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The dynamics of the focusing NLH with a potential beyond the mass-energy threshold

Shuang Ji College of Science, China Agricultural University,  Beijing,  China,  100193 [email protected] ,  Jing Lu College of Science, China Agricultural University,  Beijing,  China,  100193 [email protected]  and  Fanfei Meng Qiyuan Lab,  Tsinghua University,  Beijing,  China,  100095 [email protected]
Abstract.

In this paper, we study the dynamics of the focusing nonlinear Hartree equation with a Kato potential

i∂tu+Δu−Vu=−(|⋅|−γ∗|u|2)u,x∈ℝdi\partial_{t}u+\Delta u-Vu=-(|\cdot|^{-\gamma}\ast|u|^{2})u,\quad x\in{\mathbb{R}}^{d}

under some assumptions on the potential VV. We prove the blow up versus global existence dichotomy for solutions beyond the threshold, based on the method from Duyckaerts-Roudenko [6]. Furthermore, our result compensates for the one of in [13] below that threshold.

Key words and phrases:
Kato potential, global existence, blow up, virial identity.

1. Introduction

In this paper, we study the Cauchy problem of the focusing nonlinear Hartree equation with a potential

{i∂tu+Δu−Vu=−(|⋅|−γ∗|u|2)u,u​(0)=u0∈Hx1​(ℝd),(t,x)∈ℝ×ℝd,\displaystyle\left\{\begin{aligned} &i\partial_{t}u+\Delta u-Vu=-(|\cdot|^{-\gamma}\ast|u|^{2})u,\\ &u(0)=u_{0}\in{H}_{x}^{1}({\mathbb{R}}^{d}),\end{aligned}\right.\qquad(t,x)\in{\mathbb{R}}\times{\mathbb{R}}^{d}, (NLHV\text{NLH}_{\text{V}})

where u:ℝ×ℝd→ℂu:{\mathbb{R}}\times{\mathbb{R}}^{d}\rightarrow{\mathbb{C}} is the wave function, V:ℝd→ℝV:{\mathbb{R}}^{d}\to{\mathbb{R}} is a real-valued potential and ∗\ast denotes the convolution of spacial variable. Here we consider the energy-subcritical case, that is 2<γ<min⁡{4,d}2<\gamma<\min\{4,d\}.

We denote that ℋ:=−Δ+V\mathcal{H}:=-\Delta+V and

∥f∥H˙V1​(ℝd)2:=<ℋf,f>=∫ℝd|∇f|2dx+∫ℝdV|f|2dx,f∈Hx1(ℝd).\|f\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}:=<\mathcal{H}f,f>=\int_{{\mathbb{R}}^{d}}|\nabla f|^{2}dx+\int_{{\mathbb{R}}^{d}}V|f|^{2}dx,\quad f\in H_{x}^{1}({\mathbb{R}}^{d}).

The solution to (NLHV\text{NLH}_{\text{V}}) satisfies the laws of mass conservation and energy conservation, which can be expressed respectively by

M​(u)=∫ℝd|u​(t,x)|2​𝑑x=M​(u0),M(u)=\int_{{\mathbb{R}}^{d}}|u(t,x)|^{2}dx=M(u_{0}), (1.1)
EV​(u)=12​‖u‖H˙V1​(ℝd)2−14​P​(u)=EV​(u0),E_{V}(u)=\frac{1}{2}\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}-\frac{1}{4}P(u)=E_{V}(u_{0}), (1.2)

where

P​(u)=∫ℝd∫ℝd|u​(x)|2​|u​(y)|2|x−y|γ​𝑑x​𝑑y=∫ℝd(|x|−γ∗|u|2)​|u|2​𝑑x.P(u)=\int_{{\mathbb{R}}^{d}}\int_{{\mathbb{R}}^{d}}\frac{|u(x)|^{2}|u(y)|^{2}}{|x-y|^{\gamma}}dxdy=\int_{{\mathbb{R}}^{d}}(|x|^{-\gamma}\ast|u|^{2})|u|^{2}dx. (1.3)

Before stating our main results, we first recall the literature of the focusing nonlinear Hartree equations with no potential

i∂tu+Δu=μ(|⋅|−γ∗|u|2)u,2<γ<min{4,d},x∈ℝd,\displaystyle i\partial_{t}u+\Delta u=\mu(|\cdot|^{-\gamma}\ast|u|^{2})u,\quad 2<\gamma<\min\{4,d\},\quad x\in{\mathbb{R}}^{d}, (NLH0\text{NLH}_{\text{0}})

which is called focusing if μ<0\mu<0 and defocusing if μ>0\mu>0.

In [7], Gao-Wu investigated the focusing (​NLH0​)\eqref{NLH0} for γ=3\gamma=3 with u0∈Hx1​(ℝd)u_{0}\in H_{x}^{1}({\mathbb{R}}^{d}) by concentration compactness method. Denote the mass-energy threshold as M​(Q)​E0​(Q),M(Q)E_{0}(Q), where QQ is the ground state of the equation (NLH0\text{NLH}_{\text{0}}) and

E0​(u)=12​∫ℝd|∇u|2​𝑑x−14​P​(u)=E0​(u0).E_{0}(u)=\frac{1}{2}\int_{{\mathbb{R}}^{d}}|\nabla u|^{2}dx-\frac{1}{4}P(u)=E_{0}(u_{0}).

Their results are as follows.

Theorem 1.1 (Sub-threshold dynamics for (​NLH0​)\eqref{NLH0}, Gao-Wu [7]).

For (γ,d)=(3,5)(\gamma,d)=(3,5), let the radial u0∈Hx1​(ℝd)u_{0}\in H_{x}^{1}({\mathbb{R}}^{d}) and satisfy M​(u0)​E0​(u0)<M​(Q)​E0​(Q)M(u_{0})E_{0}(u_{0})<M(Q)E_{0}(Q).

  1. (i)

    If ‖∇u0‖Lx2​(ℝd)​‖u0‖Lx2​(ℝd)<‖∇Q‖Lx2​(ℝd)​‖Q‖Lx2​(ℝd)\|\nabla u_{0}\|_{L_{x}^{2}({\mathbb{R}}^{d})}||u_{0}||_{L_{x}^{2}({\mathbb{R}}^{d})}<||\nabla Q||_{L_{x}^{2}({\mathbb{R}}^{d})}||Q||_{L_{x}^{2}({\mathbb{R}}^{d})}, then the solution is global and scatters.

  2. (ii)

    If ‖∇u0‖Lx2​(ℝd)​‖u0‖Lx2​(ℝd)>‖∇Q‖Lx2​(ℝd)​‖Q‖Lx2​(ℝd)\|\nabla u_{0}\|_{L_{x}^{2}({\mathbb{R}}^{d})}||u_{0}||_{L_{x}^{2}({\mathbb{R}}^{d})}>||\nabla Q||_{L_{x}^{2}({\mathbb{R}}^{d})}||Q||_{L_{x}^{2}({\mathbb{R}}^{d})}, then the solution will blow up in finite time.

For the case beyond the threshold, Yang-Li [27] have established a dichotomy of blow up versus global existence for (​NLH0​)\eqref{NLH0} by the method from Duyckaerts-Roudenko [6]. Their results are as follows.

Theorem 1.2 (Super-threshold dynamics for (​NLH0​)\eqref{NLH0}, Yang-Li [27]).

For sc=γ−22s_{c}=\frac{\gamma-2}{2} and d⩾3d\geqslant 3, assume I​(0)<∞I(0)<\infty, M​(u0)​E0​(u0)>M​(Q)​E0​(Q)M(u_{0})E_{0}(u_{0})>M(Q)E_{0}(Q) and

(M​(u0)​E0​(u0)M​(Q)​E0​(Q))​(1−(I′​(0))232​E0​(u0)​I​(0))⩽1.\displaystyle\biggl{(}\frac{M(u_{0})E_{0}(u_{0})}{M(Q)E_{0}(Q)}\biggr{)}\biggl{(}1-\frac{(I^{\prime}(0))^{2}}{32E_{0}(u_{0})I(0)}\biggr{)}\leqslant 1.
  1. (i)

    If [M(u0)]1−sc[P(u0)]sc>[M(Q)]1−sc[P(Q)]sc\bigl{[}M(u_{0})\bigr{]}^{1-s_{c}}\bigl{[}P(u_{0})\bigr{]}^{s_{c}}>\bigl{[}M(Q)\bigr{]}^{1-s_{c}}\bigr{[}P(Q)\bigl{]}^{s_{c}} and I′​(0)⩽0I^{\prime}(0)\leqslant 0, then the solution blows up in finite time.

  2. (ii)

    If [M(u0)]1−sc[P(u0)]sc<[M(Q)]1−sc[P(Q)]sc\bigl{[}M(u_{0})\bigr{]}^{1-s_{c}}\bigl{[}P(u_{0})\bigr{]}^{s_{c}}<\bigl{[}M(Q)\bigr{]}^{1-s_{c}}\bigr{[}P(Q)\bigl{]}^{s_{c}} and I′​(0)⩾0I^{\prime}(0)\geqslant 0, then the solution exists globally.

For (​NLH0​)\eqref{NLH0}, Miao-Xu-Zhao studied the well-posedness, ill-posedness, the sharp local well-posedness and the global existence in [22]. After that, they further established the global well-posedness and scattering criteria when γ=2\gamma=2 below the threshold [23]. Li-Miao-Zhang [17] proved that the solution scatters in both time directions when γ=4\gamma=4 and d⩾5d\geqslant 5. A similar result was obtained for spherically symmetric initial data in [21]. Meng [19] used the radial Sobolev embedding and a virial-Morawetz type estimate to study the scattering result of (​NLH0​)\eqref{NLH0} with (γ,d)=(3,5)(\gamma,d)=(3,5), instead of Kenig-Merle’s concentrated compactness methods [14, 15]. In particular, Miao-Wu-Xu [20] continued the study on the dynamics of the radial solutions at the energy threshold.

Recently, more and more mathematicians have been considerably interested in the dispersive equations with different potentials, which are of paramount physical importance. For example, the Coulomb potential describes the coulomb force between two charged particles in quantum mechanical terms, and the inverse-square potential reveals that the intensity of the action between particles decays linearly with the square of the distance.

