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Research article

Existence and nonexistence of global solutions for logarithmic hyperbolic equation

  • Received: 18 January 2022 Revised: 15 February 2022 Accepted: 20 February 2022 Published: 08 March 2022
  • This article is concerned with the initial-boundary value problem for a equation of quasi-hyperbolic type with logarithmic nonlinearity. By applying the Galerkin method and logarithmic Sobolev inequality, we prove the existence of global weak solutions for this problem. In addition, by means of the concavity analysis, we discuss the nonexistence of global solutions in the unstable set and give the lifespan estimation of solutions.

    Citation: Yaojun Ye, Qianqian Zhu. Existence and nonexistence of global solutions for logarithmic hyperbolic equation[J]. Electronic Research Archive, 2022, 30(3): 1035-1051. doi: 10.3934/era.2022054

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  • This article is concerned with the initial-boundary value problem for a equation of quasi-hyperbolic type with logarithmic nonlinearity. By applying the Galerkin method and logarithmic Sobolev inequality, we prove the existence of global weak solutions for this problem. In addition, by means of the concavity analysis, we discuss the nonexistence of global solutions in the unstable set and give the lifespan estimation of solutions.



    In this paper, we study the initial-boundary value problem for logarithmic hyperbolic equation of pLaplacian type

    utt+Δpu=|u|p2uln|u|,  (x,t)Ω×R+,  (1.1)
    u(x,0)=u0(x), ut(x,0)=u1(x),  xΩ, (1.2)
    u(x,t)=0, (x,t)Ω×R+, (1.3)

    where

    Δpu=ni=1xi(|uxi|p2uxi), p>2. (1.4)

    ΩRN is a bounded domain with smooth boundary Ω, and parameter p fulfills

    2<p<+, Np;  2<pNpNp, N>p. (1.5)

    The logarithmic nonlinearity arises in a lot of different areas of physics such as quantum mechanics and inflation cosmology, and is applied to nuclear physics, optics and geophysics [1,2,3,4,5,6,7,8]. Because of these special physical meanings, the research of evolution equations with logarithmic nonlinearity has attracted much attention.

    When the logarithmic term in Eq (1.1) is replaced by nonlinear source term |u|r2u, the Eq (1.1) becomes

    utt+Δpu=|u|r2u,  (x,t)Ω×R+.  (1.6)

    In the case of p=2 and r>2, J. Ball [9] and M. Tsustsumi [10] obtained the blow-up solutions with negative initial energy. By using the concavity method, H. A. Levine and L. E. Payne [11] established the nonexistence of global weak solutions for Eq (1.6) with the conditions (1.2) and (1.3). Y. J. Ye [12] proved the global existence and blow-up result of solutions to the Eq (1.6) with initial-boundary value conditions. S. Ibrahim and A. Lyaghfouri [13] considered the Cauchy problem of (1.6), under appropriate assumptions, they established the finite time blow-up of solutions and, hence, extended a result by V. A. Galaktionov and S. I. Pohozaev [14]. For the Eq (1.6) with dissipative term, Y. J. Ye [15,16] studied the global solutions by constructing a stable set in W1,p0(Ω) and the decay property of solution by using an integral inequality [17].

    S. A. Messaoudi and B. S. Said-Houari [18] considered the nonlinear hyperbolic type equation

    utt+Δαu+ΔβutΔut+a|ut|m2ut=b|u|p2u,

    where a,b>0,α,β,m,p>2 and Ω is a bounded domain in RN(N1). Under suitable conditions on α,β,m,p, they proved a global nonexistence result of solutions with negative initial energy. For the nonlinear wave equation of p-Laplacian type

    utt+ΔpuΔut+q(x,u)=f(x).

    C. Chen et al.[19] obtained the global existence and uniqueness of solutions and established the long-time behavior of solutions.

    L. C. Nhan and T. X. Le [20] studied the existence and nonexistence of global weak solutions for a class of p-Laplacian evolution equations with logarithmic nonlinearity and gave sufficient conditions for the large time decay and blow-up of solutions. Later, Y. Z. Han et al. [21] also considered this problem. They studied global solutions and blow-up solution for arbitrarily high initial energy. For a mixed pseudo-parabolic p-Laplacian type equation with logarithmic term, under various assumptions about initial values, H. Ding and J. Zhou [22] proved the solution exists globally and blow up in finite time. Moreover, T. Boudjeriou [23] was concerned with the fractional p-Laplacian with logarithmic nonlinearity, by applying the potential well theory and a differential inequality, he proved the existence and decay estimates of global solutions and obtained the blow-up result of solutions.

