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Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy

  • Nonlinear Klein-Gordon equation with combined power type nonlinearity and critical initial energy is investigated. The qualitative properties of a new ordinary differential equation are studied and the concavity method of Levine is improved. Necessary and sufficient conditions for finite time blow up and global existence of the solutions are proved. New sufficient conditions on the initial data for finite time blow up, based on the necessary and sufficient ones, are obtained. The asymptotic behavior of the global solutions is also investigated.

    Citation: Milena Dimova, Natalia Kolkovska, Nikolai Kutev. Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy[J]. Electronic Research Archive, 2020, 28(2): 671-689. doi: 10.3934/era.2020035

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  • Nonlinear Klein-Gordon equation with combined power type nonlinearity and critical initial energy is investigated. The qualitative properties of a new ordinary differential equation are studied and the concavity method of Levine is improved. Necessary and sufficient conditions for finite time blow up and global existence of the solutions are proved. New sufficient conditions on the initial data for finite time blow up, based on the necessary and sufficient ones, are obtained. The asymptotic behavior of the global solutions is also investigated.



    The aim of this paper is to study the global behavior of the solutions to the Cauchy problem for the nonlinear Klein-Gordon equation

    uttΔu+u=f(u),(t,x)R×Rn,u(0,x)=u0(x),ut(0,x)=u1(x),xRn,u0(x)H1(Rn),u1(x)L2(Rn) (1)

    with critical initial energy E(0)=d. The nonlinear term f(u) has one of the following forms

    f(u)=lk=1ak|u|pk1usj=1bj|u|qj1u,f(u)=a1|u|p1+lk=2ak|u|pk1usj=1bj|u|qj1u, (2)

    where the constants ak, pk (k=1,2,,l) and bj, qj (j=1,2,,s) fulfill the conditions

    a1>0,ak0,bj0fork=2,,l,j=1,,s,1<qs<qs1<<q1<p1<p2<<pl1<pl,pl<forn=1,2;pl<n+2n2forn3. (3)

    The combined power type nonlinearity (2) appears in numerous models of quantum mechanics, field theory, nonlinear optics and others. For example, the quadratic-cubic nonlinearity f(u)=u2+u3 describes the dislocation of crystals, see [16], while the the cubic-quintic nonlinearity f(u)=u3+u5 arises in particles physics, see e.g. [21,14].

    The global existence or finite time blow up of the solutions to (1) - (3) is fully investigated for nonpositive energy E(0)0 and for subcritical energy 0<E(0)<d by means of the potential well method. Here d is the critical energy constant, defined in (8). Potential well method is suggested in [20] for the wave equation and further on is applied for wide class of nonlinear dispersive equations, e.g. for nonlinear Klein-Gordon equations see [1,18,19,24,30]. Within this method the sign of the Nehari functional I(0), see (7), is crucial for the global behavior of the solutions to (1) - (3). More precisely, for 0<E(0)<d the solutions blow up for a finite time if I(0)<0 and they are globally defined if I(0)0.

    The case of critical initial energy, i.e. E(0)=d, is treated in [5,9,26,28] for the wave and damped wave equations in bounded domains and for the Klein-Gordon equation – in [8,16,19,24]. In the above papers the global existence is proved under conditions I(0)>0 or u(t,)H1(Rn)=0 without any restrictions on the sign of the scalar product (u0,u1) of the initial data. In the same papers the finite time blow up is obtained when I(0)<0 and (u0,u1)0. The case E(0)=d, I(0)<0 and (u0,u1)<0 is investigated only in [9,16]. The asymptotic behavior of the global solutions to the wave equation in bounded domains is studied in [9], while for Klein-Gordon equation similar results are given in [16].

    In the case of supercritical initial energy, i.e. E(0)>d, only partial results for global behavior of the solutions to (1) - (3) are reported in the literature. There are a few sufficient conditions on the initial data u0 and u1, which guarantee finite time blow up, see [3,4,8,11,12,13,19,22,23,27,29]. In these sufficient conditions the nonnegative sign of (u0,u1) is crucial.

    In our previous paper [4] we prove a necessary and sufficient condition for finite time blow up of the solutions to (1) - (3) for arbitrary positive initial energy E(0)>0. More precisely, if u(t,x) is a solution to (1) - (3) defined in the maximal existence time interval [0,Tm), 0<Tm, then u(t,x) blows up for a finite time Tm< if and only if there exists b[0,Tm) such that

    (u(b,),ut(b,))>0and0<E(0)p112(p1+1)u(b,)2L2(Rn). (4)

    Let us emphasize once again that the sign condition (u0,u1)0 plays very important role in all known sufficient conditions for finite time blow up of the solutions to (1) - (3) with supercritical energy, as well as in the necessary and sufficient condition (4).

    In the present paper we focus on the global behavior of the solutions to (1) - (3) with critical initial energy E(0)=d. We give new necessary and sufficient conditions for finite time blow up and global existence, which are based on the study of the qualitative properties to a new ordinary differential equation. This approach improves the concavity method of Levine. As a consequence of the necessary and sufficient conditions for finite time blow up, we get new, more general sufficient conditions on the initial data for finite time blow up. In the case I(0)<0, new necessary conditions on the initial data for global existence are proved. The asymptotic behavior of the global solutions with I(0)<0 is studied in a similar way as in [9], where the wave equation in bounded domains is considered.