There are also many studies on the focusing nonlinear Schrödinger equations with a potential VV. Firstly, Rodnianski-Schlag [25] established the dispersive estimates for solutions to the linear Schrödinger equation in 3D with a time-dependent potential. Later, more and more mathematicians studied the nonlinear Schrödinger equation with a real-valued potential VV

i​∂tu+Δ​u−V​u+|u|p−1​u=0,x∈ℝd.\displaystyle i\partial_{t}u+\Delta u-Vu+|u|^{p-1}u=0,\quad x\in{\mathbb{R}}^{d}. (NLSV\text{NLS}_{\text{V}})

For the case that VV is the inverse-square potential a/|x|2a/|x|^{2} in (NLSV\text{NLS}_{\text{V}}), Killip-Murphy-Visan-Zheng [16] obtained the scattering and blow up dichotomy below the threshold with a>−14a>-\frac{1}{4} and p=3p=3 in ℝ3{\mathbb{R}}^{3}. Then Lu-Miao-Murphy [18] extended their results to dimensions 3⩽d⩽63\leqslant d\leqslant 6. Based on the method from Duyckaerts-Roudenko [6], Deng-Lu-Meng [3] further extended the results of [16] to beyond the threshold. For the case that VV is the Coulomb potential K​|x|−1K|x|^{-1} in (NLSV\text{NLS}_{\text{V}}), Miao-Zhang-Zheng [24] proved the global existence when K>0K>0, and the scattering theory when K<0K<0 for in ℝ3{\mathbb{R}}^{3}.

In particular, we focus on the case that VV is the Kato potential with a small negative part denoted as

V−​(x):=min⁡{V​(x),0}.V_{-}(x):=\min\{V(x),0\}.

To be precise, we define that the potential class 𝒦0\mathcal{K}_{0} is the closed space of bounded and compactly supported functions endowed with the Kato norm

∥V∥𝒦:=∥(−Δ)−1V)∥Lx∞.\|V\|_{\mathcal{K}}:=\|(-\Delta)^{-1}V)\|_{L_{x}^{\infty}}.

In order to prove the scattering result for (NLSV\text{NLS}_{\text{V}}) with (p,d)=(3,3)(p,d)=(3,3) below the mass-energy threshold, Hong used the concentration compactness method and introduced the following customized assumptions of Kato potential V in [11]

V∈𝒦0∩Lx32​(ℝ3),V\in\mathcal{K}_{0}\cap L_{x}^{\frac{3}{2}}({\mathbb{R}}^{3}), (1.4)

and

‖V−​(x)‖𝒦<4​π,\|V_{-}(x)\|_{\mathcal{K}}<4\pi, (1.5)

Hamano-Ikeda [10] studied the energy scattering below the threshold from Dodson-Murphy [4] and the blow up criteria based on an argument of Du-Wu-Zhang [5] when p>1p>1. Later, Wang [26] investigated the blow up and scattering results of (NLSV\text{NLS}_{\text{V}}) beyond the mass-energy threshold when p=3p=3.

Inspired by their results, we want to expand the range of VV to higher dimensions d⩾3d\geqslant 3. We assume that the Kato potential VV satisfies

V∈𝒦0∩Lxd2​(ℝd),V\in\mathcal{K}_{0}\cap L_{x}^{\frac{d}{2}}({\mathbb{R}}^{d}), (1.6)

and

‖V−​(x)‖𝒦<1Cd,\|V_{-}(x)\|_{\mathcal{K}}<\frac{1}{C_{d}}, (1.7)

where Cd=Γ​(d2)/[(2−d)​2​πd2]C_{d}=\Gamma(\frac{d}{2})\big{/}\bigl{[}(2-d)2\pi^{\frac{d}{2}}\bigr{]}.

Remark 1.3.

For the sake of clarity, we explain why the Kato potential VV need to satisfy (1.6) and (1.7). V∈𝒦0V\in\mathcal{K}_{0} and ‖V−​(x)‖𝒦<1/Cd\|V_{-}(x)\|_{\mathcal{K}}<1/C_{d} ensure that the operator ℋ\mathcal{H} is positive, which implies that ℋ\mathcal{H} is equivalent to −Δ-\Delta in the sense of

(1−Cd​‖V−‖𝒦)​‖∇u‖Lx2​(ℝd)2⩽‖ℋ12​u‖Lx2​(ℝd)2⩽(1+Cd​‖V‖𝒦)​‖∇u‖Lx2​(ℝd)2\displaystyle(1-C_{d}\|V_{-}\|_{\mathcal{K}})\|\nabla u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}\leqslant\|\mathcal{H}^{\frac{1}{2}}u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}\leqslant(1+C_{d}\|V\|_{\mathcal{K}})\|\nabla u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}

(see the specific proof in Lemma 2.2). V∈Lxd/2​(ℝd)V\in L_{x}^{d/2}({\mathbb{R}}^{d}) is an additional condition for proving such the equivalence of the Sobolev norm in the sense of ‖f‖H˙V1​(ℝd)∼‖f‖H˙x1​(ℝd)\|f\|_{{\dot{H}}_{V}^{1}({\mathbb{R}}^{d})}\sim\|f\|_{\dot{H}_{x}^{1}({\mathbb{R}}^{d})} when d⩾3d\geqslant 3 (see details in [11]). Then from Beceanu-Goldberg [1], the operator ℋ\mathcal{H} has no eigenvalues or nonnegative resonance. Therefore the dispersive estimates and Strichartz estimates are valid by Ionescu-Jerison [12], providing a robust foundation for the dichotomy of blow up versus global existence.

Next we review our previous work and demonstrate our main results in this paper. For completeness and logic, let’s start with a discussion of the Kato potential VV and the mass-energy threshold. Hong [11] has shown a stimulating discussion about the existence of ground state in (NLSV\text{NLS}_{\text{V}}). Later, for the nonlinear Hartree equation with the inverse-potential, Chen-Lu-Meng [2] also discussed the existence of ground state. Based on their methods, we give the similar variational characterization of (NLHV\text{NLH}_{\text{V}}) as follows.

Proposition 1.4 (Variational properties).

Suppose VV satisfies (1.6) and (1.7).

  1. (i)

    If V−=0V_{-}=0, then the sequence Q(⋅−n)n∈ℕ{Q(\cdot-n)}_{n\in{\mathbb{N}}} maximizes W​(u)W(u), where QQ is the ground state of the elliptic equation

    ΔQ−Q+(|⋅|−γ∗|Q|2)Q=0,\Delta Q-Q+(|\cdot|^{-\gamma}\ast|Q|^{2})Q=0, (1.8)

    and

    W​(u)=P​(u)‖u‖H˙V1​(ℝd)γ​‖u‖Lx2​(ℝd)4−γ.W(u)=\frac{P(u)}{\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{\gamma}\|u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{4-\gamma}}.
  2. (ii)

    If V−≠0V_{-}\neq 0, then there exists a maximizer 𝒬∈Hx1​(ℝd)\mathcal{Q}\in H_{x}^{1}({\mathbb{R}}^{d}) solving the elliptic equation

    (−Δ+V)𝒬+w𝒬2𝒬−(|⋅|−γ∗|𝒬|2)|𝒬=0,w𝒬2=(4−γ)​‖𝒬‖H˙V1​(ℝd)2γ​‖𝒬‖Lx2​(ℝd)2.\displaystyle(-\Delta+V)\mathcal{Q}+w_{\mathcal{Q}}^{2}\mathcal{Q}-(|\cdot|^{-\gamma}\ast|\mathcal{Q}|^{2})|\mathcal{Q}=0,\quad w_{\mathcal{Q}}^{2}=\frac{(4-\gamma)\|\mathcal{Q}\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}}{\gamma\|\mathcal{Q}\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}}. (1.9)

We denote that

ℳℰ={M​(u)1−sc​EV​(u)scM​(Q)1−sc​E0​(Q)sc,i​f​V−=0,M​(u)1−sc​EV​(u)scM​(𝒬)1−sc​EV​(𝒬)sc,i​f​V−≠0.\mathcal{M}\mathcal{E}=\left\{\begin{aligned} &\frac{M(u)^{1-s_{c}}E_{V}(u)^{s_{c}}}{M(Q)^{1-s_{c}}E_{0}(Q)^{s_{c}}},\quad&if~{}V_{-}=0,\\ &\frac{M(u)^{1-s_{c}}E_{V}(u)^{s_{c}}}{M(\mathcal{Q})^{1-s_{c}}E_{V}(\mathcal{Q})^{s_{c}}},\quad&if~{}V_{-}\neq 0.\end{aligned}\right.

In [13], we have proved the scattering theory and the blow up result of (NLH) with Kato potential below the threshold. The results are as follows:

Theorem 1.5 ([13]).

For (γ,d)=(3,5)(\gamma,d)=(3,5) in (NLHV\text{NLH}_{\text{V}}), we assume V⩾0V\geqslant 0 and satisfies (1.6) and (1.7). Let u0∈Hx1​(ℝd)u_{0}\in H_{x}^{1}({\mathbb{R}}^{d}) be radial and satisfy ℳ​ℰ<1\mathcal{M}\mathcal{E}<1.

  1. (i)

    If the initial data u0u_{0} satisfies ‖u0‖H˙V1​(ℝd)​‖u0‖Lx2​(ℝd)<‖Q‖H˙x1​(ℝd)​‖Q‖Lx2​(ℝd),\|u_{0}\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}||u_{0}||_{L_{x}^{2}({\mathbb{R}}^{d})}<\|Q\|_{\dot{H}_{x}^{1}({\mathbb{R}}^{d})}||Q||_{L_{x}^{2}({\mathbb{R}}^{d})}, then ‖u‖H˙V1​(ℝd)​‖u‖Lx2​(ℝd)<‖Q‖H˙x1​(ℝd)​‖Q‖Lx2​(ℝd).\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}||u||_{L_{x}^{2}({\mathbb{R}}^{d})}<\|Q\|_{\dot{H}_{x}^{1}({\mathbb{R}}^{d})}||Q||_{L_{x}^{2}({\mathbb{R}}^{d})}. Moreover, if x⋅∇V⩽0\ x\cdot\nabla V\leqslant 0 and x⋅∇V∈Lx52​(ℝd)x\cdot\nabla V\in L_{x}^{\frac{5}{2}}({\mathbb{R}}^{d}), then the global solution scatters in Hx1​(ℝd)H_{x}^{1}({\mathbb{R}}^{d}) in both time directions.

