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Convergence for global curve diffusion flows

  • In this note we establish exponentially fast smooth convergence for global curve diffusion flows, and discuss open problems relating embeddedness to global existence (Giga's conjecture) and the shape of Type I singularities (Chou's conjecture).

    Citation: Glen Wheeler. Convergence for global curve diffusion flows[J]. Mathematics in Engineering, 2022, 4(1): 1-13. doi: 10.3934/mine.2022001

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  • In this note we establish exponentially fast smooth convergence for global curve diffusion flows, and discuss open problems relating embeddedness to global existence (Giga's conjecture) and the shape of Type I singularities (Chou's conjecture).



    This article discusses the Cauchy problem for the following inhomogeneous nonlinear Schrödinger equation (NLS) with inverse-square potential:

    {iφt=(Δc|x|2)φK(x)|φ|p2φ,t>0,xRN,φ(0,x)=φ0H1(RN),xRN, (1.1)

    where N3, 0<T, φ:[0,T)×RNC, K(x)C1(RN,R), 2<pp=2NN2 and 0<cc, where c=(N2)24 represents the sharp constant of the following Hardy's inequality:

    cRN|x|2|φ|2dxRN|φ|2dx,φH1(RN).

    The model of inhomogeneous NLS (1.1) with inverse-square potential can be applied to a variety of physical environments, such as black hole solutions of Einstein's equations or quantum field equations (see e.g., [1,2,3]) and quantum gas theory (see e.g., [4,5,6]).

    In the present work, our interest focuses on the optimal criteria of global existence and finite-time blow-up, as well as the L2-concentration property and the dynamics of blow-up solutions for Eq (1.1), which are pursued in both mathematics and physics.

    Before going further, we recall some existing results. When c=0, Eq (1.1) becomes the following common inhomogeneous NLS:

    iφ=ΔφK(x)|φ|p2φ, (1.2)

    which is widely used in fields such as quantum mechanics, nonlinear optics and Bose-Einstein condensation. In the past several decades, this kind of NLS has garnered a great deal of interest. Under the condition of K(x)=1 in Eq (1.2), Weinstein [7] not only applied the ground-state solution to the scalar equation

    Δφφ+|φ|4N=0 (1.3)

    to establish the sharp criterion of global and blow-up solutions, he also showed the existence of the unstable standing wave solutions in the L2-critical case (p=2+4N). Merle, in [8], proved the mass concentration of explosive solutions and classified the minimal-mass blow-up solutions on non-radial data in the L2-critical setting by utilizing the concentration lemma and pseudo-conformal transformation, together with the variational characterization of the ground-state solution to Eq (1.3). In [9], Hmidi and Keraani proposed a refined compactness lemma by applying the profile decomposition to bounded sequences in H1(RN). In view of the obtained compactness result, they also gave the simpler proofs to the concentration property of solutions blowing up in finite time, the limiting profile and determination of the minimal blow-up solutions to the homogeneous L2-critical NLS.

    Under the condition that K(x)constant and satisfies some appropriate assumptions, Merle [10] explored, in detail, the existence or nonexistence for the minimal-mass blow-up solutions, as well as the blow-up dynamical properties of solutions to Eq (1.2). Shu and Zhang [11] derived a sharp condition for the existence of a global solution to Eq (1.2) with the mass critical exponent p=2+4N by constructing some cross-invariant manifolds and variational problems. It is also worth noting that, for p=2+4N2 and K(x)constant satisfying (A1)(A2) (see Section 2), Liu [12] established a new sharp criterion for the explosive solutions to the H1-critical inhomogeneous Eq (1.2) with p=2+4N2 for the non-radial case by using the potential well method. These studies are strongly dependent on the hypotheses imposed on the inhomogeneous coefficient K(x). Then, a natural problem arises: Are these results valid for the inhomogeneous NLS (1.1) with inverse-square potential (c0) and variable coefficient K(x)constant? It is one of our starting points to study problem (1.1) in the current article.

    For the case that c0 and K(x)=1, Eq (1.1) corresponds to the homogeneous NLS with inverse-square potential, which has been extensively discussed due to the singular property of inverse-square potential c|x|2. In [13,14], the authors researched the global scattering and blow-up problems for the focusing and defocusing NLS in the intercritical and energy-critical settings, respectively. By making full use of the ground state to the elliptic equation

    Δφc|x|2φ|φ|p2φ+φ=0,φH1(RN), (1.4)

    Dinh [15] showed the finite-time blow-up and global existence of radially symmetric and non-radial solutions of Eq (1.1) in the mass-critical and mass-supercritical settings with c<0 or 0<c<c, as well as the criterion of blow-up solutions for the energy-critical case with c0 and c<N2+4N(N+2)2c, respectively. By replacing the power-type nonlinearity by (Iα|φ|p)|φ|p2φ in Eq (1.1), in light of Dinh [15], Li [16] investigated the criteria of global existence versus blow-up for non-radial and radial solutions to the L2-critical and L2-supercritical Choquard equation in the cases of 0<c<c and c<0, respectively. Under the condition that N3, p=2+4N and 0<c<c in Eq (1.1), Csobo and Genoud [17] showed that the ground-state solution to the mass critical Eq (1.4) exists, and they obtained an optimal condition for global existence. In addition, the ground-state solution of Eq (1.4) and pseudo-conformal transformation were applied to classify the explosive solutions with minimal-mass for Eq (1.1) in [17], and a detailed characterization of minimal blow-up solutions was given via the variational characteristic of the ground-state solution to Eq (1.4) and the concentration compactness principle related to the Hardy functional. For the critical case of c=c in Eq (1.1), Mukherjee et al. [18] was the first to verify the existence and uniqueness of the ground-state solution to Eq (1.4) for 2<p<p, which was applied to establish the criteria of blow-up versus global existence dichotomy for the homogeneous Eq (1.1). Moreover, by taking advantage of the variational characterization to the ground-state solution of Eq (1.4) and the corresponding concentration compactness principle, they studied the mass concentration phenomenon of explosive solutions and gave a complete characterization of the minimal-mass blow-up solutions. Regarding the case that p=2+4N and 0<c<c, Bensouilah [19] proposed a refined compactness lemma related to the Hardy functional by using the profile decomposition technique in H1(RN), and they applied it to investigate the L2-concentration property of solutions blowing up at time 0<T<. Based on [17] and [19], Pan and Zhang [20] demonstrated the accurate L2-concentration property for the explosive solutions with a minimal mass by using the variational characterization to the ground state of Eq (1.4) and the compactness lemma proposed by [19]. For 0<c<c or c<0 in Eq (1.1), the authors of [21,22] studied the stability and strong instability of standing waves for different values of p. For the homogeneous NLS including combined nonlinearities and inverse-square potential, Xia [23], Cao [24] and Li and Zou [25] researched the global existence and scattering, the existence of stable standing waves or the existence and properties of normalized solutions, respectively. Regarding the standing waves, Goubet and Manoubi [26] researched the existence and orbital stability of standing waves for a class of NLS involving a discontinuous dispersion. Zuo et al. [27] applied a variational approach based on the scaling function method to investigate the existence of normalized solutions for a kind of fractional NLS with bounded parametric potential; the solutions have attracted widespread attention due to their important applications in many settings, such as physics.

    For K(x)=λ|x|b with λ=±1, 0<b<2 and 0<c<c, Campos and Guzmán [28] and An et al. [29] considered the blow-up and global solutions to Eq (1.1). Pan and Zhang [30] studied the existence of a ground-state solution to the corresponding L2-critical inhomogeneous elliptic equation with K(x)=|x|b, and they proved the uniqueness of the minimal-mass blow-up solutions by using the concentration compactness principle, with respect to the Hardy functional and inhomogeneous nonlinearity, as well as variational characterization to the ground state. To the best of our knowledge, there is no literature concerning the inhomogeneous Eq (1.1) that includes inverse-square potential and the general C1 variable coefficient K(x), which has significant differences with the cases of [15,17,18,19,20]. Therefore, it is particularly meaningful for us to study Eq (1.1).

