Processing math: 40%
Review

Internet of Things for smart energy systems: A review on its applications, challenges and future trends

  • Internet of Things (IoT) is a terminology used for a mixed connection of heterogeneous objects to the internet and to each other with the employment of recent technological and communication infrastructures. Its incorporation into engineering systems have gradually become very popular in recent times as it promises to transform and ease the life of end users. The use of IoT in smart energy systems (SES) facilitates an ample offer of variety of applications that transverses through a wide range of areas in energy systems. With the numerous benefits that includes unmatched fast communication between subsystems, the maximization of energy use, the decrease in environmental impacts and a boost in the dividends of renewable energies, IoT has grown into an emerging innovative technology to be integrated into smart energy systems. In this work, we have provided an overview of the link between SES, IoT and Internet of Energy (IoE). The main applications of IoT in smart energy systems consisting of smart industries, smart homes and buildings, and smart cities are explored and analyzed. The paper also explores the challenges limiting the employment of IoT technologies in SES and the possible remedies to these challenges. In addition, the future trends of this technology, its research direction and reasons why industry should adopt it are also addressed. The aim of this work is to furnish researchers in this field, decision and energy policy makers, energy economist and energy administrators with a possible literature outline on the roles and impacts of IoT technology in smart energy systems.

    Citation: Efe Francis Orumwense, Khaled Abo-Al-Ez. Internet of Things for smart energy systems: A review on its applications, challenges and future trends[J]. AIMS Electronics and Electrical Engineering, 2023, 7(1): 50-74. doi: 10.3934/electreng.2023004

    Related Papers:

    [1] Jun Zhou . Initial boundary value problem for a inhomogeneous pseudo-parabolic equation. Electronic Research Archive, 2020, 28(1): 67-90. doi: 10.3934/era.2020005
    [2] Yang Cao, Qiuting Zhao . Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations. Electronic Research Archive, 2021, 29(6): 3833-3851. doi: 10.3934/era.2021064
    [3] Qianqian Zhu, Yaojun Ye, Shuting Chang . Blow-up upper and lower bounds for solutions of a class of higher order nonlinear pseudo-parabolic equations. Electronic Research Archive, 2024, 32(2): 945-961. doi: 10.3934/era.2024046
    [4] Xu Liu, Jun Zhou . Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity. Electronic Research Archive, 2020, 28(2): 599-625. doi: 10.3934/era.2020032
    [5] Hui Jian, Min Gong, Meixia Cai . Global existence, blow-up and mass concentration for the inhomogeneous nonlinear Schrödinger equation with inverse-square potential. Electronic Research Archive, 2023, 31(12): 7427-7451. doi: 10.3934/era.2023375
    [6] Shuting Chang, Yaojun Ye . Upper and lower bounds for the blow-up time of a fourth-order parabolic equation with exponential nonlinearity. Electronic Research Archive, 2024, 32(11): 6225-6234. doi: 10.3934/era.2024289
    [7] Yitian Wang, Xiaoping Liu, Yuxuan Chen . Semilinear pseudo-parabolic equations on manifolds with conical singularities. Electronic Research Archive, 2021, 29(6): 3687-3720. doi: 10.3934/era.2021057
    [8] Yaning Li, Yuting Yang . The critical exponents for a semilinear fractional pseudo-parabolic equation with nonlinear memory in a bounded domain. Electronic Research Archive, 2023, 31(5): 2555-2567. doi: 10.3934/era.2023129
    [9] Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li . Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28(1): 369-381. doi: 10.3934/era.2020021
    [10] Lianbing She, Nan Liu, Xin Li, Renhai Wang . Three types of weak pullback attractors for lattice pseudo-parabolic equations driven by locally Lipschitz noise. Electronic Research Archive, 2021, 29(5): 3097-3119. doi: 10.3934/era.2021028
  • Internet of Things (IoT) is a terminology used for a mixed connection of heterogeneous objects to the internet and to each other with the employment of recent technological and communication infrastructures. Its incorporation into engineering systems have gradually become very popular in recent times as it promises to transform and ease the life of end users. The use of IoT in smart energy systems (SES) facilitates an ample offer of variety of applications that transverses through a wide range of areas in energy systems. With the numerous benefits that includes unmatched fast communication between subsystems, the maximization of energy use, the decrease in environmental impacts and a boost in the dividends of renewable energies, IoT has grown into an emerging innovative technology to be integrated into smart energy systems. In this work, we have provided an overview of the link between SES, IoT and Internet of Energy (IoE). The main applications of IoT in smart energy systems consisting of smart industries, smart homes and buildings, and smart cities are explored and analyzed. The paper also explores the challenges limiting the employment of IoT technologies in SES and the possible remedies to these challenges. In addition, the future trends of this technology, its research direction and reasons why industry should adopt it are also addressed. The aim of this work is to furnish researchers in this field, decision and energy policy makers, energy economist and energy administrators with a possible literature outline on the roles and impacts of IoT technology in smart energy systems.



    In this paper, we consider the following initial-boundary value problem

    {utΔutΔu=|x|σ|u|p1u,xΩ,t>0,u(x,t)=0,xΩ,t>0,u(x,0)=u0(x),xΩ (1)

    and its corresponding steady-state problem

    {Δu=|x|σ|u|p1u,xΩ,u=0,xΩ, (2)

    where ΩRn (n1 is an integer) is a bounded domain with boundary Ω and u0H10(Ω); the parameters p and σ satisfy

    1<p<{,n=1,2;n+2n2,n3,σ>{n,n=1,2;(p+1)(n2)2n,n3. (3)

    (1) was called homogeneous (inhomogeneous) pseudo-parabolic equation when σ=0 (σ0). The concept "pseudo-parabolic" was proposed by Showalter and Ting in 1970 in the paper [20], where the linear case was considered. Pseudo-parabolic equations describe a variety of important physical processes, such as the seepage of homogeneous fluids through a fissured rock [1], the unidirectional propagation of nonlinear, dispersive, long waves [2,23], and the aggregation of populations [17].

    The homogeneous problem, i.e. σ=0, was studied in [3,4,5,7,9,10,13,15,16,21,24,25,26,27,28,29]. Especially, for the Cauchy problem (i.e. Ω=Rn and there is no boundary condition), Cao et al. [4] showed the critical Fujita exponent pc (which was firstly introduced by Fujita in [8]) is 1+2/n, i.e. if 1<ppc, then any nontrivial solution blows up in finite time, while global solutions exist if p>pc. In [28], Yang et al. proved that for p>pc, there is a secondary critical exponent αc=2/(p1) such that the solution blows up in finite time for u0 behaving like |x|α at {|x|} if α=(0,αc); and there are global solutions for for u0 behaving like |x|α at {|x|} if α=(αc,n). For the zero Dirichlet boundary problem in a bounded domain Ω, in [13,25,26], the authors studied the properties of global existence and blow-up by potential well method (which was firstly introduced by Sattinger [19] and Payne and Sattinger [18], then developed by Liu and Zhao in [14]), and they showed the global existence, blow-up and asymptotic behavior of solutions with initial energy at subcritical, critical and supercritical energy level. The results of [13,25,26] were extended by Luo [15] and Xu and Zhou [24] by studying the lifespan (i.e. the upper bound of the blow-up time) of the blowing-up solutions. Recently, Xu et al. [27] and Han [9] extended the previous studies by considering the problem with general nonlinearity.

    Li and Du [12] studied the Cauchy problem of equation in (1) with σ>0. They got the critical Fujita exponent (pc) and second critical exponent (αc) by the integral representation and comparison principle. The main results obtained in [12] are as follows:

    (1) If 1<ppc:=1+(2+σ)/n, then every nontrivial solution blows up in finite time.

    (2) If p>pc, the distribution of the initial data has effect on the blow-up phenomena. More precisely, if u0Φα and 0<α<αc:=(2+σ)/(p1) or u0 is large enough, then the solution blows up in finite time; if u0=μϕ(x), ϕΦα with αc<α<n, 0<μ<μ1, then the solution exists globally, where μ1 is some positive constant,

    Φα:={ξ(x)BC(Rn):ξ(x)0,lim inf|x||x|αξ(x)>0},

    and

    Φα:={ξ(x)BC(Rn):ξ(x)0,lim sup|x||x|αξ(x)<}.

    Here BC(Rn) is the set of bounded continuous functions in Rn.

    In view of the above introductions, we find that

    (1) for Cauchy problem in Rn, only the case σ0 was studied;

    (2) for zero Dirichlet problem in a bounded domain Ω, only the case σ=0 was studied.

    The difficulty of allowing σ to be less than 0 is the term |x|σ become infinity at x=0. In this paper, we consider the problem in a bounded domain Ω with zero Dirichlet boundary condition, i.e. problem (1), and the parameters satisfies (3), which allows σ to be less than 0. To overcome the singularity of |x|σ at x=0, we use potential well method by introducing the |x|σ weighed-Lp+1(Ω) space and assume there is a lower bound of σ, i.e,

    σ>(p+1)(n2)2n<0 if n3

    for n3.

    The main results of this paper can be summarized as follows: Let J and I be the functionals given in (12) and (13), respectively; d be the mountain-pass level given in (14); Sρ and Sρ be the sets defined in (20).

    (1) (the case J(u0)d, see Fig. 1) If u0H10(Ω) such that (I(u0),J(u0)) is in the dark gray region (BR), then the solution blows up in finite time; if u0H10(Ω) such that (I(u0),J(u0)) is in the light gray region (GR), then the solution exists globally; if u0H10(Ω) such that (I(u0),J(u0))=(0,d), then u0 is a ground-state solution and (1) admits a global solution uu0; there is no u0H10(Ω) such that (I(u0),J(u0)) is in the dotted part (ER).

    Figure 1.  The results for J(u0)d.

    (2) (the case J(u0)>d) If u0Sρ for some ρJ(u0)>d, then the solution exists globally and goes to 0 in H10(Ω) as times goes to infinity; if u0Sρ for some ρJ(u0)>d, then the solution blows up in finite time.

    (3) (arbitrary initial energy level) For any MR, there exits a u0H10(Ω)) satisfying J(u0)=M such that the corresponding solution blows up in finite time.

    (4) Moreover, under suitable assumptions, we show the exponential decay of global solutions and lifespan (i.e. the upper bound of blow-up time) of the blowing-up solutions.

    The organizations of the remain part of this paper are as follows. In Section 2, we introduce the notations used in this paper and the main results of this paper; in Section 3, we give some preliminaries which will be used in the proofs; in Section 4, we give the proofs of the main results.

    Throughout this paper we denote the norm of Lγ(Ω) for 1γ by Lγ. That is, for any ϕLγ(Ω),

    ϕLγ={(Ω|ϕ(x)|γdx)1γ, if 1γ<;esssupxΩ|ϕ(x)|, if γ=.

    We denote the |x|σ-weighted Lp+1(Ω) space by Lp+1σ(Ω), which is defined as

    Lp+1σ(Ω):={ϕ:ϕ is measurable on Ω and uLp+1σ<}, (4)

    where

    ϕLp+1σ:=(Ω|x|σ|ϕ(x)|p+1dx)1p+1,ϕLp+1σ(Ω). (5)

    By standard arguments as the space Lp+1(Ω), one can see Lp+1σ(Ω) is a Banach space with the norm Lp+1σ.

    We denote the inner product of H10(Ω) by (,)H10, i.e.,

    (ϕ,φ)H10:=Ω(ϕ(x)φ(x)+ϕ(x)φ(x))dx,ϕ,φH10(Ω). (6)

    The norm of H10(Ω) is denoted by H10, i.e.,

    ϕH10:=(ϕ,ϕ)H10=ϕ2L2+ϕ2L2,ϕH10(Ω). (7)

    An equivalent norm of H10(Ω) is ()L2, and by Poincaré's inequality, we have

    ϕL2ϕH10λ1+1λ1ϕL2,ϕH10(Ω), (8)

    where λ1 is the first eigenvalue of Δ with zero Dirichlet boundary condition, i.e,

    λ1=infϕH10(Ω)ϕ2L2ϕ2L2. (9)

    Moreover, by Theorem 3.2, we have

    for p and σ satisfying (4), H10(Ω)Lp+1σ(Ω) continuously and compactly. (10)

    Then we let Cpσ as the optimal constant of the embedding H10(Ω)Lp+1σ(Ω), i.e.,

    Cpσ=supuH10(Ω){0}ϕLp+1σϕL2. (11)

    We define two functionals J and I on H10(Ω) by

    J(ϕ):=12ϕ2L21p+1ϕp+1Lp+1σ (12)

    and

    I(ϕ):=ϕ2L2ϕp+1Lp+1σ. (13)

    By (3) and (10), we know that J and I are well-defined on H10(Ω).