  2. (ii)

    If the initial data u0u_{0} satisfies ‖u0‖H˙V1​(ℝd)​‖u0‖Lx2​(ℝd)>‖Q‖H˙x1​(ℝd)​‖Q‖Lx2​(ℝd),\|u_{0}\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}||u_{0}||_{L_{x}^{2}({\mathbb{R}}^{d})}>\|Q\|_{\dot{H}_{x}^{1}({\mathbb{R}}^{d})}||Q||_{L_{x}^{2}({\mathbb{R}}^{d})}, then ‖u‖H˙V1​(ℝd)​‖u‖Lx2​(ℝd)>‖Q‖H˙x1​(ℝd)​‖Q‖Lx2​(ℝd)\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}||u||_{L_{x}^{2}({\mathbb{R}}^{d})}>\|Q\|_{\dot{H}_{x}^{1}({\mathbb{R}}^{d})}||Q||_{L_{x}^{2}({\mathbb{R}}^{d})} during the maximal existence time. Moreover, if x⋅∇V∈Lx52​(ℝd)\ x\cdot\nabla V\in L_{x}^{\frac{5}{2}}({\mathbb{R}}^{d}), 2​V+x⋅∇V⩾02V+x\cdot\nabla V\geqslant 0, then either T∗<+∞T^{*}<+\infty and

    limt→T∗‖∇u​(t)‖Lx2​(ℝd)=∞,\lim_{t\to T^{*}}\|\nabla u(t)\|_{L_{x}^{2}({\mathbb{R}}^{d})}=\infty,

    or T∗=+∞T^{*}=+\infty and there exists a time sequence tn→+∞t_{n}\to+\infty such that

    limn→+∞‖∇u​(tn)‖Lx2​(ℝd)=∞.\lim_{n\to+\infty}\|\nabla u(t_{n})\|_{L_{x}^{2}({\mathbb{R}}^{d})}=\infty.
Remark 1.6.

In [13], we dismiss the influence of V−V_{-} for the assumption that V⩾0V\geqslant 0. So the ground state 𝒬\mathcal{Q} in the result here satisfies the elliptic equation (1.8), which implies that 𝒬\mathcal{Q} is equivalent to QQ under the condition V⩾0V\geqslant 0. Thus we can rewrite the condition ℳ​ℰ<1\mathcal{M}\mathcal{E}<1 as EV​(u0)​M​(u0)<E0​(Q)​M​(Q)E_{V}(u_{0})M(u_{0})<E_{0}(Q)M(Q) in Theorem 1.5.

Inspired by this result, we want to know what will happen when beyond the threshold. Here we define that

I​(t)=∫ℝd|x|2​|u​(t,x)|2​𝑑x,x​u∈Lx2​(ℝd).\displaystyle I(t)=\int_{{\mathbb{R}}^{d}}|x|^{2}|u(t,x)|^{2}dx,\quad xu\in L_{x}^{2}({\mathbb{R}}^{d}).

The main theorem of this paper can be concluded as follows.

Theorem 1.7.

Assume that the Kato potential V satisfies (1.6) and (1.7) in (NLHV\text{NLH}_{\text{V}}) with d⩾3d\geqslant 3. Suppose u0∈Σ:={f∈Hx1​(ℝd):x​f∈Lx2​(ℝd)}u_{0}\in\Sigma:=\left\{f\in H_{x}^{1}({\mathbb{R}}^{d}):xf\in L_{x}^{2}({\mathbb{R}}^{d})\right\}, I​(0)<∞I(0)<\infty, ℳ​ℰ>1\mathcal{M}\mathcal{E}>1 and

ℳ​ℰ​(1−(I′​(0))232​EV​(u0)​I​(0))⩽1,\displaystyle\mathcal{M}\mathcal{E}\biggl{(}1-\frac{(I^{\prime}(0))^{2}}{32E_{V}(u_{0})I(0)}\biggr{)}\leqslant 1, (1.10)
  1. (i)

    (Blow up) If I′​(0)⩽0I^{\prime}(0)\leqslant 0, 2​V+x⋅∇V⩾02V+x\cdot\nabla V\geqslant 0, and

    [M(u0)]1−sc[P(u0)]sc>[M(𝒬)]1−sc[P(𝒬)]sc,\displaystyle\bigl{[}M(u_{0})\bigr{]}^{1-s_{c}}\bigl{[}P(u_{0})\bigr{]}^{s_{c}}>\bigl{[}M(\mathcal{Q})\bigr{]}^{1-s_{c}}\bigr{[}P(\mathcal{Q})\bigl{]}^{s_{c}}, (1.11)

    then u​(t,x)u(t,x) blows up in finite time.

  2. (ii)

    (Global existence) If I′​(0)⩾0I^{\prime}(0)\geqslant 0, 2​V+x⋅∇V⩽02V+x\cdot\nabla V\leqslant 0, and

    [M(u0)]1−sc[P(u0)]sc<[M(𝒬)]1−sc[P(𝒬)]sc,\displaystyle\bigl{[}M(u_{0})\bigr{]}^{1-s_{c}}\bigl{[}P(u_{0})\bigr{]}^{s_{c}}<\bigl{[}M(\mathcal{Q})\bigr{]}^{1-s_{c}}\bigr{[}P(\mathcal{Q})\bigl{]}^{s_{c}}, (1.12)

    then u​(t,x)u(t,x) exists globally. Moreover,

    limt→∞sup[M​(u)]1−sc​[P​(u)]sc<[M​(𝒬)]1−sc​[P​(𝒬)]sc.\displaystyle\lim_{t\to\infty}\sup\bigl{[}M(u)\bigr{]}^{1-s_{c}}\bigl{[}P(u)\bigr{]}^{s_{c}}<\bigl{[}M(\mathcal{Q})\bigr{]}^{1-s_{c}}\bigl{[}P(\mathcal{Q})\bigr{]}^{s_{c}}. (1.13)

The sketch of blow up: Based on the classical argument from Glassey in [9], if the initial data u0∈Σu_{0}\in\Sigma, then the solution will blow up in finite time. Then we simplify the problem to prove that for x​u∈Lx2​(ℝd)xu\in L_{x}^{2}({\mathbb{R}}^{d}),

d2d​t2​I​(t)=d2d​t2​‖x​u‖Lx2​(ℝd)2<0,∀t∈[0,T+​(u)).\displaystyle\frac{d^{2}}{dt^{2}}I(t)=\frac{d^{2}}{dt^{2}}\|xu\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}<0,\quad\forall~{}t\in\left[0,T_{+}(u)\right).

We make use of z​(t)=I​(t)z(t)=\sqrt{I(t)}, and prove the claim that z′′​(t)<0z^{\prime\prime}(t)<0 in finite time (see details in Section 3.1). So we derive that uu will blow up in finite time.

The sketch of global existence: By the contradiction argument, we prove the lower bound of z′​(t)z^{\prime}(t) (see details in Section 3.2). Then we use the claim to prove the boundedness (1.13). Combing it with the mass and energy conservation, we eventually obtain the boundedness of ‖u‖H˙V1​(ℝd)\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}, which implies that the solution uu exists globally.

Remark 1.8.

The condition I​(0)<∞I(0)<\infty ensures the existence of I′​(0)I^{\prime}(0) and I′′​(0)I^{\prime\prime}(0) of virial identity, which play important role of proving our main results. The condition ℳ​ℰ>1\mathcal{M}\mathcal{E}>1 implies that our paper is based on (NLHV\text{NLH}_{\text{V}}) beyond the threshold. The condition (1.10) provides an unified estimate of z′​(0)z^{\prime}(0), which facilitates the proof of blow up and global existence. The condition I′​(0)⩽0I^{\prime}(0)\leqslant 0 is instrumental in the proof of blow up. The condition I′​(0)⩾0I^{\prime}(0)\geqslant 0 is used to prove the claim, which is essential for demonstrating global existence. The various forms of the expression 2​V+x⋅∇V2V+x\cdot\nabla V are used to estimate the potential term in I′′​(t)I^{\prime\prime}(t) in different conditions.

Outline of this paper: In Section 2, we introduce the local well-posedness, the positivity of the operator ℋ\mathcal{H}, variational characterization and the virial identity. In Section 3, with several claims and analysis, we prove our main results, including blow up and global existence.

2. Preliminaries

In this section, we introduce the notation and several fundamental lemmas needed in this paper. The notation A≲BA\lesssim B means that A⩽C​B{A}\leqslant{CB} for some constant C>0C>0. Likewise, if A≲B≲AA\lesssim B\lesssim A, we say that A∼BA\sim B. We use Lxr​(ℝd)L_{x}^{r}(\mathbb{R}^{d}) to denote the Lebesgue space of functions f:ℝd→ℂf:\mathbb{R}^{d}\rightarrow{\mathbb{C}} whose norm

‖f‖Lxr:=(∫ℝd|f​(x)|r​𝑑x)1r\|f\|_{L_{x}^{r}}:=\Big{(}\int_{\mathbb{R}^{d}}|f(x)|^{r}dx\Big{)}^{\frac{1}{r}}

is finite, with the usual modifications when r=∞r=\infty.

2.1. Local well-posedness

In [13], we have introduced the Strichartz estimates of (NLHV\text{NLH}_{\text{V}}) and proved the local well-posedness by Banach contraction mapping principle. Here we only demonstrate the local well-posedness as follows.

Lemma 2.1 (Local well-posedness, [13]).

Let  V:ℝd→ℝV:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}} satisfy (1.6) and (1.7). Then the equation (NLHV\text{NLH}_{\text{V}}) is locally well-posed in Hx1​(ℝd)H_{x}^{1}({\mathbb{R}}^{d}) for (γ,d)=(3,5)(\gamma,d)=(3,5).

With the depiction of local well-posedness, we can further discuss the global existence and blow up of (NLHV\text{NLH}_{\text{V}}). Indeed, if we want to discuss the local well-posedness when 2<γ<42<\gamma<4 rather than γ=3\gamma=3 in [13], we only need to make some minor alterations to the proof. Then we can also obtain the similar result. Here we omit the proof.

2.2. The positivity of ℋ\mathcal{H}

In this subsection, we discuss the Kato potential VV and the operator ℋ\mathcal{H}. Indeed, the operator ℋ\mathcal{H} is positive definite when the negative part of VV is sufficiently small. We summarize this as the lemma below.

Lemma 2.2.

For Cd=Γ​(d2)/[(2−d)​2​πd2]C_{d}=\Gamma(\frac{d}{2})\big{/}\bigl{[}(2-d)2\pi^{\frac{d}{2}}\bigr{]}, if V∈𝒦V\in\mathcal{K}, then

∫ℝdV​|u|2​𝑑x⩽Cd​‖V‖𝒦​‖∇u‖Lx2​(ℝd)2.\displaystyle\int_{{\mathbb{R}}^{d}}V|u|^{2}dx\leqslant C_{d}\|V\|_{\mathcal{K}}\|\nabla u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}.

In particular, if ‖V−‖𝒦<1Cd\|V_{-}\|_{\mathcal{K}}<\frac{1}{C_{d}}, we have

(1−Cd​‖V−‖𝒦)​‖∇u‖Lx2​(ℝd)2⩽‖ℋ12​u‖Lx2​(ℝd)2=∫ℝdℋ​u​u¯​𝑑x⩽(1+Cd​‖V‖𝒦)​‖∇u‖Lx2​(ℝd)2.\displaystyle(1-C_{d}\|V_{-}\|_{\mathcal{K}})\|\nabla u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}\leqslant\|\mathcal{H}^{\frac{1}{2}}u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}=\int_{{\mathbb{R}}^{d}}\mathcal{H}u\bar{u}dx\leqslant(1+C_{d}\|V\|_{\mathcal{K}})\|\nabla u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}.
Proof.