    Motivated by the works mentioned above, the aims of this paper are to gain the criteria for the existence of global or finite-time blow-up solutions to Eq (1.1) in the L2-critical, L2-supercritical cases and the sharp criterion of blow-up in the energy-critical case, as well as the blow-up dynamics of solutions in finite time. The main difficulty stems from the presence of inverse-square potential c|x|2 and variable coefficient K(x), leading to the loss of pseudo-conformal symmetry and scaling invariance, which play a vital role in the research on the blow-up dynamics; see [9,18] for example. To overcome the difficulty, we utilize the unique ground-state solution of Eq (3.1) to establish the global existence and blow-up results to the mass-critical and mass-supercritical inhomogeneous NLS and apply the ground state to characterize the blow-up dynamic behavior of solutions to Eq (1.1). To be more precise, enlightened by [15,18], we first obtain some sharp thresholds for global existence and finite-time blow-up in the L2-critical and L2-supercritical cases for 0<cc in terms of the ground state for Eq (3.1). It is worth mentioning that, in this work, for 0<c<c and p=2+4N, the argument regarding the existence of explosive solutions is established through scaling techniques, which differs from the methods of [15,18]. Then, under the assumptions (A1)(A2) on the inhomogeneous coefficient K(x) (see Section 2), following the ideas of [12] and [15], we derive the sharp criterion of blow-up in the energy-critical case with 0<c<N2+4N(N+2)2c by utilizing the potential well method and the sharp Sobolev constant. Finally, in light of [10,19,20], we establish the mass concentration property of explosive solutions, as well as the dynamic behaviors of the minimal-mass blow-up solutions in the L2-critical setting for 0<c<c. The main ingredients we use in the proofs of the dynamics are the variational characterization of the ground state for Eq (3.1), scaling techniques and a refined compactness lemma proposed by Bensouilah [19]. Our results generalize and supplement the results of [12,15,18,19,20].

    The remaining parts of the present article is structured as follows. Section 2 gives some notations and important hypotheses, as well as some useful lemmas. Section 3 considers the criteria for the global existence and finite-time blow-up of Eq (1.1) in the L2-critical and L2-supercritical cases, as well as the sharp blow-up criterion in the energy-critical case, respectively. The last section focuses on the blow-up dynamics of solutions in the L2-critical setting with 0<c<c.

    To simplify the symbols, we use the abbreviation  dx to represent RN dx, and we denote ||||Lp (1p<) by ||||p. C represents a positive constant, which may vary from line to line.

    Hereafter, we assume that the inhomogeneous coefficient K(x) satisfies some of the following hypotheses: there exist K2K1>0 such that

    (A1) xRN, K1=infxRNK(x)K(x)supxRNK(x)=K2<;

    (A2) xRN,xK(x)0 and |xK(x)|C;

    (A3) thereisx0RNsatisfyingthatK(x0)=K2.

    In accordance with Dinh [15] and Okazawa et al. [31], we have the following argument regarding the local well-posedness of solutions to Eq (1.1).

    Proposition 2.1. Let φ0H1(RN), 2<p<p and 0<cc or p=p and 0<c<N2+4N(N+2)2c, and assume that (A1) holds; then, for T(0,] (maximal existence time), the unique solution φ(t,x)C([0,T),H1(RN)) for Eq (1.1) exists. Meanwhile, one has the alternative T= (global existence), or else T< and limtT||φ(t,x)||H1(RN) = (blow-up). Furthermore, for all t[0,T), the solution φ(t,x) possesses the conserved quantities of mass and energy as shown below:

    |φ(t,x)|2dx=|φ0|2dx, (2.1)
    E(φ0)=E(φ(t,x))=12|φ|2dxc2|x|2|φ|2dx1pK(x)|φ|pdx. (2.2)

    Define the Hardy functional as below:

    H(φ)=|φ|2dxc|x|2|φ|2dx,

    which is of great importance in analyzing the dynamical properties for blow-up solutions. Taking account of the hypothesis on c, the semi-norm defined by H(φ) on H1(RN) is equivalent to ||φ||22. Thus, the solution φ(t,x) to Eq (1.1) blows up at T>0 if and only if limtTH(φ)=.

    To go further, we review some useful lemmas.

    Lemma 2.2. (Hardy-Gagliardo-Nirenberg inequality ([18,32])) Let N3, 0<cc and 2<p<p. Then, we obtain

    φp1CHGNH(φ)θ2φ1θ2,θ=N2Np, (2.3)

    for all φH1(RN), with the sharp constant

    CHGN=Q(x)p2p2(1θ)1p(θ1θ)N(p2)4p,

    where Q(x) is the unique radial positive solution to the elliptic equation (1.4).

    Lemma 2.3. ([7]) Assume that φH1(RN); then, we get

    |φ|2dx2N(|φ|2dx)12(|x|2|φ|2dx)12.

    Lemma 2.4. (Sharp Sobolev embedding ([33])) Let N3 and 0<c<c; then, we have

    ||f||pCSE(c)H(f)12,

    where the sharp Sobolev constant CSE(c) is as follows:

    CSE(c)=sup{||f||p÷H(f)12:fH1(RN)}. (2.4)

    In the current section, we are devoted to researching the criteria for the global existence and finite-time blow-up to Eq (1.1) in the L2-critical and L2-supercritical settings, as well as the sharp threshold of blow-up in the energy-critical case, respectively. As we know, the ground state has a crucial role in the criteria for blow-up versus global existence, and it is the unique positive radially symmetric solution of the following elliptic equation with power nonlinearity and inverse-square potential:

    Δφc|x|2φK2|φ|p2φ+φ=0,φH1(RN), (3.1)

    where K2 is same as the hypotheses (A1)(A3). Furthermore, we will apply the ground-state solution of Eq (3.1) to characterize the dynamical properties of explosive solutions in the next section.

    Assume that Q(x) is the unique radial positive solution to Eq (1.4); by the scaling transformation QK2(x)=K1p22Q(x), it is easy to obtain that QK2(x) is the positive ground state of Eq (3.1). Combining Eqs (1.4) and (3.1), we immediately get the scaling identity and Poho˘zaev identities as follows:

    ||QK2||22=K2p22||Q||22, (3.2)
    ||QK2||22=4(N2)(p2)N(p2)H(QK2),||QK2||pp=2pNK2(p2)H(QK2). (3.3)

    Define the following functionals:

    EK2(φ)=12H(φ)K2p||φ||pp,L(c)=EK2(QK2)||QK2||2σ2,G(c)=H(QK2)12||QK2||σ2, (3.4)

    where σ=2N+2pNpNp2N4 when 2+4N<p<2+4N2. From Eqs (2.3), (3.2) and (3.3), one has

    CpHGN=2pNp+2N2pK2||QK2||p22(Np2N2pNp+2N)N(p2)4=N(p2)2pK2H(QK2)p22[4(N2)(p2)N(p2)]2p+2NNp4=Kp22||QK2||p(p2)2p[4(N2)(p2)]2p+2NNp4[N(p2)]Np2N4(2p)p2 (3.5)

    and

    EK2(QK2)=Np2N42[4(N2)(p2)]||QK2||22=Np2N42N(p2)H(QK2). (3.6)

    It follows from Eqs (3.2)–(3.6) that

    L(c)=Np2N42(Np2N)[2pCpHGN(Np2N)K2]4Np2N4

    and

    G(c)=[2pCpHGN(Np2N)K2]2Np2N4. (3.7)

    In particular, we have

    L(c)=Np2N42(Np2N)G2(c). (3.8)

    Now, we consider the virial-type identities, which play a key role in the research of the existence of explosive solutions to Eq (1.1). Let

    Σ={φH1(RN):xφL2(RN)},

    and for φ(t,x)Σ, we introduce the variance functional

    V(t)=|x|2|φ(t,x)|2dx.

    Then, we are able to derive the following conclusion.

