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Research article

Some new concepts in fuzzy calculus for up and down λ-convex fuzzy-number valued mappings and related inequalities

  • In recent years, numerous scholars have investigated the relationship between symmetry and generalized convexity. Due to this close relationship, generalized convexity and symmetry have become new areas of study in the field of inequalities. With the help of fuzzy up and down relation, the class of up and down λ-convex fuzzy-number valued mappings is introduced in this study; and weighted Hermite-Hadamard type fuzzy inclusions are demonstrated for these functions. The product of two up and down λ-convex fuzzy-number valued mappings also has Hermite-Hadamard type fuzzy inclusions, which is another development. Additionally, by imposing some mild restrictions on up and down λ-convex (λ-concave) fuzzy number valued mappings, we have introduced two new significant classes of fuzzy number valued up and down λ-convexity (λ-concavity), referred to as lower up and down λ-convex (lower up and down λ-concave) and upper up and down λ-convex (λ-concave) fuzzy number valued mappings. Using these definitions, we have amassed many classical and novel exceptional cases that implement the key findings. Our proven results expand and generalize several previous findings in the literature body. Additionally, we offer appropriate examples to corroborate our theoretical findings.

    Citation: Muhammad Bilal Khan, Hakeem A. Othman, Gustavo Santos-García, Muhammad Aslam Noor, Mohamed S. Soliman. Some new concepts in fuzzy calculus for up and down λ-convex fuzzy-number valued mappings and related inequalities[J]. AIMS Mathematics, 2023, 8(3): 6777-6803. doi: 10.3934/math.2023345

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  • In recent years, numerous scholars have investigated the relationship between symmetry and generalized convexity. Due to this close relationship, generalized convexity and symmetry have become new areas of study in the field of inequalities. With the help of fuzzy up and down relation, the class of up and down λ-convex fuzzy-number valued mappings is introduced in this study; and weighted Hermite-Hadamard type fuzzy inclusions are demonstrated for these functions. The product of two up and down λ-convex fuzzy-number valued mappings also has Hermite-Hadamard type fuzzy inclusions, which is another development. Additionally, by imposing some mild restrictions on up and down λ-convex (λ-concave) fuzzy number valued mappings, we have introduced two new significant classes of fuzzy number valued up and down λ-convexity (λ-concavity), referred to as lower up and down λ-convex (lower up and down λ-concave) and upper up and down λ-convex (λ-concave) fuzzy number valued mappings. Using these definitions, we have amassed many classical and novel exceptional cases that implement the key findings. Our proven results expand and generalize several previous findings in the literature body. Additionally, we offer appropriate examples to corroborate our theoretical findings.



    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 a 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.



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