We first consider the fundamental solution K​(x)K(x) of the Laplace equation −Δ​u=0-\Delta u=0, that is Δ​K​(x)=δ​(x)\Delta K(x)=\delta(x). Then for the equation −Δ​u=f-\Delta u=f, we have

−Δ​u=−Δ​(−Δ)−1​f=−Δ​(−K∗f)=−Δ​(−K)∗f=f.\displaystyle-\Delta u=-\Delta(-\Delta)^{-1}f=-\Delta(-K\ast f)=-\Delta(-K)\ast f=f.

For d⩾3d\geqslant 3, the fundamental solution of the Laplace equation is K​(x)=Cd​|x|2−dK(x)=C_{d}|x|^{2-d} , where

Cd=Γ​(d2)(d−2)​2​πd2.\displaystyle C_{d}=\frac{\Gamma(\frac{d}{2})}{(d-2)2\pi^{\frac{d}{2}}}.

Thus we find the relation (−Δ)−1​f=−K∗f=∫ℝdCd​f​(y)|x−y|d−2​𝑑y(-\Delta)^{-1}f=-K\ast f=\int_{{\mathbb{R}}^{d}}C_{d}\frac{f(y)}{|x-y|^{d-2}}dy. Know that

∥|V|12​(−Δ)−1​|V|12​u∥Lx2​(ℝd)2\displaystyle\bigl{\|}|V|^{\frac{1}{2}}(-\Delta)^{-1}|V|^{\frac{1}{2}}u\bigr{\|}_{L_{x}^{2}({\mathbb{R}}^{d})}^{2} =∫ℝd|V​(x)|​|Cd​|V​(y)|12|x−y|d−2​u​(y)​d​y|2​𝑑x\displaystyle=\int_{{\mathbb{R}}^{d}}|V(x)|\biggl{|}C_{d}\frac{|V(y)|^{\frac{1}{2}}}{|x-y|^{d-2}}u(y)dy\biggr{|}^{2}dx
⩽∫ℝd|V​(x)|​(Cd​∫ℝd|V​(y)||x−y|d−2​𝑑y)​(Cd​∫ℝd|u​(y)|2|x−y|d−2​𝑑y)​𝑑x\displaystyle\leqslant\int_{{\mathbb{R}}^{d}}|V(x)|\biggl{(}C_{d}\int_{{\mathbb{R}}^{d}}\frac{|V(y)|}{|x-y|^{d-2}}dy\biggr{)}\biggl{(}C_{d}\int_{{\mathbb{R}}^{d}}\frac{|u(y)|^{2}}{|x-y|^{d-2}}dy\biggr{)}dx
⩽Cd​‖V‖𝒦​∫ℝdCd​|V​(x)||x−y|d−2​|u​(y)|2​𝑑y​𝑑x\displaystyle\leqslant C_{d}\|V\|_{\mathcal{K}}\int_{{\mathbb{R}}^{d}}\frac{C_{d}|V(x)|}{|x-y|^{d-2}}|u(y)|^{2}dydx
⩽(Cd​‖V‖𝒦)2​‖u‖Lx2​(ℝd)2.\displaystyle\leqslant\bigl{(}C_{d}\|V\|_{\mathcal{K}}\bigr{)}^{2}\|u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}.

From the T​T∗TT^{*} argument [8], we set T=|V|12​|∇|−1T=|V|^{\frac{1}{2}}|\nabla|^{-1}. Then we find

∫ℝdV​|u|2​𝑑x=∥|V|12​|∇|−1​∇u∥Lx2​(ℝd)2⩽Cd​‖V‖𝒦​‖∇u‖Lx2​(ℝd)2.\displaystyle\int_{{\mathbb{R}}^{d}}V|u|^{2}dx=\bigl{\|}|V|^{\frac{1}{2}}|\nabla|^{-1}\nabla u\bigr{\|}_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}\leqslant C_{d}\|V\|_{\mathcal{K}}\|\nabla u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}. (2.1)

In particular, if ‖V−‖𝒦<1Cd\|V_{-}\|_{\mathcal{K}}<\frac{1}{C_{d}}, combining with (2.1), we have

(1−Cd​‖V−‖𝒦)​‖∇u‖Lx2​(ℝd)2⩽‖∇u‖Lx2​(ℝd)2+‖V12​u‖Lx2​(ℝd)2⩽(1+Cd​‖V‖𝒦)​‖∇u‖Lx2​(ℝd)2.\displaystyle(1-C_{d}\|V_{-}\|_{\mathcal{K}})\|\nabla u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}\leqslant\|\nabla u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}+\|V^{\frac{1}{2}}u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}\leqslant(1+C_{d}\|V\|_{\mathcal{K}})\|\nabla u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}.

Then the proof is completed. ∎

2.3. Variational analysis

In view of the proof in [11], we can compute the sharp constant CG​NC_{GN} for Gagliardo-Nirenberg inequality, which is crucially used to our later proof. It can be estimated as follows.

P​(u)⩽CG​N​‖u‖H˙V1​(ℝd)γ​‖u‖Lx2​(ℝd)4−γ,P(u)\leqslant C_{GN}\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{\gamma}\|u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{4-\gamma}, (2.2)

where

CG​N=supu∈Hx1​(ℝd)\{0}W​(u)=supu∈Hx1​(ℝd)\{0}P​(u)‖u‖H˙V1​(ℝd)γ​‖u‖Lx2​(ℝd)4−γ.\displaystyle C_{GN}=\sup_{u\in H_{x}^{1}({\mathbb{R}}^{d})\backslash\{0\}}W(u)=\sup_{u\in H_{x}^{1}({\mathbb{R}}^{d})\backslash\{0\}}\frac{P(u)}{\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{\gamma}\|u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{4-\gamma}}.

We know that 𝒬\mathcal{Q} is a strong solution in [11]. Rewrite (2.2) as

[P​(u)]2γ⩽C𝒬​‖u‖H˙V1​(ℝd)2​(‖u‖Lx2​(ℝd)2)4−γγ,\displaystyle\bigl{[}P(u)\bigr{]}^{\frac{2}{\gamma}}\leqslant C_{\mathcal{Q}}\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}\bigl{(}\|u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}\bigr{)}^{\frac{4-\gamma}{\gamma}},

where

C𝒬=(CG​N)2γ=[P​(𝒬)]2γ‖𝒬‖H˙V1​(ℝd)2​‖𝒬‖Lx2​(ℝd)2​(4−γ)γ=42γ(4−γ)2−γγ​γ​‖𝒬‖Lx2​(ℝd)4γ.\displaystyle C_{\mathcal{Q}}=(C_{GN})^{\frac{2}{\gamma}}=\frac{\bigl{[}P(\mathcal{Q})\bigr{]}^{\frac{2}{\gamma}}}{\|\mathcal{Q}\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}\|\mathcal{Q}\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{\frac{2(4-\gamma)}{\gamma}}}=\frac{4^{\frac{2}{\gamma}}}{(4-\gamma)^{\frac{2-\gamma}{\gamma}}\gamma\|\mathcal{Q}\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{\frac{4}{\gamma}}}.

Since our paper focus on the long-time dynamical behavior of (NLHV\text{NLH}_{\text{V}}) beyond the mass-energy threshold, the ground state 𝒬\mathcal{Q} is vital to the proof. We show the properties of 𝒬\mathcal{Q} as follows.

Proposition 2.3 (Pohozhaev identities).

For 𝒬\mathcal{Q} in the (1.9), we have

‖𝒬‖H˙V1​(ℝd)2=γ​w𝒬24−γ​‖𝒬‖Lx2​(ℝd)2,P​(𝒬)=4​w𝒬24−γ​‖𝒬‖Lx2​(ℝd)2,\displaystyle\|\mathcal{Q}\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}=\frac{\gamma w_{\mathcal{Q}}^{2}}{4-\gamma}\|\mathcal{Q}\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2},\quad P(\mathcal{Q})=\frac{4w_{\mathcal{Q}}^{2}}{4-\gamma}\|\mathcal{Q}\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2},

and then

P​(𝒬)=8γ−2​EV​(𝒬).\displaystyle P(\mathcal{Q})=\frac{8}{\gamma-2}E_{V}(\mathcal{Q}). (2.3)
Proof.

Let QwQ_{w} be a strong solution to the equation

(Δ−V)Qw−w2Qw+(|⋅|−γ∗|Qw|2)Qw=0,(\Delta-V)Q_{w}-w^{2}Q_{w}+(|\cdot|^{-\gamma}\ast|Q_{w}|^{2})Q_{w}=0, (2.4)

(i) Multiplying (2.4) by QwQ_{w} and integrating by parts, we have

∫ℝd(Δ−V)Qw⋅Qwdx−w2∫ℝdQw2dx+∫ℝd(|⋅|−γ∗|Qw|2)Qw2dx=0.\int_{{\mathbb{R}}^{d}}(\Delta-V)Q_{w}\cdot Q_{w}dx-w^{2}\int_{{\mathbb{R}}^{d}}Q_{w}^{2}dx+\int_{{\mathbb{R}}^{d}}(|\cdot|^{-\gamma}\ast|Q_{w}|^{2})Q_{w}^{2}dx=0.

We find

∫ℝd(Δ−V)​Qw⋅Qw​𝑑x=−∫ℝd|∇Qw|2​𝑑x−∫ℝdV​Qw2​𝑑x=−‖Qw‖H˙V1​(ℝd)2,\displaystyle\int_{{\mathbb{R}}^{d}}(\Delta-V)Q_{w}\cdot Q_{w}dx=-\int_{{\mathbb{R}}^{d}}|\nabla Q_{w}|^{2}dx-\int_{{\mathbb{R}}^{d}}VQ_{w}^{2}dx=-||Q_{w}||_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2},

Then we obtain

P​(Qw)=‖Qw‖H˙V1​(ℝd)2+w2​‖Qw‖Lx2​(ℝd)2P(Q_{w})=||Q_{w}||_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}+w^{2}||Q_{w}||_{L_{x}^{2}({\mathbb{R}}^{d})}^{2} (2.5)

(ii) Multiplying (2.4) by x⋅∇Qwx\cdot\nabla Q_{w} and integrating by parts, we have

∫ℝd(Δ−V)Qw⋅x⋅∇Qwdx−∫ℝdQw⋅x⋅∇Qwdx+∫ℝd(|⋅|−γ∗|Qw|2)Qw⋅x⋅∇Qwdx=0.\int_{{\mathbb{R}}^{d}}(\Delta-V)Q_{w}\cdot x\cdot\nabla Q_{w}dx-\int_{{\mathbb{R}}^{d}}Q_{w}\cdot x\cdot\nabla Q_{w}dx+\int_{{\mathbb{R}}^{d}}(|\cdot|^{-\gamma}\ast|Q_{w}|^{2})Q_{w}\cdot x\cdot\nabla Q_{w}dx=0.

