLT(L))][0l|γ2|ν(2l+2σ)R2l(R+E[N2]<|μ|MymμujLT(L2)+M<|μ|N+M0+MμujLT(L2))]nj,k=1γ=γ1+γ2[Rσ+1l|γ1|ν(2lσ)R2l1(R+E[N2]<|μ||γ1|ym(32σ)μukLT(L))][0l|γ2|ν(2l+2σ)R1+2l]. (3.6)

    By Sobolev injection, we have

    ym(32σ)μukLT(L)μukLT(L)μukL((0,T),˙H1+E[N2]).

    Thus, by (2.2), we get

    (A|γ1|<N)nj,k=1γ=γ1+γ2[Rσ+1l|γ1|ν(2lσ)R2l1(R+1+2E[N2]<|μ|1+N+E[N2]μukLT(L2))][0l|γ2|ν(2l+2σ)R1+2l]nj,k=1γ=γ1+γ2[Rσ+1lNν(2lσ)R2l][0lN+M0+Mν(2l+2σ)R1+2l]. (3.7)

    Now, with Lemma 2.3, we have

    (AN|γ1|M)nj,k=1γ=γ1+γ2ym(2σ)γ1(|uk|p)LT(L2)ym(2σ)γ2(|uj|σ2uj)LT(L)nj,k=1γ=γ1+γ2[Rσy2m(1σ)+1l|γ1|ν(2lσ)R2l1(Ry2m(1σ)+E[N2]<|μ||γ1|ym(32σ)μukLT(L2))][0l|γ2|ν(2l+2σ)R2l(Rym+E[N2]<|μ||γ2|μujLT(L)].

    Moreover, (2.2) gives y2m(1σ)L2, and so

    (AN|γ1|M)nj,k=1γ=γ1+γ2[Rσ+1l|γ1|ν(2lσ)R2l1(R+E[N2]<|μ|Mym(32σ)μukLT(L2))][0l|γ2|ν(2l+2σ)R2l(R+E[N2]<|μ||γ2|μujLT(L))]nj,k=1γ=γ1+γ2[Rσ+1l|γ1|ν(2lσ)R2l][0l|γ2|ν(2l+2σ)R2l(R+E[N2]<|μ|1+E[N2]+|γ2|μujLT(L2))]. (3.8)

    Since N+|γ2||γ1|+|γ2|N+M0+M, we have |γ2|M+M02N, and

    (AN|γ1|M)nj,k=1γ=γ1+γ2[Rσ+1l|γ1|ν(2lσ)R2l][0l|γ2|ν(2l+2σ)R2l(R+E[N2]<|μ|1+E[N2]+|γ2|μujLT(L2))]nj,k=1γ=γ1+γ2[Rσ+1l|γ1|ν(2lσ)R2l][0l|γ2|ν(2l+2σ)R2l(R+E[N2]<|μ|N+M0+MμujLT(L2))]nj,k=1γ=γ1+γ2[Rσ+1lMν(2lσ)R2l][0lN+M0+Mν(2l+2σ)R1+2l]. (3.9)

    Finally, assume that M<|γ1|N+M0+M. Thus, |γ2|M0NME[N2]1. So, with Sobolev injections via Lemma 2.3, we have

    (AM<|γ1|N+M0+M)nj,k=1γ=γ1+γ2γ1(|uk|p)LT(L2)γ2(|uj|σ2uj)LT(L)[Rσymσ+1k|γ1|ν(2kσ)R2k1(Rymσ+E[N2]<|γ||γ1|ym(1σ)γuLT(L2))][0k|γ2|ν(2k+2σ)R2k(Rym(σ1)+E[N2]<|γ||γ2|ym(2σ)γuLT(L))][Rσ+1k|γ1|ν(2kσ)R2k1(R+E[N2]<|γ||γ1|ym(1σ)γuLT(L2))][0k|γ2|ν(2k+2σ)R2k(R+E[N2]<|γ||γ2|+1+E[N2]ymγuLT(L2))].