    T. Cazenave and A. Haraux [8] considered the following logarithmic wave equation

    uttΔu=uln|u|, (1.7)

    they gave the existence of solutions for the Cauchy problem of Eq (1.7). P. Gorka [4] obtained the global existence of weak solutions for the initial-boundary value problem of Eq (1.7). K. Bartkowski and P. Gorka [7] proved the existence of classical solutions and weak solutions for the corresponding one dimensional Cauchy problem of Eq (1.7). In the case of 0<E(0)d, W. Lian et al. [24] proved the global existence of solution and obtained the blow-up of solution for the Eq (1.7) with the conditions (1.2) and (1.3).

    In this paper, by means of the potential well theory and the concavity analysis method [25,26,27,28,29], we prove the global existence and blow-up of solutions of the problem (1.1)–(1.3).

    For simplicity, we denote Lp(Ω) and L2(Ω) norm by p and by respectively. The space W1,p0(Ω) norm W1,p0(Ω) is replaced by p.

    At first, we define the weak solutions of the problem (1.1)–(1.3) and give a few known lemmas.

    Definition 2.1. [30] If

    uC([0,T),W1,p0(Ω)), utC([0,T),L2(Ω)), uttC([0,T),W1,p(Ω))

    and satisfies

    Ωuttφdx+ΩΔpuφdx=Ω|u|p2uln|u|φdx,

    then u(t) is called a weak solution of (1.1)–(1.3) on [0,T), where φW1,p0(Ω).

    Lemma 2.1. [31] Let q be a real number with 2q<+ if 2np and 2qnpnp if 2<p<n. Then there exists a positive constant C depending on Ω,p and q such that uqCup.

    Lemma 2.2. [30] Let B0,B,B1 be Banach spaces with B0BB1 and

    X={u: uLp([0,T];B0), utLq[0,T];B1)}, 1p,q+.

    Suppose that B0 is compactly embedded in B and that B is continuously embedded in B1, then (i) the embedding of X into Lp(0,T;B) is compact if p<+. (ii) the embedding of X into C([0,T];B) is compact if p=+ and q>1.

    Lemma 2.3. [32] Assume that un(x) is a bounded sequence in Lq(Ω),1q<+, un(x)u(x) a.e.. Then u(x)Lq(Ω) and un(x)u(x) weakly converges in Lq(Ω).

    Lemma 2.4. [33,34,35] (L2logarithmic Sobolev inequality) If vH10(Ω), then

    Ω|v|2ln|v|dxv2lnv+a22πv2n2(1+lna)v2,  a>0. (2.1)

    In order to deal with the logarithmic term |u|p2uln|u| in Eq (1.1), we introduce the following Lplogarithmic Sobolev inequality.

    Lemma 2.5. [36] Let uW1,p0(Ω), then one has the inequality

    Ω|u|pln|u|dxupplnup+(p2)a24πupp+a22πuppnp(1+lna)upp, (2.2)

    where a>0 is a constant.

    For convenience, in the following we are going to give the proof of Lemma 2.5.

    Proof. By (2.1) in Lemma 2.4, we have

    Ω|v|2ln|v|dxv2lnv+a22πv2n2(1+lna)v2=Ω|v|2dxln(Ω|v|2dx)12+a22πΩ|v|2dxn2(1+lna)Ω|v|2dx. (2.3)

    Let v=up2 in (2.3), then we obtain

    12Ω|u|pln|u|pdxΩ|u|pdxln(Ω|u|pdx)12+a22πΩ|up2|2dxn2(1+lna)Ω|u|pdx=12upplnupp+a22πΩ|up2|2dxn2(1+lna)upp. (2.4)

    By direct computation, we get

    up2=p2up21u=p2up22u. (2.5)

    From (2.5) and Hölder inequality, we receive

    Ω|up2|2dx=p24Ω|u|p2|u|2dxp24(Ω|u|pdx)p2p(Ω|u|pdx)2p=p24up2pu2p. (2.6)

    By Young inequality XYXαα+Yββ with α=pp2, β=p2, we conclude that

    p24up2pu2pp24(p2pupp+2pupp)=p(p2)4upp+p2upp. (2.7)

    It follows from (2.4), (2.6) and (2.7) that

    12Ω|u|pln|u|pdx12upplnupp+a22π(p(p2)4upp+p2upp)n2(1+lna)upp,

    which implies

    Ω|u|pln|u|dxupplnup+(p2)a24πupp+a22πuppnp(1+lna)upp.

