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Review

Genetic etiology of cleft lip and cleft palate

  • Genetic studies in humans have demonstrated that Cleft lip with or without cleft palate (CL/P) have a diverse genetic background and probably environmental factors influencing these malformations. CL/P is one of the most common congenital birth defects in the craniofacial region with complex etiology involving multiple genetic factors, environmental factors and gene-environment interaction. Children born with these defects suffer from various difficulties such as difficulty in speech, hearing, feeding and other psychosocial problems, and their rehabilitation involves a multidisciplinary approach. The article describes the brief introduction of CL/P, epidemiology and general concepts, etiological factors, and the genes implicated in the etiology of nonsyndromic CL/P (NSCL/P) as suggested by different human genetic studies, animal models, and other expression studies.

    Citation: AN Mahamad Irfanulla Khan, CS Prashanth, N Srinath. Genetic etiology of cleft lip and cleft palate[J]. AIMS Molecular Science, 2020, 7(4): 328-348. doi: 10.3934/molsci.2020016

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  • Genetic studies in humans have demonstrated that Cleft lip with or without cleft palate (CL/P) have a diverse genetic background and probably environmental factors influencing these malformations. CL/P is one of the most common congenital birth defects in the craniofacial region with complex etiology involving multiple genetic factors, environmental factors and gene-environment interaction. Children born with these defects suffer from various difficulties such as difficulty in speech, hearing, feeding and other psychosocial problems, and their rehabilitation involves a multidisciplinary approach. The article describes the brief introduction of CL/P, epidemiology and general concepts, etiological factors, and the genes implicated in the etiology of nonsyndromic CL/P (NSCL/P) as suggested by different human genetic studies, animal models, and other expression studies.


    In this paper, we deal with the following plate equation with nonlinear damping and a logarithmic source term

    {utt+Δ2u+|ut|m2ut=|u|p2ulog|u|k,(x,t)Ω×R+,u=uν=0,(x,t)Ω×R+,u(x,0)=u0(x),ut(x,0)=u1(x),xΩ, (1)

    where Ω is a bounded domain in Rn (n1) with sufficiently smooth boundary Ω, ν is the unit outer normal to Ω and k is a positive real number, u0(x), and u1(x) are given initial data. The parameter m2 and p satisfies

    2<p<2(n2)n4ifn5;2<p<+ifn4. (2)

    The logarithmic nonlinearity is of much interest in many branches of physics such as nuclear physics, optics and geophysics (see [5,6,15] and references therein). It has also been applied in quantum field theory, where this kind of nonlinearity appears naturally in cosmological inflation and in super symmetric field theories [4,13].

    Let us review somework with logarithmic term which is closely related to the problem (1). Birula and Mycielski[6,7] studied the following problem

    {uttuxx+uεulog|u|2=0,(x,t)[a,b]×(0,T),u(a,t)=u(b,t)=0,t(0,T),u(x,0)=u0(x),ut(x,0)=u1(x),x[a,b], (3)

    which is a relativistic version of logarithmic quantum mechanics and can also be obtained by taking the limit p1 for the p-adic string equation [16,36]. Cazenave and Haraux [8] established the existence and uniqueness of a solution to the Cauchy problem for the following equation

    uttΔu=ulog|u|k, (4)

    in R3. Using some compactness method, Gorka [15] established the global existence of weak solutions for all (u0,u1)H10×L2 to the initial boundary value problem of equation (4) in the one-dimensional case. In [5], Bartkowski and Gorka obtained the existence of classical solutions and investigated weak solutions for the corresponding Cauchy problem of equation (4) in the one-dimensional case. Recently, using potential well combined with logarithmic Sobolev inequality, Lian et al. [25] derived the global existence and infinite time blow up of the solution to the initial boundary value problem of (4) in finite dimensional case under suitable assumptions on initial data. Similar results were obtained by Lian et al. [26] for nonlinear wave equation with weak and strong damping terms and logarithmic source term. Hiramatsu et al. [19] also introduced the following equation

    uttΔu+u+ut+|u|2u=ulog|u| (5)

    to study the dynamics of Q-ball in theoretical physics. A numerical research was given in that work, while, there was no theoretical analysis for this problem. For the initial boundary value problem of(5), Han [17] obtained the global existence of weak solution in R3, and Zhang et al. [40] obtained the decay estimate of energy for the problem (5) in finite dimensional case. Later, the authors in [20] considered the initial boundary problem of (5) in ΩR3, they proved that the solution will grow exponentially as time goes to infinity if the solution lies in unstable set or the initial energy is negative; the decay rate of the energy was also obtained if the solution lies in a smaller set compared with the stable set. Peyravi[35] extended the results obtained in [20] to the following logarithmic wave equation

    uttΔu+u+(gΔu)(t)+h(ut)ut+|u|2u=ulog|u|k.

    Recently, Al-Gharabli and Messaoudi [1] considered the following plate equation with logarithmic source term

    {utt+Δ2u+u+h(ut)=ulog|u|k,(x,t)Ω×R+,u=uν=0,(x,t)Ω×R+,u(x,0)=u0(x),ut(x,0)=u1(x),xΩ, (6)

    where Ω is a bounded domain in R2, and obtained the global existence and decay rate of the solution using the multiplier method. As the special case, i.e. h(ut)=ut in (6), the same authors [2] established the global existence and the decay estimate by constructing a Lyapunov function. Moreover, Al-Gharabli et al. [3] considered the following initial boundary value problem of viscoelastic plate equation with with logarithmic source term

    utt+Δ2u+ut0g(ts)Δ2u(s)ds=ulog|u|k,(x,t)Ω×R+, (7)

    they established the existence of solutions and proved an explicit and general decay rate result. However, there is no information on the finite or infinite blow up results in these researches [1,2,3].