    Thus,

    (AM<|γ1|N+M0+M)[Rσ+1k|γ1|ν(2kσ)R2k1(R+E[N2]<|γ|MymγuLT(L2)+M<|γ|N+M0+MγuLT(L2))][0kME[N2]1ν(2k+2σ)R1+2k][Rσ+1kN+M0+Mν(2kσ)R2k][0kME[N2]1ν(2k+2σ)R1+2k]. (3.10)

    Collecting (3.7), (3.9), and (3.10), we get

    IM+M0NFLT(L2)[Rσ+1kN+M0+Mν(2kσ)R2k][0kN+M0+Mν(2k+2σ)R1+2k]. (3.11)

    Thus, with (3.1)–(3.3) and (3.11), we get

    ymαfj(u)LT(L)TE[N2]+1+m(|γ|E[N2]ymγu0+IN+M0+Mu0+E[N2]<|γ|Mymγu0)+TTE[N2]+1+m[Rσ+1lN+M0+Mν(2lσ)R2l][0lN+M0+Mν(2l+2σ)R1+2l].

    Now, by Lemma 2.1, (3.3), and (3.11), we write

    ymαfj(u)LT(L2)Tm(IN+M0+Mu0+ymαu0)+TTm(IN+M0+MFLT(L2)+ymαFLT(L2))Tm(IN+M0+Mu0+ymαu0)+TTm[Rσ+1lN+M0+Mν(2lσ)R2l][0lN+M0+Mν(2l+2σ)R1+2l]. (3.12)

    Moreover, with Lemma 2.1 and (3.11), for |α|N+M0+M, yields

    αfj(u)LT(L2)IN+M0+Mu0+TαFLT(L2)u0HN+M0+M+T[Rσ+1lN+M0+Mν(2lσ)R2l][0lN+M0+Mν(2l+2σ)R1+2l].

    So, taking R:=2cu0Y, we get

    f(u)LT(Y)cTE[N2]+1+mu0Y+cTTE[N2]+1+m[Rσ+1lN+M0+Mν(2lσ)R2l][0lN+M0+Mν(2l+2σ)R1+2l]TE[N2]+1+mR2+cTTE[N2]+1+m[Rσ+1lN+M0+Mν(2lσ)R2l][0lN+M0+Mν(2l+2σ)R1+2l]:=TE[N2]+1+mR2+cTTE[N2]+1+mF1(ν,R)F2(ν,R). (3.13)

    Then, choosing 0<T<<1, it follows that f(BT(R))BT(R). Now, we prove that

    2inf(t,y)[0,T]×RN|ymf(u(t,y))|ν.

    Using the time derivative identity (eitΔ)(k)=(iΔ)keitΔ, we get

    eitΔ=0kE[N2](it)kk!Δk+i1+E[N2](E[N2])!t0(ts)E[N2]Δ1+E[N2](eisΔ)ds.

    Thus, with Lemma 2.1 and Sobolev injections via (2.2), we write

    ym(eitΔu0u0)1jn1kE[N2]tkymΔku0,j+t0(ts)E[N2]ymΔ1+E[N2](eisΔu0,j)dsttE[N2](1|α|E[N2]ymαu0+E[N2]<|α|Nymαu0)+t0(ts)E[N2]ymΔ1+E[N2](eisΔu0)dsttE[N2](1|α|E[N2]ymαu0+E[N2]<|α|Mymαu0)+t0(ts)E[N2]ymΔ1+E[N2](eisΔu0)ds.

    Now, by Lemma 2.2, for s=5+m+5E[N2] and K=2(1+E[N2]),

    ymΔ1+E[N2](eisΔu0)tE[N2]|γ|E[N2]ymγu+t1+E[N2]+m(Isu+E[N2]<|γ|2(1+E[N2])+3+3E[N2]ymγu).

    Now, since M4E[N2]+5+m and M0N>m>N2, we get

    ymΔ1+E[N2](eisΔu0)tE[N2]|γ|E[N2]ymγu+t1+E[N2]+m(IN+M0+Mu+E[N2]<|γ|5+5E[N2]ymγu)tE[N2]|γ|E[N2]ymγu+t1+E[N2]+m(IN+M0+Mu+E[N2]<|γ|Mymγu).

    Thus,

    ym(eitΔu0u0)ttE[N2](1|α|E[N2]ymαu0+E[N2]<|α|Mymαu0)+ttE[N2][tE[N2]|γ|E[N2]ymγu+t1+E[N2]+m(IN+M0+Mu+E[N2]<|γ|Mymγu)].

    Now, with Lemma 2.2 with (3.13),

    ym(f(u)eitΔu0)LT(L)cTTE[N2]+1+mF1(ν,R)F2(ν,R).