    We denote the mountain-pass level d by

    d:=infϕNJ(ϕ), (14)

    where N is the Nehari manifold, which is defined as

    N:={ϕH10(Ω){0}:I(ϕ)=0}. (15)

    By Theorem 3.3, we have

    d=p12(p+1)C2(p+1)p1pσ, (16)

    where Cpσ is the positive constant given in (11).

    For ρR, we define the sub-level set Jρ of J as

    Jρ={ϕH10(Ω):J(ϕ)<ρ}. (17)

    Then, we define the set Nρ:=NJρ. In view of (15), (12), (17), we get

    Nρ={ϕN:ϕ2L2<2(p+1)ρp1},ρ>d. (18)

    For ρ>d, we define two constants

    λρ:=infϕNρϕH10,Λρ:=supϕNρϕH10 (19)

    and two sets

    Sρ:={ϕH10(Ω):ϕH10λρ,I(ϕ)>0},Sρ:={ϕH10(Ω):ϕH10Λρ,I(ϕ)<0}. (20)

    Remark 1. There are two remarks on the above definitions.

    (1) By the definitions of Nρ, λρ and Λρ, it is easy to see λρ is non-increasing with respect to ρ and Λρ is non-decreasing with respect to ρ.

    (2) By Theorem 3.4, we have

    2(p+1)dp1λρΛρ2(p+1)(λ1+1)ρλ1(p1). (21)

    Then the sets Sρ and Sρ are both nonempty. In fact, for any ϕH10(Ω){0} and s>0,

    sϕH102(p+1)dp1sδ1:=2(p+1)dp1ϕ1H10,I(sϕ)=s2ϕ2L2sp+1ϕp1Lp+1σ>0s<δ2:=(ϕ2L2ϕp+1Lp+1σ)1p1,sϕH102(p+1)(λ1+1)ρλ1(p1)sδ3:=2(p+1)(λ1+1)ρλ1(p1)ϕ1H10,I(sϕ)=s2ϕ2L2sp+1ϕp1Lp+1σ<0s>δ2.

    So,

    {sϕ:0<s<min{δ1,δ2}}Sρ,{sϕ:s>max{δ2,δ3}}Sρ.

    In this paper we consider weak solutions to problem (1), local existence of which can be obtained by Galerkin's method (see for example [22,Chapter II,Sections 3 and 4]) and a standard limit process and the details are omitted.

    Definition 2.1. Assume u0H10(Ω) and (3) holds. Let T>0 be a constant. A function u=u(x,t) is called a weak solution of problem (1) on Ω×[0,T] if u(,t)L(0,T;H10(Ω)), ut(,t)L2(0,T;H10(Ω)) and the following equality

    Ω(utv+utv+uv|x|σ|u|p1uv)dx=0 (22)

    holds for any vH10(Ω) and a.e. t[0,T]. Moreover,

    u(,0)=u0() in H10(Ω). (23)

    Remark 2. There are some remarks on the above definition.

    (1) Since u(,t)L(0,T;H10(Ω))L2(0,T;H10(Ω)), ut(,t)L2(0,T;H10(Ω)), we have uH1(0,T;H10(Ω)). According to [6], uC([0,T];H10(Ω)), then (23) makes sense. Moreover, by (10), all terms in (22) make sense for uC([0,T];H10(Ω)) and utL2(0,T;H10(Ω)).

    (2) Denote by Tmax the maximal existence of u, then u(,t)L(0,T;H10(Ω))C([0,T];H10(Ω)), ut(,t)L2(0,T;H10(Ω)) for any T<Tmax.

    (3) Taking v=u in (22), we get

    u(,t)2H10=u02H102t0I(u(,s))ds,0tT, (24)

    where H10 is defined in (7) and I is defined in (13).

    (4) Taking v=ut in (22), we get

    J(u(,t))=J(u0)t0us(,s)2H10ds,0tT, (25)

    where J is defined in (12).

    Definition 2.2. Assume (3) holds. A function uH10(Ω) is called a weak solution of (2) if

    Ω(uv|x|σ|u|p1uv)dx=0 (26)

    holds for any vH10(Ω).

    Remark 3. There are some remarks to the above definition.

    (1) By (10), we know all the terms in (26) are well-defined.

    (2) If we denote by Φ the set of weak solutions to (2), then by the definitions of J in (12) and N in (15), we have

    Φ={ϕH10(Ω):J(ϕ)=0 in H1(Ω)}(N{0}), (27)

    where J(ϕ)=0 in H1(Ω) means J(ϕ),ψ=0 for all ψH10 and , means the dual product between H1(Ω) and H10(Ω).

    With the set Φ defined above, we can defined the ground-state solution to (2).

    Definition 2.3. Assume (3) holds. A function uH10(Ω) is called a ground-state solution of (2) if uΦ{0} and

    J(u)=infϕΦ{0}J(ϕ).

    With the above preparations, now we can state the main results of this paper. Firstly, we consider the case J(u0)d. By the sign of I(u0), we can classify the discussions into three cases:

    (1) J(u0)d, I(u0)>0 (see Theorem 2.4);

    (2) J(u0)d, I(u0)<0 (see Theorem 2.5);

    (3) J(u0)d, I(u0)=0. In this case, by the definition of d in (14), we have u0=0 or J(u0)=d and I(u0)=0. In Theorem 2.6, we will show problem (1) admits a global solution u(,t)u0.

    Theorem 2.4. Assume (3) holds and u=u(x,t) is a weak solution to (1) with u0V, then u exists globally and

    u(,t)L22(p+1)J(u0)p1,0t<, (28)

    where

    V:={ϕH10(Ω):J(ϕ)d,I(ϕ)>0}. (29)

    In, in addition, J(u0)<d, we have the following decay estimate:

    u(,t)H10u0H10exp[λ1λ1+1(1(J(u0)d)p12)t]. (30)

    Remark 4. Since u0V, we have I(u0)>0. Then it follows from the definitions of J in (12) and I in (13) that

    J(u0)>p12(p+1)u02L2>0.

    So the equality (28) makes sense.

    Theorem 2.5. Assume (3) holds and u=u(x,t) is a weak solution to (1) with u0W. Then Tmax< and u blows up in finite time in the sense of

    limtTmaxt0u(,s)2H10ds=,

    where

    W:={ϕH10(Ω):J(ϕ)d,I(ϕ)<0} (31)

    and Tmax is the maximal existence time of u. If, in addition, J(u0)<d, then

    Tmax4pu02H10(p1)2(p+1)(dJ(u0)). (32)

    Remark 5. There are two remarks.

    (1) If J(ϕ)<0, then we can easily get from the definitions of J and I in (12) and (13) respectively that I(ϕ)<0. So we have ϕW if J(ϕ)<0.

    (2) The sets V and W defined in (29) and (31) respectively are both nonempty. In fact for any ϕH10(Ω){0}, we let

    f(s)=J(sϕ)=s22ϕ2L2sp+1p+1ϕp+1Lp+1σ,g(s)=I(sϕ)=s2ϕ2L2sp+1ϕp+1Lp+1σ.

    Then (see Fig. 2)

    Figure 2.  The graphs of f and g.

    (a) f(0)=f(s4)=0, f(s) is strictly increasing for s(0,s3), strictly decreasing for s(s3,), limsf(s)=, and

    maxs[0,)f(s)=f(s3)=p12(p+1)(ϕL2ϕLp+1σ)2(p+1)p1dBy (14) since s3ϕN, (33)

    (b) g(0)=g(s3)=0, g(s) is strictly increasing for s(0,s1), strictly decreasing for s(s1,), limsg(s)=, and

    maxs[0,)g(s)=g(s1)=p1p+1(2p+1)2p1(ϕL2ϕLp+1σ)2(p+1)p1,

    (c) f(s)<g(s) for 0<s<s2, f(s)>g(s) for s>s2, and

    f(s2)=g(s2)=p12p(p+12p)2p1(ϕL2ϕLp+1σ)2(p+1)p1,

    where

    s1:=(2ϕ2L2(p+1)ϕp+1Lp+1σ)1p1<s2:=((p+1)ϕ2L22pϕp+1Lp+1σ)1p1<s3:=(ϕ2L2ϕp+1Lp+1σ)1p1<s4:=((p+1)ϕ2L22ϕp+1Lp+1σ)1p1.

    So, {sϕ:s1<s<s3}V, {sϕ:s3<s<}W.

    Theorem 2.6. Assume (3) holds and u=u(x,t) is a weak solution to (1) with u0G. Then problem (1) admits a global solution u(,t)u0(), where

    G:={ϕH10(Ω):J(ϕ)=d,I(ϕ)=0}. (34)

    Remark 6. There are two remarks on the above theorem.

    (1) Unlike Remark 5, it is not easy to show G. In fact, if we use the arguments as in Remark 5, we only have J(s3ϕ)d and I(s3ϕ)=0 (see Fig. 2 and (33)). In Theorem 2.7, we will use minimizing sequence argument to show G.

    (2) To prove the above Theorem, we only need to show G is the set of the ground-state solution of (2), which is done in Theorem 2.7.

    Theorem 2.7. Assume (3) holds and let G be the set defined in (34), then G and G is the set of the ground-state solution of (2).

    Secondly, we consider the case J(u0)>d, and we have the following theorem.

    Theorem 2.8. Assume (3) holds and the initial value u0H10(Ω) satisfying J(u0)>d.

    (i): If u0Sρ with ρJ(u0), then problem (1) admits a global weak solution u=u(x,t) and u(,t)H100 as t.

    (ii): If u0Sρ with ρJ(u0), then the weak solution u=u(x,t) of problem (1) blows up in finite time.

    Here Sρ and Sρ are the two sets defined in (20).

    Next, we show the solution of the problem (1) can blow up at arbitrary initial energy level (Theorem 2.10). To this end, we firstly introduce the following theorem.

    Theorem 2.9. Assume (3) holds and u=u(x,t) is a weak solution to (1) with u0ˆW. Then

    Tmax8pu02H10(p1)2(λ1(p1)λ1+1u02H102(p+1)J(u0)) (35)

    and u blows up in finite time in the sense of

    limtTmaxt0u(,s)2H10ds=,

    where

    ˆW:={ϕH10(Ω):J(ϕ)<λ1(p1)2(λ1+1)(p+1)ϕ2H10}. (36)

    and Tmax is the maximal existence time of u.

    By using the above theorem, we get the following theorem.

    Theorem 2.10. For any MR, there exists u0H10(Ω) satisfying J(u0)=M such that the corresponding weak solution u=u(x,t) of problem (1) blows up in finite time.

    The following lemma can be found in [11].

    Lemma 3.1. Suppose that 0<T and suppose a nonnegative function F(t)C2[0,T) satisfies

    F(t)F(t)(1+γ)(F(t))20

    for some constant γ>0. If F(0)>0, F(0)>0, then

    TF(0)γF(0)<

    and F(t) as tT.

    Theorem 3.2. Assume p and σ satisfy (3). Then H10(Ω)Lp+1σ(Ω) continuously and compactly.

    Proof. Since ΩRn is a bounded domain, there exists a ball B(0,R):={xRn:|x|=x21+x2n<R}Ω.

    We divide the proof into three cases. We will use the notation ab which means there exits a positive constant C such that a\leq Cb .

    Case 1. \sigma\geq0 . By the assumption on p in (3), one can see

    \begin{equation} \hbox{$H_0^1( \Omega)\hookrightarrow L^{p+1}( \Omega)$ continuously and compactly.} \end{equation} (37)

    Then we have, for any u\in H_0^1( \Omega) ,

    \begin{equation*} \|u\|_{L^{p+1}_{\sigma}}^{p+1} = \int_ \Omega|x|^\sigma|u|^{p+1}dx\leq R^\sigma\|u\|_{L^{p+1}}^{p+1}\lesssim\|u\|_{H_0^1}^{p+1}, \end{equation*}

    which, together with (37), implies H_0^1( \Omega)\hookrightarrow L^{p+1}_{\sigma}( \Omega) continuously and compactly.