    The paper is organized in the following way. In Section 2 some preliminary results are given. Section 3 deals with the global behavior of the solutions to a new ordinary differential equation. The results are an improvement of the concavity method of Levine and allow us to formulate necessary and sufficient conditions for finite time blow up. The main results of the paper are formulated and proved in Section 4 and Section 5. In Section 4 the finite time blow up is treated, while Section 5 deals with the global existence of the solutions and their asymptotic behavior.

    We will use the following short notations for the functions u(t,x) and v(t,x) depending on t and x

    u=u(t,)L2(Rn),u1=u(t,)H1(Rn),(u,v)=(u(t,),v(t,))=Rnu(t,x)v(t,x)dx.

    We have the following local existence result to the Cauchy problem (1) - (3), see e.g. [2,6,7].

    Theorem 2.1. Problem (1) - (3) admits a unique local weak solution

    u(t,x)C((0,Tm);H1(Rn))C1((0,Tm);L2(Rn))C2((0,Tm);H1(Rn))

    in the maximal existence time interval [0,Tm). Moreover,

    (ⅰ)

    iflim suptTm,t<Tmu1<,thenTm=;

    (ⅱ) for every t[0,Tm) the solution u(t,x) satisfies the conservation law

    E(0)=E(t), (5)

    where the energy functional E(t) is defined by

    E(t):=E(u(t,),ut(t,))=12(ut2+u21)Rnu0f(y)dydx. (6)

    Definition 2.2. The solution u(t,x) to (1) - (3), defined in the maximal existence time interval [0,Tm), Tm, blows up at Tm if

    lim suptTm,t<Tmu1=.

    In order to prove a necessary and sufficient condition for finite time blow up of the solutions to (1) - (3), we use the following equivalence between the blow up of the H1 and L2 norms of u(t,x).

    Lemma 2.3. Suppose u(t,x) is the solution to (1) - (3) with pl<(n+4)/n in the maximal existence time interval [0,Tm), 0<Tm. Then the blow up of H1 norm of u(t,x) is equivalent to the blow up of the L2 norm of u(t,x) at Tm, i.e.

    lim suptTm,t<Tmu1=if and only iflim suptTm,t<Tmu=.

    The proof of Lemma 2.3 is based on the Gagliardo - Nirenberg inequality. In one-dimensional case it is given in [4]. The multidimensional case is treated in a similar way and we omit the proof.

    Let us recall some important functionals - the Nehari functional I(u(t,)) and the potential energy functional J(u(t,)), as well as the critical energy constant d. When u depends on x and t we use the short notations I(u(t,))=I(t) and J(u(t,))=J(t), i.e.

    I(t):=I(u(t,))=u21Rnf(u)udx, (7)
    J(t):=J(u(t,))=12u21Rnu0f(y)dydx,
    d=infuNJ(u(t,)),N={uH1(Rn): u10, I(u(t,))=0}. (8)

    In the framework of the potential well method there are two important subsets of H1(Rn):

    W={uH1(Rn):I(u)>0}{0},V={uH1(Rn):I(u)<0}.

    In the following theorem we formulate the sign preserving properties of I(u), i.e. the invariance of V and W under the flow of (1) - (3) when E(0)=d.

    Theorem 2.4. Suppose u(t,x) is the weak solution of (1) - (3) defined in the maximal existence time interval [0,Tm) and E(0)=d.

    (ⅰ) If u0W, then u(t,x)W for every t[0,Tm);

    (ⅱ) If u0V, then u(t,x)V for every t[0,Tm).

    Proof. (ⅰ) Suppose u0W but the result in (ⅰ) fails. Then for some t0(0,Tm) we have u(t,x)W for t[0,t0) and u(t0,x)W, i.e. I(t)>0 or u1=0 for t[0,t0) but I(t0)=0 and u(t0,)10. Hence u(t0,)N and from (5), (6), (8) it follows that J(t0)d and the following inequalities hold

    d=E(t0)=12ut(t0,)2+J(t0)d.

    Hence

    ut(t0,)=0,J(t0)=dandI(t0)=0. (9)

    If ˆu(x) is a ground state solution of (1), then ˆu(x) satisfies the equation

    Δˆu+ˆuf(ˆu)=0forxRn.

    Consequently, condition (9) means that the function u(t0,x) coincides with some ground state solution of (1). Without loss of generality we assume that u(t0,x)=ˆu(x). Since tˆu(x)=tu(t0,x)=0, from the uniqueness of the weak solution to (1) - (3) we get u(t,x)=u(t0,x)=ˆu(x) for every t[0,Tm), xRn. Hence from (9) it follows that I(t)=I(t0)=0 for every t[0,Tm) and for t=0 we get I(0)=0, u01=ˆu10, which contradicts the assumption u0W. Thus statement (ⅰ) in Theorem 2.4 is proved.

    (ⅱ) Suppose u0V. If u(t0,x)V for some t0(0,Tm), then either u(t0,x)W or I(t0)=0 and u(t0,)10. If u(t0,x)W, then from (ⅰ) it follows that u(t,x)W for every t[0,Tm). When t=0 we get u0(x)W, which contradicts our assumption u0V. If I(t0)=0 and u(t0,)10, then u(t0,x) coincides with some ground state solution ˆu(x) of (1). Since u(t0,)10 from the uniqueness result it follows that u(t,x)=ˆu(x) for every t0. Hence I(u(t,))=I(ˆu)=0 for every t0, which contradicts our assumption u0V. Thus (ii) in Theorem 2.4 is proved.