    Thus, there is C(T)0 as T0 such that

    |ymf(u)||ymeitΔu0||ym(eitΔu0u0)||ym(f(u)eitΔu0)|νC(T). (3.14)

    So, taking 0<T<<1, we get

    inf{(t,y)[0,T]×RN}|ymf(u)|ν2.

    Thus, f(Bν,T(R))Bν,T(R). Now, we prove that f is a contraction. For u,vCT(Y) and w:=uv, we have

    f(u)f(v)LT(Y)=0ei(s)Δ[F(u)F(v)]dsLT(Y)0ei(s)Δ[(|uk|σ|u1|σ2u1|vk|σ|v1|σ2v1,,|uk|σ|un|σ2un|vk|σ|vn|σ2vn)]dsLT(Y)1jn(|α|E[N2]0ymαei(s)Δ[(|uk|σ|uj|σ2uj|vk|σ|vj|σ2vj)]LT(L)+E[N2]<|α|M0ymαei(s)Δ[(|uk|σ|uj|σ2uj|vk|σ|vj|σ2vj)]LT(L2)+M<|α|N+M0+M0αei(s)Δ[(|uk|σ|uj|σ2uj|vk|σ|vj|σ2vj)]LT(L2)).

    Let us control the three above terms. By Lemma 2.2 via (2.4), we have

    (I):=|α|E[N2]0ymαei(s)Δ[(|uk|σ|uj|σ2uj|vk|σ|vj|σ2vj)]LT(L)TTE[N2]+1+m(|γ|E[N2]ymγ(Fj(u)Fj(v))LT(L)+IN+M0+M(Fj(u)Fj(v))LT(L2)+E[N2]<|γ|Mymγ(Fj(u)Fj(v))LT(L2)).

    Let |γ|E[N2], and write

    (I1):=ymγ(Fj(u)Fj(v))LT(L)=ymγ((|uk|σ|vk|σ)|uj|σ2uj+|vk|σ(|uj|σ2uj|vj|σ2vj))LT(L)γ=γ1+γ2(ymγ1(|uk|σ|vk|σ)γ2(|uj|σ2uj)LT(L)+ymγ1|vk|σγ2(|uj|σ2uj|vj|σ2vj)LT(L))γ=γ1+γ2(ym(2σ)γ1(|uk|σ|vk|σ)LT(L)ym(σ1)γ2(|uj|σ2uj)LT(L)+ym(2σ)γ1|vk|σLT(L)ym(σ1)γ2(|uj|σ2uj|vj|σ2vj)LT(L)).

    Taking account of Lemma 2.4, we have

    ym(2σ)γ1(|uk|σ|vk|σ)LT(L)(Rσ1+ν2(3σ)R5σ)y2m(1σ)wLT(Y)+1kE[N2]ν2(2kσ)R4kσ1y2m(1σ)wLT(Y)+1kE[N2]ν(2kσ)R2k1y2m(1σ)wLT(Y)+1kE[N2]ν(2kσ)R2(k1)wLT(Y)[Rσ1+ν2(3σ)R5σ+1kE[N2](ν2(2kσ)R4kσ1+ν(2kσ)R2k1)]wLT(Y).

    Moreover, also with Lemma 2.4, we have

    ym(σ1)γ2(|uk|σ2uk|vk|σ2vk)LT(L)[ν2(3σ)R4σ+1kE[N2](ν2(2(1+k)σ)R4kσ+2+ν(2(1+k)σ)R2k)]wLT(Y).

    Now, by (3.1), it follows that

    (I1)([Rσ1+ν2(3σ)R5σ+1kE[N2](ν2(2kσ)R4kσ1+ν(2kσ)R2k1)]F2(ν,R)+[ν2(3σ)R4σ+1kE[N2](ν2(2(1+k)σ)R4kσ+2+ν(2(1+k)σ)R2k)]F1(ν,R))wLT(Y). (3.15)

    Let E[N2]<γM. Then,

    (I2):=ymγ(Fj(u)Fj(v))LT(L2)=ymγ((|uk|σ|vk|σ)|uj|σ2uj+|vk|σ(|uj|σ2uj|vj|σ2vj))LT(L2)γ=γ1+γ2(ymγ1(|uk|σ|vk|σ)γ2(|uj|σ2uj)LT(L2)+ymγ1|vk|σγ2(|uj|σ2uj|vj|σ2vj)LT(L2))γ=γ1+γ2(ymαγ1(|uk|σ|vk|σ)LT(L2)ym(1α)γ2(|uj|σ2uj)LT(L)+ymαγ1|vk|σLT(L2)ym(1α)γ2(|uj|σ2uj|vj|σ2vj)LT(L)).