    Case 2. -n<\sigma<0 and n = 1 or 2 . We can choose r\in\left(1,-\frac{n}{\sigma}\right) . Then by Hölder's inequality and

    \begin{equation} \hbox{$H_0^1( \Omega)\hookrightarrow L^{\frac{(p+1)r}{r-1}}( \Omega)$ continuously and compactly,} \end{equation} (38)

    for any u\in H_0^1( \Omega) , we have

    \begin{align*} \|u\|_{L^{p+1}_{\sigma}}^{p+1}& = \int_ \Omega|x|^\sigma|u|^{p+1}dx\\&\leq \left(\int_{B(0,R)}|x|^{\sigma r}dx\right)^{\frac1r}\left(\int_ \Omega|u|^{\frac{(p+1)r}{r-1}}dx\right)^{\frac{r-1}{r}}\\ &\leq\left\{ \begin{array}{ll} \left(\frac{2}{\sigma r+1}R^{\sigma r+1}\right)^{\frac 1r}\|u\|_{L^{\frac{(p+1)r}{r-1}}}^{p+1}\lesssim\|u\|_{H_0^1}^{p+1}, & n = 1;\\ \left(\frac{2\pi}{\sigma r+2}R^{\sigma r+2}\right)^{\frac 1r}\|u\|_{L^{\frac{(p+1)r}{r-1}}}^{p+1}\lesssim\|u\|_{H_0^1}^{p+1}, & n = 2, \end{array} \right. \end{align*}

    which, together with (38), implies H_0^1( \Omega)\hookrightarrow L^{p+1}_{\sigma}( \Omega) continuously and compactly.

    Case 3. \frac{(p+1)(n-2)}{2}-n<\sigma<0 and n\geq 3 . Then there exists a constant r>1 such that

    \begin{equation*} -\frac{\sigma}{n} < \frac1r < 1-\frac{(p+1)(n-2)}{2n}. \end{equation*}

    By the second inequality of the above inequalities, we have

    \begin{equation*} \frac{(p+1)r}{r-1} = \frac{p+1}{1-\frac1r} < \frac{p+1}{\frac{(p+1)(n-2)}{2n}} = \frac{2n}{n-2}. \end{equation*}

    So,

    \begin{equation} \hbox{$H_0^1( \Omega)\hookrightarrow L^{\frac{(p+1)r}{r-1}}( \Omega)$ continuously and compactly.} \end{equation} (39)

    Then by Hölder's inequality, for any u\in H_0^1( \Omega) , we have

    \begin{align*} \|u\|_{L^{p+1}_{\sigma}}^{p+1}& = \int_ \Omega|x|^\sigma|u|^{p+1}dx\\&\leq \left(\int_{B(0,R)}|x|^{\sigma r}dx\right)^{\frac1r}\left(\int_ \Omega|u|^{\frac{(p+1)r}{r-1}}dx\right)^{\frac{r-1}{r}}\\ &\leq\left(\frac{\omega_{n-1}}{\sigma r+n}R^{\sigma r+n}\right)^{\frac1r}\|u\|_{L^{\frac{(p+1)r}{r-1}}}^{p+1}\lesssim\|u\|_{H_0^1}^{p+1}, \end{align*}

    which, together with (39), implies H_0^1( \Omega)\hookrightarrow L^{p+1}_{\sigma}( \Omega) continuously and compactly. Here \omega_{n-1} denotes the surface area of the unit ball in \mathbb{R}^n .

    Theorem 3.3. Assume p and \sigma satisfy (3). Let d be the constant defined in (14), then

    \begin{equation*} d = \frac{p-1}{2(p+1)}C_{p\sigma}^{\frac{2(p+1)}{p-1}}, \end{equation*}

    where C_{p\sigma} is the positive constant defined in (11).

    Proof. Firstly, we show

    \begin{equation} \inf\limits_{\phi\in N}J(\phi) = \min\limits_{\phi\in H_0^1( \Omega)\setminus\{0\}}J(s_\phi^*\phi), \end{equation} (40)

    where N is the set defined in (15) and

    \begin{equation} s_\phi^*: = \left(\frac{\|\nabla \phi\|_{L^2}^2}{\|\phi\|_{L_\sigma^{p+1}}^{p+1}}\right)^{\frac{1}{p-1}}. \end{equation} (41)

    By the definition of N in (15) and s_\phi^* in (41), one can easily see that s_\phi^* = 1 if \phi\in N and s_\phi^*\phi\in N for any \phi\in H_0^1( \Omega)\setminus\{0\} .

    On one hand, since N\subset H_0^1( \Omega) and s_\phi^* = 1 for \phi\in N , we have

    \begin{equation*} \min\limits_{\phi\in H_0^1( \Omega)\setminus\{0\}}J(s_\phi^*\phi)\leq\min\limits_{\phi\in N}J(s_\phi^*\phi) = \min\limits_{\phi\in N}J(\phi). \end{equation*}

    On the other hand, since \{s_\phi^*\phi: \phi\in H_0^1( \Omega)\setminus\{0\}\}\subset N , we have

    \begin{equation*} \inf\limits_{\phi\in N}J(\phi)\leq \inf\limits_{\phi\in H_0^1( \Omega)\setminus\{0\}}J(s_\phi^*\phi). \end{equation*}

    Then (40) follows from the above two inequalities.

    By (40), the definition of d in (14), the definition of J in (12), and the definition of C_{p\sigma} in (11), we have

    \begin{align*} d& = \min\limits_{\phi\in H_0^1( \Omega)\setminus\{0\}}J(s_\phi^*\phi)\\ = &\frac{p-1}{2(p+1)}\min\limits_{\phi\in H_0^1( \Omega)\setminus\{0\}}\left(\frac{\|\nabla \phi\|_{L^2}}{\|\phi\|_{L_\sigma^{p+1}}}\right)^{\frac{2(p+1)}{p-1}}\\ = &\frac{p-1}{2(p+1)}C_{p\sigma}^{-\frac{2(p+1)}{p-1}}. \end{align*}

    Theorem 3.4. Assume (3) holds. Let \lambda_\rho and \Lambda_\rho be the two constants defined in (19). Here \rho>d is a constant. Then

    \begin{equation} \sqrt{\frac{2(p+1)d}{p-1}}\leq \lambda_\rho\leq \Lambda_\rho\leq \sqrt{\frac{2(p+1)( \lambda_1+1)\rho}{ \lambda_1(p-1)}}. \end{equation} (42)

    Proof. Let \rho>d and N^\rho be the set defined in (18). By the definitions of \lambda_\rho and \Lambda_\rho in (19), it is obvious that

    \begin{equation} \lambda_\rho\leq\Lambda_\rho. \end{equation} (43)

    Since N^\rho\subset N , it follows from the definitions of d , J , I in (14), (12), (13), respectively, that

    \begin{align*} d& = \inf\limits_{\phi\in N}J(\phi)\\ & = \frac{p-1}{2(p+1)}\inf\limits_{\phi\in N}\|\nabla\phi\|_{L^2}^2\\ &\leq\frac{p-1}{2(p+1)}\inf\limits_{\phi\in N^\rho}\|\phi\|_{H_0^1}^2\\ & = \frac{p-1}{2(p+1)} \lambda_\rho^2, \end{align*}

    which implies

    \begin{equation*} \lambda_\rho\geq\sqrt{\frac{2(p+1)d}{p-1}} \end{equation*}

    On the other hand, by (8) and (18), we have

    \begin{equation*} \begin{split} \Lambda_\rho& = \sup\limits_{\phi\in N^\rho}\|\phi\|_{H_0^1}\\&\leq\sqrt{\frac{ \lambda_1+1}{ \lambda_1}}\sup\limits_{\phi\in N^\rho}\|\nabla\phi\|_{L^2}\\&\leq\sqrt{\frac{ \lambda_1+1}{ \lambda_1}}\sqrt{\frac{2(p+1)\rho}{p-1}}. \end{split} \end{equation*}

    Combining the above two inequalities with (43), we get (42), the proof is complete.

    Theorem 3.5. Assume (3) holds and u = u(x,t) is a weak solution to (1). Then the sets W and V , defined in (31) and (29) respectively, are both variant for u , i.e., u(\cdot,t)\in W ( u(\cdot,t)\in V ) for 0\leq t< {T_{\max}} when u_0\in W ( u_0\in V ), where {T_{\max}} is the maximal existence time of u .

    Proof. We only prove the invariance of W since the proof of the invariance of V is similar.

    For any \phi\in W , since I(\phi)<0 , it follows from the definition of I (see (13)) and (11) that

    \begin{equation*} \|\nabla\phi\|_{L^2}^2 < \|\phi\|_{L_\sigma^{p+1}}^{p+1}\leq C_{p\sigma}^{p+1}\|\nabla \phi\|_{L^2}^{p+1}, \end{equation*}

    which implies

    \begin{equation} \|\nabla\phi\|_{L^2} > C_{p\sigma}^{-\frac{p+1}{p-1}}. \end{equation} (44)

    Let u(x,t) be the weak solution of problem (1) with u_0\in W . Since I(u_0)<0 and u\in C([0, {T_{\max}}), H_0^1( \Omega)) , there exists a constant \varepsilon>0 small enough such that

    \begin{equation} I(u(\cdot,t)) < 0,\; \; \; t\in[0, \varepsilon]. \end{equation} (45)

    Then by (24), \frac{d}{dt}\|u(\cdot,t)\|_{H_0^1}^2>0 for t\in[0, \varepsilon] , and then by (25) and J(u_0)\leq d , we get

    \begin{equation} J(u(\cdot,t)) < d\hbox{ for }t\in(0, \varepsilon]. \end{equation} (46)

    We argument by contradiction. Since u(\cdot,t)\in C([0, {T_{\max}}),H_0^1( \Omega)) , if the conclusion is not true, then there exists a t_0\in(0, {T_{\max}}) such that u(\cdot,t)\in W for 0\leq t<t_0 , but I(u(\cdot,t_0)) = 0 and

    \begin{equation} J(u(\cdot,t_0)) < d \end{equation} (47)

    (note (25) and (46), J(u(\cdot,t_0)) = d cannot happen). By (44), we have u(\cdot,t)\in C([0, {T_{\max}}),H_0^1( \Omega)) and u(\cdot,t_0)\in \overline{W} , then

    \begin{equation*} \|\nabla u(\cdot,t_0)\|_{L^2}\geq C_{p\sigma}^{-\frac{p+1}{p-1}} > 0, \end{equation*}

    which, together with I(u(\cdot,t_0)) = 0 , implies u(\cdot,t_0)\in N . Then it follows from the definition of d in (14) that

    \begin{equation*} J(u(\cdot,t_0))\geq d, \end{equation*}

    which contradicts (47). So the conclusion holds.

    Theorem 3.6. Assume (3) holds and u = u(x,t) is a weak solution to (1) with u_0\in W . Then

    \begin{equation} \|\nabla u(\cdot,t)\|_{L^2}^2\geq\frac{2(p+1)}{p-1}d,\; \; \; 0\leq t < {T_{\max}}, \end{equation} (48)

    where W is defined in (31) and {T_{\max}} is the maximal existence time of u .

    Proof. Let N^-: = \{\phi\in H_0^1( \Omega):I(\phi)<0\} . Then by Theorem 3.5, u(\cdot,t)\in N^- for 0\leq t< {T_{\max}} .

    By the proof in Theorem 3.3,

    \begin{align*} d& = \min\limits_{\phi \in H_0^1( \Omega)\setminus\{0\}}J(s^*_\phi\phi)\\ &\leq\min\limits_{\phi\in N^-}J(s_\phi^*\phi)\\ &\leq J(s_u^*u(\cdot,t))\\ & = \frac{(s_u^*)^2}{2}\|\nabla u(\cdot,t)\|_{L^2}^2-\frac{(s_u^*)^{p+1}}{p+1}\|u(\cdot,t)\|_{L_\sigma^{p+1}}^{p+1}\\ &\leq\left(\frac{(s_u^*)^2}{2}-\frac{(s_u^*)^{p+1}}{p+1}\right)\|\nabla u(\cdot,t)\|_{L^2}^2, \end{align*}

    where we have used I(u(\cdot,t))<0 in the last inequality. Since I(u(\cdot,t))<0 , we get from (41) that

    \begin{align*} d& = \min\limits_{\phi \in H_0^1( \Omega)\setminus\{0\}}J(s^*_\phi\phi)\\ &\leq\min\limits_{\phi\in N^-}J(s_\phi^*\phi)\\ &\leq J(s_u^*u(\cdot,t))\\ & = \frac{(s_u^*)^2}{2}\|\nabla u(\cdot,t)\|_{L^2}^2-\frac{(s_u^*)^{p+1}}{p+1}\|u(\cdot,t)\|_{L_\sigma^{p+1}}^{p+1}\\ &\leq\left(\frac{(s_u^*)^2}{2}-\frac{(s_u^*)^{p+1}}{p+1}\right)\|\nabla u(\cdot,t)\|_{L^2}^2, \end{align*}

    Then

    \begin{equation*} \begin{split} d&\leq\max\limits_{0\leq s\leq 1}\left(\frac{s^2}{2}-\frac{s^{p+1}}{p+1}\right)\|\nabla u(\cdot,t)\|_{L^2}^2\\ & = \left(\frac{s^2}{2}-\frac{s^{p+1}}{p+1}\right)_{s = 1}\|\nabla u(\cdot,t)\|_{L^2}^2\\ & = \frac{p-1}{2(p+1)}\|\nabla u(\cdot,t)\|_{L^2}^2, \end{split} \end{equation*}

    and (48) follows from the above inequality.