For each part, we have

(a)​∫ℝd(Δ−V)​Qw⋅x⋅∇Qw​d​x=d−22​‖∇Qw‖Lx2​(ℝd)2+12​∫ℝd(x⋅∇V+2​V)​|Qw|2​𝑑x,\displaystyle(a)~{}\int_{{\mathbb{R}}^{d}}(\Delta-V)Q_{w}\cdot x\cdot\nabla Q_{w}dx=\frac{d-2}{2}||\nabla Q_{w}||_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}+\frac{1}{2}\int_{{\mathbb{R}}^{d}}(x\cdot\nabla V+2V)|Q_{w}|^{2}dx,
(b)​∫ℝdw2​Qw⋅x⋅∇Qw​d​x=−d​w22​‖Qw‖Lx2​(ℝd)2,\displaystyle(b)~{}\int_{{\mathbb{R}}^{d}}w^{2}Q_{w}\cdot x\cdot\nabla Q_{w}dx=-\frac{dw^{2}}{2}||Q_{w}||_{L_{x}^{2}({\mathbb{R}}^{d})}^{2},
(c)∫ℝd(|⋅|−γ∗|Qw|2)Qw⋅x⋅∇Qwdx=(γ4−d2)P(Qw).\displaystyle(c)~{}\int_{{\mathbb{R}}^{d}}(|\cdot|^{-\gamma}\ast|Q_{w}|^{2})Q_{w}\cdot x\cdot\nabla Q_{w}dx=(\frac{\gamma}{4}-\frac{d}{2})P(Q_{w}).

Collecting them all, we have

(d−γ2)​P​(Qw)=(d−2)​‖Qw‖H˙V1​(ℝd)2+d​w2​‖Qw‖Lx2​(ℝd)2+∫ℝd(2​V+x⋅∇V)​|Qw|2​𝑑x.\displaystyle(d-\frac{\gamma}{2})P(Q_{w})=(d-2)||Q_{w}||_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}+dw^{2}||Q_{w}||_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}+\int_{{\mathbb{R}}^{d}}(2V+x\cdot\nabla V)|Q_{w}|^{2}dx. (2.6)

Combining (2.4) and (2.6) together, we find

{‖Qw‖H˙V1​(ℝd)2=γ4−γ​w2​‖Qw‖Lx2​(ℝd)2+24−γ​∫ℝd(2​V+x⋅∇V)​|Qw|2​𝑑x,P​(Qw)=44−γ​w2​‖Qw‖Lx2​(ℝd)2+24−γ​∫ℝd(2​V+x⋅∇V)​|Qw|2​𝑑x.\left\{\begin{aligned} &||Q_{w}||_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}=\frac{\gamma}{4-\gamma}w^{2}||Q_{w}||_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}+\frac{2}{4-\gamma}\int_{{\mathbb{R}}^{d}}(2V+x\cdot\nabla V)|Q_{w}|^{2}dx,\\ &P(Q_{w})=\frac{4}{4-\gamma}w^{2}||Q_{w}||_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}+\frac{2}{4-\gamma}\int_{{\mathbb{R}}^{d}}(2V+x\cdot\nabla V)|Q_{w}|^{2}dx.\end{aligned}\right. (2.7)

Indeed, if V=0V=0, then

{‖∇Qw‖Lx2​(ℝd)2=γ4−γ​w2​‖Qw‖Lx2​(ℝd)2,P​(Qw)=44−γ​w2​‖Qw‖Lx2​(ℝd)2.\left\{\begin{aligned} &||\nabla Q_{w}||_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}=\frac{\gamma}{4-\gamma}w^{2}||Q_{w}||_{L_{x}^{2}({\mathbb{R}}^{d})}^{2},\\ &P(Q_{w})=\frac{4}{4-\gamma}w^{2}||Q_{w}||_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}.\end{aligned}\right. (2.8)

Let 𝒬\mathcal{Q} be the ground state given in Proposition 1.4, and

w𝒬2=(4−γ)​‖𝒬‖H˙V1​(ℝd)2γ​‖𝒬‖Lx2​(ℝd)2.w_{\mathcal{Q}}^{2}=\frac{(4-\gamma)~{}||\mathcal{Q}||_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}}{\gamma~{}||\mathcal{Q}||_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}}.

From (2.7) we find

∫ℝd(2​V+x⋅∇V)​|𝒬|2​𝑑x=0.\int_{{\mathbb{R}}^{d}}(2V+x\cdot\nabla V)|\mathcal{Q}|^{2}dx=0.

Thus we have

‖𝒬‖H˙V1​(ℝd)2=γ​w𝒬24−γ​‖𝒬‖Lx2​(ℝd)2,P​(𝒬)=4​w𝒬24−γ​‖𝒬‖Lx2​(ℝd)2.\displaystyle\|\mathcal{Q}\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}=\frac{\gamma w_{\mathcal{Q}}^{2}}{4-\gamma}\|\mathcal{Q}\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2},\quad P(\mathcal{Q})=\frac{4w_{\mathcal{Q}}^{2}}{4-\gamma}\|\mathcal{Q}\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}.

∎

Remark 2.4.

If V−=0V_{-}=0, according to Proposition 1.4, the ground state QQ satisfies (1.8). By the similar method, we find the pohozhaev identity

‖Q‖H˙V1​(ℝd)2=γ4−γ​‖Q‖Lx2​(ℝd)2,P​(Q)=44−γ​‖Q‖Lx2​(ℝd)2.\displaystyle\|Q\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}=\frac{\gamma}{4-\gamma}\|Q\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2},\quad P(Q)=\frac{4}{4-\gamma}\|Q\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}.

and then

P​(Q)=8γ−2​EV​(Q).\displaystyle P(Q)=\frac{8}{\gamma-2}E_{V}(Q).

2.4. Virial identity

We will discuss the virial identity for that our main results are closely based on the property of it. We have

I​(t)=∫ℝd|x|2​|u​(t,x)|2​𝑑x,\displaystyle I(t)=\int_{{\mathbb{R}}^{d}}|x|^{2}|u(t,x)|^{2}dx,

for x​u∈Lx2​(ℝd)xu\in L_{x}^{2}({\mathbb{R}}^{d}).

A natural question is that we want to make some estimates for I​(t)I(t). By the accurate calculation, we summarize the first and second derivatives of I​(t)I(t) as follows, which will be used throughout our proof.

Lemma 2.5.

Assume that u​(t,x)u(t,x) is the solution to (NLHV\text{NLH}_{\text{V}}). Then we have

I′​(t)=4​Im∫ℝdx​u¯​∇u​d​x,\displaystyle I^{\prime}(t)=4\mathop{\mathrm{Im}}\int_{{\mathbb{R}}^{d}}x\bar{u}\nabla udx,

and

I′′​(t)=\displaystyle I^{\prime\prime}(t)= 8​‖∇u‖Lx2​(ℝd)2−4​∫ℝdx⋅∇V​|u|2​d​x−2​γ​P​(u)\displaystyle~{}8\|\nabla u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}-4\int_{{\mathbb{R}}^{d}}x\cdot\nabla V|u|^{2}dx-2\gamma P(u)
=\displaystyle= 8​‖u‖H˙V1​(ℝd)2−4​∫ℝdx⋅∇V​|u|2​d​x−8​∫ℝdV​|u|2​𝑑x−2​γ​P​(u)\displaystyle~{}8\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}-4\int_{{\mathbb{R}}^{d}}x\cdot\nabla V|u|^{2}dx-8\int_{{\mathbb{R}}^{d}}V|u|^{2}dx-2\gamma P(u)
=\displaystyle= 8​‖u‖H˙V1​(ℝd)2−2​γ​P​(u)−e​(t),\displaystyle~{}8\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}-2\gamma P(u)-e(t),

where

e​(t):=4​∫ℝd(2​V+x⋅∇V)​|u|2​𝑑x.e(t):=4\int_{{\mathbb{R}}^{d}}(2V+x\cdot\nabla V)|u|^{2}dx.

Indeed, we incorporate all terms involving the potential VV into e​(t)e(t). And e​(t)e(t) is entirely dependent on 2​V+x⋅∇V2V+x\cdot\nabla V, which have been provided in Theorem 1.7. So we can primarily focus on estimating the rest terms. We denote that

I′′​(t)~=I′′​(t)+e​(t)=8​‖u‖H˙V1​(ℝd)2−2​γ​P​(u).\displaystyle\widetilde{I^{\prime\prime}(t)}=I^{\prime\prime}(t)+e(t)=8\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}-2\gamma P(u). (2.9)

Rewrite (1.2) as

EV​(u)=12​‖u‖H˙V1​(ℝd)2−14​P​(u).E_{V}(u)=\frac{1}{2}\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}-\frac{1}{4}P(u).

Then combining the above two equations, we find

P​(u)=12​(γ−2)​(16​EV​(u)−I′′​(t)~)\displaystyle P(u)=\frac{1}{2(\gamma-2)}\bigl{(}16E_{V}(u)-\widetilde{I^{\prime\prime}(t)}\bigr{)} (2.10)

and

‖u‖H˙V1​(ℝd)2=8​γ​EV​(u)−I′′​(t)~4​(γ−2).\displaystyle\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}=\frac{8\gamma E_{V}(u)-\widetilde{I^{\prime\prime}(t)}}{4(\gamma-2)}. (2.11)

Next, according to (2.10) and (2.11), we come to find the relation between I′​(t)I^{\prime}(t), P​(u)P(u) and ‖u‖H˙V1​(ℝd)2\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}, which is important to our proof of Theorem 1.7.

Lemma 2.6.
(Im∫x​u¯​∇u​d​x)2⩽∫|x|2​|u|2​𝑑x​(‖u‖H˙V1​(ℝd)2−[P​(u)]2γC𝒬​[M​(u)]4−γγ).\displaystyle\biggl{(}\mathop{\mathrm{Im}}\int x\bar{u}\nabla udx\biggr{)}^{2}\leqslant\int|x|^{2}|u|^{2}dx\biggl{(}\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}-\frac{[P(u)]^{\frac{2}{\gamma}}}{C_{\mathcal{Q}}[M(u)]^{\frac{4-\gamma}{\gamma}}}\biggr{)}.
Proof.