    Now, by Lemma 2.4 and using (2.1), we get

    ymαγ1(|uk|σ|vk|σ)LT(L2)(Rσ1+ν2(3σ)R5σ)ym(ασ)wLT(Y)+1kMν2(2kσ)R4kσ2(Rym(ασ)+E[N2]<|μ||γ1|ym(1+ασ)μukLT(L2))wLT(Y)+1kMν(2kσ)(R2k1(ym(ασ)wLT(Y)+E[N2]<|μ||γ1|ym(1+ασ)μ(ukvk)LT(L2))+R2(k1)(E[N2]<|μ||γ1|ym(1+ασ)μukLT(L2)+E[N2]<|μ||γ1|ym(1+ασ)μvkLT(L2))wLT(Y))(Rσ1+ν2(3σ)R5σ+1kMν2(2kσ)R4kσ1+1kMν(2kσ)R2k1)wLT(Y). (3.16)

    Assume that |γ2|N, thus, MN+E[N2]+1 gives by Sobolev embedding and Lemma 2.3 via (2.1), we get

    ym(1α)γ2(|uj|σ2uj)LT(L)0l|γ2|ν(2l+2σ)R2l(Rym(2ασ)+E[N2]<|μ||γ2|ym(3ασ)μujLT(L))0l|γ2|ν(2l+2σ)R2l(R+E[N2]<|μ||γ2|+1+E[N2]ymμujLT(L2))0lMν(2l+2σ)R1+2lF2(ν,R).

    Moreover, with Lemma 2.4, we write

    ym(1α)γ2(|uj|σ2uj|vj|σ2vj)LT(L)ν3(2σ)R4σym(2ασ)wLT(Y)+1kMν2(2(1+k)σ)R4kσ+1(Rym(2ασ)+E[N2]<|μ||γ2|ym(3ασ)μukLT(L))wLT(Y)+1kMν(2(1+k)σ)(R2k(ym(2ασ)wLT(Y)+E[N2]<|μ||γ2|ym(3ασ)μ(ukvk)LT(L)+R2k1(E[N2]<|μ||γ2|ym(3ασ)μukLT(L)+E[N2]<|μ||γ2|ym(3ασ)μvkLT(L))wLT(Y)).

    So, with Sobolev embeddings,

    ym(1α)γ2(|uj|σ2uj|vj|σ2vj)LT(L)ν3(2σ)R4σwLT(Y)+1kMν2(2(1+k)σ)R4kσ+1(R+E[N2]<|μ||γ2|+1+E[N2]ymμukLT(L2))wLT(Y)+1kMν(2(1+k)σ)(R2k(wLT(Y)+E[N2]<|μ||γ2|+1+E[N2]ymμ(ukvk)LT(L2)+R2k1(E[N2]<|μ||γ2|+1+E[N2]ymμukLT(L2)+E[N2]<|μ||γ2|+1+E[N2]ymμvkLT(L2))wLT(Y))(ν3(2σ)R4σ+1kMν2(2(1+k)σ)R4kσ+2+1kMν(2(1+k)σ)R2k)wLT(Y). (3.17)

    Finally, with Lemma 2.3, we have

    ymαγ1|vk|σLT(L2)Rσym(ασ)+1k|γ1|ν(2kσ)R2k1(Rym(ασ)+E[N2]<|γ||γ1|ym(1+ασ)γuLT(L2))Rσ+1k|γ1|ν(2kσ)R2k1(R+E[N2]<|γ||γ1|ym(1+ασ)γuLT(L2))Rσ+1k|γ1|ν(2kσ)R2kF1(ν,R).