    Proposition 3.1. Assume that 2<p<p and 0<cc or p=p and 0<c<N2+4N(N+2)2c, and let φ(t,x) be a solution of problem (1.1) on t[0,T). If φ0H1(RN) and |x|φ0L2(RN), then φ(t,x)Σ for any t[0,T) and V(t) satisfies the following identities:

    V(t)=4Imxφ¯φdx

    and

    V(t)=8H(φ)+8N4NppK(x)|φ|pdx+8p|φ|pxK(x)dx=16E(φ0)+4(4+2NNp)pK(x)|φ|pdx+8p|φ|pxK(x)dx. (3.9)

    Proof. Based on the work of Csobo and Genoud [17] (see also Cazenave [34]), by a formal computation, it is easy to obtain that

    V(t)=2Re|x|2ˉφφtdx=2Re(i)|x|2ˉφ(Δφc|x|2φK(x)|φ|p2φ)dx=4Imxφˉφdx

    and

    V(t)=4Imddtxφˉφdx=4(ImNφtˉφdx+2Imxφ¯φtdx)=4(I1+I2), (3.10)

    where

    I1=ImNˉφφtdx=NReˉφ(Δφc|x|2φK(x)|φ|p2φ)dx=N(|φ|2c|x|2|φ|2K(x)|φ|p)dx, (3.11)
    I2=2Imxφ¯φtdx=2Imxˉφφtdx=2Rexˉφ(Δφc|x|2φK(x)|φ|p2φ)dx=(N2)|φ|2dx(2N)c|x|2|φ|2dx+2NpK(x)|φ|pdx+2p|φ|pxK(x)dx. (3.12)

    From Eqs (3.10)–(3.12), we claim that Eq (3.9) holds.

    From Proposition 3.1 and Lemma 2.3, we get the following sufficient conditions for blow-up.

    Corollary 3.2. Assume that 2+4Npp and (A1) and (A2) hold; if φ0Σ and φ0 meets one of the three conditions below:

    Case 1): E(φ0)<0;

    Case 2): E(φ0)=0 and Imxφ0¯φ0dx<0;

    Case 3): E(φ0)>0 and Imxφ0¯φ0dx+(2V(0)E(φ0))120,

    then blow-up of the solution φ(t,x) to Eq (1.1) occurs in finite time.

    Proof. Since K(x) satisfies (A1) and (A2), using Proposition 3.1, we have

    V(t)16E(φ0).

    Thus

    0V(t)=V(0)+V(0)t+t0(ts)V(s)dsV(0)+V(0)t+8E(φ0)t2.

    Then, for each of the cases 1, 2 or 3, one can deduce that 0<T<+ must exist and satisfy

    limtTV(t)=limtT|x|2|φ(t)|2dx=0.

    This, together with Lemma 2.3, implies that

    limtTφ(t)2=+.

    Therefore, the explosion of the solution φ(t,x) to Eq (1.1) happens in the time period of 0<T<+.

    The first argument of our work is about the global existence and blow-up of Eq (1.1) in the L2-critical setting (i.e., p=2+4N).

    Theorem 3.3. Assume that p=2+4N and φ0H1(RN). Let QK2(x) be the positive ground state of Eq (3.1).

    (i) Global existence: Assume that (A1) holds true. If 0<cc and φ02<QK22, then the solution φ(t,x) of Eq (1.1) exists globally in t[0,+).

    (ii) Blow-up: Assume that (A1) and (A2) hold.

    (a) Then, for 0<c<c, any λ>0 and any real constant C1 satisfying that |C1|(K2K1)N41,

    there exists φ0=C1λN2QK2(λx)Σ such that

    ||φ0||22||QK2||22,

    and blow-up of the corresponding solution φ(t,x) to problem (1.1) occurs in 0<T<+.

    (b) For c=c, if E(φ0)<0 and |x|φ0L2(RN), then the blow-up of the solution φ(t,x) for Eq (1.1)

    happens in the time period of 0<T<+.

    Proof. (ⅰ) Since p=2+4N, we have that pθ=2 and pCpHGN=2Q4N2. Thus, from Eq (2.2), (A1), Eqs (2.3) and (2.1), we have the following estimate:

    E(φ0)=E(φ)12H(φ)K22Q4N2H(φ)pθ2φ0p(1θ)2=12H(φ)K22H(φ)(φ02Q2)4N=12H(φ)12H(φ)(φ02QK22)4N=12(1φ02QK22)4NH(φ).

    Since φ02<QK22, H(φ) is bounded uniformly for t[0,+). From Proposition 2.1, we claim that φ(t,x) must be a global solution.

    Next, we prove part (ⅱ) of Theorem 3.3. For 0<c<c, let

    φ0=C1λN2QK2(λx)

    for any λ>0, and C1R will be determined later. Based on the scaling arguments, one has that

    |φ0|2dx=|C1|2Q2K2dx; (3.13)
    |φ0|2dx=|C1|2λ2|QK2|2dx; (3.14)
    |φ0|pdx=|C1|pλN(p2)2QpK2dx; (3.15)
    |x|2|φ0|2dx=|C1|2λ2|x|2Q2K2dx. (3.16)

    Take

    |C1|(K2K1)N41;

    then, we have that φ0H1(RN) and |x|φ0L2(RN). Indeed, according to Bensouilah-Dinh-Zhu's conclusion in [21], one has that Q(x)H1(RN) and |x|Q(x)L2(RN). Thus, we obtain that QK2(x)H1(RN) and |x|QK2(x)L2(RN), which yields that φ0=C1λN2QK2(λx)Σ. Moreover, it follows from Eq (3.13) that

    ||φ0||22=|C1|2||QK2||22(K2K1)N2||QK2||22||QK2||22.

    From Eq (2.2), (A1), Eqs (3.14)–(3.16) and the Poho˘zaev identities given by Eq (3.3), we have

    E(φ0)=12H(φ0)1pK(x)|φ0|pdx12H(φ0)K1p|φ0|pdx=12|C1|2λ2H(QK2)K1pCp1λN(p2)2QpK2dx=12|C1|2λ2(1K1K2Cp21λN(p2)22)H(QK2)<0,

    where the last inequality is based on the fact that p=2+4N and |C1|(K2K1)N41. Thus, we infer from Corollary 3.2 that blow-up of the corresponding solution φ(t,x) to problem (1.1) occurs in 0<T<+.

    For c=c, if E(φ0)<0 and |x|φ0L2(RN), then, by Corollary 3.2, it is easy to derive that blow-up of the solution φ(t,x) for Eq (1.1) happens in the time period of 0<T<+.

    Then, we analyze the L2-supercritical case (i.e., 2+4N<p<p=2+4N2). For this case, we obtain the threshold for global existence as shown below.

    Theorem 3.4. Suppose that N3, 0<cc and 2+4N<p<p. Let φ0H1(RN) and φ(t,x) be the corresponding solution of Eq (1.1). Suppose that

    E(φ0)||φ0||2σ2<L(c). (3.17)

    (i) Global existence: Assume that (A1) holds. If

    H(φ0)12||φ0||σ2<G(c), (3.18)

    then the global solution φ(t,x) to Eq (1.1) exists. In addition,

    H(φ)12||φ||σ2<G(c)foranyt>0.

    (ii) Blow-up: Assume that (A1)(A2) hold. If

    H(φ0)12||φ0||σ2>G(c) (3.19)

    and |x|φ0L2(RN), then the finite-time blow-up solution φ(t,x) of Eq (1.1) exists and

    H(φ)12||φ||σ2>G(c) (3.20)

    for any t[0,T). Furthermore, the finite-time blow-up result still holds true if we assume that E(φ0)<0, in place of Eqs (3.17) and (3.19).