    At the same time, there are many results concerning the existence and nonexistence on evolution equation with polynomial source term. For example, for plate equation with polynomial source term |u|p2u, Messaoudi [31] considered the following problem

    {utt+Δ2u+|ut|m2ut=|u|p2u,(x,t)Ω×R+,u=uν=0,(x,t)Ω×R+,u(x,0)=u0(x),ut(x,0)=u1(x),xΩ,

    established an existence result and showed that the solution continues to exist globally if mp and blows up in finite time if m<p and the initial energy is negative. This result was later improved by Chen and Zhou [12], see also Wu and Tsai [37]. Here, we also mention that there are a lot of results on the global well-posedness of solutions to the initial boundary value problem of nonlinear wave equations can be found [30,39] and papers cited therein by using of potential well method.

    To the best of our knowledge, there are few results on the evolution equation with the nonlinear logarithmic source term |u|p2ulog|u|k (p>2). Kafini and Messaoudi [22] studied the following delayed wave equation with nonlinear logarithmic source

    uttΔu+μ1ut(x,t)+μ2ut(x,tτ)=|u|p2ulog|u|k, (8)

    obtained the local existence by using the semigroup theory and proved a finite time blow-up result when the initial energy is negative. Of course, these results also hold for the equation (8) without delay term (i.e. μ2=0). However, there are no results on general decay and blow-up for positive initial energy. Chen et al. [9,10] studied parabolic type equations with logarithmic nonlinearity ulog|u|k, and obtained the global existence of solution and the solutions cannot blow up in finite time. Recently, Chen and Xu [11] study the initial-boundary value problem for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity and obtain the similar results. Nhan and Truong studied parabolic p-Laplacian equation [23] and pseudo-parabolic p-Laplacian equation [24] with logarithmic nonlinearity |u|p2ulog|u|k where they need the p-Laplacian term to control the logarithmic nonlinearity. We also refer to [18], where pseudo-parabolic p-Laplacian equation with logarithmic nonlinearity |u|q2ulog|u|k was considered.

    Motivated by the above mentioned papers, our purpose in this research is to investigative the existence, energy decay and finite time blow-up of the solution to the initial boundary value problem (1). We note here that (i) the term u plays an important role in the studying the problem (6)(see [1,2]) and (7) (see [3]) when the Logarithmic Sobolev inequality is used, while we do not care the term u in this paper; (ii) The constant k in (6) and (7) should satisfy 0<k<k0, where k0 is defined by 2πk0cp=e32k0 (see details in [1,2,3]), while we only need k>0.

    The rest of this article is organized as follows: Section 2 is concerned with some notation and some properties of the potential well. In Sect. 3, we present the existence and uniqueness of local solutions to (1) by using the contraction mapping principle. In Sect. 4, we prove the global existence and energy decay results. The proof of global existence result is based on the potential well theory and the continuous principle; while for energy decay result, the proof is based on the Nakao's inequality and some techniques on logarithmic nonlinearity. In Sect. 5, we prove the the finite time blow-up when the initial energy is negative. In Sect. 6, we establish the finite time blow-up result for problem (1) with m=2 under the arbitrarily high initial energy level (E(0)>0).

    We give some material needed in the proof of our results. We use the standard Lebesgue space Lp(Ω) and Sobolev space H20(Ω) with their usual scalar products and norms. In particular, we denote .=.2. By Poincarˊe 's inequality [27], we have that Δ is equivalent to H20 and we will use Δ as the norm of H20, the corresponding duality between H20 and H2 is denote by ,. We also use C and Ci to denote various positive constant that may have different values in different places.

    Firstly, we introduce the Sobolev's embedding inequality : assume that p be a constant such that 1p2nn4 if n5; p1 if n4, then H20(Ω)Lp(Ω) continuously, and

    upCpΔu2,for uH20(Ω) (9)

    where Cp denotes the best embedding constant.

    Suppose (2) holds, we define

    α:={2nn4p if n5,+ if n4

    for any α[0,α), then H20(Ω)Lp+α(Ω) continuously. And we denote Cp+α by C.

    Definition 2.1. A function uC([0,T],H20(Ω))C1([0,T],L2(Ω))C2([0,T],H2(Ω)) with utLm([0,T],Lm(Ω)) is called a weak solution to (1) if the following conditions hold

    u(0)=u0,ut(0)=u1

    and

    utt,v+(Δu,Δv)+Ω|ut|m2utvdx=Ω(|u|p2ulog|u|k)vdx (10)

    for any vH20(Ω) and a.e. t[0,T].

    Now, we introduce the energy functional J and the Nehari functional I defined on H20(Ω){0} by

    J(u)=J(u(t))=J(t)=12Δu21pΩ|u|plog|u|kdx+kp2upp, (11)

    and

    I(u)=I(u(t))=I(t)=Δu2Ω|u|plog|u|kdx. (12)

    From the definitions (11) and (12), we have

    J(u)=1pI(u)+(121p)Δu2+kp2upp. (13)

    The following lemmas play an important role in the studying the properties of the potential well.

    Lemma 2.2. Let uH20(Ω){0} and g(λ)=J(λu). Then we have

    (ⅰ): limλ0+J(λu)=0,limλ+J(λu)=;

    (ⅱ): there exists a unique λ=λ(u)>0 such that ddλJ(λu)λ=λ=0, and J(λu) is increasing on (0,λ), decreasing on (λ,+) and taking the maximum at λ.

    (ⅲ): I(λu)>0 for 0<λ<λ,I(λu)<0 for λ<λ<+ and I(λu)=0.

    Proof. We know

    g(λ)=J(λu)==12λ2Δu2kpλpΩ|u|plog|u|dxkpλplogλupp+kp2λpupp.

    It is obvious that (i) holds due to p2 and up0. Taking derivative of g(λ) we obtain

    g(λ)=λ(Δu2kλp2Ω|u|plog|u|dxkλp2logλupp) (14)

    and

    g(λ)=Δu2k(p1)λp2Ω|u|plog|u|dxk(p1)λp2logλuppkλp2upp.

    From (14) and p2, we see that there exists a unique positive λ such that

    g(λ)|λ=λ=0,

    then we obtain

    Δu2=kλp2Ωu2log|u|dx+kλp2logλupp.