    Collecting the above estimates, we get

    γ=γ1+γ2,|γ2|N(ymαγ1(|uk|σ|vk|σ)LT(L2)ym(1α)γ2(|uj|σ2uj)LT(L)+ymαγ1|vk|σLT(L2)ym(1α)γ2(|uj|σ2uj|vj|σ2vj)LT(L))[(Rσ1+ν2(3σ)R5σ+1kMν2(2kσ)R4kσ1+1kMν(2kσ)R2k1)F2(ν,R)+F1(ν,R)(ν3(2σ)R4σ+1kMν2(2(1+k)σ)R4kσ+2+1kMν(2(1+k)σ)R2k)]wLT(Y). (3.18)

    Now, assume that M|γ2|N. Thus, arguing as in (3.3), we have

    γ=γ1+γ2,M|γ2|N(ymαγ2(|uk|σ|vk|σ)LT(L2)ym(1α)γ1(|uj|σ2uj)LT(L)+ymαγ2|vk|σLT(L2)ym(1α)γ1(|uj|σ2uj|vj|σ2vj)LT(L))γ=γ1+γ2,M|γ2|N(ymαγ2(|uk|σ|vk|σ)LT(L2)F2(ν,R)+ym(1α)γ1(|uj|σ2uj|vj|σ2vj)LT(L)F1(ν,R)).

    Moreover, since |γ1|MNME[N2]1, by (3.16) and (3.17), we write

    γ=γ1+γ2,M|γ2|N(ymαγ2(|uk|σ|vk|σ)LT(L2)ym(1α)γ1(|uj|σ2uj)LT(L)+ymαγ2|vk|σLT(L2)ym(1α)γ1(|uj|σ2uj|vj|σ2vj)LT(L))γ=γ1+γ2((Rσ1+ν2(3σ)R5σ+1kMν2(2kσ)R4kσ1+1kMν(2kσ)R2k1)F2(ν,R)wLT(Y)+(ν3(2σ)R4σ+1kMν2(2(1+k)σ)R4kσ+2+1kMν(2(1+k)σ)R2k)F1(ν,R))wLT(Y). (3.19)

    Now, with the Leibniz rule via Lemma 2.3, for |γ|N+M0+M, we have

    (I3):=γ(Fj(u)Fj(v))LT(L2)γ=γ1+γ2(ym(2σ)γ1(|uk|σ|vk|σ)LT(L)ym(2σ)γ2(|uj|σ2uj)LT(L2)+ym(2σ)γ1|vk|σLT(L2)ym(2σ)γ2(|uj|σ2uj|vj|σ2vj)LT(L)).

    Let us start with estimating the first term. By (3.6),

    (A1):=ym(2σ)γ1(|uk|σ|vk|σ)LT(L)ym(2σ)γ2(|uj|σ2uj)LT(L2)ym(2σ)γ1(|uk|σ|vk|σ)LT(L)F2(ν,R).

    With Lemma 2.4, we get

    ym(2σ)γ1(|uk|σ|vk|σ)LT(L)(Rσ1+ν2(3σ)R5σ)y2m(1σ)wLT(Y)+1kMν2(2kσ)R4kσ2(Ry2m(1σ)+E[N2]<|μ||γ1|ym(32σ)μukLT(L))wLT(Y)+1kMν(2kσ)(R2k1(y2m(1σ)wLT(Y)+E[N2]<|μ||γ1|ym(32σ)μ(ukvk)LT(L)+R2(k1)(E[N2]<|μ||γ1|ym(32σ)μukLT(L)+E[N2]<|μ||γ1|ym(32σ)μvkLT(L))wLT(Y)).

    Since σ>32, we get

    ym(2σ)γ1(|uk|σ|vk|σ)LT(L)(Rσ1+ν2(3σ)R5σ)wLT(Y)+1kMν2(2kσ)R4kσ2(R+E[N2]<|μ||γ1|ym(32σ)μukLT(L))wLT(Y)+1kMν(2kσ)(R2k1(wLT(Y)+E[N2]<|μ||γ1|ym(32σ)μ(ukvk)LT(L)+R2(k1)(E[N2]<|μ||γ1|ym(32σ)μukLT(L)+E[N2]<|μ||γ1|ym(32σ)μvkLT(L))wLT(Y)).