    Proof. (ⅰ) From Eq (2.2), (A1) and Eq (2.3), one has

    E(φ0)||φ0||2σ2=E(φ)||φ||2σ212(H(φ)12||φ||σ2)2K2p||φ||pp||φ||2σ212(H(φ)12||φ||σ2)2K2pCpHGNH(φ)Np2N4||φ||2p+2NNp22||φ||2σ2=12(H(φ)12||φ||σ2)2K2pCpHGN(H(φ)12||φ||σ2)Np2N2=f(H(φ)12||φ||σ2),

    where

    f(x)=12x2K2pCpHGNxNp2N2. (3.21)

    Enlightened by the idea of [35] (the function f in [35] differs from ours; however, this change does not make a significant difference), we will utilize an important fact that f strictly increases in [0,G(c)] and strictly decreases in [G(c),). Moreover, from Eqs (3.21), (3.7) and (3.8), we get

    f(G(c))=12G2(c)K2pCpHGNGNp2N2(c)=(122Np2N)G2(c)=L(c). (3.22)

    Thus, combining this with Eqs (3.17), (3.21) and (3.22), we have

    f(H(φ)12||φ||σ2)E(φ0)||φ0||2σ2<L(c)=f(G(c)). (3.23)

    From the above inequality, Eq (3.18) and the continuity argument, we deduce that

    H(φ)12||φ||σ2<G(c)foranyt>0.

    From Eq (2.1), we obtain the boundedness of H(φ)12, which implies that the solution φ(t,x) of Eq (1.1) exists globally.

    We next treat part (ⅱ) of Theorem 3.4. For the case that E(φ0)0, we claim that Eq (3.20) holds. Indeed, from Eqs (3.23) and (3.19) and the continuity argument, one has

    H(φ)12||φ||σ2>G(c)foranyt<T,

    which means that Eq (3.20) holds true.

    On the other hand, from Eq (3.17) and the continuity argument, we can take δ>0 small enough such that

    E(φ0)||φ0||2σ2(1δ)L(c), (3.24)

    which yields that

    f(H(φ)12||φ||σ2)(1δ)L(c). (3.25)

    Applying Eqs (3.21), (3.7) and (3.8) to Eq (3.25), one has that

    Np2NNp2N4(H(φ)12||φ||σ2G(c))24Np2N4(H(φ)12||φ||σ2G(c))Np2N21δ.

    By making use of the continuity argument again, we deduce from Eq (3.19) that there exists δ>0 that relies upon δ satisfying

    H(φ)12||φ||σ2(1+δ)G(c). (3.26)

    Moreover, taking

    LHS=8H(φ)+8N4NppK(x)|φ|pdx+ε,

    then, for any t[0,T), we claim that

    LHS<C<0

    for ε>0 small enough. In fact, multiplying LHS by ||φ||2σ2, we obtain

    LHS×||φ||2σ2=(4pN8N)E(φ)||φ||2σ2+(8+4N2Np+ε)H(φ)||φ||2σ2.

    Utilizing Eqs (2.1), (2.2), (3.24), (3.26) and (3.8), one has

    LHS×||φ0||2σ2(4pN8N)(1δ)L(c)+(8+4N2Np+ε)(1+δ)2G2(c)=[2(Np2N4)(1δ)+(8+4N2Np+ε)(1+δ)2]G2(c)=[2(Np2N4)(1δ(1+δ)2)+ε(1+δ)2]G2(c).

    We readily obtain LHSC<0 by taking ε>0 small enough. Then, it follows from the virial identity (3.9) and (A2) that

    V(t)=8H(φ)+8N4NppK(x)|φ|pdx+8p|φ|pxK(x)dx<C<0.

    This yields that blow-up of the solution φ(t,x) must occur in the time period of 0<T<+.

    The case that E(φ0)<0 is easy. By Corollary 3.2, we conclude that the finite-time blow-up solution φ(t,x) of Eq (1.1) exists.

    Finally, the energy-critical setting (i.e., p=p=2+4N2) for problem (1.1) is considered in this subsection. According to Dinh [15], taking account the sharp constant (see also Eq (2.4))

    S=1C2SE(c)=infφH1(RN){0}H(φ)(|φ|2NN2dx)N2N,

    then we are able to get the optimal blow-up criterion.

    Theorem 3.5. Assume that p=p, 0<c<N2+4N(N+2)2c, φ0H1(RN) and E(φ0)<SN2NKN222. Let φ(t,x) be the corresponding solution of Eq (1.1), defined on [0,T)×RN, 0<T.

    (i) Uniform boundedness: Assume that K(x) satisfies (A1). If H(φ0)<SN2KN222, then the solution φ(t,x) of Eq (1.1) is bounded in H1(RN) for t[0,T) and H(φ(t))<SN2KN222.

    (ii) Blow-up: Assume that K(x) satisfies (A1)(A2). If H(φ0)>SN2KN222 and |x|φ0L2(RN), then there exists 0<T< such that the solution φ(t,x) of Eq (1.1) blows up at T.

    Proof. (ⅰ) Assume that φ0H1(RN) satisfies that E(φ0)<SN2NKN222 and H(φ0)<SN2KN222. We claim that

    H(φ)<SN2KN222for any t[0,T). (3.27)

    We demonstrate Eq (3.27) by contradiction. Suppose that there exists t1(0,T) such that H(φ(t1))=SN2KN222 by using the continuity of the solution φ(t,x) in H1(RN) at time t. By Eq (2.2), (A1) and Lemma 2.4, we found that

    SN2NKN222>E(φ0)=E(φ(t1))12H(φ(t1))1pK2|φ(t1)|pdx12H(φ(t1))K2pSNN2H(φ(t1))NN2=12SN2KN222K2pSNN2(SN2KN222)NN2=SN2NKN222,

    which contradicts the fact that H(φ(t1))=SN2KN222. Therefore, Eq (3.27) holds. This means that the solution φ(t,x) is bounded in H1(RN) for t[0,T).

    Now, we prove part (ⅱ) of Theorem 3.5. Since φ0H1(RN) satisfies that E(φ0)<SN2NKN222 and H(φ0)>SN2KN222. It remains to be proven that

    H(φ)>SN2KN222for any t[0,T). (3.28)

    If otherwise, since φ(t,x) is continuous with respect to time t in H1(RN), we deduce that there exists t2(0,T) satisfying that H(φ(t2))=SN2KN222. Combining Eq (2.2) with (A1), and by Lemma 2.4, one has the following estimate:

    SN2NKN222>E(φ0)=E(φ(t2))12H(φ(t2))1pK2|φ(t2)|pdx12H(φ(t2))K2pSNN2H(φ(t2))NN2=12SN2KN222K2pSNN2(SN2KN222)NN2=SN2NKN222,

    which gives a contradiction. Thus, Eq (3.28) holds. Keeping in mind that p=2+4N2 and φ0H1(RN) with |x|φ0L2(RN), we have

    V(t)=16E(φ0)16NK(x)|φ|2NN2dx+4(N2)NxK(x)|φ|2NN2dx.

    Then, we infer from Eqs (3.28) and (2.2) that

    K(x)|φ0|2NN2dx=2NN2(E(φ0)12H(φ0))<2NN2(SN2NKN222SN22KN222)=SN2KN222. (3.29)

    Inserting Eq (3.29) into V(t), and then from (A2), one has the estimate

    V(t)16E(φ0)16SN2NKN222=16(E(φ0)SN2NKN222)<0,

    from which we know that explosion of the solution φ(t,x) to Eq (1.1) must happen within the time period of 0<T<.

    Remark 3.6. (i) Under the conditions that K(x)=1, c=c and 2<p<p in Eq (1.1), Mukherjee et al. [18] studied the criterion of blow-up versus global existence in the L2-critical and L2-supercritical cases (see [18], Theorem 3); Dinh [15] obtained the blow-up and global existence results for Eq (1.1) for non-radial data in the mass-critical and mass-supercritical cases with 0<c<c, as well as the sharp blow-up criterion in the energy-critical case with 0<c<N2+4N(N+2)2c (see [15], Theorems 1.3, 1.6 and 1.12). Our results (Theorems 3.3–3.5) generalize the results of [15,18] to the case of the inhomogeneous NLS involving inverse-square potential and a bounded positive variable coefficient.