    Substituting the above equation into g(λ), we have

    g(λ)=k(p2)λp2Ω|u|plog|u|dxk(p2)λp2logλuppkλp2upp=(p2)Δu2kλp2upp<0.

    From these and (i), we can yield that g(λ) has a maximum value at λ=λ and J(λu) increasing on 0<λλ and decreasing on λλ<+. So (ii) holds.

    From (12) and (14), we have

    I(λu)=λddλJ(λu)=λg(λ){>0,0<λ<λ,=0,λ=λ,<0,λ<λ<+.

    Then, we could define the potential well depth of the functional J (also known as mountain pass level) by

    d=inf{supλ0J(λu)|uH20(Ω){0}}. (15)

    We also define the well-known Nehari manifold

    N={u|uH20(Ω){0},I(u)=0}.

    As in [29,34], that the mountain pass level d defined in (15) can also be characterized as

    d=infuNJ(u).

    It is easy to see that d0 from (13). Now, we will prove that d is strictly positive.

    Lemma 2.3. Assume that p>2 holds. Let α(0,α), and

    r(α):=(αkCp+α)1p+α2.

    Then, for any uH20(Ω){0}, we have

    (ⅰ) : if u2r(α), then I(u)>0

    (ⅱ) : if I(u)0, then Δu2>r(α).

    Proof. Since logy<y for any constant y>0, using (9) and the definition of I(u) in (12), we obtain that

    I(u)=u22kΩ|u|plog|u|dx>u22kαup+αp+αu22kCp+ααup+α2=kCp+ααu22(rp+α2(α)up+α22). (16)

    Obviously, the results can be obtained from the above inequality (16).

    Lemma 2.4. Assume the notations in Lemma 2.2 hold, we have

    0<r:=supα(0,α)=(αkCp+α)1p+α2r:=supα(0,α)(αkBp+α)1p+α2|Ω|αp(p+α2)<+,

    where B=Cp as in (9) and |Ω| is the measure of Ω.

    Proof. It is obvious that r>0 (if exists), hence, we only need to prove r(α)γ(α), r exists and r<+, where

    γ(α)=(αkBp+α)1p+α2|Ω|αp(p+α2),α(0,+).

    For any uH20(Ω), using the Hölder's inequality, we have

    up|Ω|αp(p+α)up+α.

    Then, noticing C=Cp+α and B=Cp, we deduce

    C=supuH20{0}up+αΔu2|Ω|αp(p+α)supuH20{0}upΔu2|Ω|αP(p+α)B,

    which implies

    (αkCp+α)1p+α2(αkBp+α)1p+α2|Ω|αp(p+α2),

    that is r(α)γ(α).

    Now, we will prove r exists and r<+. For this purpose, we divide the proof into two cases.

    Case a. If n5, we see that α(0,α)=(0,2nn4p) and γ(α) is continuous on closed interval [0,2nn4p]. Hence, we have r exists and

    r=supα(0,2nn4p)γ(α)maxα[0,2nn4p]γ(α)<+

    Case b. If n4, we define the following auxiliary function

    h(α):=log[γ(α)]=1p+α2[logαlogk(p+α)logB]+αp(p+α2)log|Ω|.

    Hence

    h(α)=p2+pα2p+pαlogkpαlogα+2pαlogB+pαlog|Ω|2αlog|Ω|pα(p+α2)2.

    For simplicity, we set

    g(α):=p2+pα2p+pαlogkpαlogα+2pαlogB+pαlog|Ω|2αlog|Ω|,

    then

    g(α)=p+plogkplogαp+2plogB+plog|Ω|2log|Ω|=plogkB2|Ω|12pα,

    which yields that the function g(α) is strictly increasing on (0,kB2|Ω|12p) and strictly decreasing on (kB2|Ω|12p,+).

    On the one hand, due to p>2, it is easy to see that

    limα0+g(α)=p22p>0

    which implies that g(α)>0 for α(0,kB2|Ω|12p) by g(α) is strictly increasing on (0,kB2|Ω|12p).

    While on the other hand, we can deduce that

    limα+g(α)=limα+(p22p+pα[1+log(kB2|Ω|12p)]logα)=,

    which together with g(kB2|Ω|12p)>0 and g(α) is strictly decreasing on (kB2|Ω|12p,+), implies that there exists a unique α(kB2|Ω|12p,+) such that g(α)=0.

    Noting the relation between h(α) and g(α), we deduce that h(α)>0 for α(0,α), and h(α)<0 for α(α,+). Therefore, h(α) achieves its maximum value at α=α, that is

    r=supα(0,+)σ(α)=eh(α)<+.

    Making using of the Lemmas 2.2 and 2.3, we obtain the following corollary.

    Corollary 1. Assume that p>2 holds. Then, we have

    (ⅰ): if u2<r, then I(u)>0;

    (ⅱ): if I(u)0, then u2r

    for any uH20(Ω){0}, where r is the positive constant defined in Lemma 2.3.

    Lemma 2.5. Assume that p2 holds. Then the constant d defined in (15) is strictly positive.

    Proof. (ⅰ) For the case p=2, we have dk4(2πk)n2en(see [9,20,25] for details).

    (ⅱ) For the case p>2, by (ii) of Corollary 2.1, we get that u2r if uN. Then, making using of (13) with I(u)>0, we obtian

    J(u)=(121p)Δu22+kp2upp(p22p)r2>0.

    We define energy for the problem (1), which obeys the following energy equality of the weak solution u

    E(t)+t0uτmmdτ=E(0), for all t[0,T) (17)

    where

    E(t)=12ut2+12Δu21pΩ|u|plog|u|kdx+kp2upp,
    E(0)=12u12+12Δu021pΩ|u0|plog|u0|kdx+kp2u0pp.

    It is obvious that

    E(t)=12ut2+J(u).

    Taking v=ut in (10), after a simple calculation, we get

    ddtE(t)=utmm. (18)

    Now, we define the subsets of H20(Ω) related to problem (1). Set

    W={uH10(Ω)|J(u)<d,I(u)>0},V={uH10(Ω)|J(u)<d,I(u)<0}, (19)

    where W and V are called the stable and unstable set, respectively [21].