    Theorem 3.7. Assume (3) holds and u = u(x,t) is a weak solution to (1) with u_0\in\hat W . Then I(u(\cdot,t))<0 for 0\leq t< {T_{\max}} , where {T_{\max}} is the maximal existence time of u and \hat W is defined in (36)

    Proof. Firstly, we show I(u_0)<0 . In fact, by the definition of J in (12), u_0\in\hat W , and (8), we get

    \begin{align*} \frac12\|\nabla u_0\|_{L^2}^2-\frac{1}{p+1}\|u_0\|_{L_\sigma^{p+1}}^{p+1}& = J(u_0)\\ & < \frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\|u_0\|_{H_0^1}^2\\ &\leq\frac{p-1}{2(p+1)}\|\nabla u_0\|_{L^2}^2, \end{align*}

    which implies

    \begin{equation*} I(u_0) = \|\nabla u_0\|_{L^2}^2-\|u_0\|_{L_\sigma^{p+1}}^{p+1} < 0. \end{equation*}

    Secondly, we prove I(u(\cdot,t))<0 for 0<t< {T_{\max}} . In fact, if it is not true, in view of u\in C([0, {T_{\max}}),H_0^1( \Omega)) , there must exist a t_0\in(0, {T_{\max}}) such that I(u(\cdot,t))<0 for t\in[0,t_0) but I(u(\cdot,t_0)) = 0 . Then by (24), we get \|u(\cdot,t_0)\|_{H_0^1}^2> \|u_0\|_{H_0^1}^2 , which, together with u_0\in\hat W and (8), implies

    \begin{equation} \begin{split} J(u_0)& < \frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\|u_0\|_{H_0^1}^2\\ & < \frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\|u(\cdot,t_0)\|_{H_0^1}^2\\ &\leq \frac{p-1}{2(p+1)}\|\nabla u(\cdot,t_0)\|_{L^2}^2. \end{split} \end{equation} (49)

    On the other hand, by (24), (12), (13) and I(u(\cdot,t_0)) = 0 , we get

    \begin{equation*} J(u_0)\geq J(u(\cdot,t_0)) = \frac{p-1}{2(p+1)}\|\nabla u(\cdot,t_0)\|_{L^2}^2, \end{equation*}

    which contradicts (49). The proof is complete.

    Proof of Theorem 2.4. Let u = u(x,t) be a weak solution to (1) with u_0\in V and {T_{\max}} be its maximal existence time. By Theorem 3.5, u(\cdot,t)\in V for 0\leq t< {T_{\max}} , which implies I(u(\cdot,t))>0 for 0\leq t< {T_{\max}} . Then it follows from (25), (12) and (13) that

    \begin{align*} J(u_0)&\geq J(u(\cdot,t))\geq\frac{p-1}{2(p+1)}\|\nabla u(\cdot,t)\|_{L^2}^2,\; \; \; 0\leq t < {T_{\max}}, \end{align*}

    which implies u exists globally (i.e. {T_{\max}} = \infty ) and

    \begin{equation} \|\nabla u(\cdot,t)\|_{L^2}\leq \sqrt{\frac{2(p+1)J(u_0)}{p-1}},\; \; \; 0\leq t < \infty. \end{equation} (50)

    Next, we prove \|u(\cdot,t)\|_{H_0^1} decays exponentially, if in addition, J(u_0)<d . By (24), (13), (11), (50), (16) we have

    \begin{align*} \frac{d}{dt}\left(\|u(\cdot,t)\|_{H_0^1}^2\right) = &-2I(u(\cdot,t)) = -2\left(\|\nabla u(\cdot,t)\|_{L^2}^2-\|u(\cdot,t)\|_{L_\sigma^{p+1}}^{p+1}\right)\\ \leq &-2\left(1-C_{p\sigma}^{p+1}\|\nabla u(\cdot,t)\|_{L^2}^{p-1}\right)\|\nabla u(\cdot,t)\|_{L^2}^2\\ \leq &-2\left(1-C_{p\sigma}^{p+1}\left(\sqrt{\frac{2(p+1)J(u_0)}{p-1}}\right)^{p-1}\right)\|\nabla u(\cdot,t)\|_{L^2}^2\\ = &-2\left(1-\left(\frac{J(u_0)}{d}\right)^{\frac{p-1}{2}}\right)\|\nabla u(\cdot,t)\|_{L^2}^2\\ \leq&-\frac{2 \lambda_1}{ \lambda_1+1}\left(1-\left(\frac{J(u_0)}{d}\right)^{\frac{p-1}{2}}\right)\|u(\cdot,t)\|_{H_0^1}^2, \end{align*}

    which leads to

    \begin{equation*} \|u(\cdot,t)\|_{H_0^1}^2\leq\|u_0\|_{H_0^1}^2\exp\left[-\frac{2 \lambda_1}{ \lambda_1+1}\left(1-\left(\frac{J(u_0)}{d}\right)^{\frac{p-1}{2}}\right)t\right]. \end{equation*}

    The proof is complete.

    Proof of Theorem 2.5. Let u = u(x,t) be a weak solution to (1) with u_0\in W and {T_{\max}} be its maximal existence time.

    Firstly, we consider the case J(u_0)<d and I(u_0)<0 . By Theorem 3.5, u(\cdot,t)\in W for 0\leq t< {T_{\max}} . Let

    \begin{equation} \xi(t): = \left(\int_0^t\|u(\cdot,s)\|_{H_0^1}^2ds\right)^{\frac 12},\; \; \; \eta(t): = \left(\int_0^t\|u_s(\cdot,s)\|_{H_0^1}^2ds\right)^{\frac 12},\; \; \; \; 0\leq t < {T_{\max}}. \end{equation} (51)

    For any T^*\in(0, {T_{\max}}) , \beta>0 and \alpha>0 , we let

    \begin{equation} F(t): = \xi^2(t)+(T^*-t)\|u_0\|_{H_0^1}^2+ \beta(t+ \alpha)^2,\; \; \; 0\leq t\leq T^*. \end{equation} (52)

    Then

    \begin{equation} F(0) = T^*\|u_0\|_{H_0^1}^2+ \beta \alpha^2 > 0, \end{equation} (53)
    \begin{equation} \begin{split} F'(t)& = \|u(\cdot,t)\|_{H_0^1}^2-\|u_0\|_{H_0^1}^2+2 \beta(t+ \alpha)\\ & = 2\left(\frac12\int_0^t\frac{d}{ds}\|u(\cdot,s)\|_{H_0^1}^2ds+ \beta(t+ \alpha)\right),\; \; \; 0\leq t\leq T^*, \end{split} \end{equation} (54)

    and (by (24), (12), (13), (48), (25))

    \begin{equation} \begin{split} F''(t) = &-2I(u(\cdot,t))+2 \beta\\ = &(p-1)\|\nabla u(\cdot,t)\|_{L^2}^2-2(p+1)J(u(\cdot,t))+2 \beta\\ \geq&2(p+1)(d-J(u_0))+2(p+1)\eta^2(t)+2 \beta,\; \; \; 0\leq t\leq T^*. \end{split} \end{equation} (55)

    Since I(u(\cdot,t))<0 , it follow from (24) and the first equality of (54) that

    \begin{equation*} F'(t)\geq2 \beta(t+ \alpha). \end{equation*}

    Then

    \begin{equation} F(t) = F(0)+\int_0^tF'(s)ds\geq T^*\|u_0\|_{H_0^1}^2+ \beta \alpha^2+2 \alpha \beta t+ \beta t^2,\; \; \; 0\leq t\leq T^*. \end{equation} (56)

    By (6), Schwartz's inequality and Hölder's inequality, we have

    \begin{align*} \frac12\int_0^t\frac{d}{ds}\|u(\cdot,s)\|_{H_0^1}^2ds& = \int_0^t(u(\cdot,s),u_s(\cdot,s))_{H_0^1}ds\\ &\leq\int_0^t\|u(\cdot,s)\|_{H_0^1}\|u_s(\cdot,s)\|_{H_0^1}ds\leq \xi(t)\eta(t),\; \; \; 0\leq t\leq T^*, \end{align*}

    which, together with the definition of F(t) , implies

    \begin{align*} &\ \ \ \left(F(t)-(T^*-t)\|u_0\|_{H_0^1}^2\right)\left(\eta^2(t)+ \beta\right)\\ & = \left(\xi^2(t)+ \beta(t+ \alpha)^2\right)\left(\eta^2(t)+ \beta\right)\\ & = \xi^2(t)\eta^2(t)+ \beta\xi^2(t)+ \beta(t+ \alpha)^2\eta^2(t)+ \beta^2(t+ \alpha)^2\\ &\geq\xi^2(t)\eta^2(t)+2\xi(t)\eta(t) \beta(t+ \alpha)+ \beta^2(t+ \alpha)^2\\ &\geq\left(\xi(t)\eta(t)+ \beta(t+ \alpha)\right)^2\\ &\geq\left(\frac12\int_0^t\frac{d}{ds}\|u(\cdot,s)\|_{H_0^1}^2ds+ \beta(t+ \alpha)\right)^2,\; \; \; 0\leq t\leq T^*. \end{align*}

    Then it follows from (54) and the above inequality that

    \begin{equation} \begin{split} \left(F'(t)\right)^2& = 4\left(\frac12 \int\limits_0^t\frac{d}{ds}\|u(s)\|_{H_0^1}^2ds+ \beta(t+ \alpha)\right)^2\\ &\leq 4F(t)\left(\eta^2(t)+ \beta\right),\; \; \; \; \; \; 0\leq t\leq T^*. \end{split} \end{equation} (57)

    In view of (55), (56), and (57), we have

    \begin{align*} F(t)F''(t)-\frac{p+1}{2}(F'(t))^2\geq F(t)\left(2(p+1)(d-J(u_0))-2p \beta\right),\; \; \; 0\leq t\leq T^*. \end{align*}

    If we take \beta small enough such that

    \begin{equation} 0 < \beta\leq\frac{p+1}{p}(d-J(u_0)), \end{equation} (58)

    then F(t)F''(t)-\frac{p+1}{2}(F'(t))^2\geq0 . Then, it follows from Lemma 3.1 that

    \begin{equation*} T^*\leq\frac{F(0)}{\left(\frac{p+1}{2}-1\right) F'(0)} = \frac{T^*\|u_0\|_{H_0^1}^2+ \beta \alpha^2}{(p-1) \alpha \beta}. \end{equation*}

    Then for

    \begin{equation} \alpha\in\left(\frac{\|u_0\|_{H_0^1}^2}{(p-1) \beta},\infty\right), \end{equation} (59)

    we get

    \begin{equation*} T^*\leq \frac{ \beta \alpha^2}{(p-1) \alpha \beta-\|u_0\|_{H_0^1}^2}. \end{equation*}

    Minimizing the above inequality for \alpha satisfying (59), we get

    \begin{equation*} T^*\leq\left.\frac{ \beta \alpha^2}{(p-1) \alpha \beta-\|u_0\|_{H_0^1}^2}\right|_{ \alpha = \frac{2\|u_0\|_{H_0^1}^2}{(p-1) \beta}} = \frac{4\|u_0\|_{H_0^1}^2}{(p-1)^2 \beta}. \end{equation*}

    Minimizing the above inequality for \beta satisfying (58), we get

    \begin{equation*} T^*\leq\frac{4p\|u_0\|_{H_0^1}^2}{(p-1)^2(p+1)(d-J(u_0))}. \end{equation*}

    By the arbitrariness of T^*< {T_{\max}} it follows that

    \begin{equation*} {T_{\max}}\leq\frac{4p\|u_0\|_{H_0^1}^2}{(p-1)^2(p+1)(d-J(u_0))}. \end{equation*}

    Secondly, we consider the case J(u_0) = d and I(u_0)<0 . By the proof of Theorem 3.5, there exists a t_0>0 small enough such that J(u(\cdot,t_0))<d and I(u(\cdot,t_0))<0 . Then it follows from the above proof that u will blow up in finite time. The proof is complete.