    (ii) Given c=0, Liu [12] verified the sharp existence of non-radial finite-time blow-up solutions to the inhomogeneous Eq (1.2) in the energy-critical case p=2NN2 (see [12], Theorem 1.2). We extend this conclusion to the case of the inhomogeneous NLS with inverse-square potential for 0<c<N2+4N(N+2)2c (see Theorem 3.5).

    (iii) For 0<c<N2+4N(N+2)2c and p=p, we can derive that the global solution of Eq (1.1) exists for some initial value small enough.

    In the present part, we investigate the dynamics of blow-up solutions in the L2-critical setting (p=2+4N) with 0<c<c, including the mass concentration phenomenon of blow-up solutions and the dynamical properties of blow-up solutions with a minimal mass for Eq (1.1). To achieve these goals, we first recall a key compactness lemma established by Bensouilah [19].

    Lemma 4.1. Assume that {vn}n=1 is a bounded sequence in H1(RN) satisfying

    lim supnH(vn)M,  lim supnvnpm.

    Then, there exists {xn}n=1RN such that, up to a subsequence,

    vn(x+xn)U  weakly in  H1(RN),

    with U2(NN+2)N4mN2+1MN4Q(x)2, where Q(x) is the ground-state solution to Eq (1.4).

    With Lemma 4.1 in hand, we are able to obtain the following concentration property of the explosive solutions to Eq (1.1).

    Theorem 4.2. (L2-concentration) Assume that K(x) satisfies (A1) and (A2). Suppose that φ(t,x) is a solution to Eq (1.1) blowing up at finite time T, and that s(t) is a nonnegative real-valued function on [0,T) such that s(t)φ(t)2+ as tT. Then, there exists a function x(t)RN for t<T satisfying

    lim inftT|xx(t)|s(t)|φ(t,x)|2dxQ2K2dx, (4.1)

    where QK2(x) denotes the ground state for Eq (3.1).

    Proof. Take

    ρ(t)=[H(QK2)H(φ)]12and  v(t,x)=ρ(t)N2φ(t,ρ(t)x). (4.2)

    Suppose that {tn}n=1 is an arbitrary time sequence satisfying that tnT as n, and denote ρn=ρ(tn) and vn(x)=v(tn,x). It follows from Eq (2.1) and the definition of vn that

    vn2=φ(tn)2=φ02,H(vn)=ρ2nH(φ)=H(QK2). (4.3)

    For v(x)H1(RN), we define the functional

    F(v)=H(v)2K2p||v||pp.

    From (A1) and Eqs (4.3), (2.2) and (4.2), we find that

    12F(vn)=12H(vn)K2p||vn||pp12H(vn)1pK(x)|vn|pdx=ρ2n(12H(φ)1pK(x)|φ|pdx)=ρ2nE(φ0)0  since  ρn0  as n.

    Thus,

    limn|vn|pdxp2K2H(QK2)  as n. (4.4)

    Take M=H(QK2) and m2+4N=p2K2H(QK2). Thanks to Lemma 4.1, it is sufficient to demonstrate that there exist U(x)H1(RN) and {xn}n=1RN such that, up to a subsequence,

    vn(+xn)=ρN2nφ(tn,ρn+xn)U  weakly  in H1(RN), (4.5)

    with

    ||U||2(NN+2)N4mN2+1MN4||Q||2=(NN+2)N4[(N+2NK2)H(QK2)]N4H(QK2)N4||Q||2=||QK2||2, (4.6)

    which leads to

    vn(+xn)U  weakly  in L2(RN).

    This, together with the lower semi-continuity of the L2-norm, yields

    lim infn|x|AρNn|φ(tn,ρnx+xn)|2dx|x|A|U|2dxforanyA>0. (4.7)

    Since

    limns(tn)ρn=limns(tn)H(φ)12H(QK2)12=,

    there exists n0>0 such that for any n>n0, we obtain that Aρn<s(tn). Combining this result and Eq (4.7), one has

    lim infnsupyRN|xy|s(tn)|φ(tn,x)|2dxlim infn|xxn|Aρn|φ(tn,x)|2dx=lim infn|x|AρNn|φ(tn,ρnx+xn)|2dx|x|A|U|2dx,  for any A>0,

    which means that

    lim infnsupyRN|xy|s(tn)|φ(tn,x)|2dx|U|2dx=U22.

    According to the arbitrariness of the sequence {tn}n=1, using the fact that U2QK22, we obtain

    lim inftTsupyRN|xy|s(t)|φ(t,x)|2dxQK222. (4.8)

    For any fixed t[0,T), it is simple to infer that the function g(y):=|xy|s(t)|φ(t,x)|2dx is continuous on yRN and lim|y|g(y)=0. Thus, for any 0t<T, there exists a function x(t)RN that satisfies

    supyRN|xy|s(t)|φ(t,x)|2dx=|xx(t)|s(t)|φ(t,x)|2dx. (4.9)

    Therefore, from Eqs (4.8) and (4.9) we infer that Eq (4.1) holds true.

    Corollary 4.3. Suppose that φ(t,x) is a solution of Eq (1.1) blowing up within the time period of 0<T<. Then, for any l>0, there exists x(t)RN for 0<t<T satisfying that

    lim inftTB(x(t),l)|φ(t,x)|2dxQ2K2dx,

    where QK2(x) denotes the ground state for Eq (3.1) and B(x(t),l)={xRN||xx(t)|l}.

    Now, in terms of Theorem 4.2 and Corollary 4.3, we are concerned with the dynamics of blow-up solutions with the mass ||φ0||2=||QK2||2.

    Theorem 4.4. (Limiting profile) Let K(x) satisfy (A1)(A2) and assume that ||φ0||2=||QK2||2. Assume that φ(t,x) is a corresponding solution to problem (1.1) blowing up within the time period of 0<T<; then, there exist two functions θ(t)[0,2π) and x(t)RN that satisfy

    ρ(t)N2eiθ(t)φ(t,ρ(t)x+x(t))QK2  strongly in H1(RN)whentT,

    where QK2(x) denotes the ground state for Eq (3.1) and ρ(t)=[H(QK2)H(φ)]12.

    Proof. From Theorem 4.2, we have that U2QK22 (see Eq (4.6)). Then, by using the assumption that φ02=QK22 and Eq (2.1), we obtain

    \begin{equation*} \|Q_{K_2}\|_{2}^{2}\leq\|U\|_{2}^{2}\leq \liminf\limits_{n\rightarrow \infty}\|v_{n}\|_{2}^{2} = \liminf\limits_{n\rightarrow \infty}\|\varphi(t_n)\|_{2}^{2} = \|\varphi_{0}\|_{2}^{2} = \|Q_{K_{2}}\|_{2}^{2}, \end{equation*}

    which implies that

    \begin{equation} \lim\limits_{n\rightarrow \infty}\|v_{n}\|_{2}^{2} = \|U\|_{2}^{2} = \|Q_{K_{2}}\|_{2}^{2}. \end{equation} (4.10)

    It follows from Eqs (4.1) and (4.10) that

    \begin{equation} v_n(\cdot+x_n) = \rho_{n}^{\frac{N}{2}}\varphi(t_n, \rho_{n}\cdot+ x_n)\rightarrow U \ \ strongly \ \ in \ L^2(\mathbb{R}^{N})\ \ as \ \ n\rightarrow \infty . \end{equation} (4.11)