For ei​λ​|x|2​ue^{i\lambda|x|^{2}}u, we can compute that

|∇(ei​λ​|x|2​u)|2=\displaystyle\bigl{|}\nabla(e^{i\lambda|x|^{2}}u)\bigr{|}^{2}= [(∇ei​λ​|x|2)​u+ei​λ​|x|2​∇u]2\displaystyle~{}\bigl{[}(\nabla e^{i\lambda|x|^{2}})u+e^{i\lambda|x|^{2}}\nabla u\bigr{]}^{2}
=\displaystyle= [(∇ei​λ​|x|2)​u]2+2​(∇ei​λ​|x|2)​u⋅ei​λ​|x|2​∇u+[ei​λ​|x|2​∇u]2\displaystyle~{}\bigl{[}(\nabla e^{i\lambda|x|^{2}})u\bigr{]}^{2}+2(\nabla e^{i\lambda|x|^{2}})u\cdot e^{i\lambda|x|^{2}}\nabla u+\bigl{[}e^{i\lambda|x|^{2}}\nabla u\bigr{]}^{2}
=\displaystyle= [ei​λ​|x|2⋅∇(i​λ​|x|2)⋅u]2+2​ei​λ​|x|2⋅∇(i​λ​|x|2)⋅u​∇u+(∇u)2\displaystyle~{}\bigl{[}e^{i\lambda|x|^{2}}\cdot\nabla(i\lambda|x|^{2})\cdot u\bigr{]}^{2}+2e^{i\lambda|x|^{2}}\cdot\nabla(i\lambda|x|^{2})\cdot u\nabla u+(\nabla u)^{2}
=\displaystyle= [i​λ​(∇|x|2)​u]2+2​i​λ​∇|x|2⋅u⋅∇u+(∇u)2\displaystyle~{}\bigl{[}i\lambda(\nabla|x|^{2})u\bigr{]}^{2}+2i\lambda\nabla|x|^{2}\cdot u\cdot\nabla u+(\nabla u)^{2}
=\displaystyle= [2​λ​(Im∇x⋅x¯)⋅u]2+4​λ​(Im∇x⋅x¯)⋅u⋅∇u+(∇u)2.\displaystyle~{}\bigl{[}2\lambda(\mathop{\mathrm{Im}}\nabla x\cdot\bar{x})\cdot u\bigr{]}^{2}+4\lambda(\mathop{\mathrm{Im}}\nabla x\cdot\bar{x})\cdot u\cdot\nabla u+(\nabla u)^{2}.

Then we have the H˙V1\dot{H}_{V}^{1} norm of ei​λ​|x|2​ue^{i\lambda|x|^{2}}u, that is

‖ei​λ​|x|2​u‖H˙V1​(ℝd)=\displaystyle\|e^{i\lambda|x|^{2}}u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}= ∫|∇(ei​λ​|x|2​u)|2​𝑑x+∫V​(x)​|ei​λ​|x|2​u|2​𝑑x\displaystyle~{}\int\bigl{|}\nabla(e^{i\lambda|x|^{2}}u)\bigr{|}^{2}dx+\int V(x)\bigl{|}e^{i\lambda|x|^{2}}u\bigr{|}^{2}dx
=\displaystyle= 4​λ2​∫|x|2​|u|2​𝑑x+4​λ​Im∫x​u¯​∇u​d​x+∫(|∇u|2+V​|u|2)​𝑑x\displaystyle~{}4\lambda^{2}\int|x|^{2}|u|^{2}dx+4\lambda\mathop{\mathrm{Im}}\int x\bar{u}\nabla udx+\int(|\nabla u|^{2}+V|u|^{2})dx
=\displaystyle= 4​λ2​∫|x|2​|u|2​𝑑x+4​λ​Im∫x​u¯​∇u​d​x+‖u‖H˙V1​(ℝd)2.\displaystyle~{}4\lambda^{2}\int|x|^{2}|u|^{2}dx+4\lambda\mathop{\mathrm{Im}}\int x\bar{u}\nabla udx+\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}.

We substitute ei​λ​|x|2​ue^{i\lambda|x|^{2}}u for uu in (2.2), that is

P​(ei​λ​|x|2​u)⩽CG​N​∥ei​λ​|x|2​u∥H˙V1​(ℝd)γ​∥ei​λ​|x|2​u∥Lx2​(ℝd)4−γ.\displaystyle P(e^{i\lambda|x|^{2}}u)\leqslant C_{GN}\bigl{\|}e^{i\lambda|x|^{2}}u\bigr{\|}_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{\gamma}\bigl{\|}e^{i\lambda|x|^{2}}u\bigr{\|}_{L_{x}^{2}({\mathbb{R}}^{d})}^{4-\gamma}.

Thus, according to C𝒬=(CG​N)2γC_{\mathcal{Q}}=(C_{GN})^{\frac{2}{\gamma}},

C𝒬​‖u‖Lx2​(ℝd)2​(4−γ)γ​[4​λ2​∫|x|2​|u|2​𝑑x+4​λ​Im∫x​u¯​∇u​d​x+‖u‖H˙V1​(ℝd)2]−[P​(u)]2γ⩾0\displaystyle C_{\mathcal{Q}}\|u\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{\frac{2(4-\gamma)}{\gamma}}\bigl{[}4\lambda^{2}\int|x|^{2}|u|^{2}dx+4\lambda\mathop{\mathrm{Im}}\int x\bar{u}\nabla udx+\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}^{2}\bigr{]}-[P(u)]^{\frac{2}{\gamma}}\geqslant 0 (2.12)

for any λ∈ℝ\lambda\in{\mathbb{R}}. We find that the left side of (2.12) is a quadratic function in λ\lambda. The discriminant of this function in λ\lambda must be negative, which yields Lemma 2.6. ∎

Remark 2.7.

In order to describe more visually, we rewrite Lemma 2.6 by (2.10) and (2.11). It follows that

[I′​(t)]2⩽16​I​(t)​[8​γ​EV​(u)−I′′​(t)~4​(γ−2)−1C𝒬​[M​(u)]4−γγ​(16​EV​(u)−I′′​(t)~2​(γ−2))2γ].\displaystyle[I^{\prime}(t)]^{2}\leqslant 16I(t)\biggl{[}\frac{8\gamma E_{V}(u)-\widetilde{I^{\prime\prime}(t)}}{4(\gamma-2)}-\frac{1}{C_{\mathcal{Q}}[M(u)]^{\frac{4-\gamma}{\gamma}}}\biggl{(}\frac{16E_{V}(u)-\widetilde{I^{\prime\prime}(t)}}{2(\gamma-2)}\biggr{)}^{\frac{2}{\gamma}}\biggr{]}. (2.13)

We denote that

f​(x)=2​γγ−2​EV​(u)−14​(γ−2)​x−1C𝒬​[M​(u)]4−γγ​(16​EV​(u)−x2​(γ−2))2γ\displaystyle f(x)=\frac{2\gamma}{\gamma-2}E_{V}(u)-\frac{1}{4(\gamma-2)}x-\frac{1}{C_{\mathcal{Q}}[M(u)]^{\frac{4-\gamma}{\gamma}}}\biggl{(}\frac{16E_{V}(u)-x}{2(\gamma-2)}\biggr{)}^{\frac{2}{\gamma}}

for any x∈(−∞,16​EV​(u)]x\in\left(-\infty,16E_{V}(u)\right]. Thus, (2.13) can be simplified as

[I′​(t)]2⩽16​I​(t)​f​(I′′​(t)~).\displaystyle[I^{\prime}(t)]^{2}\leqslant 16I(t)f(\widetilde{I^{\prime\prime}(t)}). (2.14)

3. Proof of the Main Theorem

In this section, we will prove the main results in Theorem 1.7.

Let z​(t)=I​(t)z(t)=\sqrt{I(t)}. By (2.14), we have

[z′​(t)]2=[I′​(t)]24​I​(t)⩽4​f​(I′′​(t)~).\displaystyle[z^{\prime}(t)]^{2}=\frac{[I^{\prime}(t)]^{2}}{4I(t)}\leqslant 4f(\widetilde{I^{\prime\prime}(t)}). (3.1)

It indicates that z′​(0)z^{\prime}(0) can be estimated by f​(I′′​(t)~)f(\widetilde{I^{\prime\prime}(t)}). Actually, the condition I′​(0)I^{\prime}(0) in Theorem 1.7 is closely related to z′​(0)z^{\prime}(0). Thus, by (3.1), we first need to discuss the property of f​(x)f(x) to pave the way for proving our main results.

We can compute that

f′​(x)=−14​(γ−2)+1C𝒬​[M​(u)]4−γγ⋅2γ⋅[12​(γ−2)]2γ​[16​EV​(u)−x]2γ−1.\displaystyle f^{\prime}(x)=-\frac{1}{4(\gamma-2)}+\frac{1}{C_{\mathcal{Q}}[M(u)]^{\frac{4-\gamma}{\gamma}}}\cdot\frac{2}{\gamma}\cdot\bigl{[}\frac{1}{2(\gamma-2)}\bigr{]}^{\frac{2}{\gamma}}\bigl{[}16E_{V}(u)-x\bigr{]}^{\frac{2}{\gamma}-1}.

Since 2γ−1<0​(sc>0)\frac{2}{\gamma}-1<0~{}(s_{c}>0), f​(x)f(x) is decreasing on (−∞,x0)(-\infty,x_{0}) and increasing on (x0,16​EV​(u))(x_{0},16E_{V}(u)), where x0x_{0} satisfies

14​(γ−2)=1C𝒬​[M​(u)]4−γγ⋅2γ⋅[12​(γ−2)]2γ​[16​EV​(u)−x0]2γ−1.\displaystyle\frac{1}{4(\gamma-2)}=\frac{1}{C_{\mathcal{Q}}[M(u)]^{\frac{4-\gamma}{\gamma}}}\cdot\frac{2}{\gamma}\cdot\bigl{[}\frac{1}{2(\gamma-2)}\bigr{]}^{\frac{2}{\gamma}}\bigl{[}16E_{V}(u)-x_{0}\bigr{]}^{\frac{2}{\gamma}-1}. (3.2)

Then

f​(x0)=\displaystyle f(x_{0})= 2​γγ−2​EV​(u)−14​(γ−2)​x0−1C𝒬​[M​(u)]4−γγ​(16​EV​(u)−x02​(γ−2))2γ\displaystyle~{}\frac{2\gamma}{\gamma-2}E_{V}(u)-\frac{1}{4(\gamma-2)}x_{0}-\frac{1}{C_{\mathcal{Q}}[M(u)]^{\frac{4-\gamma}{\gamma}}}\biggl{(}\frac{16E_{V}(u)-x_{0}}{2(\gamma-2)}\biggr{)}^{\frac{2}{\gamma}}
=\displaystyle= 2​γγ−2​EV​(u)−14​(γ−2)​x0−14​(γ−2)⋅γ2⋅(16​EV​(u)−x0)\displaystyle~{}\frac{2\gamma}{\gamma-2}E_{V}(u)-\frac{1}{4(\gamma-2)}x_{0}-\frac{1}{4(\gamma-2)}\cdot\frac{\gamma}{2}\cdot(16E_{V}(u)-x_{0})
=\displaystyle= x08.\displaystyle~{}\frac{x_{0}}{8}.