    Proof of Theorems 2.6 and 2.7. Since Theorem 2.6 follows from Theorem 2.7 directly, we only need to prove Theorem 2.7.

    Firstly, we show G\neq\emptyset . By the definition of d in (14), we get

    \begin{equation*} d = \inf\limits_{\phi\in N}J(\phi) = \frac{p-1}{2(p+1)}\inf\limits_{\phi\in N}\|\nabla\phi\|_{L^2}^2. \end{equation*}

    Then a minimizing sequence \{\phi_k\}_{k = 1}^\infty\subset N exists such that

    \begin{equation} \lim\limits_{k\uparrow\infty}J(\phi_k) = \frac{p-1}{2(p+1)}\lim\limits_{k\uparrow\infty}\|\nabla\phi_k\|_{L^2}^2 = d, \end{equation} (60)

    which implies \{\phi_k\}_{k = 1}^\infty is bounded in H_0^1( \Omega) . Since H_0^1( \Omega) is reflexive and H_0^1( \Omega)\hookrightarrow L_\sigma^{p+1} continuously and compactly (see (10)), there exists \varphi\in H_0^1( \Omega) such that

    (1) \phi_k\rightharpoonup \varphi in H_0^1( \Omega) weakly;

    (2) \phi_k\rightarrow \varphi in L_\sigma^{p+1}( \Omega) strongly.

    Now, in view of \|\nabla(\cdot)\|_{L^2} is weakly lower continuous in H_0^1( \Omega) , taking \liminf_{k\uparrow\infty} in the equality \|\nabla\phi_k\|_{L^2}^2 = \|\phi_k\|_{L_\sigma^{p+1}}^{p+1} (sine \phi_k\in N ), we get

    \begin{equation} \lim\limits_{k\uparrow\infty}J(\phi_k) = \frac{p-1}{2(p+1)}\lim\limits_{k\uparrow\infty}\|\nabla\phi_k\|_{L^2}^2 = d, \end{equation} (60)

    We claim

    \begin{equation} \|\nabla\varphi\|_{L^2}^2 = \|\varphi\|_{L_\sigma^{p+1}}^{p+1}\hbox{ i.e. }I(\varphi) = 0. \end{equation} (62)

    In fact, if the claim is not true, then by (61),

    \begin{equation*} \|\nabla\varphi\|_{L^2}^2 < \|\varphi\|_{L_\sigma^{p+1}}^{p+1}. \end{equation*}

    By the proof of Theorem 3.3, we know that s^*_\varphi\varphi\in N , which, together with the definition of d in (14), implies

    \begin{equation} J(s^*_\varphi\varphi)\geq d, \end{equation} (63)

    where

    \begin{equation} J(s^*_\varphi\varphi)\geq d, \end{equation} (63)

    On the other hand, since s^*_\varphi\varphi\in N , we get from the definitions of J in (12) and I in (13), s^*_\varphi\in(0,1) , \|\nabla(\cdot)\|_{L^2} is weakly lower continuous in H_0^1( \Omega) , (60) that

    \begin{equation*} \begin{split} J(s^*_\varphi\varphi)& = \frac{p-1}{2(p+1)}(s^*_\varphi)^2\|\nabla \varphi\|_{L^2}^2\\ & < \frac{p-1}{2(p+1)}\|\nabla \varphi\|_{L^2}^2\\ &\leq\frac{p-1}{2(p+1)}\liminf\limits_{k\uparrow\infty}\|\nabla \phi_k\|_{L^2}^2\\& = d, \end{split} \end{equation*}

    which contradicts to (63). So the claim is true, i.e.

    \begin{equation*} \lim\limits_{k\uparrow\infty}\|\nabla\phi_k\|_{L^2}^2 = \|\varphi\|_{L_\sigma^{p+1}}^{p+1}, \end{equation*}

    which, together with H_0^1( \Omega) is uniformly convex and \phi_k\rightharpoonup \varphi in H_0^1( \Omega) weakly, implies \phi_k\rightarrow \varphi strongly in H_0^1( \Omega) . Then by (60), J(\varphi) = d , which, together with (62) and the definition of G in (34), implies \varphi\in G , i.e., G\neq\emptyset .

    Second, we prove G\subset\Phi , where \Phi is the set defined in (27). For any \varphi\in G , we need to show \varphi\in\Phi , i.e. \varphi satisfies (26). Fix any v\in H_0^1( \Omega) and s\in(- \varepsilon, \varepsilon) , where \varepsilon>0 is a small constant such that \|\varphi+sv\|_{L_\sigma^{p+1}}^{p+1}>0 for s\in(- \varepsilon, \varepsilon) . Let

    \begin{equation*} \lim\limits_{k\uparrow\infty}\|\nabla\phi_k\|_{L^2}^2 = \|\varphi\|_{L_\sigma^{p+1}}^{p+1}, \end{equation*}

    Then I(\tau(s)(\varphi+sv)) = 0 . So by the definition of N in (15), the set

    \begin{equation*} A: = \{\tau(s)(\varphi+sv):s\in(- \varepsilon, \varepsilon)\} \end{equation*}

    is a curve on N , which passes \varphi when s = 0 . The function \tau(s) is differentiable and

    \begin{equation*} A: = \{\tau(s)(\varphi+sv):s\in(- \varepsilon, \varepsilon)\} \end{equation*}

    where

    \begin{align*} \xi&: = 2\int_ \Omega\nabla(\varphi+sv)\cdot\nabla vdx\|\varphi+sv\|_{L_\sigma^{p+1}}^{p+1},\\ \eta&: = (p+1)\int_ \Omega|x|^\sigma|\varphi+sv|^{p-1}(\varphi+sv)vdx\|\nabla(\varphi+sv)\|_{L^2}^2. \end{align*}

    Since (62), we get \tau(0) = 1 and

    \begin{equation} \tau'(0) = \frac{1}{(p-1)\|\varphi\|_{L_\sigma^{p+1}}^{p+1}}\left(2\int_ \Omega\nabla\varphi\nabla vdx-(p+1)\int_ \Omega|x|^\sigma|\varphi|^{p-1}\varphi vdx\right). \end{equation} (65)

    Let

    \begin{equation*} \varrho(s): = J(\tau(s)(\varphi+sv)) = \frac{\tau^2(s)}{2}\|\nabla(\varphi+sv)\|_{L^2}^2-\frac{\tau^{p+1}(s)}{p+1}\|\varphi+sv\|_{L^{p+1}_{\sigma}}^{p+1},\; \; \; s\in(- \varepsilon, \varepsilon). \end{equation*}

    Since \tau(s)(\varphi+sv)\in N for s\in(- \varepsilon, \varepsilon) , \tau(s)(\varphi+sv)|_{s = 0} = \varphi , \varrho(0) = J(\varphi) = d , it follows from the definition of d that \varrho(s) (s\in(- \varepsilon, \varepsilon)) achieves its minimum at s = 0 , then \varrho'(0) = 0 . So,

    \begin{align*} 0 = &\varrho'(0) = \left.\tau(s)\tau'(s)\|\nabla(\varphi+sv)\|_{L^2}^2+\tau^2(s)\int_ \Omega\nabla(\varphi+sv)\cdot\nabla vdx\right|_{s = 0}\\ &\left.-\tau^p(s)\tau'(s)\|\varphi+sv\|_{L_\sigma^{p+1}}^{p+1}-\tau^{p+1}(s)\int_ \Omega|x|^\sigma|\varphi+sv|^{p-1}(\varphi+sv)vdx\right|_{s = 0}\\ = &\int_ \Omega\nabla\varphi\cdot\nabla vdx-\int_ \Omega|x|^\sigma|\varphi|^{p-1}\varphi vdx. \end{align*}

    So, \varphi\in\Phi , i.e. G\subset \Phi . Moreover, we have G\subset (\Phi\setminus\{0\}) since \varphi\neq0 for any \varphi\in G .

    Finally, in view of Definition 2.3 and J(\varphi) = d (\forall \varphi\in G) , to complete the proof, we only need to show

    \begin{equation} d = \inf\limits_{\phi\in\Phi\setminus\{0\}}J(\phi). \end{equation} (66)

    In fact, by the above proof and (27), we have G\subset\Phi\setminus\{0\}\subset N . Then, in view of the definition of d in (14), i.e.,

    \begin{equation*} d = \inf\limits_{\phi\in N}J(\phi) \end{equation*}

    and J(\varphi) = d for any \varphi\in G , we get (66). The proof is complete.

    Proof of Theorem 2.8. Let u = u(x,t) be the solution of problem (1) with initial value u_0 satisfying J(u_0)>d . We denote by {T_{\max}} the maximal existence of u . If u is global, i.e. {T_{\max}} = \infty , we denote by

    \begin{equation*} \omega(u_0) = \cap_{t\geq0} \overline{\{u(\cdot,s):s\geq t\}}^{H_0^1( \Omega)} \end{equation*}

    the \omega -limit set of u_0 .

    (i) Assume u_0\in S_\rho = \{\phi\in H_0^1( \Omega):\|\phi\|_{H_0^1}\leq \lambda_\rho, I(\phi)>0\} (see (20)) with \rho\geq J(u_0) . Without loss of generality, we assume u(\cdot,t)\not = 0 for 0\leq t< {T_{\max}} . In fact it there exists a t_0 such that u(\cdot,t_0) = 0 , then it is easy to see the function v defined as

    \begin{equation*} v(x,t) = \left\{ \begin{array}{ll} u(x,t), & \hbox{ if }0\leq t\leq t_0; \\ 0, & \hbox{ if }t > t_0 \end{array} \right. \end{equation*}

    is a global weak solution of problem (1), and the proof is complete.

    We claim that

    \begin{equation} I(u(\cdot,t)) > 0,\; \; \; 0\leq t < {T_{\max}}. \end{equation} (67)

    Since I(u_0)>0 , if the claim is not true, there exists a t_0\in(0, {T_{\max}}) such that

    \begin{equation} I(u(\cdot,t)) > 0,\; \; \; 0\leq t < t_0 \end{equation} (68)

    and

    \begin{equation} I(u(\cdot,t_0)) = 0, \end{equation} (69)

    which together with the definition of N in (15) and the assumption that u(\cdot,t)\neq0 for 0\leq t< {T_{\max}} , implies u(\cdot,t_0)\in N . Moreover, by using (68), similar to the proof of (46), we have J(u(\cdot,t_0))<J(u_0) , i.e. u(\cdot,t_0)\in J^{J(u_0)} (see (17)). Then u(\cdot,t_0)\in N^{J(u_0)} (since N^{J(u_0)} = N\cap J^{J(u_0)} ) and then \|u(\cdot,t_0)\|_{H_0^1}\geq \lambda_{J(u_0)} (see (19)). By monotonicity (see Remark 1) and \rho\geq J(u_0) , we get

    \begin{equation} \|u(\cdot,t_0)\|_{H_0^1}\geq \lambda_{\rho}. \end{equation} (70)

    On the other hand, it follows from (24), (68) and u_0\in S_\rho that

    \begin{equation*} \|u(\cdot,t)\|_{H_0^1} < \|u_0\|_{H_0^1}\leq \lambda_\rho, \end{equation*}

    which contradicts (70). So (67) is true. Then by (24) again, we get

    \begin{equation*} \|u(\cdot,t)\|_{H_0^1}\leq\|u_0\|_{H_0^1},\; \; \; 0\leq t < {T_{\max}}, \end{equation*}

    which implies u exists globally, i.e. {T_{\max}} = \infty .