    From Eq (2.3), one has

    \begin{eqnarray} \|v_n(x+x_{n})-U\|^{2+\frac{4}{N}}_{2+\frac{4}{N}} &\leq&\frac{1}{C_{HGN}^{p}}H(v_n(x+x_{n})-U)^{\frac{\theta p}{2}}||v_n(x+x_{n})-U||_{2}^{(1-\theta)p} \\ & = &\frac{p}{2||Q||_{2}^{\frac{4}{N}}}H(v_n(x+x_{n})-U)||v_n(x+x_{n})-U||_{2}^{\frac{4}{N}}\\ &\leq&C||v_n(x+x_{n})-U||_{2}^{\frac{4}{N}}(||\nabla v_n(x+x_{n})||_{2}^{2}+||\nabla U||_{2}^{2}) \\ &\leq& C\|v_n(x+x_{n})-U\|^{\frac{4}{N}}_{2}, \end{eqnarray} (4.12)

    where the last inequality is based on the boundedness of v_n(x+x_{n}) in H^1(\mathbb{R}^{N}) . Thus, Eqs (4.11) and (4.12) give us that

    \begin{equation} v_n(\cdot+x_n)\rightarrow U \ \ in \ L^{2+\frac{4}{N}}(\mathbb{R}^{N}) \ \ as \ \ n\rightarrow \infty. \end{equation} (4.13)

    Now, we claim that

    \begin{equation} v_n(\cdot+x_n)\rightarrow U \ \ strongly \ \ in \ H^1(\mathbb{R}^{N})\ \ when \ \ n\rightarrow \infty. \end{equation} (4.14)

    Indeed, we deduce from Eqs (4.3) and (4.4) that

    \begin{equation} \lim\limits_{n \rightarrow \infty}\int |v_n|^{p}dx\geq \frac{p}{2K_{2}}H(Q_{K_2}) = \frac{p}{2K_{2}}\lim\limits_{n \rightarrow \infty} H(v_n). \end{equation} (4.15)

    Then, from Eqs (4.5), (4.15), (4.13), (2.3) and (4.10), one has

    \begin{eqnarray*} H(U) &\leq&\liminf\limits_{n \rightarrow \infty} H(v_n) = H(Q_{K_{2}}) \\ &\leq&\frac{2K_{2}}{p}\lim\limits_{n \rightarrow \infty}\int |v_n|^{p}dx = \frac{2K_{2}}{p}\int |U|^{p}dx \\ &\leq&\frac{K_{2}}{||Q||_{2}^{\frac{4}{N}}}H(U)||U||_{2}^{\frac{4}{N}}\\ & = & \frac{1}{||Q_{K_{2}}||_{2}^{\frac{4}{N}}}H(U)||U||_{2}^{\frac{4}{N}} = H(U), \end{eqnarray*}

    which implies that

    \begin{equation*} \liminf\limits_{n \rightarrow \infty} H(v_n(x+x_n)) = H(U) = H(Q_{K_{2}}) = \frac{2K_{2}}{p}\int |U|^{p}dx. \end{equation*}

    From this and Eq (4.5), we get Eq (4.14) and

    \begin{equation*} F(U) = H(U)-\frac{2K_{2}}{p}\int |U|^{p}dx = 0. \end{equation*}

    In summary, we identify the properties of the profile U as below:

    \begin{equation*} \| U\|_{2}^{2} = \|Q_{K_2}\|_{2}^{2}, \ \ \ H(U) = H(Q_{K_2}), \ \ \ F(U) = 0, \end{equation*}

    from which we deduce that there exist \theta\in[0, 2\pi) and x_{0}\in\mathbb{R}^{N} such that

    \begin{equation*} {U(x) = e^{i\theta}Q_{K_2}(x+x_{0})} \end{equation*}

    and

    \begin{equation*} \rho_{n}^{\frac{N}{2}}\varphi(t_{n}, \rho_{n}\cdot+x_{0})\rightarrow e^{i\theta}Q_{K_2}(\cdot+x_{0}) \ \ strongly \ in \ H^1(\mathbb{R}^{N}) \ \ as \ \ n\rightarrow \infty, \end{equation*}

    where we utilize the variational characterization to the ground state Q_{K_2} . Since the sequence \{t_n\}_{n = 1}^{\infty} is arbitrary and tends to T as n\rightarrow \infty , one can infer that there exist two functions \theta(t)\in[0, 2\pi) and x(t)\in \mathbb{R}^{N} such that

    \begin{equation*} \lambda(t)^{\frac{N}{2}}e^{i\theta(t)}\varphi(t, \lambda(t)x+x(t))\rightarrow Q_{K_{2}} \ \ strongly \ in \ H^1(\mathbb{R}^{N})\ \ when \ \ t \rightarrow T, \end{equation*}

    with \rho(t) = \bigg[\frac{H(Q_{K_2})}{H(\varphi)}\bigg]^{\frac{1}{2}}\rightarrow0 as t \rightarrow T .

    Theorem 4.5. Assume that K(x) satisfies (A_{1}) (A_{3}) and that \|\varphi_{0}\|_{2} = \|Q_{K_{2}}\|_{2} . Denote W = \{x\in \mathbb{R}^{N}|K(x) = K_{2}\} . Let \varphi(t, x) be the corresponding solution to Eq (1.1) blowing up within the time period of 0 < T < \infty ; then the following holds true:

    (i) (Location of L^2 -concentration point) There exists x_{0}\in W such that

    \begin{equation} {\lim\limits_{t\rightarrow T}\; x(t) = x_{0} \ and \ |\varphi(t, x)|^{2}\rightarrow \|Q_{K_{2}}\|_{2}^{2}\delta_{x = x_{0}} \ in \; the\; distribution \; sense\; as\; t\; \rightarrow\; T, } \end{equation} (4.16)

    where Q_{K_{2}} is the ground state for Eq (3.1).

    (ii) (Blow-up rate) There is a positive constant C > 0 that satisfies

    \begin{equation} {\|\nabla \varphi(t)\|_{2}\geq \frac{C}{T-t}\; for\; all\; t \in [0, T).} \end{equation} (4.17)

    Proof. (ⅰ) From Eq (2.1) and \|\varphi_{0}\|_{2} = \|Q_{K_{2}}\|_{2} , we deduce that

    \begin{equation} {\|\varphi\|_{2} = \|\varphi_{0}\|_{2} = \|Q_{K_{2}}\|_{2}\; for\; t < T.} \end{equation} (4.18)

    On the other side, according to Theorem 4.2 and Corollary 4.3, for any l > 0 , one has that

    \begin{equation} \|Q_{K_{2}}\|_{2}^{2}\leq\liminf\limits_{t\rightarrow T}\int_{|x-x(t)|\leq l} |\varphi(t, x)|^{2}dx \leq \liminf\limits_{t\rightarrow T}\int |\varphi(t, x)|^{2}dx\leq \|\varphi_{0}\|_{2}^{2}. \end{equation} (4.19)

    From Eqs (4.18) and (4.19), we have

    \begin{equation*} \liminf\limits_{t \rightarrow T}\int_{|x-x(t)| < l}|\varphi(t, x)|^{2}dx = \|Q_{K_{2}}\|_{2}^{2}, \end{equation*}

    which shows that

    \begin{equation} {|\varphi(t, x+x(t))|^{2} \rightarrow \|Q_{K_{2}}\|_{2}^{2}\delta_{x = 0} \; in \; the\; distribution\; sense\; as\; t\rightarrow T.} \end{equation} (4.20)

    We shall demonstrate in what follows that there exists x_{0}\in W satisfying that

    \begin{equation*} |\varphi(t, x)|^{2}\rightarrow \|Q_{K_{2}}\|_{2}^{2}\delta_{x = x_{0}}\ in\; the\; sense\; of\; distribution\; as\; t \rightarrow T, \end{equation*}

    which implies that

    \begin{equation*} \lim\limits_{t\rightarrow T}\int w(x)|\varphi(t, x)|^{2}dx = w(x_{0})||Q_{K_{2}}||^{2}_{2}, \; for\; any\; w(x)\in C_{0}^{\infty}(\mathbb{R}^{N}). \end{equation*}