Using (2.10), (2.11) and the expression of C𝒬C_{\mathcal{Q}}, we rewrite (3.2) as

(M​(u)M​(𝒬))1−sc​(EV​(u)−116​x0EV​(𝒬))sc=ℳ​ℰ​(1−x016​EV​(u))=1\displaystyle\biggl{(}\frac{M(u)}{M(\mathcal{Q})}\biggr{)}^{1-s_{c}}\biggl{(}\frac{E_{V}(u)-\frac{1}{16}x_{0}}{E_{V}(\mathcal{Q})}\biggr{)}^{s_{c}}=\mathcal{M}\mathcal{E}\biggl{(}1-\frac{x_{0}}{16E_{V}(u)}\biggr{)}=1 (3.3)

for x0∈(−∞,16​EV​(u)]x_{0}\in\left(-\infty,16E_{V}(u)\right]. Then ℳ​ℰ>1\mathcal{M}\mathcal{E}>1 is equivalent to x0⩾0x_{0}\geqslant 0. We can also rewrite (1.10) as

[z′​(0)]2⩾x02=4​f​(x0).\displaystyle[z^{\prime}(0)]^{2}\geqslant\frac{x_{0}}{2}=4f(x_{0}). (3.4)

3.1. The proof of blow up.

Using the Hardy inequality

∫0∞[1x​∫0xg​(ξ)​𝑑ξ]p​𝑑x⩽(pp−1)p​∫0∞gp​(x)​𝑑x,g​(x)⩾0,p>1,\int_{0}^{\infty}\biggl{[}\frac{1}{x}\int_{0}^{x}g(\xi)d\xi\biggr{]}^{p}dx\leqslant\biggl{(}\frac{p}{p-1}\biggr{)}^{p}\int_{0}^{\infty}g^{p}(x)dx,\quad g(x)\geqslant 0,\quad p>1,

and the conservation of mass, we have

‖u0‖Lx2​(ℝd)2=∫ℝd|u​(x,t)|2​𝑑x⩽‖x​u​(t)‖Lx2​(ℝd)​‖u​(t)‖H˙x1​(ℝd)→0,t→T+​(u),\displaystyle\|u_{0}\|_{L_{x}^{2}({\mathbb{R}}^{d})}^{2}=\int_{{\mathbb{R}}^{d}}|u(x,t)|^{2}dx\leqslant\|xu(t)\|_{L_{x}^{2}({\mathbb{R}}^{d})}\|u(t)\|_{\dot{H}_{x}^{1}({\mathbb{R}}^{d})}\to 0,\quad t\to T_{+}(u),

where we have used the fact that u0∈Σu_{0}\in\Sigma implies the corresponding solution uu belongs to Σ\Sigma. On the basis of Glassey’s classical argument [9], our proof of blow up is to prove

I′′​(t)<0,∀t∈[0,T+​(u)).\displaystyle I^{\prime\prime}(t)<0,\quad\forall~{}t\in\left[0,T_{+}(u)\right).

To that end, we demonstrate z′′​(t)z^{\prime\prime}(t) for that z​(t)=I​(t)z(t)=\sqrt{I(t)}.

Firstly, we make some equivalent transformation to the conditions about the blow up in Theorem 1.7. The assumption I′​(0)⩽0I^{\prime}(0)\leqslant 0 implies z′​(0)⩽0z^{\prime}(0)\leqslant 0. According to (2.3) and (3.3), the assumption (1.11) implies that

[M​(u0)M​(𝒬)]1−sc[γ−28​P​(u0)EV​(𝒬)]>1=[M​(u)M​(𝒬)]1−sc[EV​(u)−116​x0EV​(𝒬)]sc.\displaystyle\biggr{[}\frac{M(u_{0})}{M(\mathcal{Q})}\biggl{]}^{1-s_{c}}\biggr{[}\frac{\frac{\gamma-2}{8}P(u_{0})}{E_{V}(\mathcal{Q})}\biggl{]}>1=\biggr{[}\frac{M(u)}{M(\mathcal{Q})}\biggl{]}^{1-s_{c}}\biggl{[}\frac{E_{V}(u)-\frac{1}{16}x_{0}}{E_{V}(\mathcal{Q})}\biggr{]}^{s_{c}}.

Then by (2.10), we get

I′′​(0)~<x0.\displaystyle\widetilde{I^{\prime\prime}(0)}<x_{0}. (3.5)

Since 2​V+x⋅∇V⩾02V+x\cdot\nabla V\geqslant 0, we find e​(0)⩾0e(0)\geqslant 0. Then I′′​(0)~=I′′​(0)+e​(0)<x0\widetilde{I^{\prime\prime}(0)}=I^{\prime\prime}(0)+e(0)<x_{0} yields that I′′​(0)<x0I^{\prime\prime}(0)<x_{0}. Thus, we have

z′′​(0)=\displaystyle z^{\prime\prime}(0)= 1z​(0)​(I′′​(0)2−(z′​(0))2)\displaystyle~{}\frac{1}{z(0)}\bigl{(}\frac{I^{\prime\prime}(0)}{2}-(z^{\prime}(0))^{2}\bigr{)}
<\displaystyle< 1z​(0)​(x02−(z′​(0))2)\displaystyle~{}\frac{1}{z(0)}\bigl{(}\frac{x_{0}}{2}-(z^{\prime}(0))^{2}\bigr{)}
⩽\displaystyle\leqslant 1z​(0)​(x02−x02)=0.\displaystyle~{}\frac{1}{z(0)}\bigl{(}\frac{x_{0}}{2}-\frac{x_{0}}{2}\bigr{)}=0.

We  claim  that

𝒛′′​(𝒕)<𝟎,∀𝒕∈[𝟎,𝑻+​(𝒖)).\displaystyle\bm{z^{\prime\prime}(t)<0,\quad\forall~{}t\in\left[0,T_{+}(u)\right)}. (3.6)

Indeed, if the claim holds, we assume that T+​(u)=∞T_{+}(u)=\infty. By z′​(0)<0z^{\prime}(0)<0 and z′′​(t)<0z^{\prime\prime}(t)<0, we have z′​(1)<0z^{\prime}(1)<0. Then

z​(t)=\displaystyle z(t)= ∫1tz′​(s)​𝑑s+z​(1)\displaystyle~{}\int_{1}^{t}z^{\prime}(s)ds+z(1)
⩽\displaystyle\leqslant ∫1tz′​(1)​𝑑s+z​(1)\displaystyle~{}\int_{1}^{t}z^{\prime}(1)ds+z(1)
⩽\displaystyle\leqslant z′​(1)​(t−1)+z​(1).\displaystyle~{}z^{\prime}(1)(t-1)+z(1).

For t→∞t\to\infty, z​(t)<0z(t)<0 is contradicted to z​(t)>0z(t)>0. Thus T+​(u)<∞T_{+}(u)<\infty, which implies that z​(t)z(t) will approach 0 in a finite time.

Now we come to prove the claim (3.6). If the claim does not hold, there exists t0=sup{t∈(0,T+​(u)),z′′​(0)⩾0}t_{0}=\sup\{t\in(0,T_{+}(u)),z^{\prime\prime}(0)\geqslant 0\} satisfying z′′​(t)<0z^{\prime\prime}(t)<0 for any t∈[0,t0)t\in\left[0,t_{0}\right). And by the continuity of z′′​(t)z^{\prime\prime}(t), we have

z′′​(t0)=0.z^{\prime\prime}(t_{0})=0.

Using z′​(0)⩽0z^{\prime}(0)\leqslant 0 and (3.4), we have

z′​(t)<z′​(0)⩽0,∀t∈[0,t0],\displaystyle z^{\prime}(t)<z^{\prime}(0)\leqslant 0,\quad\forall~{}t\in[0,t_{0}],

and

4​f​(x0)⩽[z′​(0)]2⩽[z′​(t)]2,∀t∈[0,t0].\displaystyle 4f(x_{0})\leqslant[z^{\prime}(0)]^{2}\leqslant[z^{\prime}(t)]^{2},\quad\forall~{}t\in[0,t_{0}].

Combining with (3.1), we obtain f​(I′′​(t)~)>f​(x0)f(\widetilde{I^{\prime\prime}(t)})>f(x_{0}) for any t∈[0,t0]t\in[0,t_{0}]. Then I′′​(t)~≠x0\widetilde{I^{\prime\prime}(t)}\neq x_{0} for any t∈[0,t0]t\in[0,t_{0}]. According to (3.5) and the continuity of I′′​(t)~\widetilde{I^{\prime\prime}(t)}, we have

I′′​(t)~<x0,∀t∈[0,t0].\displaystyle\widetilde{I^{\prime\prime}(t)}<x_{0},\quad\forall~{}t\in[0,t_{0}]. (3.7)

Furthermore, 2​V+x⋅∇V⩾02V+x\cdot\nabla V\geqslant 0 yields that e​(t)⩾0e(t)\geqslant 0. From (2.9), We find

I′′​(t)=I′′​(t)~−e​(t)<x0,∀t∈[0,t0].\displaystyle I^{\prime\prime}(t)=\widetilde{I^{\prime\prime}(t)}-e(t)<x_{0},\quad\forall~{}t\in[0,t_{0}].

Therefore,

z′′​(t0)=\displaystyle z^{\prime\prime}(t_{0})= 1z​(t0)​(I′′​(t0)2−(z′​(t0))2)\displaystyle~{}\frac{1}{z(t_{0})}\bigl{(}\frac{I^{\prime\prime}(t_{0})}{2}-(z^{\prime}(t_{0}))^{2}\bigr{)}
<\displaystyle< 1z​(t0)​(x02−(z′​(t0))2)\displaystyle~{}\frac{1}{z(t_{0})}\bigl{(}\frac{x_{0}}{2}-(z^{\prime}(t_{0}))^{2}\bigr{)}
⩽\displaystyle\leqslant 1z​(t0)​(x02−x02)=0,\displaystyle~{}\frac{1}{z(t_{0})}\bigl{(}\frac{x_{0}}{2}-\frac{x_{0}}{2}\bigr{)}=0,

which is contradicted to z′′​(t0)=0z^{\prime\prime}(t_{0})=0. Thus the claim holds.

Consequently, the proof of Part (i)(i) in Theorem 1.7 is completed.

3.2. Global existence.

We first convert the conditions related to the global existence equivalently. The assumption I′​(0)⩾0I^{\prime}(0)\geqslant 0 implies z′​(0)⩾0z^{\prime}(0)\geqslant 0. The assumption (1.12) implies

I′′​(0)~>x0.\widetilde{I^{\prime\prime}(0)}>x_{0}.