    By (24) and (67), \|u(\cdot,t)\|_{H_0^1} is strictly decreasing for 0\leq t<\infty , so a constant c\in[0,\|u_0\|_{H_0^1}) exists such that

    \begin{equation*} \lim\limits_{t\uparrow\infty}\|u(\cdot,t)\|_{H_0^1} = c. \end{equation*}

    Taking t\uparrow\infty in (24), we get

    \begin{equation*} \int_0^\infty I(u(\cdot,s))ds\leq\frac12\left(\|u_0\|_{H_0^1}^2-c\right) < \infty. \end{equation*}

    Note that I(u(\cdot,s))>0 for 0\leq s<\infty , so, for any sequence \{t_n\} satisfying t_n\uparrow\infty as n\uparrow\infty , if the limit \lim_{n\uparrow\infty}I(u(\cdot,t_n)) exists, it must hold

    \begin{equation} \lim\limits_{n\uparrow\infty}I(u(\cdot,t_n)) = 0. \end{equation} (71)

    Let \omega be an arbitrary element in \omega(u_0) . Then there exists a sequence \{t_n\} satisfying t_n\uparrow\infty as n\uparrow\infty such that

    \begin{equation} u(\cdot,t_n)\rightarrow\omega \hbox{ in }H_0^1( \Omega)\hbox{ as }n\uparrow\infty. \end{equation} (72)

    Then by (71), we get

    \begin{equation} I(\omega) = \lim\limits_{n\uparrow\infty}I(u(\cdot,t_n)) = 0. \end{equation} (73)

    As the above, one can easily see

    \begin{equation*} \|\omega\|_{H_0^1} < \lambda_\rho\leq \lambda_{J(u_0)}, \; \; \; \underbrace{J(\omega) < J(u_0)}_{\Rightarrow\omega\in J^{J(u_0)}}, \end{equation*}

    which implies \omega\notin N^{J(u_0)} . In fact, if \omega \in N^{J(u_0)} , by (19), \lambda_{J(u_0)}\leq \|\omega\|_{H_0^1} , a contradiction. Since N^{J(u_0)} = N\cap J^{u_0} and \omega\in J^{J(u_0)} , we get \omega\notin N . Therefore, by the definition of N in (15) and (73), \omega = 0 , then it follows from \|u(\cdot,t)\|_{H_0^1} is strictly decreasing and (72) that

    \begin{equation*} \lim\limits_{t\uparrow\infty}\|u(\cdot,t)\|_{H_0^1} = \lim\limits_{n\uparrow\infty}\|u(\cdot,t_n)\|_{H_0^1} = \|\omega\|_{H_0^1} = 0. \end{equation*}

    (ⅱ) Assume u_0\in S^\rho = \{\phi\in H_0^1( \Omega):\|\phi\|_{H_0^1}\geq\Lambda_\rho, I(\phi)<0\} (see (20)) with \rho\geq J(u_0) . We claim that

    \begin{equation} I(u(\cdot,t)) < 0,\; \; \; 0\leq t < {T_{\max}}. \end{equation} (74)

    Since I(u_0)<0 , if the claim is not true, there exists a t_0\in(0, {T_{\max}}) such that

    \begin{equation} I(u(\cdot,t)) < 0,\; \; \; 0\leq t < t_0 \end{equation} (75)

    and

    \begin{equation} I(u(\cdot,t_0)) = 0. \end{equation} (76)

    Since (75), by (44) and u\in C([0, {T_{\max}}),H_0^1( \Omega)) , we get

    \begin{equation*} \|\nabla u(\cdot,t_0)\|_{L^2}\geq C_{p\sigma}^{-\frac{p+1}{p-1}}, \end{equation*}

    which, together with the definition of N in (15), implies u(\cdot,t_0)\in N . Moreover, by using (75), similar to the proof of (46), we have J(u(\cdot,t_0))<J(u_0) , i.e. u(\cdot,t_0)\in J^{J(u_0)} (see (17)). Then u(\cdot,t_0)\in N^{J(u_0)} (since N^{J(u_0)} = N\cap J^{J(u_0)} ) and then \|u(\cdot,t_0)\|_{H_0^1}\leq \Lambda_{J(u_0)} (see (19)). By monotonicity (see Remark 1) and \rho\geq J(u_0) , we get

    \begin{equation} \|u(\cdot,t_0)\|_{H_0^1}\leq \Lambda_{\rho}. \end{equation} (77)

    On the other hand, it follows from (24), (75) and u_0\in S^\rho that

    \begin{equation*} \|u(\cdot,t)\|_{H_0^1} > \|u_0\|_{H_0^1}\geq\Lambda_\rho, \end{equation*}

    which contradicts (77). So (74) is true.

    Suppose by contradiction that u does not blow up in finite time, i.e. {T_{\max}} = \infty . By (24) and (74), \|u(\cdot,t)\|_{H_0^1} is strictly increasing for 0\leq t<\infty . If the limit \lim_{t\uparrow\infty}\|u(t)\|_{H_0^1} exists, i.e. there exists a constant \tilde c\in[\|u_0\|_{H_0^1},\infty) such that

    \begin{equation*} \lim\limits_{t\uparrow\infty}\|u(\cdot,t)\|_{H_0^1} = \tilde c, \end{equation*}

    Taking t\uparrow\infty in (24), we get

    \begin{equation*} -\int_0^\infty I(u(\cdot,s))ds\leq\frac12\left( \tilde c-\|u_0\|_{H_0^1}^2\right) < \infty. \end{equation*}

    Note -I(u(\cdot,s))>0 for 0\leq s<\infty , so, for any sequence \{t_n\} satisfying t_n\uparrow\infty as n\uparrow\infty , if the limit \lim_{n\uparrow\infty}I(u(\cdot,t_n)) exists, it must hold

    \begin{equation} \lim\limits_{n\uparrow\infty}I(u(\cdot,t_n)) = 0. \end{equation} (78)

    Let \omega be an arbitrary element in \omega(u_0) . Then there exists a sequence \{t_n\} satisfying t_n\uparrow\infty as n\uparrow\infty such that

    \begin{equation} u(\cdot,t_n)\rightarrow\omega \hbox{ in }H_0^1( \Omega)\hbox{ as }n\uparrow\infty. \end{equation} (79)

    Since \|u(\cdot,t)\|_{H_0^1} is strictly increasing, \lim_{t\uparrow\infty}\|u(\cdot,t)\|_{H_0^1} exists and

    \begin{equation*} \lim\limits_{t\uparrow\infty}\|u(\cdot,t)\|_{H_0^1} = \lim\limits_{n\uparrow\infty}\|u(\cdot,t_n)\|_{H_0^1} = \|\omega\|_{H_0^1}. \end{equation*}

    Then by (78), we get

    \begin{equation} I(\omega) = \lim\limits_{n\uparrow\infty}I(u(\cdot,t_n)) = 0. \end{equation} (80)

    By (24), (25) and (74), one can easily see

    \begin{equation*} \|\omega\|_{H_0^1} > \|u_0\|_{H_0^1}\geq\Lambda_\rho\geq\Lambda_{J(u_0)},\; \; \; \underbrace{J(\omega) < J(u_0)}_{\Rightarrow\omega\in J^{J(u_0)}}, \end{equation*}

    which implies \omega\notin N^{J(u_0)} . In fact, if \omega \in N^{J(u_0)} , by (19), \Lambda_{J(u_0)}\geq \|\omega\|_{H_0^1} , a contradiction. Since N^{J(u_0)} = N\cap J^{u_0} and \omega\in J^{J(u_0)} , we get \omega\notin N . Therefore, by the definition of N in (15) and (80), \omega = 0 . However, this contradicts \|\omega\|_{H_0^1}>\Lambda_{J(u_0)}>0 . So u blows up in finite time. The proof is complete.

    Proof of Theorem 2.9. Let u = u(x,t) be a weak solution of (1) with u_0\in\hat W and {T_{\max}} be its maximal existence time, where \hat W is defined in (36). By Theorem 3.7, we know that I(u(\cdot,t))<0 for 0\leq t< {T_{\max}} . Then by (8) and (24), we get

    \begin{equation} \|\nabla u(\cdot,t)\|_{L^2}^2\geq \frac{ \lambda_1}{ \lambda_1+1}\|u(\cdot,t)\|_{H_0^1}^2\geq\frac{ \lambda_1}{ \lambda_1+1}\|u_0\|_{H_0^1}^2,\; \; \; 0\leq t < {T_{\max}}. \end{equation} (81)

    The remain proofs are similar to the proof of Theorem 2.9. For any T^*\in(0, {T_{\max}}) , \beta>0 and \alpha>0 , we consider the functional F(t) again (see (52)). We also have (53), (54), but there are some differences in (55), in fact, by (81) and (25), we have

    \begin{equation} \begin{split} F''(t) = &-2I(u(\cdot,t))+2 \beta\\ = &(p-1)\|\nabla u(\cdot,t)\|_{L^2}^2-2(p+1)J(u(\cdot,t))+2 \beta\\ \geq&\frac{ \lambda_1(p-1)}{ \lambda_1+1}\|u_0\|_{H_0^1}^2-2(p+1)J(u_0)+2(p+1)\eta^2(t)+2 \beta,\; \; \; 0\leq t\leq T^*. \end{split} \end{equation} (82)

    We also have (56) and (57). Then it follows from (56), (57) and (82) that

    \begin{align*} &F(t)F''(t)-\frac{p+1}{2}(F'(t))^2\\ \geq& F(t)\left(\frac{ \lambda_1(p-1)}{ \lambda_1+1}\|u_0\|_{H_0^1}^2-2(p+1)J(u_0)-2p \beta\right),\; \; \; 0\leq t\leq T^*. \end{align*}

    If we take \beta small enough such that

    \begin{equation} 0 < \beta\leq\frac{1}{2p}\left(\frac{ \lambda_1(p-1)}{ \lambda_1+1}\|u_0\|_{H_0^1}^2-2(p+1)J(u_0)\right), \end{equation} (83)

    then F(t)F''(t)-\frac{p+1}{2}(F'(t))^2\geq0 . Then, it follows from Lemma 3.1 that

    \begin{equation*} T^*\leq\frac{F(0)}{\left(\frac{p+1}{2}-1\right) F'(0)} = \frac{T^*\|u_0\|_{H_0^1}^2+ \beta \alpha^2}{(p-1) \alpha \beta}. \end{equation*}

    Then for

    \begin{equation} \alpha\in\left(\frac{\|u_0\|_{H_0^1}^2}{(p-1) \beta},\infty\right), \end{equation} (84)

    we get

    \begin{equation*} T^*\leq \frac{ \beta \alpha^2}{(p-1) \alpha \beta-\|u_0\|_{H_0^1}^2}. \end{equation*}

    Minimizing the above inequality for \alpha satisfying (84), we get

    \begin{equation*} T^*\leq\left.\frac{ \beta \alpha^2}{(p-1) \alpha \beta-\|u_0\|_{H_0^1}^2}\right|_{ \alpha = \frac{2\|u_0\|_{H_0^1}^2}{(p-1) \beta}} = \frac{4\|u_0\|_{H_0^1}^2}{(p-1)^2 \beta}. \end{equation*}

    Minimizing the above inequality for \beta satisfying (58), we get

    \begin{equation*} T^*\leq\frac{8p\|u_0\|_{H_0^1}^2}{(p-1)^2\left(\frac{ \lambda_1(p-1)}{ \lambda_1+1}\|u_0\|_{H_0^1}^2-2(p+1)J(u_0)\right)}. \end{equation*}

    By the arbitrariness of T^*< {T_{\max}} it follows that

    \begin{equation*} {T_{\max}}\leq\frac{8p\|u_0\|_{H_0^1}^2}{(p-1)^2\left(\frac{ \lambda_1(p-1)}{ \lambda_1+1}\|u_0\|_{H_0^1}^2-2(p+1)J(u_0)\right)}. \end{equation*}

    Proof of Theorem 2.10. For any M\in \mathbb{R} , let \Omega_1\subset \Omega and \Omega_2\subset \Omega be two arbitrary disjoint open domains. Let \psi\in H_0^1( \Omega_2)\setminus\{0\} , extending \psi to \Omega by letting \psi = 0 in \Omega\setminus \Omega_2 , then \psi\in H_0^1( \Omega) . We choose \alpha large enough such that

    \begin{equation} \| \alpha\psi\|_{H_0^1}^2 > \frac{2( \lambda_1+1)(p+1)}{ \lambda_1(p-1)}M. \end{equation} (85)

    For such \alpha and \psi , we take a \phi\in H_0^1( \Omega_1)\setminus\{0\} (which is extended to \Omega by letting \phi = 0 in \Omega\setminus \Omega_1 i.e. \phi\in H_0^1( \Omega) ) such that

    \begin{equation} J(s_3^*\phi)\geq M-J( \alpha \psi), \end{equation} (86)

    where (see Remark 5)

    \begin{equation} J(s_3^*\phi)\geq M-J( \alpha \psi), \end{equation} (86)

    which can be done since

    \begin{equation*} J(s_3^*\phi) = \frac{p-1}{2(p+1)}\left(\frac{\|\nabla \phi\|_{L^2}}{\|\phi\|_{L_\sigma^{p+1}}}\right)^{\frac{2(p+1)}{p-1}} \end{equation*}

    and \phi can be chosen such that \|\nabla \phi\|_{L^2}\gg\|\phi\|_{L_\sigma^{p+1}} .