    As a matter of fact, for any real-valued function \theta(x)\in\mathbb{R}^{N} and any \beta\in \mathbb{R} , from (A_{1}) and Eq (2.3), one has the following estimate

    \begin{eqnarray*} \; \; \; \; \; E(e^{i\beta\theta(x)}\varphi)& = &\frac{1}{2}H(e^{i\beta\theta(x)}\varphi) -\frac{1}{p}\int K(x)|e^{i\beta\theta(x)}\varphi|^{p}dx \\ &\geq&\frac{1}{2}H(e^{i\beta\theta(x)}\varphi) -\frac{K_{2}}{p}||e^{i\beta\theta(x)}\varphi||_{p}^{p} \\ &\geq&\frac{1}{2}H(e^{i\beta\theta(x)}\varphi) -\frac{K_{2}}{p}\frac{1}{C_{HGN}^{p}}H(e^{i\beta\theta(x)}\varphi)^{\frac{\theta p}{2}}||e^{i\beta\theta(x)}\varphi||_{2}^{(1-\theta)p}\\ & = &\frac{1}{2}H(e^{i\beta\theta(x)}\varphi) -\frac{K_{2}}{2}\bigg(\frac{||\varphi_{0}||_{2}}{||Q||_{2}}\bigg)^{\frac{4}{N}}H(e^{i\beta\theta(x)}\varphi)\\ & = &\frac{1}{2}H(e^{i\beta\theta(x)}\varphi) -\frac{K_{2}}{2}\bigg(\frac{||Q_{K_{2}}||_{2}}{||Q||_{2}}\bigg)^{\frac{4}{N}}H(e^{i\beta\theta(x)}\varphi)\\ & = &0, \end{eqnarray*}

    where \theta is defined in Lemma 2.2. Here in the last equality, we have used the fact that ||Q_{K_{2}}||_{2} = K_{2}^{-\frac{N}{4}}||Q||_{2} . Therefore, from Eqs (2.1) and (2.2), for any \beta\in\mathbb{R} , we obtain

    \begin{eqnarray*} 0\leq E(e^{i\beta\theta(x)}\varphi)& = & \frac{1}{2}H(e^{i\beta\theta(x)}\varphi) -\frac{1}{p}\int K(x)|e^{i\beta\theta(x)}\varphi|^{p}dx \cr & = &\frac{1}{2}\bigg[\int \beta^{2}|\nabla\theta(x)\cdot\varphi|^{2}dx+2\beta Im\int \nabla\theta(x)\nabla\varphi\overline{\varphi}dx+\int |\nabla\varphi|^{2}dx\bigg]\cr &\; &\; \; \; \; \; -\frac{1}{2}\int c|x|^{-2}|e^{i\beta\theta(x)}\varphi|^{2}dx - \frac{1}{p}\int K(x)|e^{i\beta\theta(x)}\varphi|^{p}dx \cr & = &\frac{1}{2}\beta^{2}\int|\nabla\theta(x)|^{2}|\varphi|^{2}dx + \beta Im\int\nabla\theta(x) \cdot\nabla \varphi \cdot\overline{\varphi}dx+E(\varphi_0), \end{eqnarray*}

    which implies that

    \begin{equation} \bigg|Im\int\nabla\theta(x) \cdot\nabla \varphi \cdot\overline{\varphi}dx\bigg|\leq \bigg[2E(\varphi_0)\int|\nabla\theta(x)|^{2}|\varphi|^{2}dx\bigg]^{\frac{1}{2}}. \end{equation} (4.21)

    Then, choosing \theta_{j}(x) = x_{j} for j = 1, 2, \cdot\cdot\cdot, N in Eq (4.21), it follows from Eqs (1.1), (2.1) and (2.2) that

    \begin{eqnarray} \bigg|\frac{d}{dt}\int|\varphi(t, x)|^{2}x_{j}dx\bigg| & = &\bigg|2Im\int i\varphi_{t}\cdot\overline{\varphi}\cdot x_{j}dx\bigg|\\ & = &\bigg|2Im\int [(- \Delta -|x|^{-2}) \varphi -K(x)|\varphi|^{p-2}\varphi]\overline{\varphi}\cdot x_{j}dx\bigg|\\ & = &\bigg|2Im\int \nabla \varphi\cdot\overline{\varphi}\cdot \nabla x_{j}dx\bigg|\\ &\leq& 2\bigg(2E(\varphi_0)\int |\varphi_0|^{2} dx\bigg)^{\frac{1}{2}} = C. \end{eqnarray} (4.22)

    Take any two sequences \{t_n\}_{n = 1}^{\infty} , \{t_m\}_{m = 1}^{\infty}\subset[0, T) such that \lim\limits_{n\rightarrow \infty}t_n = \lim\limits_{m\rightarrow \infty}t_m = T . Thus, for all j = 1, 2, \cdot\cdot\cdot, N , from Eq (4.22), we have

    \begin{eqnarray*} \bigg|\int|\varphi(t_n, x)|^{2}x_{j}dx-\int|\varphi(t_m, x)|^{2}x_{j}dx\bigg| &\leq&\int^{t_n}_{t_m} \bigg|\frac{d}{dt}\int|\varphi(t, x)|^{2}x_{j}dx\bigg|dt\\ &\leq& C|t_{n}-t_{m}|\rightarrow 0 \ \ as \ \ n, \ m\rightarrow \infty, \end{eqnarray*}

    which means that

    \begin{equation*} \lim\limits_{t\rightarrow T}\int|\varphi(t, x)|^{2}x_{j}dx \; \; exists \; for\; any \; j = 1, 2, \cdot\cdot\cdot, N. \end{equation*}

    Thus,

    \begin{equation*} \lim\limits_{t\rightarrow T}\int|\varphi(t, x)|^{2}xdx \ \ exists. \end{equation*}

    Let x_{0} = \frac{\lim\limits_{t\rightarrow T}\int|\varphi(t, x)|^{2}xdx}{\|Q_{K_{2}}\|_{2}^{2}} ; then, one has that x_{0}\in \mathbb{R}^{N} and

    \begin{equation} \lim\limits_{t\rightarrow T}\int|\varphi(t, x)|^{2}xdx = x_{0}\|Q_{K_{2}}\|_{2}^{2}. \end{equation} (4.23)

    On the other hand, notice that p = 2+\frac{4}{N} ; then, from Proposition 3.1 and (A_{2}) , we have

    \begin{equation*} V^{{\prime}{\prime}}(t) = 16E(\varphi_0)+\frac{4(4+2N-Np)}{p}\int K(x)|\varphi|^{p}dx+\frac{8}{p}\int|\varphi|^{p}\cdot x\cdot\nabla K(x)dx \leq 16E(\varphi_0). \end{equation*}

    Therefore, there is a positive constant c_{0} > 0 satisfying that

    \begin{equation*} { V(t)\leq c_{0}\; for\; each\; t\in[0, T).} \end{equation*}

    Thus, one has the following estimate:

    \begin{eqnarray} \int|x|^{2}|\varphi(t, x+x(t))|^{2}dx &\leq&2\int|x+x(t)|^{2}|\varphi(t, x+x(t))|^{2}dx \\ && + \ 2\int|x(t)|^{2}|\varphi(t, x+x(t))|^{2}dx \\ &\leq&2c_{0}+2\|\varphi_{0}\|_{2}^{2} |x(t)|^{2}\\ & = &2c_0+2\|Q_{K_{2}} \|_{2}^{2}|x(t)|^{2}. \end{eqnarray} (4.24)

    From Eq (4.20), one has

    \begin{eqnarray} \limsup\limits_{t\rightarrow T}|x(t)|^{2}\|Q_{K_{2}}\|_{2}^{2}& = &\limsup\limits_{t\rightarrow T}\int_{|x| < 1}|x+x(t)|^{2}|\varphi(t, x+x(t))|^{2}dx\\ &\leq&\int|x|^{2}|\varphi(t, x)|^{2}dx\leq c_0. \end{eqnarray} (4.25)

    We derive from Eq (4.25) that

    \begin{equation} \limsup\limits_{t\rightarrow T}|x(t)|\leq\frac{\sqrt{c_0}}{\|Q_{K_{2}}\|_{2}}. \end{equation} (4.26)