Furthermore, 2​V+x⋅∇V⩽02V+x\cdot\nabla V\leqslant 0 ensures that e​(t)⩽0e(t)\leqslant 0. Then

I′′​(x0)=I′′​(0)~−e​(0)>x0.I^{\prime\prime}(x_{0})=\widetilde{I^{\prime\prime}(0)}-e(0)>x_{0}.

Using (3.4) and z′​(0)⩾0z^{\prime}(0)\geqslant 0, we derive

z′​(0)⩾x02.\displaystyle z^{\prime}(0)\geqslant\sqrt{\frac{x_{0}}{2}}. (3.8)

This leads to the existence of t1⩾0t_{1}\geqslant 0 such that

z′​(t1)>x02=2​f​(x0).\displaystyle z^{\prime}(t_{1})>\sqrt{\frac{x_{0}}{2}}=2\sqrt{f(x_{0})}. (3.9)

If z′​(0)z^{\prime}(0) strictly exceeds x02\sqrt{\frac{x_{0}}{2}}, we can choose t1=0t_{1}=0. If z′​(0)=x02z^{\prime}(0)=\sqrt{\frac{x_{0}}{2}}, we find

z′′​(0)=1z​(0)​(I′′​(0)2−(z′​(0))2)>1z​(0)​(x02−(z′​(0))2)=0.\displaystyle z^{\prime\prime}(0)=~{}\frac{1}{z(0)}\bigl{(}\frac{I^{\prime\prime}(0)}{2}-(z^{\prime}(0))^{2}\bigr{)}>~{}\frac{1}{z(0)}\bigl{(}\frac{x_{0}}{2}-(z^{\prime}(0))^{2}\bigr{)}=0.

Then for small t1>0t_{1}>0, we can select a small parameter ε1>0{\varepsilon}_{1}>0 such that

z′​(t1)=2​f​(x0)+2​ε1.\displaystyle z^{\prime}(t_{1})=2\sqrt{f(x_{0})}+2{\varepsilon}_{1}.

We claim that

𝒛′​(𝒕)>𝟐​𝒇​(𝒙𝟎)+𝟐​𝜺𝟏,∀𝒕⩾𝒕𝟏.\displaystyle\bm{z^{\prime}(t)>2\sqrt{f(x_{0})}+2{\varepsilon}_{1},\quad\forall~{}t\geqslant t_{1}}. (3.10)

Indeed, if the claim does not hold, there exists t2=inf{t>t1:z′​(t)⩽2​f​(x0)+ε1}t_{2}=\inf\{t>t_{1}:z^{\prime}(t)\leqslant 2\sqrt{f(x_{0})}+{\varepsilon}_{1}\}. By the continuity of z′​(t)z^{\prime}(t), we have

z′​(t2)=2​f​(x0)+ε1\displaystyle z^{\prime}(t_{2})=2\sqrt{f(x_{0})}+{\varepsilon}_{1} (3.11)

and

z′​(t)⩾2​f​(x0)+ε1,∀t∈[t1,t2].\displaystyle z^{\prime}(t)\geqslant 2\sqrt{f(x_{0})}+{\varepsilon}_{1},\quad\forall~{}t\in[t_{1},t_{2}]. (3.12)

According to (3.1), we rewrite (3.12) as

(2​f​(x0)+ε1)2⩽(z′​(t))2⩽4​f​(I′′​(t)~),∀t∈[t1,t2],\displaystyle(2\sqrt{f(x_{0})}+{\varepsilon}_{1})^{2}\leqslant(z^{\prime}(t))^{2}\leqslant 4f(\widetilde{I^{\prime\prime}(t)}),\quad\forall~{}t\in[t_{1},t_{2}], (3.13)

we find that f​(I′′​(t)~)>f​(x0)f(\widetilde{I^{\prime\prime}(t)})>f(x_{0}) for all t∈[t1,t2]t\in[t_{1},t_{2}]. Hence

I′′​(t)~≠x0,∀t∈[t1,t2].\displaystyle\widetilde{I^{\prime\prime}(t)}\neq x_{0},\quad\forall~{}t\in[t_{1},t_{2}].

Combining with I′′​(0)~>x0\widetilde{I^{\prime\prime}(0)}>x_{0}, we get

I′′​(t)~>x0,∀t∈[t1,t2].\displaystyle\widetilde{I^{\prime\prime}(t)}>x_{0},\quad\forall~{}t\in[t_{1},t_{2}].

Then again, there could exist a constant CC such that

I′′​(t)~⩾x0+ε1C,∀t∈[t1,t2].\displaystyle\widetilde{I^{\prime\prime}(t)}\geqslant x_{0}+\frac{\sqrt{{\varepsilon}_{1}}}{C},\quad\forall~{}t\in[t_{1},t_{2}]. (3.14)

If I′′​(t)~⩾x0+1\widetilde{I^{\prime\prime}(t)}\geqslant x_{0}+1, then (3.14) holds for CC large enough. If x0<I′′​(t)~⩽x0+1x_{0}<\widetilde{I^{\prime\prime}(t)}\leqslant x_{0}+1, by the Taylor equation of ff around x=x0x=x_{0}, there exists a>0a>0 such that

f​(x)=f​(x0)+a​(x−x0)2​when​|x−x0|⩽1.\displaystyle f(x)=f(x_{0})+a(x-x_{0})^{2}~{}\rm{when}~{}|\mathit{x-x}_{0}|\leqslant 1.

Substituting this equality for (3.13) with x=I′′​(t)~x=\widetilde{I^{\prime\prime}(t)}, we have

(2​f​(x0)+ε1)2⩽(z′​(t))2⩽4​f​(x0)+4​a​(I′′​(t)~−x0)2.\displaystyle(2\sqrt{f(x_{0})}+{\varepsilon}_{1})^{2}\leqslant(z^{\prime}(t))^{2}\leqslant 4f(x_{0})+4a(\widetilde{I^{\prime\prime}(t)}-x_{0})^{2}. (3.15)

Combining (3.14) and (3.15), we obtain C=2​a​(4​(f​(x0))12+ε1)−12C=2\sqrt{a}(4(f(x_{0}))^{\frac{1}{2}}+{\varepsilon}_{1})^{-\frac{1}{2}}. Thus (3.14) holds.

By (3.14) and e​(t)⩽0e(t)\leqslant 0, we find

x0+ε1C⩽I′′​(t)~=I′′​(t)+e​(t)⩽I′′​(t),∀t∈[t1,t2].\displaystyle x_{0}+\frac{\sqrt{{\varepsilon}_{1}}}{C}\leqslant\widetilde{I^{\prime\prime}(t)}=I^{\prime\prime}(t)+e(t)\leqslant I^{\prime\prime}(t),\quad\forall~{}t\in[t_{1},t_{2}]. (3.16)

However, by (3.11) and (3.16) we have

z′′​(t2)=\displaystyle z^{\prime\prime}(t_{2})= 1z​(t2)​(I′′​(t2)2−(z′​(t2))2)\displaystyle~{}\frac{1}{z(t_{2})}\bigl{(}\frac{I^{\prime\prime}(t_{2})}{2}-(z^{\prime}(t_{2}))^{2}\bigr{)}
⩾\displaystyle\geqslant 1z​(t2)​(ε12​C−4​ε1​f​(x0)−ε12)\displaystyle~{}\frac{1}{z(t_{2})}\bigl{(}\frac{\sqrt{{\varepsilon}_{1}}}{2C}-4{\varepsilon}_{1}\sqrt{f(x_{0})}-{\varepsilon}_{1}^{2}\bigr{)}
>\displaystyle> 1z​(t2)​ε4​C,\displaystyle~{}\frac{1}{z(t_{2})}\frac{\sqrt{{\varepsilon}}}{4C},

where ε<ε1{\varepsilon}<{\varepsilon}_{1} is small enough. Then we get z′′​(t2)>0z^{\prime\prime}(t_{2})>0, which contradicts with (3.11) and (3.12). So we obtain the claim.

Next we use the claim (3.10) to prove (1.13) in Theorem 1.7. We note that (3.14) holds for all t∈[t1,T+​(u))t\in\left[t_{1},T_{+}(u)\right). Hence, we obtain

[M​(u)]1−sc​[P​(u)]sc=\displaystyle[M(u)]^{1-s_{c}}[P(u)]^{s_{c}}= [M​(u)]1−sc​[12​(γ−2)​(16​EV​(u))−I′′​(t)~]sc\displaystyle~{}[M(u)]^{1-s_{c}}[\frac{1}{2(\gamma-2)}(16E_{V}(u))-\widetilde{I^{\prime\prime}(t)}]^{s_{c}}
⩽\displaystyle\leqslant [M​(u)]1−sc​[12​(γ−2)​(16​EV​(u))−x0−ε1C]sc\displaystyle~{}[M(u)]^{1-s_{c}}[\frac{1}{2(\gamma-2)}(16E_{V}(u))-x_{0}-\frac{\sqrt{{\varepsilon}_{1}}}{C}]^{s_{c}}
<\displaystyle< [M​(u)]1−sc​[12​(γ−2)​(16​EV​(u)−x0)]sc\displaystyle~{}[M(u)]^{1-s_{c}}[\frac{1}{2(\gamma-2)}(16E_{V}(u)-x_{0})]^{s_{c}}
=\displaystyle= [M​(𝒬)]1−sc​[P​(𝒬)]sc.\displaystyle~{}[M(\mathcal{Q})]^{1-s_{c}}[P(\mathcal{Q})]^{s_{c}}.

Then by mass and energy conservation, we have

‖u‖H˙V1​(ℝd)=2​EV​(u)+12​P​(u)⩽2​EV​(u)+M​(𝒬)1−scsc​P​(𝒬)M​(u0)1−scsc<A\displaystyle\|u\|_{\dot{H}_{V}^{1}({\mathbb{R}}^{d})}=2E_{V}(u)+\frac{1}{2}P(u)\leqslant 2E_{V}(u)+\frac{M(\mathcal{Q})^{\frac{1-s_{c}}{s_{c}}}P(\mathcal{Q})}{M(u_{0})^{\frac{1-s_{c}}{s_{c}}}}<A

for all t∈[t1,T+​(u))t\in\left[t_{1},T_{+}(u)\right), where constant AA depending on M​(u0)M(u_{0}), EV​(u0)E_{V}(u_{0}), M​(𝒬)M(\mathcal{Q}) and EV​(𝒬)E_{V}(\mathcal{Q}). So u​(t,x)u(t,x) exists globally.

Acknowledgement The author Jing Lu was supported by the National Natural Science Foundation of China (No. 12101604).

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