    In order to establish the global existence and blow-up results of solution, we have to prove the following invariance sets of W and V.

    Lemma 2.6. If u0H20,u1L2,p2,E(0)<d, and u is a weak solution of problem (1) on [0,T), where T is the maximal existence time of weak solution, then

    (ⅰ): uW if I(u0)>0;

    (ⅱ): uVifI(u0)<0.

    Proof. It follows from the definition of weak solution and (17) that

    12ut2+J(u)12u12+J(u0)<d, for any t[0,T). (20)

    (ⅰ) Arguing by contradiction, we assume that there exists a number t0(0,T) such that u(t)W on [0,t0) and u(t0)W. Then, by the continuity of J(u(t)) and I(u(t)) with respect to t, we have either (a)J(u(t0))=d or (b) I(u(t0))=0 and u(t0)0.

    It follows from (20) that (a) is impossible. If (b) holds, then by the definition of d, we have J(u(t0))d, which contradicts (20) again. Thus, we have u(t)W for all t[0,T).

    (ⅱ) The proof is similar to the proof of (ⅰ). We omit it.

    In this section, we are concerned with the local existence and uniqueness for the solution of the problem (1). The idea comes from [14,28,38], where the source term is polynomial. First, we give a technical lemma given in [22] which plays an important role in the uniqueness of the solution.

    Lemma 3.1. ([22]) For every ε>0, there exists A>0, such that the real function

    j(s)=|s|p2log|s|,p>2

    satisfies

    |j(s)|A+|s|p2+ε.

    Theorem 3.2. Suppose that u0H20(Ω),u1L2(Ω), and p>2, then there is T>0, such that the problem (1) admits a unique local weak solution on [0,T].

    Proof. For every T>0, we consider the space

    H:=C([0,T];H20(Ω))C1([0,T];L2(Ω))

    endowed with the norm

    u(t)H=(maxt[0,T](Δu(t)22+ut(t)22))12.

    For every given uH, we consider the following initial boundary value problem

    {vtt+Δ2v+|vt|m2vt=|u|p2ulog|u|k,(x,t)Ω×R+,v=vν=0,(x,t)Ω×R+,v(x,0)=u0(x),vt(x,0)=u1(x),xΩ. (21)

    We shall prove that the problem (21) admits a unique solution vHC2(|0,T|,H2(Ω)) with vtLm(|0,T|,Lm(Ω)).

    Let Wh=Span{w1,,wh}, where {ωi}i=1 is the orthogonal complete system of eigenfunctions of Δ2 in H20(Ω) with wi=1 for all i. Then, {wi} is orthogonal and complete in L2(Ω) and in H20(Ω); denote by {λi} the related eigenvalues to their multiplicity. Let

    u0h=hi=1(ΩΔu0Δwi)wi and u1h=hi=1(Ωu1wi)wi

    such that u0hWh,u1kWh,u0hu0 in H20(Ω) and u1hu1 in L2(Ω) as h. For each h>0 we seek h functions γ1h,,γhhC2[0,T] such that

    vh(t)=hi=1γih(t)ωi, (22)

    solves the following problem

    {Ω(vh+Δ2vh+|vh|m2vh|u|p2ulog|u|k)ηdx=0,vk(0)=u0h,vh(0)=u1h. (23)

    For i=1,,h, taking η=wi in (23) yields the following Cauchy problem for a nonlinear ordinary differential equation with unknown γih

    {γih(t)+λiγih(t)+ci|γih(t)|m2γih(t)=ψi(t),γih(0)=Ωu0ωi,γih(0)=Ωu1ωi,

    where

    ci=ωimm,ψi(t)=Ω|u(t)p2u(t)log|u|kωidxC[0,T].

    Then the above problem admits a unique local solution γihC2[0,T] for all i, which in turn implies a unique vh defined by (22) satisfying (23).

    Taking η=vh(t) into (23) and integrating over [0,t][0,T], we have

    vh(t)2+Δvh(t)2+2t0vh(τ)mmdτ=v1h2+Δv0h2+2t0Ω|u|p2ulog|u|kvh (24)

    for every h1. We estimate the last term in the right-hand side of (24) thanks to Young's and Sobolev's inequalities

    2t0Ω|u|p2ulog|u|kvh
    2t0Ω||u|p1log|u|k||vh|t0Ω(C||u|p1log|u||mm1+t0vhmm). (25)

    In order to estimate (25), we focus on the logarithmic term. Here we denote Ω:=Ω1Ω2, where Ω1={xΩ||u(x)|<1} and Ω2={xΩ||u(x)|1}. Then we have

    Ωu|p1log|u||mm1dx=Ω1u|p1log|u||mm1dx+Ω2u|p1log|u||mm1dx.

    By a simple calculation, we obtain

    infs(0,1)sp1logs=1e(p1),

    which implies

    Ω1u|p1log|u||mm1dx[e(p1)]mm1|Ω|.

    Let

    ρ=2nn4m1mp+1>0forn5; any positive ρ,n4.

    By the Sobolev embedding from H20(Ω) to L2nn4(Ω) if n5 and to Lq(Ω) for any q1 if n4, recalling uH:=C([0,T];H20(Ω)), we have

    Ω2u|p1log|u||mm1dxρm1mΩ2(|u|p1+ρ)m1mdxρm1mΩ2|u|2nn4dxρm1mΩ|u|2nn4dx=ρm1mu2nn42nn4Cu2nn4H20C.

    The proof of the case n4 is similar. From the above discussion, (25) yields

    2t0Ω|u|p2ulog|u|kvhCT+t0vhmm. (26)

    Substituting this inequality into (24), we obtain

    vh(t)2+Δvh(t)2+t0vh(τ)mmdτC, (27)

    where C>0 is independent of h. It follows from (27) that

    vh(t) is bounded in L([0,T],H20(Ω)),vh(t) is bounded in Lm[(0,T],Lm(Ω))L([0,T],L2(Ω)),[2mm]vh(t) is bounded in L2([0,T],H2(Ω)). (28)

    Hence, up to a subsequence, we could pass to the limit in (23) and obtain a weak solution v of (21) with regularity (28). Then, we have vC([0,T],H20(Ω))C1([0,T],L2(Ω)) with vtLm([0,T],Lm(Ω)). Finally, from (21), we obtain vC0([0,T],H2(Ω)). Then the weak local solution of problem (21) has been obtained.