    Remark 1. We rewrite the conservation law (5), (6) by means of (7) in the following way

    E(0)=12ut2+1p1+1I(t)+p112(p1+1)u21+B(t), (10)

    where from (2) and (3)

    B(t)=lk=2ak(pkp1)(pk+1)(p1+1)Rn|u|pk+1dx+sj=1bj(p1qj)(qj+1)(p1+1)Rn|u|qj+1dx0. (11)

    Remark 2. If E(0)=d then condition I(0)<0 is equivalent to

    u02>2(p1+1)p11dp1+1p11u12u022(p1+1)p11B(0), (12)

    while condition I(0)0 is equivalent to

    u022(p1+1)p11dp1+1p11u12u022(p1+1)p11B(0).

    For the proofs of our main results in Section 4 and Section 5 we need the following auxiliary statement.

    Lemma 2.5. Suppose u(t,x) is the weak solution of (1) - (3) in the maximal existence time interval [0,Tm), 0<Tm and E(0)=d. If I(0)<0, then

    I(t)<(p1+1)(J(t)d)fort[0,Tm). (13)

    The proof of Lemma 2.5 is identical with the proof of Lemma 2.3 in [25] and we omit it.

    In the last decades the concavity method, introduced by Levine [15], is one of the powerful methods in the investigation of the finite time blow up of the solutions to nonlinear dispersive equations. The main idea of the concavity method is one to prove finite time blow up of the solutions to the ordinary differential inequality

    ψ(t)ψ(t)γψ2(t)0,t0,γ>1, (14)

    where ψ(t) is a nonnegative, twice differentiable function for t>0. When

    ψ(0)>0,ψ(0)>0 (15)

    then the solution ψ(t) of (14) blows up for a finite time T and

    Tψ(0)(γ1)ψ(0).

    In the applications to nonlinear dispersive equations usually ψ(t) is some functional of the solution. For example, ψ(t)=Rnu2(t,x)dx for Klein-Gordon equation. For fourth and sixth order double dispersive equations ψ(t) is more complicated functional, including the L2 norm of the solution and some additional terms.

    Let us mention, that condition (15) is only sufficient one for finite time blow up of the solution ψ(t) to (14). The question, which naturally arises, is whether a necessary and sufficient condition for blow up of ψ(t) exists.

    In order to give a satisfactory answer of this question, instead of inequality (14) we consider the following nonlinear ordinary differential equation

    ψ(t)ψ(t)γψ2(t)=Q(t),t[0,Tm),0<Tm,γ>1, (16)
    Q(t)C([0,Tm)),Q(t)0,t[0,Tm). (17)

    Here the nonnegative, twice differentiable function ψ(t) is defined in the maximal existence time interval [0,Tm), 0<Tm. In the applications to nonlinear dispersive problems, equation (16) naturally appears instead of inequality (14). Since the nonnegative term Q(t) can not be expressed by means of ψ(t), this term has been neglected and (16) has been reduced to (14).

    We recall the definition of blow up of a nonnegative function ψ(t)C2([0,Tm)) at Tm.

    Definition 3.1. The nonnegative function ψ(t)C2([0,Tm)) blows up at Tm if

    lim suptTm,t<Tmψ(t)=. (18)

    Theorem 3.2. Suppose ψ(t)C2([0,Tm)) is a nonnegative solution to (16), (17) in the maximal existence time interval [0,Tm), 0<Tm. If ψ(t) blows up at Tm, then Tm<.

    Proof. Step 1. First we will show that

    there exists b[0,Tm)such thatψ(b)>0. (19)

    If not, then ψ(t)0 for every t[0,Tm) and the estimate

    0ψ(t)ψ(0)

    holds for every t[0,Tm). Hence we get

    lim suptTm,t<Tmψ(t)ψ(0),

    which contradicts (18). Thus (19) holds.

    Step 2. Now we will prove that ψ(t)>0 for every t[b,Tm). From (19) we have that ψ(b)>0. Otherwise from (16) it follows that γψ2(b)0, which contradicts (19). In order to prove that ψ(t)>0 for every t[b,Tm) we suppose by contradiction that there exists t0(b,Tm) such that

    ψ(t)>0fort[b,t0)andψ(t0)=0. (20)

    From (16), (17) and (20) we get

    ψ(t)=(γψ2(t)+Q(t))ψ1(t)0fort[b,t0),

    i.e. ψ(t) is a convex function for t[b,t0). Hence ψ(t)ψ(b)>0 for t[b,t0) and ψ(t) is a strictly increasing function for t[b,t0). From the monotonicity of ψ(t) we obtain the following impossible chain of inequalities 0=ψ(t0)>ψ(b)>0. Thus ψ(t) is a positive function satisfying the estimate

    ψ(t)ψ(b)>0for everyt[b,Tm). (21)

    Additionally, from (16), (17) and (21) it follows that ψ(t) is a convex function satisfying the inequality

    ψ(t)ψ(b)>0for t[b,Tm). (22)

    Step 3. Let us prove that Tm<. For this purpose we introduce the new function

    z(t)=ψ1γ(t)fort[b,Tm).