    Combining Eq (4.24) with Eq (4.26), one has

    \begin{equation*} \limsup\limits_{t\rightarrow T}\int|x|^{2}|\varphi(t, x+x(t))|^{2}dx\leq C. \end{equation*}

    Thus, we have the following for any l_0 > 0 :

    \begin{equation*} \limsup\limits_{t\rightarrow T}\int_{|x|\geq l_0}l_0|x||\varphi(t, x+x(t))|^{2}dx\leq\limsup\limits_{t\rightarrow T}\int_{|x|\geq l_0}|x|^{2}|\varphi(t, x+x(t))|^{2}dx\leq C. \end{equation*}

    Therefore, for any \varepsilon > 0 , there exists a large enough l_0 = l_0(\varepsilon) > 0 satisfying that

    \begin{equation} \limsup\limits_{t\rightarrow T}\bigg|\int_{|x|\geq l_0} x|\varphi(t, x+x(t))|^{2}dx\bigg| \leq \frac{C}{l_0} < \varepsilon . \end{equation} (4.27)

    Then, using Eqs (4.27) and (4.20), we infer that, for any \varepsilon > 0 ,

    \begin{eqnarray} \limsup\limits_{t\rightarrow T}\bigg|\int x|\varphi(t, x)|^{2}dx-x(t)\|Q_{K_{2}}\|_{2}^{2}\bigg|& = &\limsup\limits_{t\rightarrow T}\bigg|\int x|\varphi(t, x)|^{2}dx- x(t)\int |\varphi(t, x)|^{2} dx\bigg|\\ & = &\limsup\limits_{t\rightarrow T}\bigg|\int|\varphi(t, x)|^{2}(x-x(t))dx\bigg|\\ &\leq&\limsup\limits_{t\rightarrow T}\bigg|\int_{|x|\leq l_0} |\varphi(t, x+x(t))|^{2}xdx\bigg| \ + \varepsilon \\ & = &\varepsilon. \end{eqnarray} (4.28)

    Combining Eqs (4.23) and (4.28), we immediately get

    \begin{equation} { \lim\limits_{t\rightarrow T}x(t) = x_{0} \ \ and \ \ \limsup\limits_{t\rightarrow T}\int x|\varphi(t, x)|^{2}dx = x_{0}\|Q_{K_{2}}\|_{2}^{2}.} \end{equation} (4.29)

    Thus, there exists x_{0}\in \mathbb{R}^{N} (see Eq (4.23)) such that

    \begin{equation*} |\varphi(t, x)|^{2}\rightarrow \|Q_{K_{2}}\|_{2}^{2}\delta_{x = x_{0}} \; \; in\; the\; sense\; of \; distribution\; when\; t \rightarrow T. \end{equation*}

    Now, we claim that x_{0}\in W = \{x\in\mathbb{R}^{N}|K(x) = K_{2}\} . Namely, x_{0} is a global maximal point of K(x) . Indeed, since K(x)\in C^{1} , we deduce from Corollary 4.3 and Eq (4.29) that

    \begin{eqnarray*} 1\leq\liminf\limits_{t\rightarrow T} \frac {\|\varphi(t, x)\|_{L^2(B(x(t), l))}^{2}}{\|Q_{K(x(t))}\|_{2}^{2}} &\leq&\liminf\limits_{t\rightarrow T} \frac {K_{2}^{-\frac{N}{2}}\|\varphi_{0}\|_{2}^{2}}{[K(x(t))]^{-\frac{N}{2}}\|Q_{K_{2}}\|_{2}^{2}} \\ & = &\liminf\limits_{t\rightarrow T} \bigg[\frac {K(x(t))}{K_{2}}\bigg]^{\frac{N}{2}}\\ & = &\bigg[\frac {K(x_{0})}{K_{2}}\bigg]^{\frac{N}{2}}\leq1, \end{eqnarray*}

    where Q_{K(x(t))} = K(x(t))^{-\frac{N}{4}}Q(x(t)) . This implies that K(x_{0}) = K_{2} . Thus,

    \begin{equation*} x_{0}\in W = \{x\in\mathbb{R}^{N}|K(x) = K_{2}\}. \end{equation*}

    Therefore, it is clear that Eq (4.16) holds true.

    (ⅱ) Choosing \theta(x) = |x-x_{0}|^{2} in Eq (4.21), we get

    \begin{eqnarray*} \bigg|\frac{d}{dt}\int|\varphi(t, x)|^{2}|x-x_{0}|^{2}dx\bigg| & = &\bigg|2Im\int - \Delta \varphi\cdot\overline{\varphi}\cdot|x-x_{0}|^{2}dx\bigg|\\ &\leq& 4\bigg(2E(\varphi_0)\int|\varphi(t, x)|^{2}|x-x_{0}|^{2}dx\bigg)^{\frac{1}{2}}\\ &\leq& C\bigg(\int|\varphi(t, x)|^{2}|x-x_{0}|^{2}dx\bigg)^{\frac{1}{2}}, \end{eqnarray*}

    which indicates that

    \begin{equation*} \bigg|\frac{d}{dt}\bigg(\int|\varphi(t, x)|^{2}|x-x_{0}|^{2}dx\bigg)^{\frac{1}{2}}\bigg|\leq C. \end{equation*}

    Therefore, for any t\in [0, T) , by integrating on both sides of this inequality from t to T , we derive

    \begin{equation*} \bigg(\int|\varphi(t, x)|^{2}|x-x_{0}|^{2}dx\bigg)^{\frac{1}{2}}\leq C(T-t). \end{equation*}

    Based on the uncertainty principle and Hölder's inequality, we obtain

    \begin{eqnarray*} \|\varphi_{0}\|_{2}^{2} = \int|\varphi(t, x)|^{2}dx& = &-\frac{2}{N}Re\int \nabla\varphi\cdot\overline{\varphi}\cdot (x-x_{0})dx\\ &\leq&C\bigg(\int|\varphi(t, x)|^{2}|x-x_{0}|^{2}dx\bigg)^{\frac{1}{2}} \bigg(\int|\nabla\varphi|^{2}dx\bigg)^{\frac{1}{2}}\cr &\leq& C(T-t)\|\nabla\varphi(t)\|_{2}, \end{eqnarray*}

    which means that

    \begin{equation*} {\|\nabla\varphi(t)\|_{2}\geq \frac {C}{T-t} \ \ for \ \ \forall \ t\in [0, T).} \end{equation*}

    Therefore, the conclusion Eq (4.17) holds.

    Remark 4.6. (i) Given K(x) = 1 and 0 < c < c^{*} , Bensouilah [19] verified the L^{2} -concentration of explosive solutions to Eq (1.1) in the L^{2} -critical case (see [19], Theorem 1). Furthermore, Pan and Zhang [20] obtained the precise concentration behavior for blow-up solutions with a critical mass (see [20], Theorem 1.1). Our conclusions generalize and supplement the corresponding results of [19,20] for the inhomogeneous NLS with inverse-square potential and the coefficient 0 < K(x)\neq constant (see Theorems 4.2, 4.4 and 4.5).

    (ii) For the case that c = 0 , K(x)\neq constant and (A_{1}) (A_{3}) and other proper conditions are satisfied, the mass concentration property of blow-up solutions and the dynamical behaviors of the L^{2} -minimal blow-up solutions have been derived (see [10], Theorems 1 and 2). Our results extend the results of [10] to the inhomogeneous NLS involving inverse-square potential for 0 < c < c^{*} (see Theorems 4.2 and 4.5).

    The authors declare that they have not used artificial intelligence tools in the creation of this article.

    This work was partially supported by the Jiangxi Provincial Natural Science Foundation (Grant Nos. 20212BAB211006 and 20224BAB201005) and National Natural Science Foundation of China (Grant No. 11761032). The authors are also greatly thankful to the referees and editors for their helpful comments and advice leading to the improvement of this manuscript.

    The authors make the declaration that there are no competing interests existing.



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