    To prove the uniqueness, arguing by contradiction: if w and v were two solutions of (21) which have the same initial data. Subtracting these two equations and testing the result by wtvt, we could obtain

    wtvt2+ΔwΔv2+2t0Ω(|wτ|m2wτ|vτ|m2vτ)(wτvτ)=0. (29)

    It follows from the following element inequality

    (|φ|m2φ|ψ|m2ψ)(φψ)C|φψ|m for m2,

    that (29) can make to be

    wtvt2+vw2H20+CT0wτvτmm0.

    Therefore, we have w=v, i.e. the problem (21) obeys a unique weak solution.

    Now, we are in the position to prove Theorem 3.1. For u0H20(Ω),u1L2(Ω), we denote

    R2:=2(u12+Δu02),

    and

    BRT:={uH|u(0,x)=u0(x),ut(0,x)=u1(x),uHR}

    for every T>0. From the above discussion, for any uBRT, we could introduce a map Φ:HH defined by v=Φ(u), where v is the unique solution to (21).

    Claim. Φ is a contract map satisfying Φ(BRT)BRT, for small T>0.

    In fact, assume that uBRT, the corresponding solution v=Φ(u) satisfies (21) for all t[0,T]. Thus, as did the proof (26) and (27), we have

    vt(t)2+Δv(t)2u12+Δu02+CR2nn4TR22+CR2nn4T

    for n5, For the case n4, the index 2nn4 of R in the last inequality can be replaced by any fixed positive number. If T is sufficiently small, then vHR, which implies that Φ(BRT)BRT

    Next we show that Φ is contractive in BRT. We set v1=Φ(w1),v2=Φ(w2) with w1,w2BRT, and v=v1v2, then, v satisfies

    vtt,η+(Δv,Δη)+Ω(|v1t|m2v1t|v2t|m2v2t)ηdx=Ω(|w1|p2w1log|w1|k|w2|p2w2log|w2|k)ηdx, (30)

    for any ηH20(Ω) and a.e. t[0,T].

    Taking η=vt=v1tv2t, noticing

    Ω(|v1t|m2v1t|v2t|m2v2t)(v1tv2t)dx0,

    and integrating both sides of (30) over (0,t), we have

    vt2+Δv22k|w1|p2w1log|w1||w2|p2w2log|w2|vt, (31)

    We need estimating the logarithmic term in (31) by using Lemma 3.1. By the similar argument as [22], we give the sketch of the proof.

    Making use of mean value theorem, we have, for 0<θ<1,

    ||w1|p2w1log|w1||w2|p2w2log|w2||=k|1+(p1)log|θw1+(1θ)w2|||θw1+(1θ)w2|p2|w1w2|.

    Then, it follows from Lemma 3.1 that

    ||w1|p2w1log|w1||w2|p2w2log|w2||k|θw1+(1θ)w2|p2|w1w2|+k(p1)A|w1w2|+k(p1)|θw1+(1θ)w2|p2+ε|w1w2|k(|w1|+|w2|)p2|w1w2|+k(p1)A|w1w2|+k(p1)(|w1|+|w2|)p2+ε|w1w2|.

    Since w1,w2BRT, using Hölder's inequality and the Sobolev embedding, we can obtain

    Ω[(|w1|+|w2|)p2|w1w2|]2dxC(Ω(|w1|+|w2|)2(p1)dx)(p2)/(p1)×(Ω|w1w2|2(p1)dx)1/(p1)C[w12(p1)L2(p1)+w22(p1)L2(p1)](p2)/(p1)w1w22L2(p1)C[w12(p1)H20(Ω)+w22(p1)H20(Ω)](p2)/(p1)w1w22H20(Ω)CR2(p2)w1w22H20(Ω).

    By the similar argument, we have

    Ω[(|w1|+|w2|)p2+ε|w1w2|]2dxC(Ω(|w1|+|w2|)2(p2+ε)(p1)/(p2)dx)(p2)/(p1)
    ×(Ω|w1w2|2(p1)dx)1/(p1)(Ω(|w1|+|w2|)2(p1)+2ε(p1)/(p2)dx)(p2)/(p1)w1w22L2(p1).

    Using (2), we can choose sufficiently small ε>0 such that

    ˉp=2(p1)+2ε(p1)p22nn4,

    which yields that

    Ω[(|w1|+|w2|)p2+ε|w1w2|]2dxC[w1ˉpLˉp(Ω)+w2ˉpLˉp(Ω)](p2)/(p1)w1w22L2(p1)CRˉp(p2)/(p1)w1w22H20(Ω).

    Noticing w1w2Cw1w2H20(Ω), from the above discussions, we can deduce that

    |w1|p2w1log|w1||w2|p2w2log|w2|C(Rp2+1+Rˉp(p2)/2(p1))w1w2H20(Ω).

    Thus, it follows from (31) that

    Φ(w1)Φ(w2)H=v1v2HC(Rp2+1+Rˉp(p2)/2(p1))Tw1w2H. (32)

    We choose T sufficiently small such that C(Rp2+1+Rˉp(p2)/2(p1))T<1. Thus from (32) we obtain Φ is a contract map in BRT. The contraction mapping principle then shows that there exists a unique uBRT satisfying u=Φ(u) which is a solution to problem (1). The proof is complete.

    In this section, we consider the global existence and energy decay of the solution for problem (1). First, we introduce the following lemmas which play an important role in studying the decay estimate of global solution for the problem (1).