    This completes the proof of Lemma 2.5.

    Next, we define the following functionals

    J(u)=1pupp1pΩ|u|pln|u|dx+1p2upp (2.8)

    and

    K(u)=upp1pΩ|u|pln|u|dx, (2.9)

    for uW1,p0(Ω). By (2.8) and (2.9), we have

    J(u)=1p2upp+1pK(u). (2.10)

    We denote the energy functional by

    E(t)=12ut2+1pupp1pΩ|u|pln|u|dx+1p2upp=12ut2+J(u), (2.11)

    for uW1,p0(Ω), t0.

    E(0)=12u12+1pu0pp1pΩ|u0|pln|u0|dx+1p2u0pp=12u12+J(u0) (2.12)

    is the initial total energy.

    Moreover, we define the Nehari manifold [37]

    N={uW1,p0(Ω)/{0}: K(u)=0, up0},

    the stable set

    W={uW1,p0(Ω): K(u)>0, J(u)<d}{0}

    and the unstable set

    U={uW1,p0(Ω): K(u)<0 J(u)<d},

    where

    d=inf{supθ0J(θu): uW1,p0(Ω), up0}. (2.13)

    It is readily seen that the potential well depth d defined in (2.13) can also be characterized as

    d=infuNJ(u). (2.14)

    Lemma 2.6. Let uW1,p0(Ω) and up0, then we have

    (a)  limθ0+J(θu)=0, limθ+J(θu)=;
    (b)  K(θu)=θJ(θu){>0,  0<θ<θ,=0,  θ=θ,<0,  θ<θ<+. (2.15)

    Proof. (a) For uW1,p0(Ω),

    J(θu)=θppuppθppΩ|u|pln|u|dxθppupplnθ+θpp2upp.

    It is easy to get from up0 that (a) is valid.

    (b) An elementary calculation shows that

    ddθJ(θu)=θp1(uppΩ|u|pln|u|dxupplnθ). (2.16)

    Let ddθJ(θu)=0, then we have

    θ=exp(uppΩ|u|pln|u|dxupp). (2.17)

    It follows from (2.9) that

    K(θu)=θp(uppΩ|u|pln|u|dxupplnθ). (2.18)

    From (2.16), (2.17) and (2.18), the Eq (2.15) holds.

    Lemma 2.7. Suppose that uW1,p0(Ω) and up0. Then dM, where M=1p2(2π)n2e2(n+p)p22.

    Proof. From Lemma 2.6 and Eq (2.10), one has

    supθ0J(θu)=J(θu)=1p2θupp+1pK(θu)=1p2θupp. (2.19)

    We get from Lemma 2.5 that

    K(u)=upp1pΩ|u|pln|u|dx(1a22π)upp+(npln(ae)(p2)a24πlnup)upp.

    Choosing a=2π, we have

    K(u)(npln(2π e)p22lnup)upp=[ln((2π)n2pe2(n+p)p22p)lnup]upp. (2.20)

    It follows from K(θu)=0 and (2.20) that

    ln((2π)n2e2(n+p)p22)lnθupp0,

    which implies

    θupp(2π)n2e2(n+p)p22. (2.21)

    Thus, we obtain from (2.19) and (2.21) that

    supθ0J(θu)1p2(2π)n2e2(n+p)p22. (2.22)

    Thus, by (2.13) and (2.22), we conclude that dM>0.

    In this section, we state and prove the global existence result for the problem (1.1)–(1.3).

    Theorem 3.1. Assume that p satisfies (1.5). If u0W1,p0(Ω), u1L2(Ω) and 0<E(0)<M, K(u0)0, then there is a global weak solution u(x,t) to the problem (1.1)–(1.3) which meets u(x,t)L([0,+);W1,p0(Ω)), ut(x,t)L([0,+);L2(Ω)).