    If |γ1|<N, we write by Sobolev injections,

    E[N2]<|μ||γ1|ym(32σ)μ(ukvk)LT(L)E[N2]<|μ|N+1+E[N2]μ(ukvk)LT(L2)E[N2]<|μ|N+M0+Mymμ(ukvk)LT(L2)wLT(Y). (3.20)

    Thus,

    (A1)(Rσ1+ν2(3σ)R5σ+1kMν2(2kσ)R4kσ1+1kMν(2kσ)R2k1)F2(ν,R)uvLT(Y). (3.21)

    If |γ1|N, it is sufficient to estimate the term

    (A1):=ym(2σ)γ1(|uk|σ|vk|σ)LT(L2)ym(2σ)γ2(|uj|σ2uj)LT(L)ym(2σ)γ1(|uk|σ|vk|σ)LT(L2)F2(ν,R),

    where we used (3.9). Moreover, Lemma 2.4 via (2.2) gives

    ym(2σ)γ1(|uk|σ|vk|σ)LT(L2)(Rσ1+ν2(3σ)R5σ)uvLT(Y)+1kMν2(2kσ)R4kσ2(R+E[N2]<|μ||γ1|ym(32σ)μukLT(L2))uvLT(Y)+1kMν(2kσ)(R2k1(uvLT(Y)+E[N2]<|μ||γ1|ym(32σ)μ(ukvk)LT(L2))+R2(k1)(E[N2]<|μ||γ1|ym(32σ)μukLT(L2)+E[N2]<|μ||γ1|ym(32σ)μvkLT(L2))uvLT(Y)).

    Using the fact that

    E[N2]<|μ||γ1|ym(32σ)μukLT(L2)E[N2]<|μ|MymμukLT(L2)+M<|μ|N+M0+MμukLT(L2),

    we have

    ym(2σ)γ1(|uk|σ|vk|σ)LT(L2)(Rσ1+ν2(3σ)R5σ+1kMν2(2kσ)R4kσ1+1kMν(2kσ)R2k1)uvLT(Y).

    Thus,

    (A1)(Rσ1+ν2(3σ)R5σ+1kMν2(2kσ)R4kσ1+1kMν(2kσ)R2k1)F2(ν,R)uvLT(Y). (3.22)

    By (3.9), let us estimate the term

    (A2):=ym(2σ)γ1|vk|σLT(L2)ym(2σ)γ2(|uj|σ2uj|vj|σ2vj)LT(L)F1(ν,R)ym(2σ)γ2(|uj|σ2uj|vj|σ2vj)LT(L).

    If |γ2|N, by a similar reasoning to (3.17), (for α=3σ), we have

    ym(1α)γ2(|uj|σ2uj|vj|σ2vj)LT(L)(ν3(2σ)R4σ+1kMν2(2(1+k)σ)R4kσ+2+1kMν(2(1+k)σ)R2k)uvLT(Y).

    It follows that

    (A2)(ν3(2σ)R4σ+1kMν2(2(1+k)σ)R4kσ+2+1kMν(2(1+k)σ)R2k)(Rσ+1l|γ1|ν(2lσ)R2l)uvLT(Y).

    If |γ2|N, it is sufficient to estimate the term

    (A2):=ym(2σ)γ1|vk|σLT(L)ym(2σ)γ2(|uj|σ2uj|vj|σ2vj)LT(L2). (3.23)

    Moreover, with Lemma 2.4 via (2.2), we write

    ym(σ2)γ2(|uj|σ2uj|vj|σ2vj)LT(L2)ν3(2σ)R4σuvLT(Y)+1kMν2(2(1+k)σ)R4kσ+1(R+E[N2]<|μ||γ2|ymμukLT(L2))uvLT(Y)+1kMν(2(1+k)σ)(R2k(uvLT(Y)+E[N2]<|μ||γ2|ymμ(ukvk)LT(L2)+R2k1(E[N2]<|μ||γ2|ymμukLT(L2)+E[N2]<|μ||γ2|ymμvkLT(L2))uvLT(Y))(ν3(2σ)R4σ+1kMν2(2(1+k)σ)R4kσ+2+1kMν(2(1+k)σ)R2k)uvLT(Y).

    Since N+|γ1||γ1|+|γ2|N+M0+M, we have |γ1|M+M02N. So, arguing as in (3.6), we get

    ym(2σ)γ1(|vk|p)LT(L)Rσ+1l|γ1|ν(2lσ)R2l1(R+E[N2]<|μ||γ1|ym(32σ)μvkLT(L))F1(ν,R).

    Then,

    (A2)(ν3(2σ)R4σ+1kMν2(2(1+k)σ)R4kσ+2+1kMν(2(1+k)σ)R2k)F1(ν,R)uvLT(Y). (3.24)

    Taking (3.15), (3.18), (3.19), (3.21), (3.22), (3.23), and (3.24), we get

    (I1)+(I2)+(I3)F(ν,R)uvLT(Y). (3.25)

    Now, taking Lemma 2.1, (3.2), (3.3), and (3.11) via the estimates of (I), we write

    ymα(fj(u)fj(v))LT(L2)TTm(IN+M0+M(Fj(u)Fj(v))LT(L2)+ymα(Fj(u)Fj(v))LT(L2))TTmF(ν,R)uvLT(Y).