    Lemma 4.1. [33] Let ϕ(t) be a nonincreasing and nonnegative function on [0,T], T>1, such that

    ϕ(t)1+rω0(ϕ(t)ϕ(t+1))on[0,T],

    where ω0 is a positive constant and r is a nonnegative constant. Then we have

    (ⅰ): if r>0, then

    ϕ(t)(ϕ(0)r+ω10r[t1]+)1ron[0,T];

    (ⅱ): if r=0, then

    ϕ(t)ϕ(0)eω1[t1]+on[0,T],

    where ω1=log(w0ω01), here ω0>1.

    Now, we establish the global existence and energy decay results.

    Theorem 4.2. Let u be the unique local solution to problem (1). Assume (2) and 2m<p hold. If u0W, u1L2(Ω) and E(0)<d, then u(t) is the global solution to the problem (1). Moreover it has the following decay property

    E(t)Keκt,ifm=2;

    and

    E(t)(E(0)m22+(m2)τ2[t1]+)2m2,ifm>2,

    where K and κ are positive constants, τ is given by (47).

    Proof. Step 1.. Global existence. It suffices to show that ut2+Δu2 is uniformly bounded with respect to t. It follows from Lemma 2.5 (i) that u(t)W on [0,T]. Using (13), we have the following estimate

    d>E(0)E(t)=12ut2+J(u)=12ut2+1pI(u)+(121p)Δu2+kp2upp>12ut2+p22pΔu2, (33)

    which yields that

    ut2+Δu22pp2d<+.

    The above inequality and the continuation principle imply the global existence, i.e. T=+.

    Step 2.. We claim that there exists constant θ(0,1) such that

    I(u)θΔu2. (34)

    In fact, it follows from I(u(t))>0 for all t0 and Lemma 2.1 that there exists a λ0>1 such that I(λ0u(t))=0. Making use of (33), we have

    dJ(λ0u(t))=1pI(λ0u)+(121p)Δ(λ0u)2+kp2λ0upp=p22pλ20Δu2+kp2λp0upp=λp0(p22pλ2p0Δu2+kp2upp)λp0(p22pΔu2+kp2upp)<λp0E(0),

    which implies that

    λ0>(dE(0))1p>1. (35)

    It follows from (12) that

    0=I(λ0u)=Δ(λ0u)2Ω|λ0u|plog|λ0u|kdx=λ20Δu2λp0kΩ|u|plog|u|dx(λp0klogλ0)upp=λp0I(u)λp0Δu2+λ20Δu2(λp0klogλ0)upp=λp0I(u)(λp0λ20)Δu2(λp0klogλ0)upp.

    Combining this equality with (35), we have

    λp0I(u)=(λp0λ20)Δu2+(λp0klogλ0)upp(λp0λ20)Δu2,

    which implies that

    I(u)(1λ2p0)Δu2.

    Hence, the inequality (34) holds with θ=1λ2p0.

    Step 3.. Energy decay. By integrating (18) over [t,t+1],t>0, we obtain

    E(t)E(t+1)D(t)m, (36)

    where

    D(t)m=t+1tuτmmdτ. (37)

    In view of (37) and the embedding Lm(Ω)L2(Ω), we obtain

    t+1tΩ|ut|2dxdtc(Ω)D(t)2. (38)

    Thus, from (38), there exist t1[t,t+14] and t2[t+34,t+1] such that

    ut(ti)224c(Ω)D(t)2,i=1,2. (39)

    On the other hand, multiplying (1.1)1 by u and integrating over Ω×[t1,t2], we have

    t2t1I(u)dt=t2t1ut2dt+(ut(t1),u(t1))(ut(t2),u(t2))t2t1Ω|ut|m2utudxdt. (40)

    It follows from (33) that

    |t2t1Ω|um2tutudxdt|t2t1umutm1mdtCt2t1Δuutm1mdtC(2pp2)12supt1st2E(s)12t2t1utm1mdtC(2pp2)12supt1st2E(s)12D(t)m1. (41)

    By using (33) and (39), we also have

    ut(ti)2u(ti)2C1D(t)supt1st2E(s)12,i=1,2. (42)

    Combining (38), (41) with (42), we have from (40) that

    t2t1I(u)dtc(Ω)D(t)2+2C1D(t)supt1st2E(s)12+C(2pp2)12D(t)m1supt1st2E(s)12. (43)

    Moreover, using (33) and (34), it is easy to see that

    uppCppΔupCpp(2pp2E(0))p221θI(u).

    Thus, we deduce that

    E(t)12ut2+C2I(u), (44)

    where C2=1p+p22pθ+kCppp2θ(2pp2E(0))p22. By integrating (44) over (t1,t2), we have

    t2t1E(t)dt12t2t1ut22dt+C2t2t1I(u)dt. (45)

    By integrating (18) over [t,t2], we obtain

    E(t)=E(t2)+t2tutmmds

    Since t2t112, it easy to see that

    E(t2)2t2t1E(t)dt.

    Then, in view of (36), we have

    E(t)=E(t+1)+D(t)mE(t2)+D(t)m2t2t1E(t)dt+D(t)m.

    Thus, combining (38) with (45), we get that

    E(t)(c(Ω)+2c(Ω)C2)D(t)2+D(t)m2+C2[2C1D(t)+C(2p(p2))12D(t)m1]supt1st2E(s)12(c(Ω)+2c(Ω)C2)D(t)2+D(t)m+2C2[2C1D(t)+C(2p(p2))12D(t)m1]E(t)12.

    Hence, it follows from Young's inequality that

    E(t)C3[D(t)2+D(t)m+D(t)2(m1)] (46)

    holds with some positive constant C3>1. It follows from (36) and (46), we deduce that

    E(t)C3[1+D(t)m2+D(t)2m4]D(t)2C3[1+E(0)m2m+E(0)2m4m]D(t)2,

    which implies that

    E(t)m2(C4(E(0)))m2D(t)m=(C4(E(0)))m2(E(t)E(t+1)),

    where C4(E(0))=C3[1+E(0)m2m+E(0)2m4m]. Notice that

    limE(0)0C4(E(0))=C3

    Hence, the energy decay estimates hold with

    K=E(0)eκ,κ=log3C33C31andτ=(C4(E(0)))m2. (47)

    In this section, we will establish that the solution of problem (1) blows up in finite time provided E(0)<0. For this purpose, we give some useful lemmas.