    Proof. Assume that {ωj}j=1 is a basis of space W1,p0(Ω) and that Vk is the subspace of W1,p0(Ω) generated by {ω1,ω2,,ωm}, m=1,2,. We shall look for the approximate solutions um(t)=mj=1gjm(t)ωj with gjm(t)C2[0,T], T>0. Here the functions gjm(t) fulfil the following system of equations

    (umtt,ωj)+(Δpum,ωj)=(|um|p2umln|um|,ωj), j=1,2,,m (3.1)

    with initial data

    um(0)=u0m, umt(0)=u1m. (3.2)

    Because W1,p0(Ω) is dense in L2(Ω), so there exist αjm and βjm such that

    u0m=mj=1gjm(0)ωj=mj=1αjmωju0(x) strongly in W1,p0(Ω), m, (3.3)
    u1m=mj=1gjm(0)ωj=mj=1βjmωju1(x) strongly in L2(Ω), m. (3.4)

    By Picard's iteration method, the solutions gjm(t) for the Cauchy problem (3.1)–(3.2) exist in t[0,tm), tmT. By the uniformly boundedness of functions gjm(t) and the extension theorem, these solutions gjm(t) exists in the whole interval [0,T].

    Multiplying both sides of (3.1) by gjm(t), summing on j from 1 to m and then integrating over [0,t], we obtain

    Em(t)=12umt(t)2+J(um(t))=12umt(0)2+J(um(0))=Em(0)<Md. (3.5)

    From (3.5), it is easy to verify

    um(t)W, t[0,T]. (3.6)

    Assume that there exists a time t1(0,T) such that um(t1)W, then, by the continuity of um(t) on t, we get um(t1)W. Thus, we receive either

    J(um(t1))=d, (3.7)

    or

    K(um(t1))=0, ump0. (3.8)

    From (3.5), we have J(um(t1))<d. Thus, the case (3.7) is impossible.

    If (3.8) is valid, um(t1)N. From (2.13), we obtain J(um(t1))d. This contradicts with (3.5). Therefore, the case (3.8) is also impossible as well.

    We deduce from (2.10), (3.5) and (3.6) that

    M>J(um)=1p2umpp+1pK(um)>1p2umpp, (3.9)

    which implies that

    umpp<Mp2. (3.10)

    Taking a=π in (2.2), we have from Lemma 2.5, Eqs (2.8) and (2.9) that

    umpp=2K(um)+2Ω|um|pln|um|dxumpp=2pJ(um)2pumppumpp+2Ω|um|pln|um|dx2pJ(um)2pumpp+p22umppnpln(πe2)umpp+2umpplnump2pJ(um)+p22umpp+2umpplnump<CM. (3.11)

    Here CM=2pM+(p2)p22M+2pMln(p2M). From (3.5), we have

    umt2<2M. (3.12)

    For u,vW1,p0(Ω), by (1.4), we have (Δpu,v)=Ω|u|p2|u||v|dx). Hence, from Hölder inequality and (3.11), we obtain

    ΔpuW1,p(Ω)up1p<Cp1pM. (3.13)

    It follows from (3.10)–(3.13) that the following limitations are true.

    umu weakly star in L(0,T;W1,p0(Ω)), (3.14)
    umu weakly star in L(0,T;Lp(Ω)), (3.15)
    umtut weakly star in L(0,T;L2(Ω)), (3.16)
    Δpumχ weakly star in L(0,T;W1,p(Ω)). (3.17)

    Combining (3.15) and (3.16) with Lemma 2.2 yields

    umu strongly in C([0,T];L2(Ω)), (3.18)

    which implies

    |um|p2umln|um||u|p2uln|u| almost everywhere (x,t)Ω×(0,T). (3.19)

    Let Ω1={xΩ: |um(x,t)1} and Ω2={xΩ: |um(x,t)1}, then by means of direct calculation, we know from Lemma 2.1 and Eq (3.11)

    Ω||um|p2umln|um|dx|p=Ω1||um|p2umln|um|dx|p+Ω2||um|p2umln|um|dx|p[(p1)e]p|Ω|+(npp(p1))pΩ2|um|npnp[(p1)e]p|Ω|+(npp(p1))pCnpnpumnpnppLM, (3.20)

    where 1p+1p=1 and LM=[(p1)e]p|Ω|+(npp(p1))pCnpnpCnnpM. From Lemma 2.3, Eqs (3.19) and (3.20), we receive

    |um|p2umln|um||u|p2uln|u| weakly in L(0,T;Lp(Ω)). (3.21)

    Now, we prove χ=Δpu. For this reason, multiplying both sides of (3.1) by an arbitrary smooth function φ(t)C2[0,T] and integrating over [0,T], we have

    (umt(T),φ(T)ωj)+T0(Δpum,φ(t)ωj)dt=(umt(0),φ(0)ωj)+T0(umt,φ(t)ωj)dt+T0(|um|p2umln|um|,φ(t)ωj)dt. (3.22)

    Taking the limitation of both sides of Eq (3.22) with j fixed and m, we get

    (ut(T),φ(T)ωj)+T0(χ,φ(t)ωj)dt=(ut(0),φ(0)ωj)+T0(ut,φ(t)ωj)dt+T0(|u|p2uln|u|,φ(t)ωj)dt. (3.23)

    By (3.23), we have

    (ut(T),ψ(T))+T0(χ,ψ(t))dt=(ut(0),ψ(0))+T0(ut,ψ(t))dt+T0(|u|p2uln|u|,ψ(t))dt, (3.24)

    for every ψL2(0,T;W1,p0(Ω)),  ψL2(0,T;L2(Ω)). In particular, Setting ψ=u in (3.24), we obtain

    (ut(T),u(T))+T0(χ,u)dt=(ut(0),u(0))+T0ut(t)2dt+T0(|u|p2uln|u|,u)dt. (3.25)

    On the other hand, multiplying both sides of (3.1) by gjm(t), summing on j from 1 to m and integrating over [0,T], we get

    (umt(T),um(T))+T0(Δpum,um)dt=(umt(0),um(0)+T0umt2dt+T0(|um|p2umln|um|,um)dt. (3.26)

    Taking the inferior limitation on both sides of (3.26) as m, we have

    (ut(T),u(T))+limminfT0(Δpum,um)dt(ut(0),u(0))+T0ut2dt+T0(|u|p2uln|u|,u)dt. (3.27)

    We conclude from (3.25) and (3.27) that

    limminfT0(Δpum,um)dtT0(χ,u)dt. (3.28)

    By the monotonicity of operator Δp, we have

    T0(ΔpumΔpv,umv)dt0, vL(0,T;W1,p0(Ω)). (3.29)

    We get from (3.28) and (3.29) that

    limminfT0(ΔpumΔpv,umv)dtT0(χ,uv)dtT0(Δpv,uv)dt=T0(χΔpv,uv)dt. (3.30)

    Combining (3.29) with (3.30) yields that

    T0(χΔpv,uv)dt0. (3.31)

    Let v=uλω, then, by (3.31), we obtain

    λT0(χΔp(uλω),ω)dt0, (3.32)

    for any ωLp(0,T;W1,p0(Ω)) and any real number λ.

    As λ>0,λ0, from (3.32) and the hemicontinuity of operator Δp, we conclude that

    T0(χΔpu,ω)dt0. (3.33)

    Similarly, when λ<0,λ0, we have

    T0(χΔpu,ω)dt0. (3.34)

    Thus, for all ωLp(0,T;W1,p0(Ω)), we deduce from (3.33) and (3.34) that

    T0(χΔpu,ω)dt=0, (3.35)

    which implies that χ=Δpu.

    Next, we prove above solution u(x,t) satisfies (1.2), i.e., u(x,0)=u0(x), ut(x,0)=u1(x).

    We conclude from Eqs (3.15), (3.16) and Lemma 1.2 that u(t):[0,T]L2(Ω) is continuous. Therefore, um(0)u(0) weakly in L2(Ω). According to (3.3), one has u(0)=u0.

    To prove ut(0)=u1, let ξ(t) be a smooth function with ξ(0)=1,ξ(T)=0. Noting

    T0(umtt,ξωj)dt=T0(umt,ξtωj)dt(umt(0),ξ(0)ωj).

    For given j, as m, we get in the distribution sense

    T0(utt,ξωj)dt=T0(ut,ξtωj)dt(ut(0),ξ(0)ωj) (3.36)

    in D([0,T]). On the other hand,

    T0(umtt,ξωj)dt=T0[(Δpum,ξωj)+(|um|p2umln|um|,ξωj)]dt

    converges to

    T0[(Δpu,ξωj)+(|u|p2uln|u|,ξωj)]dt=T0(utt,ξωj)dt

    as m. Therefore,

    T0(utt,ξωj)dt=T0(ut,ξtωj)dt(u1,ξ(0)ωj). (3.37)

    From (3.36) and (3.37), we have (ut(0),ωj)=(u1,ωj). By the density of{ωj}mj=1 in L2(Ω), we get ut(0)=u1. This completes the proof of Theorem 3.1.

    For the case of K(u0)0 and E(0)=Md, the global existence result of solutions to the problem (1.1)–(1.3) reads as follows:

    Theorem 3.2. Assume that p fulfils (1.5). If u0W1,p0(Ω), u1L2(Ω) and E(0)=Md, K(u0)0, then there exists a global weak solution u(x,t) for the problem (1.1)–(1.3) which satisfies u(x,t)L([0,+);W1,p0(Ω)), ut(x,t)L([0,+);L2(Ω)).

    Proof. For the case u0p0, let us suppose that ρk=11k and u0k=ρku0, k2. The problem (1.1)–(1.3) can be written as follows:

    {utt+Δpu=|u|p2uln|u|,  (x,t)Ω×R+,u(x,0)=u0k(x), ut(x,0)=u1(x),  xΩ,u(x,t)=0, (x,t)Ω×R+. (3.38)

    From K(u0)0 and Lemma 2.6, we have θ=θ(u0)1. Accordingly, we get K(u0k)>0. By (2.3), we obtain

    0<J(u0k)=1p2u0kpp+1pK(u0k)<J(u0). (3.39)

    Therefore, we receive

    0<Ek(0)=12u12+J(u0k)<12u12+J(u0)=E(0)=Md,

    which implies that u0kW.

    For each k, by Theorem 3.1, there exists a global weak solution uk(t) of the problem (3.38) such that uk(t)L([0,+);W1,p0(Ω)),  ukt(t)L([0,+);L2(Ω)) and

    (ukt,v)+t0(Δpuk,v)ds=(u1,v)+t0(|uk|p2ukln|uk|,v)ds (3.40)

    for any vW1,p0(Ω).

    In addition,

    Ek(t)=12ukt2+J(uk)=12u12+J(u0k)=Ek(0)<Md. (3.41)

    By using (3.41) and combining with the same argument as (3.6), we can prove uk(t)W.

    For the case u0p=0, we get J(u0)=0 by K(u0)0. Thus, we have E(0)=12u12+J(u0)=12u12=M. Let ρk=11k, u1k=ρku1(x), k2, we consider the following problem

    {utt+Δpu=|u|p2uln|u|,  (x,t)Ω×R+,u(x,0)=u0(x), ut(x,0)=u1k(x),  xΩ,u(x,t)=0, (x,t)Ω×R+. (3.42)

    Noting

    0<Ek(0)=12u1k2+J(u0)=12ρku12<12u12=M. (3.43)

    By Eq (3.43) and Theorem 3.1, there is a global weak solution uk(t) for the problem (3.42) such that uk(t)L(0,+;W1,p0(Ω)), ukt(t)L(0,+;L2(Ω)) and uk(t)W for each k.

    The remainder of the proof for Theorem 3.2 is the same as those of Theorem 3.1. Here, we omit them.

    Lemma 4.1. [38,39] If nonnegative function Φ(t)C2 satisfies

    Φ(t)Φ(t)(1+ρ)Φ(t)20,

    for Φ(0)>0, Φ(0)>0 and ρ>0, then there exists a time T such that 0<TΦ(0)ρΦ(0) and limtTΦ(t)=+.

    Lemma 4.2. Suppose that u(t) is a solution of (1.1)–(1.3). If u0U and E(0)<d, then u(t)U and E(t)<d, t0.

    Proof. From the conservation of energy, we obtain E(t)=E(0)<d. From (2.11), we get

    J(u)E(t)<d. (4.1)

    Assume that there is t[0,+) such that u(t)U, then by continuity of K(u(t)) on t, we obtain K(u(t))=0. That means u(t)N. From (2.14), we have J(u(t))d, which is contradiction with (4.1). Therefore, the conclusion in Lemma 4.2 holds.

    Theorem 4.1. Suppose that 0<E(0)<d and Ωu0u1dx>0, then there is no global weak solution u(t) to the problem (1.1)–(1.3). Namely, there exists a time T such that limtTu(t)2=+, where the lifespan T is estimated by 0<T<4Ψ(0)(p2)Ψ(0), Ψ(t) is given in (4.19).

    Proof. By u0U, E(0)<d and Lemma 4.2, we get uU. Thus,

    K(u)=uppΩ|u|pln|u|dx<0. (4.2)

    From (2.13) and (2.19), we have

    dsupθ0J(θu)=1p2θupp. (4.3)

    We deduce from (2.17), (4.2) and (4.3) that

    d1p2upp. (4.4)

    Let

    Ψ(t)=u(t)2=Ωu2dx. (4.5)

    Then there is a real number α>0, which satisfies

    Ψ(t)α>0. (4.6)

    By differentiating on both sides of (4.5), we get

    Ψ(t)=2Ωuutdx. (4.7)

    From (4.7), we obtain

    Ψ(t)=2ut2+2Ωuuttdx. (4.8)

    Combining (1.1) with (4.8), we get

    Ψ(t)=2(ut(t)2+Ω|u|pln|u|dxupp)=2[ut(t)2K(u)]. (4.9)

    By uU and (4.9), we receive Ψ(t)>0. Combining (4.5), (4.7) and (4.9), we get

    Ψ(t)Ψ(t)p+24Ψ(t)2=2Ψ(t)[ut(t)2+Ω|u|pln|u|dxupp](p+2)Ψ(t)ut(t)2+(p+2)Υ(t), (4.10)

    where

    Υ(t)=u(t)2ut(t)2(Ωuutdx)2. (4.11)

    By Cauchy-Schwarz inequality, we get

    (Ωuutdx)2u(t)2ut(t)2. (4.12)

    This inequality (4.12) guarantees Υ(t)0. By (4.10), we have

    Ψ(t)Ψ(t)p+24Ψ(t)2Ψ(t)Π(t), (4.13)

    where

    Π(t)=put2+2Ω|u|pln|u|dx2upp. (4.14)

    From (2.11) and (4.14), we obtain

    Π(t)=2pE(t)+2pupp. (4.15)

    By (4.4), (4.15) and E(t)=E(0)<d, we get

    Π(t)2pE(0)+2pd=2p[dE(0)]>0. (4.16)

    Therefore, there exists β>0 such that

    Π(t)β>0. (4.17)

    Combining (4.6), (4.13) and (4.17), we conclude that

    Ψ(t)Ψ(t)p+24Ψ(t)2αβ>0, t0. (4.18)

    Let ρ=p24>0, then, by the differential inequality (4.18) and Lemma 4.1, one has

    0<T<4Ψ(0)(p2)Ψ(0), (4.19)

    and

    limtTΨ(t)=+. (4.20)

    From (4.5) and (4.20), we have limtTu(t)2=+.

    This completes the proof of Theorem 4.1.

    By applying Galerkin method and Lp-Sobolev logarithmic inequality, and combining with the potential well theory, we prove the global existence result of solutions in this paper. Namely, assume that p satisfies (1.5). If u0W1,p0(Ω), u1L2(Ω) and 0<E(0)M, K(u0)0, then there is a global weak solution u(x,t) of the problem (1.1)–(1.3). Meanwhile, under the condition of positive initial energy, by using the concavity analysis method, we establish the finite time blow-up result of solutions and give the lifespan estimate of solutions. The result read as follows: If 0<E(0)<d and Ωu0u1dx>0, then the solutions of the problem (1.1)–(1.4) blows up in finite time and the lifespan T is estimated by 0<T<4Ψ(0)(p2)Ψ(0).

    The authors would like to thank the reviewers and editors for their help to improve the quality of this article. Moreover, this research was supported by Natural Science Foundation of Zhejiang Province (No. LY17A010009).

    The authors declare that there is no conflict of interests regarding the publication of this paper.



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