    Straightforward computations give us

    z(t)=(1γ)ψγ(t)ψ(t),11γψ1+γ(t)z(t)=ψ(t)ψ(t)γψ2(t)0 (23)

    and z(t) satisfies the problem

    z(t)=(γ1)Q(t)zγ+1γ1(t)fort[b,Tm),z(b)>0,z(b)<0. (24)

    Suppose that Tm is not finite, i.e. Tm=. Then from (17), (24) it follows that

    z(t)0fortb. (25)

    Integrating (25) twice from b to t we get

    z(t)z(b)(tb)+z(b).

    Consequently, there exists a constant T,

    b<Tbz(b)z(b)=b+ψ(b)(γ1)ψ(b)<, (26)

    such that z(T)=0, or equivalently ψ(T)=, which contradicts our assumption that Tm=. Theorem 3.2 is proved.

    The following necessary and sufficient condition for finite time blow up of the solution to the ordinary differential equation (16) is a key result in the investigation of the behavior of the solutions to nonlinear dispersive equations.

    Theorem 3.3. Suppose ψ(t)C2([0,Tm)) is a nonnegative solution to (16) in the maximal existence time interval [0,Tm), 0<Tm and Q(t)C([0,)), Q(t)0 for t0. Then ψ(t) blows up at Tm if and only if (19) holds, i.e. there exists b[0,Tm), such that ψ(b)>0. Moreover, the estimate

    Tmb+ψ(b)(γ1)ψ(b)< (27)

    holds.

    Proof. (Necessity) Suppose ψ(t) blows up at Tm. Then (19) holds from Step 1 in the proof of Theorem 3.2, while (27) is a consequence of the inequality (26) in Step 3.

    (Sufficiency) Suppose (19) is satisfied. From Step 2 and Step 3 in the proof of Theorem 3.2 it follows that Tm<.

    If we assume by contradiction that ψ(t) does not blow up at Tm, i.e. (18) fails, then

    lim suptTm,t<Tmψ(t)<. (28)

    From (22), (28) it follows that ψ(t) is a strictly increasing and bounded function for t[b,Tm) so that the limit of ψ(t) for tTm exists and

    limtTm,t<Tmψ(t)=ψ00,ψ0<. (29)

    Integrating (24) from b to t<Tm we get

    z(t)=z(b)(γ1)tbQ(s)zγ+1γ1(s)ds,

    or equivalently, from (23)

    ψ(t)=ψγ(t)[ψ(b)ψγ(b)+(γ1)2tbQ(s)ψγ1(s)ds].

    Thus from (21), (29) and the monotonicity of ψ(t) we have

    limtTm,t<Tmψ(t)=ψγ0[ψ(b)ψγ(b)+(γ1)2TmbQ(s)ψγ1(s)ds]=ψ1,0<ψ1<.

    The initial value problem

    φ(t)φ(t)γφ2(t)=Q(t)fortTm,φ(Tm)=ψ0,φ(Tm)=ψ1

    has a classical solution φ(t)C2([Tm,Tm+δ)) for sufficiently small δ>0. Hence the function

    ˜φ(t)={ψ(t)fort[0,Tm);φ(t)fort[Tm,Tm+δ),

    ˜φ(t)C2([0,Tm+δ)), ˜φ(t)0 for t[0,Tm+δ) is a classical, nonnegative solution of (16) in the interval [0,Tm+δ). This contradicts the choice of Tm. Hence ψ(t) blows up at Tm and Theorem 3.3 is proved.

    As a consequence of Theorem 3.2, Theorem 3.3 and Theorem 2.4 we have the following precise results for finite time blow up of the solutions to (1) - (3) in the critical case E(0)=d.

    Theorem 4.1. Suppose u(t,x) is the weak solution of (1) – (3) with initial energy E(0)=d, defined in the maximal existence time interval [0,Tm), 0<Tm. If I(0)<0 and pl<(n+4)/n then u(t,x) blows up at Tm if and only if

    there existsb[0,Tm)such that(u(b,),ut(b,))0. (30)

    Moreover, Tm is finite, i.e. Tm<.

    Proof. For the function ψ(t)=u2, simple computations give us from (10) the identities

    ψ(t)=2(u,ut),ψ(t)=2ut22I(t)=(p1+3)ut22(p1+1)E(0)+(p11)u21+2(p1+1)B(t). (31)

    Hence ψ(t) satisfies the following ordinary differential equation

    ψ(t)ψ(t)p1+34ψ2(t)=Q(t), (32)

    where

    Q(t)=(p1+3)(ut2u2(u,ut)2)+2{(p1+1)(J(t)d)I(t)}u2+2(p+1)B(t)u2. (33)

    From (11), (13) in Lemma 2.5 and the Cauchy-Schwartz inequality we have

    Q(t)0fort[0,Tm). (34)

    Thus ψ(t) is a solution to (16), (17) for γ=(p1+3)/4 and Q(t)0 defined in (33).

    (Necessity) Suppose u(t,x) blows up at Tm. From Lemma 2.3 it follows that ψ(t)=u2 blows up at Tm. Then from Step 1 in the proof of Theorem 3.2 for ψ(t)=u2, γ=(p1+3)/4 and Q(t)0 defined in (33) there exists b[0,Tm) such that ψ(b)=2(u(b,),ut(b,))>0, i.e. (30) is satisfied.