    Lemma 5.1. Assume that (2) holds. Then there exists a positive constant C such that

    (Ω|u|plog|u|kdx)s/pC[Ω|u|plog|u|kdx+Δu22],

    for any uH20(Ω) and 2sp, provided that Ω|u|plog|u|kdx0.

    Lemma 5.2. Assume that (2) holds. Then there exists a positive constant C such that

    uppC[Ω|u|plog|u|kdx+Δu2],

    for any uH20(Ω), provided that Ω|u|plog|u|kdx0.

    Lemma 5.3. Assume that (2) holds. Then there exists a positive constant C>1 such that

    uspC[upp+u22],

    for any uH20(Ω) and 2sp.

    The proof of lemma 5.1-5.3 is similar to the proof in [22], we omit the details.

    Lemma 5.4. Assume that (2) and m<p hold. Then there exists a positive constant C such that

    ummC[(Ω|u|plog|u|kdx)mp+Δu2mp],

    for any uH20(Ω), provided that Ω|u|plog|u|kdx0.

    Proof. Noting mp and using the fact that ummC(upp)mp, we can obtain the result from Lemma 5.2.

    Now we are in the position to state and prove the blow up result for E(0)<0.

    Theorem 5.5. Suppose that the conditions in Lemma 5.4 hold. Then the solution to the problem (1) blows up in finite time provided that E(0)<0.

    Proof. We denote H(t)=E(t). It follows from (17) and (18) that

    E(t)E(0)<0,H(t)=E(t)=utmm.

    and

    0<H(0)H(t)1pΩ|u|plog|u|kdx. (48)

    We define

    L(t)=H1β(t)+εΩuutdx,t0,

    where ε>0 to be determined later and

    2(pm)(m1)p2<β<pm2(m1)p<1. (49)

    By taking a derivation of L(t), we get

    L(t)=(1β)Hβ(t)H(t)+εut2εΔu2εΩ|ut|m2utudx+εΩ|u|plog|u|kdx.

    Adding and subtracting εp(1a)H(t) for some a(0,1) in the RHS of the above equation, then using the definition of H(t), we obtain

    L(t)=(1β)Hβ(t)H(t)+εp(1a)+22ut2+εp(1a)22Δu2+εp(1a)H(t)εΩ|ut|m2utudx+εaΩ|u|plog|u|kdx+ε(1a)kpupp. (50)

    In view of Young's inequality, we have

    Ω|ut|m2utudxδmmumm+m1mδm/(m1)utmm

    for any δ>0, which yields, by substitution in (50),

    L(t)[(1β)Hβ(t)m1mεδm/(m1)]utmmεδmmumm+εp(1a)+22ut2+εp(1a)22Δu2+εp(1a)H(t)+εaΩ|u|plog|u|kdx+ε(1a)kpupp. (51)

    Since the integral is taken over the x variable, (50) holds even if δ is time dependent. Thus by choosing δ so that δm/(m1)=MHβ(t), for large M to be determined later, substituting in (51), we obtain

    L(t)[(1β)m1mεM]Hβ(t)utmmεM1mmHβ(m1)umm+εp(1a)+22ut2+εp(1a)22Δu2+εp(1a)H(t)+εaΩ|u|plog|u|kdx+ε(1a)kpupp. (52)

    Making using of (48), Lemma 5.4 and Young's inequality, we find

    Hβ(m1)umm(Ω|u|plog|u|kdx)β(m1)ummC[(Ω|u|plog|u|kdx)β(m1)+mp+(Ω|u|plog|u|kdx)β(m1)Δu2mp]C[(Ω|u|plog|u|kdx)β(m1)+mp+(Ω|u|plog|u|kdx)β(m1)ppm+Δu2].

    Hence, it follows from Lemma 5.1 that

    2<β(m1)p+mpand2<β(m1)p2pmp.

    Thus, Lemma 5.1 implies

    Hβ(m1)ummC(Ω|u|plog|u|kdx+Δu2). (53)

    Combining (52) and (53), we have

    L(t)[(1β)m1mεM]Hβ(t)utmm+εp(1a)+22ut2
    +ε[p(1a)22M1mmC]Δu2+ε[aM1mmC]Ω|u|plog|u|kdx+εp(1a)H(t)+ε(1a)kpupp. (54)

    Now, we choose a>0 sufficiently small that

    p(1a)22>0

    and M sufficiently large that

    p(1a)22M1mmC>0andaM1mmC>0.

    Once M and a are fixed, we choose ε sufficiently small that

    (1β)m1mεM>0andL(0)=H1β(0)+εΩu0u1dx>0.

    Thus, for some constant γ>0, (54) has the form

    L(t)γ[H(t)+ut2+Δu2+upp+Ω|u|plog|u|kdx]. (55)

    Consequently we have

    L(t)L(0),for allt>0.

    On the other hand, using Lemma 5.3, by the same method as in [32], we can deduce

    L11β(t)C[H(t)+ut2+Δu2+upp],t0. (56)

    Combining (55) and (56), we obtain

    L(t)λL11β(t),t0. (57)

    where λ>0 is constant depending only on γ and C. By a simple integration of (57) over (0,t), we have

    Lβ/(1β)(t)1Lβ/(1β)(0)λtβ/(1β).

    which implies that L(t) blow up in finite time

    TT=1βλβLβ/(1β)(0).

    This completes the proof of Theorem 5.1.

    In this section, we consider the problem (1) with the linear damping term, i.e. m=2. We will establish the finite time blow-up result by the method of the so called concavity method. For simplicity, we denote u2B0Δu2.

    Lemma 6.1. [14] Let δ0,T>0 and h be a Lipschitzian function over [0,T). Assume that h(0)0 and h(t)+δh(t)>0 for a.e. t[0,T). Then h(t)>0 for all t(0,T).

    Lemma 6.2. Suppose that m=2. Let I(u0)<0 and u1L2(Ω) such that

    Ωu0u1dx0.