    Moreover, taking Lemma 2.1, and the estimate of (I3), for |α|N+M0+M, we get

    α(fj(u)fj(v))LT(L2)Tα(Fj(u)Fj(v))LT(L2)TF(ν,R)uvLT(Y).

    Finally, f is a contraction of BT(R) for small T>0, and the result follows with a classical Picard argument.

    In this section, we prove Theorem 2.2 and Proposition 2.1.

    Let us start with the next auxiliary result.

    Proposition 4.1. Let max{1+1N,32}<σ2, κ1, and v0Y such that infRN|v0(x)|ν. Then, there is a unique vC([0,1κ],Y) solution to

    ivj+Δvj=τ(1κt)(2N(σ1))(1knajk|vk|σ)|vj|σ2vj,j[1,n]. (4.1)

    Proof of Proposition 4.1: For simplicity and without loss of generality, let us fix τ=1. One applies a Picard fixed point argument. Let the function

    g(v):=eiΔv0+i0(1κs)(2N(σ1))ei(s)ΔFds:=(g1(v),,gm(v)),

    on the space Bν,1κ(R). Taking (3.1), (3.3), and (3.11), via σ>1+1N, we write

    g(v)LT(Y)κ1E[N2]+1+mv0Y+κ1κ1E[N2]+1+mF1(ν,R)F2(ν,R).

    Moreover, arguing as in (3.14), we write

    |ymg(v)|νcκ1κ1E[N2]+1+mv0Ycκ1κ1E[N2]+1+mF1(ν,R)F2(ν,R).

    Moreover, arguing as in (3.25), and the last lines of the previous section, we get

    g(u)g(v)LT(Y)cκ1κ1E[N2]+1+mF(ν,R)uvLT(Y).

    This implies that, for large κ1, an application of the Picard Theorem finishes the proof.

    Take the pseudo conformal transformation [26],

    vj(t,y):=(1κt)N2eiκ|x|24(1κt)uj(t1κt,x1κt),j[1,n].

    Here, 0<T<1κ, and v, given by Proposition 4.1, is a solution to (4.1). Thus, u resolves (CNLS). Now, let define u+j:=eiκ|x|24iκΔvj(1κ,). So, following [25,Section 4], we have ujC(R+,Hs)L(R+×RN,yN2dydt), and

    limt+eitΔuj(t)u+jHs=0.

    Here, one proves Proposition 2.1. Let uCT(Y) be a solution to (CNLS). Thus, m>1+N2 implies that y1mL2, and

    yuymuy1mymu.

    This implies that the solution has a finite variance RN|u(t,y)|2|x|2dy. Let us check that the energy is well-defined. Indeed,

    ˉujFj,k1=|uk|σ|uj|p1=y2mσy2mσ|uk|σ|uj|p1ymσ2ymu2σu2σLT(Y).

    Thus, one can apply [18,Theorem 2.8].

    This work examines the singular coupled non-linear Schrödinger system (CNLS) with three main objectives. First, it investigates the local existence of solutions. Second, it establishes the existence of global solutions that scatter in certain Sobolev spaces. Finally, it demonstrates the existence of non-global solutions. The primary difficulty arises from the condition σ<2, which introduces a singularity in the term |uj|σ2 near zero. This singularity renders the classical contraction method in the energy space ineffective. This paper aims to address this gap in the literature by leveraging ideas from [24]. The approach highlights that the singularity issue is localized near zero, requiring the solution to avoid this region, see assumption (2.5). This is challenging because the Schrödinger equation lacks a maximum principle. The global solutions that scatter are obtained through a pseudo-conformal transformation based on the local solutions. Lastly, the existence of non-global solutions is demonstrated using the classical variance method.

    Saleh Almuthybri and Tarek Saanouni: study conception and design, data collection, analysis and interpretation of results, and manuscript preparation. All authors have read and approved the final version of the manuscript for publication.

    The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2024-9/1).

    On behalf of all authors, the corresponding author states that there is no conflict of interest. No data-sets were generated or analyzed during the current study.



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