    (Sufficiency) Suppose (30) holds, but u(t,x) does not blow up at Tm. From Theorem 2.1(ⅰ) it follows that Tm=. From (31) and Theorem 2.4(ⅱ) the function ψ(t)=u2 is a strictly convex one, because ψ(t)=2ut22I(t)>0. Thus (30) gives us the inequality

    ψ(t)>ψ(b)0for everyt(b,Tm). (35)

    From (35) there exists b1(b,Tm), such that ψ(b1)>0. According to Theorem 3.3 in the interval [b1,Tm) for ψ(t)=u2, γ=(p1+3)/4 and Q(t)0 defined in (33) it follows that u2 blows up at Tm. Applying Theorem 3.2 we get Tm<. Theorem 4.1 is proved.

    Remark 3. From the proof of Theorem 4.1 it is clear that the restriction pl<(n+4)/n for the nonlinear term (2), (3) is used only in the proof of the (Necessity) of Theorem 4.1. Let us note that the statement in the (Sufficiency) of Theorem 4.1 holds for every pl satisfying (3), i.e. the assumption pl<(n+4)/n is superfluous.

    Remark 4. Let us compare the condition (4) and the new one (30). The careful analysis of the necessary and sufficient conditions (30) in Theorem 4.1 and (4) shows that if (4) holds then (30) is also satisfied at the same time t=b. This conclusion follows from (12) in Remark 2 in case t=b. Conversely, if (30) holds at t=b then necessarily (4) is satisfied at some time b1b.

    In the following theorem we give sufficient conditions for finite time blow up of the solutions to (1) - (3) in terms of the initial data u0, u1.

    Theorem 4.2. Suppose u(t,x) is the weak solution of (1) - (3) with initial energy E(0)=d, defined in the maximal existence time interval [0,Tm), 0<Tm. Then the weak solution u(t,x) blows up at Tm when one of the following conditions is fulfilled:

    (ⅰ) (u0,u1)0 and I(0)<0;

    (ⅱ)

    (u0,u1)<0andu022(p1+1)p11d2p11(u0,u1). (36)

    Moreover, Tm is finite, i.e. Tm<.

    Proof. (ⅰ) The proof of Theorem 4.2 (ⅰ) follows immediately from the sufficiency part of Theorem 4.1 and Remark 3 for b=0 when (u0,u1)>0. If (u0,u1)=0 then from (31) and Theorem 2.4 we obtain that ψ(0)=2u122I(0)>0 and ψ(b)>ψ(0)0 for every b>0. Since ψ(t) satisfies (32) and (34) from Theorem 3.3 it follows that ψ(t) blows up for a finite time, i.e. u(t,x) blows up for a finite time.

    (ⅱ) Suppose (36) hold. Since

    u022(p1+1)p11d2p11(u0,u1)>2(p1+1)p11dp1+1p11u12u022(p1+1)p11B(0)

    from Remark 2 it follows that I(0)<0.

    In order to prove statement (ⅱ) we suppose by contradiction that u(t,x) does not blow up at Tm. Then from the local existence result, Theorem 2.1, it follows that Tm=. Thus u(t,x) is globally defined for every t0.

    If (30) is satisfied, i.e. there exists b[0,Tm) such that (u(b,),ut(b,))0, then from Theorem 4.1 u(t,x) blows up at Tm, which contradicts our assumption. Hence u(t,x) blows up at Tm and from Theorem 3.2 it follows that Tm<. Thus statement (ⅱ) is proved when (30) is fulfilled.

    If (30) does not hold, then

    ψ(t)=2(u,ut)<0for everyt0. (37)

    From (31) the function ψ(t)=u2 is a solution to the equation

    ψ(t)=αψ(t)β+G(t)fort0. (38)

    Here α=p11>0, β=2(p1+1)E(0)>0 and

    G(t)=(p1+3)ut2+(p11)u2+2(p1+1)B(t)0,

    because B(t), given in (11), is a non negative function. Equation (38) has a unique classical solution

    ψ(t)=12(ψ(0)+1αψ(0)βα)eαt+12(ψ(0)1αψ(0)βα)eαt+βα+1αt0G(s)sinh(α(ts))ds (39)

    and

    ψ(t)=α2(ψ(0)+1αψ(0)βα)eαtα2(ψ(0)1αψ(0)βα)eαt+t0G(s)cosh(α(ts))ds. (40)

    From (39) and (40) we get

    ψ(t)+1αψ(t)βα=(ψ(0)+1αψ(0)βα+1αt0G(s)eαsds)eαt (41)

    By means of (31) the function h(t)=(u,ut) satisfies the equation

    h(t)+εh(t)=I(t)ut2ε(u,ut)=I(t)(ut+ε2u,ut+ε2u)+ε24u2=:g(t)

    Since

    g(t)ε24u2ε24u02,

    we have the estimates

    h(t)=h(0)eεt+eεtt0g(s)eεsdsh(0)eεt+ε24εu02(1eεt).

    After the limit t in the above inequality, from (37), we get the inequalities

    0lim supth(t)ε4u02.

    Thus we obtain

    limt(u,ut)=0, (42)

    because ε is an arbitrary positive constant.

    Since ψ(t) is monotone decreasing and bounded from below with zero, after the limit t in (41) we get from (42) the identity

    limtψ(t)βα=limt(ψ(0)+1αψ(0)βα+1αt0G(s)eαsds)eαt (43)

    Hence necessarily we have

    0G(s)eαsds=α(ψ(0)+1αψ(0)βα)

    and from L'Hospital's rule it follows that

    limtψ(t)βα=limt(ψ(0)+1αψ(0)βα+1αt0G(s)eαsds)eαt=1αlimtG(t)0, (44)

    i.e.

    limtψ(t)βα.

    Multiplying (38) with ψ(t) and integrating from 0 to t we obtain the identity

    ψ2(t)=α(ψ(t)βα)2+2t0G(s)ψ(s)ds+K, (45)
    K=α(ψ(0)βα)2+ψ2(0).

    Since β=2(p1+1)d, α=p11, ψ(0)=u02 and ψ(0)=2(u0,u1), then the second inequality in (36) can be rewritten as

    ψ(0)βα1αψ(0). (46)

    Thus from (37) it follows that ψ(0)<0 and (46) gives us

    ψ(0)>βα. (47)

    Let us consider the case

    limtψ(t)<βα. (48)

    Since ψ(t) is a strictly decreasing function for t[0,), from (47) and (48), there exists a point t1, t1(0,), such that

    ψ(t1)=βα.

    Then for t=t1 in (45) we get

    0<ψ2(t1)=2t10G(s)ψ(s)ds+K<K. (49)

    Now we consider the case

    limtψ(t)=βα.

    After the limit t in (45) the equality (45) becomes

    0=limtψ2(t)=20G(s)ψ(s)ds+K<K. (50)

    In both cases from (49) and (50) we have

    K=α(ψ(0)βα)2+ψ2(0)>0.

    The above inequality is satisfies if

    βα+1αψ(0)<ψ(0)<βα1αψ(0),

    or equivalently

    2(p1+1)p11d+2p11(u0,u1)<u02<2(p1+1)p11d2p11(u0,u1),

    which contradicts condition (36). Thus u(t,x) blows up at Tm and from Theorem 3.2 it follows that Tm<. Theorem 4.2 is proved.

    Remark 5. The statement of Theorem 4.2(ⅰ) has been already proved in a different way for the nonlinear wave equation in a bounded domain, see e.g. [9,26] and for nonlinear Klein-Gordon equation, see [16,19]. In the present paper the proof of Theorem 4.2(ⅰ) is a consequence of Theorem 3.3.

    Remark 6. Let the initial data satisfy conditions (36). Then from (12) it follows, that I(0)<0, i.e the assumption I(0)<0 is unnecessary in Theorem (4.2)(ⅱ).

    In the following corollary we reformulate the statements in Theorem 4.2. The requirement for the sign of the Nehari functional I(0) is replaced by the assumptions on the initial data according to Remark 2.

    Corollary 1. Suppose u(t,x) is the weak solution of (1) - (3) with initial energy E(0)=d, defined in the maximal existence time interval [0,Tm), 0<Tm. Then the weak solution u(t,x) blows up at Tm when the initial data satisfy one of the following conditions:

    (ⅰ)

    (u0,u1)0andu02>2(p1+1)p11dp1+1p11u12u022(p1+1)p11B(0);

    (ⅱ)

    (u0,u1)<0andu022(p1+1)p11d2p11(u0,u1).

    Moreover, Tm is finite, i.e. Tm<.

    Below we compare the result in Theorem 4.2 (Corollary 1) with the result in [16] for the nonlinear term

    f(u)=u2+u3. (51)

    Proposition 1. Suppose u(t,x) is the weak solution of (1) with initial energy E(0)=d, defined in the maximal existence time interval [0,Tm), 0<Tm and f(u)=u2+u3. Then u(t,x) blows up at Tm< when one of the following conditions holds:

    (ⅰ) (Theorem (4.2)(i), [16,Theorem 1.3(3)])

    (u0,u1)0andu02>6d3u12u0212Rnu40(x)dx;

    (ⅱ) (Theorem (4.2)(ii))

    (u0,u1)<0andu026d2(u0,u1);

    (ⅲ) ([16,Theorem 1.3(3)])

    (u0,u1)<0,
    12Rnu4e(x)dx+ue2<3u12+u02+12Rnu40(x)dx, (52)
    6d3u12u0212Rnu40(x)dx<u026d(12Rnu4e(x)dx+ue2), (53)

    where ue satisfies conditions I(ue)=0 and J(ue)=d.

    Proof. (ⅰ) and (ⅱ) We apply Theorem 4.2 for p1=2, p2=3, a1=1, a2=1, ak=0 for k=3,,l, bj=0 for j=1,,s, n3 and

    B(t)=112Rnu4(t,x)dx.

    According to Theorem 4.2 and Corollary 1 the solution u(t,x) of (1), (51) blows up for a finite time when the initial data satisfy one of the following conditions:

    (u0,u1)0andu02>6d3u12u0212Rnu40(x)dx;
    (u0,u1)<0andu026d2(u0,u1), (54)

    So the statements (ⅰ) and (ⅱ) are proved. Note, that for (u0,u1)0 and I(0)<0 the result in Theorem 1.3(3) in [16] coincides with the statement in Proposition 1(i).

    (ⅲ) For (u0,u1)<0 Theorem 1.3(3) says that the solution blows up for finite time if

    (u0,u1)<0,I(0)<0andu02ue2, (55)

    where ue satisfies conditions I(ue)=0 and J(ue)=d. Since the conservation law (6) gives us

    d=16ue21+112Rnu4e(x)dx,

    assumptions (55) are equivalent to

    (u0,u1)<0,6d3u12u0212Rnu40(x)dx<u026d(12Rnu4e(x)dx+ue2).

    The above inequality holds only for

    12Rnu4e(x)dx+ue2<3u12+u02+12Rnu40(x)dx.

    When the opposite inequality is satisfied, i.e.

    6d3u12u0212Rnu40(x)dx6d(12Rnu4e(x)dx+ue2),

    then the set of functions satisfying Theorem 1.3(3) in [16] is empty. In this case the finite time blow up of the solutions is possible only under conditions of Theorem (4.2)(ⅱ), i.e. when (54) is satisfied.

    However, if (52) holds, then the conditions for finite time of the solution to (1), (51) in Theorem (4.2)(ⅱ) and Theorem 1.3(3) in [16] are completely different. Indeed from the inequality

    6d2(u0,u1)>6d(12Rnu4e(x)dx+ue2),

    it follows that the intervals for u02 in assumptions (54) and (53) have no intersection points. Thus the result in Theorem (4.2)(ⅱ) is a new one.

    Theorem 5.1. Suppose u(t,x) is the weak solution of (1) - (3) with initial energy E(0)=d, defined in the maximal existence time interval [0,Tm), 0<Tm, pl<(n+4)/n and I(0)<0. Then u(t,x) is globally defined for every t0, i.e. Tm=, if and only if

    (u(t,),ut(t,))<0for everyt0. (56)

    Proof. (Necessity) Suppose u(t,x) is defined for every t0, i.e. Tm=. If (56) fails, then there exists b[0,) such that

    (u(b,),ut(b,))0.

    From Theorem 4.1 it follows that u(t,x) blows up for finite time Tm<, which contradicts our assumption Tm=.

    (Sufficiency) Suppose condition (56) is satisfied. Then u(t,) is a strictly decreasing function and the following inequality holds

    u(t,)2u02for everyt[0,Tm).

    From Lemma 2.3 it follows that

    u(t,)21K0<for everyt[0,Tm)

    and from the local existence result we get Tm=. Thus u(t,x) is defined for every t0 and Theorem 5.1 is proved.

    Remark 7. The growth condition pl<(n+4)/n for the nonlinear term (2), (3) is used only in the proof of the (Sufficiency) of Theorem 5.1. For the proof of the (Necessity) of Theorem 5.1, assumption pl<(n+2)/(n2) in (3) is enough.

    Let us formulate necessary conditions on the initial data for global existence of the solutions to (1) - (3).

    Theorem 5.2. Suppose u(t,x) is the weak solution of (1) - (3) with initial energy E(0)=d, defined in the maximal existence time interval [0,Tm), 0<Tm.

    (ⅰ) If I(0)0 then u(t,x) is globally defined for every t0, i.e. Tm=;

    (ⅱ) If I(0)<0, then a necessary condition for global existence of u(t,x) for every t0 is

    (u0,u1)<0andu02<2(p1+1)p11d2p11(u0,u1). (57)

    Proof. The statement (ⅰ) in Theorem 5.2 has been already proved in [9,16,24,27] when u0W={uH1(Rn);I(0)>0}{0}. Since the condition I(u0)0 is slightly more general then u0W, for completeness we give the proof.

    (ⅰ) If I(0)=0 and u010 then u0N and J(u0)d. From (6) and (7) we get

    d=E(0)=12u12+J(u0)d,

    i.e. u1=0 and J(u0)=d. Since the function u0(x) is a solution to (1) - (3), from the uniqueness result it follows that u(t,x)=u0(x) for every t0, i.e. u(t,x) is a global solution.

    If I(0)>0 or I(0)=0 but u01=0, i.e. u0=0 for a.e. xRn, then u0W and from Theorem 2.4 I(t)W for every t[0,Tm). From the conservation law (5), see also (10), we have the estimate

    u(t,)21=2(p1+1)(p11)d(p1+1)(p11)ut(t,)22(p11)I(t)2(p1+1)(p11)B(t)2(p1+1)(p11)d<.

    Thus from statement (ⅰ) in the local existence result Theorem 2.1 it follows that Tm=.

    (ⅱ) Suppose that u(t,x) is defined for every t0, i.e. Tm=. If (57) fails, i.e. one of the following conditions is satisfied,

    (u0,u1)0

    or

    (u0,u1)<0andu022(p1+1)p11d2p11(u0,u1),

    then from Theorem 4.2 it follows that u(t,x) blows up for finite time. Thus Tm<, which contradicts our assumption Tm=. Theorem 5.2 is proved.

    Let us mention that the asymptotic behavior of the global solution to the nonlinear wave equation in bounded domains has been studied in [9] and to the Klein-Gordon equation with quadratic-cubic nonlinearity - in [16].

    In the following theorem we study the asymptotic behavior for t of the global solutions to (1) - (3) with critical initial energy E(0)=d and I(0)<0, i.e. when condition (ⅱ) of Theorem 5.2 is satisfied.

    Theorem 5.3. Suppose u(t,x) is a weak solution of (1) - (3) with initial energy E(0)=d, defined for every t[0,), pl<(n+4)/n and I(0)<0. Then there exist a sequence of time tm, functions ˆu(x)H1(Rn) and ˆv(x)L2(Rn), such that u(tm,x)ˆu(x) weakly in H1(Rn) and ut(tm,x)ˆv(x) weakly in L2(Rn) when m. Moreover,

    (ⅰ) ˆv=0 and lim inftut(t,)=0;

    (ⅱ) limtI(u(t,))=0;

    (ⅲ) limtt(u(t,),ut(t,))=0.

    Proof. Since the solution u(t,x) is defined for every t, it follows that

    u(t,1C0for everyt0. (58)

    Indeed, if (58) fails, then from the local existence result, Theorem 2.1 (ⅰ), we get limtu(t,)1=, which contradicts the results in Lemma 2.3 and Theorem 3.2. From the embedding of H1(Rn) into Lp(Rn) for p>2 we obtain

    |Rnu0f(y)dydx|C1fort0 (59)

    and from the conservation law (6)

    ut(t,)C2fort0, (60)

    where the constants C1 and C2 depend on the parameters of the nonlinearity f(u) as well as on the initial data. As a consequence of (58), (59) and (60), there exist a sequence tm and functions ˆu(x)H1(Rn) and ˆv(x)L2(Rn), such that for every ΦH1(Rn) and every ΘL2(Rn) the following equations are true

    limtmRnu(tm,)0f(y)dy Φ(x)dx=Rnˆu(x)0f(y)dy Φ(x)dx,
    (u(tm,),Φ)(ˆu,Φ),(ut(tm,),Θ)(ˆv,Θ)whentm, (61)

    i.e. u(t,x)ˆu(x) and ut(t,x)ˆv(x) for t.

    (ⅰ) We will prove that ˆv=0. Since I(t)<0 for every t0 (see Theorem 2.4), integrating (31) for s>t0 we obtain

    0<st(ut(τ,)2I(τ))dτ=(u(s,),ut(s,))(u(t,),ut(t,)) (62)
    0<t+1tut(τ,)2dτ<t+1t(ut(τ,)2I(τ))dτ=(u(t+1,),ut(t+1,))(u(t,),ut(t,)). (63)

    After the change of the variable τ=λ+t (63) becomes

    0<10ut(λ+t,)2dλ<(u(t+1,),ut(t+1,))(u(t,),ut(t,)). (64)

    Thus for every sm from (42) and (64) it follows that

    limm10ut(sm+λ,)2dλ=0.

    As a consequence of Fatou's lemma we get

    lim infmut(sm+λ,)2=0fora.e.λ[0,1]. (65)

    By means of the weak convergence of ut(t,x) to ˆv(x) in L2(Rn), i.e. (61), and the lower semicontinuity of the norm of L2(Rn), we get for some λ0[0,1], sm=tmλ0 and (65) the final inequality

    0ˆv2lim infmut(tm,)2=0,

    i.e.

    ˆv2=0.

    Thus we proved that every weak limit of ut(tm,x) for m is zero and

    lim inftut(t,)=0.

    (ⅱ) From (44) and (38) it follows that

    limtψ(t)βα+1αlimtG(t)=0,limtψ(t)=α(limtψ(t)+1αlimtG(t)βα)=0.

    Thus (31) and (ⅰ) in Theorem 5.3 give us

    0lim suptI(t)lim inftI(t)=lim inftut(t,)212limtψ(t)=0.

    Hence limtI(t)=0 and (ⅱ) is proved.

    (ⅲ) From (42) and (62) after the limit s we obtain

    t(ut(τ,)2I(τ))dτ=(u(t,),ut(t,)) (66)

    and integrating (66) from 0 to it follows that

    0t(ut(τ,)2I(τ))dτdt=0(u(t,),ut(t,))dt=12(u02limtu(t,)2). (67)

    Applying Fubini's theorem, (67) becomes

    12(u02limtu(t,)2)=0t(ut(τ,)2I(τ))dτdt=0τ(ut(τ,)2I(τ))dτ. (68)

    After the integration of (66) from 0 to s we have

    s0t(ut(τ,)2I(τ))dτdt=12(u02u(s,)2)

    and Fubini's theorem implies

    12(u02u(s,)2)=s0τ0(ut(τ,)2I(τ))dtdτ+ss0(ut(τ,)2I(τ))dtdτ=s0τ(ut(τ,)2I(τ))dτ+ss(ut(τ,)2I(τ))dτ. (69)

    After the limit s in (69), from (66) and (68) it follows that

    0=limsss(ut(τ,)2I(τ))dτ=limss(u(s,),ut(s,),

    hence (ⅲ) holds. Theorem 5.3 is proved.

    The all authors have been partially supported by the National Scientific Program "Information and Communication Technologies for a Single Digital Market in Science, Education and Security (ICTinSES)", contract No D01205 / 23.11.2018, financed by the Ministry of Education and Science in Bulgaria. The first author has been also supported by the Bulgarian National Science Fund under grant DFNI 12/5. The second author has been also supported by the Bulgarian National Science Fund under grant KΠ-06-H22/2.



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