    Let u(t) be the solution with the initial data (u0,u1). Then the map {tu(t)2} is strictly increasing provided I(u(t))<0.

    Proof. Let F(t)=u(t)2 and G(t)=F(t)=2Ωuutdx. A direct computation yields

    utt,u=ddt(ut,u)ut2fora.e.t0,

    Moreover, by testing the equation with u(t), we have

    utt,u+Δu2+(ut,u)=Ω|u|plog|u|kdx,

    which implies

    ddt((ut,u)+12u2)=ut2I(u).

    Hence, if I(u(t))>0, we can deduce

    G(t)+G(t)=2ut22I(u(t))>0fora.e.t[0,T).

    Therefore, it follows from Lemma 6.1 with δ=1 that G(t)=F(t)>0. Hence F(t) is strictly increasing provided I(u(t))<0.

    Lemma 6.3. Let m=2. Assume that u0H20(Ω), u1L2(Ω) and (2) holds. Assume that the initial data satisfies

    u122(u1,u0)+ΛE(0)<0, (58)

    where Λ=2B0pp2. Then the solution u(t) of the problem (1) with E(0)>0 satisfies I(u(t))<0 provided I(u0)<0.

    Proof. If this was not the case, by the continuity of I(u(t)) in t, then there would exist a first time t0(0,T) such that I(u(t0))=0 and I(u(t))<0 for t[0,t0). It follows form the Cauchy-Schwarz inequality that

    (u1,u0)u1u012(u12+u02). (59)

    By Lemma 6.2, (58) and (59), we deduce that

    F(t)=u(t)2>u022(u1,u0)u12>ΛE(0)fort(0,t0), (60)

    which implies

    F(t0)=u(t0)2>ΛE(0) (61)

    by the continuity of u(t) in t. Moreover, it follows from (13) and (17) that

    E(0)E(t0)=1pI(u(t0))+(121p)Δu(t0)2+kp2u(t0)ppp22pΔu(t0)2

    that is

    Δu(t0)22pp2E(0).

    Hence, we have

    F(t0)=u(t0)2B0Δu(t0)22B0pp2E(0)=ΛE(0),

    which is a contradiction with (61). The proof is complete.

    We now present the main blow-up result for the weak solution of problem (1) with m=2 with arbitrary positive initial energy.

    Theorem 6.4. Assume the conditions of Lemma 6.3 hold. Then the weak solution u(t) of the problem (1) blows up in finite time provided that E(0)>0 and I(u0)<0.

    Proof. It follows from Lemma 6.3 that I(u(t))<0 for t[0,T). By contradiction, we assume now that u(t) is global, namely T=. Then, for any T0>0, we may consider η:[0,T0]R+ defined by

    η(t)=u2+t0u(τ)2dτ+(T0t)u02.

    Notice η(t)>0 for all t[0,T0]; hence, since η is continuous, there exists ϱ>0 (independent of the choice of T0) such that

    η(t)ϱforallt[0,T0]. (62)

    Moreover,

    η(t)=2(ut,u)+u2u02=2(ut,u)+2t0(uτ,u)dτ, (63)

    hence, we have

    η(t)=2ut2+2utt,u+2(ut,u)=2(ut2Δu2+Ω|u|plog|u|kdx)=2ut22I(u(t)). (64)

    It follows from (63) that

    (η(t))2=4((ut,u)2+2(ut,u)t0(uτ,u)dτ+(t0(uτ,u)dτ)2).

    By the Cauchy-Schwarz inequality, we obtain

    ut2u2(ut,u)2
    t0u2dτt0uτ2dτ(t0(uτ,u)dτ)2

    and

    2(ut,u)t0(uτ,u)dτ2utu(t0uτ2dτ)12(t0u2dτ)12ut2t0u2dτ+u2t0uτ2dτ.

    Combining the above inequalities, we have

    η(t)24(ut2u2+ut2t0u2dτ+u2t0uτ2dτ+t0uτ2dτt0u2dτ)=4(u2+t0u2dτ)(ut2+t0uτ2dτ)4η(t)(ut2+t0uτ2dτ). (65)

    Hence, it follows from (64) and (65) that

    η(t)η(t)p+24η(t)2η(t)(η(t)(p+2)(ut2+t0uτ2dτ))=η(t)(put22Δu2+2Ω|u|plog|u|kdx(p+2)t0uτ2dτ). (66)

    Now, we define

    ξ(t)=put22Δu2+2Ω|u|plog|u|kdx(p+2)t0uτ2dτ.

    Noticing m=2 in this section, using (13) and (17), we obtain

    ξ(t)=(p2)Δu22pE(t)(p+2)t0uτ2dτ+2kpupp=(p2)Δu22pE(0)+(p2)t0uτ2dτ+2kpupp(p2)Δu22pE(0).

    From (60) and Lemma 6.2, we deduce that

    2pE(0)<p2B0u02<p2B0u2<(p2)Δu2.

    which yields that ξ(t)ς>0. Then, (66) can be rewritten as

    η(t)p+24η(t)2ϱς,t[0,T0],

    which implies that

    (ηp24(t))p24ϱς(η(t))p+64<0.

    Hence, it follows that there exists a T>0 such that

    limtTηp24(t)=0,

    that is

    limtTη(t)=+.

    In turn, this implies that

    limtTΔu(t)2=+. (67)

    In fact, if u(t)+ as tT, then (67) immediately follows. On the contrary, if u(t) remains bounded on [0,T), then

    limtTt0u(τ)2dτ=+

    so that (67) is also satisfied. Hence (67) is a contraction with T=+. The proof is complete.

    The author would like to thank Professor Hua Chen for the careful reading of this paper and for the valuable suggestions to improve the presentation of the paper. This project is supported by NSFC (No. 11801145), Key Scientific Research Foundation of the Higher Education Institutions of Henan Province, China (Grant No.19A110004 and the Fund of Young Backbone Teacher in Henan Province (NO. 2018GGJS068, 21420048).



    Conflict of interest



    All authors declare no conflicts of interest in this paper.

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