    By Remark 5 again,

    \begin{equation} J(\{s\phi:0\leq s < \infty\}) = (-\infty, J(s_3^*\phi)]. \end{equation} (87)

    By (87) and (86), we can choose s\in[0,\infty) such that v: = s\phi satisfies J(v) = M-J( \alpha \psi) . Letting u_0: = v+ \alpha \psi\in H_0^1( \Omega) , since \Omega_1 and \Omega_2 are disjoint, we get

    \begin{equation*} J(u_0) = J(v)+J( \alpha\psi) = M \end{equation*}

    and (note (85))

    \begin{align*} J(u_0)& = M < \frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\| \alpha\psi\|_{H_0^1}^2\\ &\leq\frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\left(\| \alpha\psi\|_{H_0^1}^2+\|v\|_{H_0^1}^2\right)\\ & = \frac{ \lambda_1(p-1)}{2( \lambda_1+1)(p+1)}\|u_0\|_{H_0^1}^2. \end{align*}

    Let u = u(x,t) be the weak solution of problem (1) with initial value u_0 given above. Then by Theorem 2.9, u blows up in finite time.



    [1] International Energy Agency (IEA), 2022. Global Energy & CO2 Status Report 2022.
    [2] IEA, 2019. Global Energy and CO2 Report 2019. Available from: https://iea.org/reports/global-energy-co2-status-report-2019
    [3] Jensen M (1993) The Modern Industrial Revolution, Exit, and the Failure of Internal Control Systems. J Financ 48: 831–880. https://doi.org/10.1111/j.1540-6261.1993.tb04022.x doi: 10.1111/j.1540-6261.1993.tb04022.x
    [4] Saleem Y, Crespi N, Rehmani MH, et al. (2017) Internet of things-aided Smart Grid: technologies, architectures, applications, prototypes, and future research directions. IEEE Access 7: 62962-63003. https://doi.org/10.1109/ACCESS.2019.2913984. doi: 10.1109/ACCESS.2019.2913984
    [5] Chelloug SA, El-Zawawy MA (2017) Middleware for Internet of Things: Survey and Challenges. Intell Autom Soft Comput 3: 70–95. https://doi.org/10.1080/10798587.2017.1290328 doi: 10.1080/10798587.2017.1290328
    [6] Shrouf F, Ordieres J, Miragliotta G (2014) Smart factories in Industry 4.0: A review of the concept and of energy management approached in production based on the Internet of Things paradigm. Proceedings of the 2014 IEEE International Conference on Industrial Engineering and Engineering Management; 697–701. https://doi.org/10.1109/IEEM.2014.7058728
    [7] Wang Q, Wang YG (2018) Research on Power Internet of Things Architecture for Smart Grid Demand. 2018 2nd IEEE Conference on Energy Internet and Energy System Integration (EI2). https://doi.org/10.1109/EI2.2018.8582132
    [8] Vu TL, Le NT, Jang YM, et al. (2018) An Overview of Internet of Energy (IoE) Based Building Energy Management System. 2018 International Conference on Information and Communication Technology Convergence (ICTC), 852-855. https://doi.org/10.1109/ICTC.2018.8539513
    [9] Bin X, Qing C, Jun M, et al. (2019) Research on a Kind of Ubiquitous Power Internet of Things System for Strong Smart Power Grid. 2019 IEEE Innovative Smart Grid Technologies - Asia (ISGT Asia), 2805-2808. https://doi.org/10.1109/ISGT-Asia.2019.8881652
    [10] Wang W, Zhou Z (2020) Exploring Novel Internet-of-things Based on Free Space Optical Communications for Smart Grids. 2020 IEEE 4th Conference on Energy Internet and Energy System Integration (EI2), 4277-4281. https://doi.org/10.1109/EI250167.2020.9347166
    [11] Xiang W, Wang Y, Gao X, et al. (2021) Design and Implementation of Internet of Things System Based on Customer Electricity Behavior Analysis. 2021 IEEE 5th Conference on Energy Internet and Energy System Integration (EI2), 3411-3415. https://doi.org/10.1109/EI252483.2021.9713593
    [12] Amin SM, Wollenberg BF (2005) Toward a smart grid: power delivery for the 21st century. IEEE power and energy magazine 3: 34–41. https://doi.org/10.1109/MPAE.2005.1507024 doi: 10.1109/MPAE.2005.1507024
    [13] Orumwense EF, Abo-Al-Ez KM (2019) A systematic review to aligning research paths: Energy cyber-physical systems. Cogent Eng 6: 1700738. https://doi.org/10.1080/23311916.2019.1700738 doi: 10.1080/23311916.2019.1700738
    [14] Zeng Z, Ding T, Xu Y, et al. (2020) Reliability Evaluation for Integrated Power-Gas Systems with Power-to-Gas and Gas Storagesin. IEEE T Power Syst 35: 571-583. https://doi.org/10.1109/TPWRS.2019.2935771 doi: 10.1109/TPWRS.2019.2935771
    [15] Gahleitner G (2013) Hydrogen from renewable electricity: An international review of power-to-gas pilot plants for stationary applications. Int J Hydrogen Energy 38: 2039–2061. https://doi.org/10.1016/j.ijhydene.2012.12.010 doi: 10.1016/j.ijhydene.2012.12.010
    [16] Cui S, Wang Y, Xiao J (2019) Peer-to-Peer Energy Sharing Among Smart Energy Buildings by Distributed Transaction. IEEE T Smart Grid 10: 6491-6501. https://doi.org/10.1109/TSG.2019.2906059 doi: 10.1109/TSG.2019.2906059
    [17] Vision of Smart Energy – Research, development and Demonstration, Smart Energy Networks.
    [18] IEEE smart grid domains - IEEE smart grid. (2020) https://smartgrid.ieee.org/domains.
    [19] Mohassel R, Fung A, Mohammadi F, et al. (2014) A survey of Advanced Metering Infrastructure. International Journal of Electrical Power and Energy Systems 63: 473-484. https://doi.org/10.1016/j.ijepes.2014.06.025 doi: 10.1016/j.ijepes.2014.06.025
    [20] Pereira R, Figueiredo J, Melicio R, et al. (2015) Consumer energy management system with integration of smart meters. Energy Rep 1: 22–29. https://doi.org/10.1016/j.egyr.2014.10.001 doi: 10.1016/j.egyr.2014.10.001
    [21] Sayed K, Gabbar HA (2017) SCADA and smart energy grid control automation. Smart Energy Grid Engineering, 481–514. https://doi.org/10.1016/B978-0-12-805343-0.00018-8
    [22] Mohanty SP, Choppali U, Kougianos K (2016) Everything you wanted to know about smart cities. IEEE Consum Electron Mag 5: 60–70. https://doi.org/10.1109/MCE.2016.2556879 doi: 10.1109/MCE.2016.2556879
    [23] Bouzid AM, Guerrero JM, Cheriti A, et al. (2015) A survey on control of electric power distributed generation systems for microgrid applications. Renew Sustain Energy Rev 44: 751–766. https://doi.org/10.1016/j.rser.2015.01.016 doi: 10.1016/j.rser.2015.01.016
    [24] Haseeb K, Almogren A, Islam N, et al. (2019) An Energy-Efficient and Secure Routing Protocol for Intrusion Avoidance in IoT-Based WSN. Energies 12, 4174. https://doi.org/10.3390/en12214174
    [25] Zouinkhi A, Ayadi H, Val T, et al. (2019) Auto-management of energy in IoT networks. Int J Commun Syst 33: e4168. https://doi.org/10.1002/dac.4168.
    [26] Hö ller J, Tsiatsis V, Mulligan C, et al. (2014) From Machine-to-Machine to the Internet of Things: Introduction to a New Age of Intelligence; Elsevier: Amsterdam, The Netherlands.
    [27] Hersent O, Boswarthick D, Elloumi O (2011) The internet of things: Key Applications and Protocols. John Wiley & Sons. https://doi.org/10.1002/9781119958352
    [28] Tulemissova G (2016) The Impact of the IoT and IoE Technologies on Changes of Knowledge Management Strategy. ECIC2016-Proceedings of the 8th European Conference on Intellectual Capital: ECIC2016, 300. Academic Conferences and publishing limited.
    [29] Zhou K, Yang S, Shao Z (2016) Energy Internet: The business perspective. Appl Energy 178: 212–222. https://doi.org/10.1016/j.apenergy.2016.06.052 doi: 10.1016/j.apenergy.2016.06.052
    [30] Tahanan M, Van Ackooij W, Frangioni A, et al. (2015) Large-scale Unit Commitment under uncertainty. 4OR 13: 115–171, https://doi.org/10.1007/s10288-014-0279-y doi: 10.1007/s10288-014-0279-y
    [31] Anvari-Moghaddam A, Monsef H, Rahimi-Kian A (2015) Cost-effective and comfort-aware residential energy management under different pricing schemes and weather conditions. Energy Build 86: 782–793, https://doi.org/10.1016/j.enbuild.2014.10.017 doi: 10.1016/j.enbuild.2014.10.017
    [32] Mahmud K, Town GE, Morsalin S, et al. (2017) Integration of electric vehicles and management in the internet of energy. Renew Sustain Energy Rev 82: 4179–4203, https://doi.org/10.1016/j.rser.2017.11.004 doi: 10.1016/j.rser.2017.11.004
    [33] Orumwense EF, Abo-Al-Ez K (2019) An Energy Efficient Cognitive Radio based Smart Grid Communication Architecture. Proceedings of the 17th IEEE Industrial and Commercial Use of Energy, Cape Town, South Africa. https://doi.org/10.2139/ssrn.3638151
    [34] Patil K, Lahudkar PSL (2015) Survey of MAC Layer Issues and Application layer Protocols for Machine-to-Machine Communications. IEEE Internet Things J 2: 175–186. https://doi.org/10.1109/JIOT.2015.2394438 doi: 10.1109/JIOT.2015.2394438
    [35] Li Z, Shahidehpour M, Aminifar F (2017) Cybersecurity in Distributed Power Systems. Proc IEEE, 105: 1367–1388, https://doi.org/10.1109/JPROC.2017.2687865 doi: 10.1109/JPROC.2017.2687865
    [36] Groopman J, Etlinger S (2015) Consumer Perceptions of Privacy in the Internet of Things: What Brands Can Learn from a Concerned Citizenry. Altimeter Group: San Francisco, CA, USA, 1–25.
    [37] Zafar R, Mahmood A, Razzaq S, et al. (2018) Prosumer based energy management and sharing in smart grid. Renew Sustain Energy Rev 82: 1675–1684, https://doi.org/10.1016/j.rser.2017.07.018 doi: 10.1016/j.rser.2017.07.018
    [38] Luna AC, Diaz NL, Graells M, et al. (2016) Cooperative energy management for a cluster of households prosumers. IEEE T Consum Electron 62: 235–242. https://doi.org/10.1109/TCE.2016.7613189 doi: 10.1109/TCE.2016.7613189
    [39] Iannello F, Simeone O, Spagnolini U (2010) Energy Management Policies for Passive RFID Sensors with RF-Energy Harvesting. Proceedings of the 2010 IEEE International Conference on Communications, 1–6, https://doi.org/10.1109/ICC.2010.5502035
    [40] Ramamurthy A, Jain P (2017) The Internet of Things in the Power Sector: Opportunities in Asia and the Pacific. https://doi.org/10.22617/WPS178914-2
    [41] Sigfox, Inc. Utilities & Energy (2019) Available from: https://www.sigfox.com/en/utilities-energy/.
    [42] Immelt JR (2015) The Future of Electricity Is Digital; Technical Report; General Electric: Boston, MA, USA, 2015.
    [43] Huneria HK, Yadav P, Shaw RN, et al. (2021) AI and IOT-Based Model for Photovoltaic Power Generation. Innovations in Electrical and Electronic Engineering, 697-706. https://doi.org/10.1007/978-981-16-0749-3_55
    [44] Singh R, Akram SV, Gehlot A, et al. (2022) Energy System 4.0: Digitalization of the Energy Sector with Inclination towards Sustainability. Sensors 22: 6619. https://doi.org/10.3390/s22176619
    [45] Ejaz W, Naeem M, Shahid A, et al. (2017) Efficient energy management for the internet of things in smart cities. IEEE Commun Mag 55: 84–91. https://doi.org/10.1109/MCOM.2017.1600218CM doi: 10.1109/MCOM.2017.1600218CM
    [46] Mitchell S, Villa N, Stewart-Weeks M, et al. (2013) The Internet of Everything for Cities; Cisco: San Jose, CA, USA, 2013.
    [47] Idwan S, Mahmood I, Zubairi J, et al. (2020) Optimal Management of Solid Waste in Smart Cities using Internet of Things. Wireless Pers Commun 110: 485-501. https://doi.org/10.1007/s11277-019-06738-8 doi: 10.1007/s11277-019-06738-8
    [48] Vakiloroaya V, Samali B, Fakhar A, et al. (2014) A review of different strategies for HVAC energy saving. Energy Convers Manag 77: 738–754. https://doi.org/10.1016/j.enconman.2013.10.023 doi: 10.1016/j.enconman.2013.10.023
    [49] Arasteh H, Hosseinnezhad V, Loia V, et al. (2016) IoT-based smart cities: A survey. Proceedings of the 2016 IEEE 16th International Conference on Environment and Electrical Engineering (EEEIC), 1–6. https://doi.org/10.1109/EEEIC.2016.7555867
    [50] Lee C, Zhang S (2016) Development of an Industrial Internet of Things Suite for Smart Factory towards Re-industrialization in Hong Kong. Proceedings of the 6th International Workshop of Advanced Manufacturing and Automation, 285-289. https://doi.org/10.2991/iwama-16.2016.54
    [51] Reinfurt L, Falkenthal M, Breitenbücher U, et al. (2017) Applying IoT Patterns to Smart Factory Systems. Proceedings of the 2017 Advanced Summer School on Service Oriented Computing (Summer SOC), 66.
    [52] Cheng J, Chen W, Tao F, et al. (2018) Industrial IoT in 5G Environment towards Smart Manufacturing. J Ind Inf Integr 10: 10–19. https://doi.org/10.1016/j.jii.2018.04.001 doi: 10.1016/j.jii.2018.04.001
    [53] IoT application areas (2016) https://www.iot-analytics.com/top-10-iot-project-application-areas-q3-2016/.
    [54] Shi W, Xie X, Chu C, et al. (2015) Distributed Optimal Energy Management in Microgrids. IEEE T Smart Grid 6: 1137–1146. https://doi.org/10.1109/TSG.2014.2373150 doi: 10.1109/TSG.2014.2373150
    [55] Kamalinejad P, Mahapatra C, Sheng Z, et al. (2015) Wireless energy harvesting for the Internet of Things. IEEE Commun Mag 53: 102–108. https://doi.org/10.1109/MCOM.2015.7120024 doi: 10.1109/MCOM.2015.7120024
    [56] Song T, Li R, Mei B, et al. (2017) A privacy preserving communication protocol for IoT applications in smart homes. IEEE Internet Things J 4: 1844–1852. https://doi.org/10.1109/JIOT.2017.2707489. doi: 10.1109/JIOT.2017.2707489
    [57] Fhom HS, Kuntze N, Rudolph C, et al. (2010) A user-centric privacy manager for future energy systems. Proceedings of the 2010 International Conference on Power System Technology, 1–7. https://doi.org/10.1109/POWERCON.2010.5666447.
    [58] Pramudita R, Hariadi IF, Achmad AS (2017) Development of IoT Authentication Mechanisms for Microgrid Applications. 2017 International Symposium on Electronics and Smart Devices (ISESD), 12-17. https://doi.org/10.1109/ISESD.2017.8253297
    [59] Trappe W, Howard R, Moore RS (2015) Low-Energy Security: Limits and Opportunities in the Internet of Things. IEEE Secur Priv 13: 14-21. https://doi.org/10.1109/MSP.2015.7 doi: 10.1109/MSP.2015.7
    [60] IEEE Std 802.15.4-2015 (Revision of IEEE Std 802.15.4-2011) (2016) IEEE Standard for Low-Rate Wireless Networks. IEEE Stand, 1–708, https://doi.org/10.1109/IEEESTD.2016.7460875
    [61] Al-Qaseemi SA, Almulhim HA, Almulhim MF, et al. (2016) IoT architecture challenges and issues: Lack of standardization. Proceedings of the 2016 Future Technologies Conference (FTC), 731–738. https://doi.org/10.1109/FTC.2016.7821686
    [62] Stojmenovic I (2014) Machine-to-Machine Communications with In-Network Data Aggregation, Processing, and Actuation for Large-Scale Cyber-Physical Systems. IEEE Internet Things J 1: 122–128. https://doi.org/10.1109/JIOT.2014.2311693 doi: 10.1109/JIOT.2014.2311693
    [63] Lloret J, Tomas J, Canovas A, et al. (2016) An Integrated IoT Architecture for Smart Metering. IEomEE Commun Mag 54: 50–57. https://doi.org/10.1109/MCOM.2016.1600647CM doi: 10.1109/MCOM.2016.1600647CM
    [64] Breur T (2015) Big data and Internet of Things. J Mark Anal 3: 1–4. https://doi.org/10.1057/jma.2015.7 doi: 10.1057/jma.2015.7
    [65] Shakerighadi B, Anvari-Moghaddam A, Vasquez JC, et al. (2018) Internet of things for modern energy systems: state-of-the-art, challenges, and open issues. Energies 11: 1252. 10.3390/en11051252
    [66] Xu J, Yao J, Wang L, et al. (2017) Narrowband internet of things: evolutions, technologies and open issues. IEEE Internet of things journal 5: 1449-1462. https://doi.org/10.1109/JIOT.2017.2783374 doi: 10.1109/JIOT.2017.2783374
    [67] Venkatesh N (2015) Ensuring Coexistence of IoT Wireless Protocols Using a Convergence Module to Avoid Contention, Embedded Innovator, 12th Edition, 2015.
    [68] Bedi G, Venayagamoorthy GK, Singh R (2016) Navigating the challenges of Internet of Things (IoT) for power and energy systems. 2016 Clemson University Power Systems Conference (PSC). https://doi.org/10.1109/PSC.2016.7462853
    [69] Singh R, Akram SV, Gehlot A, et al. (2022) Energy System 4.0: Digitalization of the Energy Sector with Inclination towards Sustainability. Sensors 22: 6619. https://doi.org/10.3390/s22176619.
    [70] Rana MM, Xiang W, Wang E (2018) IoT-based state estimation for microgrids. IEEE Internet of things Journal 5: 1345-1346. https://doi.org/10.1109/JIOT.2018.2793162 doi: 10.1109/JIOT.2018.2793162
    [71] Rana MM, Xiang W, Wang E, et al. (2017) IoT Infrastructure and Potential Application to Smart Grid Communications. IEEE Global communication conference (GLOBECOM 2017). https://doi.org/10.1109/GLOCOM.2017.8254511
    [72] Naqvi SAR, Hassan SA, Hussain F (2017) IoT Applications and Business Models. Springer Briefs in Electrical and Computer Engineering, 45–61. https://doi.org/10.1007/978-3-319-55405-1_4
    [73] Research and Markets (2020) Global Internet of Things (IoT) in Energy Market Size Expected to Grow from USD 20.2 billion in 2020 to USD 35.2 billion by 2025, at a CAGR of 11.8%. https://www.globenewswire.com/news-release/2020/05/28/2040020/28124/en/Global-Internet-of-Things-IoT-in-Energy-Market-Size-Expected-to-Grow-from-USD-20-2-billion-in-2020-to-USD-35-2-billion-by-2025-at-a-CAGR-of-11-8.html.
    [74] IoT Analytics, January (2022) https://iot-analytics.com/product/list-of1600-enterprise-iot-projects-2022/.
    [75] The Insight planners, July (2022) https://www.theinsightpartners.com/reports/south-africa-iot-market/. Accessed 1o August 2022.
    [76] Growth Enabler (2017) Market pulse report, Internet of Things (IoT). 1–35. GrowthEnabler. https://growthenabler.com/flipbook/pdf/IOTReport.pdf.
    [77] Hawlitschek F, Notheisen B, Teubner T (2018) The limits of trust-free systems: A literature review on blockchain technology and trust in the sharing economy. Electron Commer Res Appl 29: 50–63. https://doi.org/10.1016/j.elerap.2018.03.005 doi: 10.1016/j.elerap.2018.03.005
    [78] Christidis K, Devetsikiotis M (2016) Blockchains and Smart Contracts for the Internet of Things. IEEE Access 4: 2292–2303. https://doi.org/10.1109/ACCESS.2016.2566339. doi: 10.1109/ACCESS.2016.2566339
    [79] LG and Samsung to Show Off New Food Identifying Smart Fridges at CES Next Week: https://thespoon.tech/lg-and-samsung-to-show-off-new-food-identifying-smart-fridges-at-ces-next-week/
    [80] IoT Statistics. https://www.statista.com/statistics/471264/iot-number-of-connected-devices-worldwide/.
    [81] Abalansa S, El Mahrad B, Icely J, et al. (2021) Electronic Waste, an Environmental Problem Exported to Developing Countries: The GOOD, the BAD and the UGLY. Sustainability 13: 5302. https://doi.org/10.3390/su13095302. doi: 10.3390/su13095302
    [82] Zhu C, Leung VCM, Shu L, et al. (2015) Green Internet of Things for Smart World. IEEE Access 3: 2151–2162. https://doi.org/10.1109/ACCESS.2015.2497312. doi: 10.1109/ACCESS.2015.2497312
    [83] Kabalci Y, Ali M (2019) Emerging LPWAN Technologies for Smart Environments: An Outlook. Proceedings of the 2019 1st Global Power, Energy and Communication Conference (GPECOM), 24–29. https://doi.org/10.1109/GPECOM.2019.8778626
    [84] Bembe M, Abu-Mahfouz A, Masonta M, et al. (2019) A survey on low-power wide area networks for IoT applications. Telecommun Syst 71: 249–274. https://doi.org/10.1007/s11235-019-00557-9. doi: 10.1007/s11235-019-00557-9
  • This article has been cited by:

    1. Yuxuan Chen, Jiangbo Han, Global existence and nonexistence for a class of finitely degenerate coupled parabolic systems with high initial energy level, 2021, 14, 1937-1632, 4179, 10.3934/dcdss.2021109
    2. Quang-Minh Tran, Hong-Danh Pham, Global existence and blow-up results for a nonlinear model for a dynamic suspension bridge, 2021, 14, 1937-1632, 4521, 10.3934/dcdss.2021135
    3. Jun Zhou, Guangyu Xu, Chunlai Mu, Analysis of a pseudo-parabolic equation by potential wells, 2021, 200, 0373-3114, 2741, 10.1007/s10231-021-01099-1
    4. Le Thi Phuong Ngoc, Khong Thi Thao Uyen, Nguyen Huu Nhan, Nguyen Thanh Long, On a system of nonlinear pseudoparabolic equations with Robin-Dirichlet boundary conditions, 2022, 21, 1534-0392, 585, 10.3934/cpaa.2021190
    5. Jinxing Liu, Xiongrui Wang, Jun Zhou, Huan Zhang, Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source, 2021, 14, 1937-1632, 4321, 10.3934/dcdss.2021108
    6. Peiqun Lin, Chenxing He, Lingshu Zhong, Mingyang Pei, Chuhao Zhou, Yang Liu, Bus timetable optimization model in response to the diverse and uncertain requirements of passengers for travel comfort, 2023, 31, 2688-1594, 2315, 10.3934/era.2023118
    7. Hang Ding, Renhai Wang, Jun Zhou, Infinite time blow‐up of solutions to a class of wave equations with weak and strong damping terms and logarithmic nonlinearity, 2021, 147, 0022-2526, 914, 10.1111/sapm.12405
    8. Le Thi Phuong Ngoc, Nguyen Anh Triet, Phan Thi My Duyen, Nguyen Thanh Long, General Decay and Blow-up Results of a Robin-Dirichlet Problem for a Pseudoparabolic Nonlinear Equation of Kirchhoff-Carrier Type with Viscoelastic Term, 2023, 0251-4184, 10.1007/s40306-023-00496-3
    9. Gang Cheng, Yijie He, Enhancing passenger comfort and operator efficiency through multi-objective bus timetable optimization, 2024, 32, 2688-1594, 565, 10.3934/era.2024028
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(5723) PDF downloads(500) Cited by(18)

Figures and Tables

Figures(7)  /  Tables(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog