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Harmonic and subharmonic solutions of quadratic Liénard type systems with sublinearity

  • In this paper, we prove the existence of harmonic solutions and infinitely many subharmonic solutions of dissipative second order sublinear differential equations named quadratic Liénard type systems. The method of the proof is based on the Poincaré-Birkhoff twist theorem.

    Citation: Chunmei Song, Qihuai Liu, Guirong Jiang. Harmonic and subharmonic solutions of quadratic Liénard type systems with sublinearity[J]. AIMS Mathematics, 2021, 6(11): 12913-12928. doi: 10.3934/math.2021747

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  • In this paper, we prove the existence of harmonic solutions and infinitely many subharmonic solutions of dissipative second order sublinear differential equations named quadratic Liénard type systems. The method of the proof is based on the Poincaré-Birkhoff twist theorem.



    Let (M,ω) be a complex n-dimensional compact Hermitian manifold and χ be a smooth real (1, 1)-form on (M,ω). Γωk is the set of all real (1, 1)-forms whose eigenvalues belong to the k-positive cone Γk. For any uC2(M), we can get a new (1, 1)-form

    χu:=χ+1¯u.

    In any local coordinate chart, χu can be expressed as

    χu=1(χi¯j+ui¯j)dzid¯zj.

    In this article, we study the following form of parabolic Hessian quotient equations

    {u(x,t)t=logCknχkuωnkClnχluωnllogϕ(x,u), (x,t)M×[0,T),u(x,0)=u0(x),  xM, (1.1)

    where 0l<kn, [0,T) is the maximum time interval in which the solution exists and ϕ(x,z)C(M×R) is a given strictly positive function.

    The study of the parabolic flows is motivated by complex equations

    χkuωnk=ClnCknϕ(x,u)χluωnl, χuΓωk. (1.2)

    Equation (1.2) include some important geometry equations, for example, complex Monge-Ampère equation and Donaldson equation [6], which have attracted extensive attention in mathematics and physics since Yau's breakthrough in the Calabi conjecture [28]. Since Eq (1.2) are fully nonlinear elliptic, a classical way to solve them is the continuity method. Using this method, the complex Monge-Ampère equation

    χnu=ϕ(x)ωn, χuΓωn

    was solved by Yau [28]. Donaldson equation

    χnu=MχnMχωn1χuωn1, χuΓωn

    was independently solved by Li-Shi-Yao [11], Collins-Szèkelyhidi [3] and Sun [17]. Equation (1.2) also include the complex k-Hessian equation and complex Hessian quotient equation, which, respectively, correspond to

    Cknχkuωnk=ϕ(x)ωn, χuΓωk,
    χkuωnk=ClnCknϕ(x)χluωnl, χuΓωk.

    Dinew and Kolodziej [7] proved a Liouville type theorem for m-subharmonic functions in Cn, and combining with the estimate of Hou-Ma-Wu [10], solved the complex k-Hessian equation by using the continuity method. Under the cone condition, Sun [16] solved the complex Hessian quotient equation by using the continuity method. There have been many extensive studies for complex Monge-Ampère equation, Donaldson equation, the complex k-Hessian equation and the complex Hessian quotient equation on closed complex manifolds, see, e.g., [4,12,20,22,23,29,30]. When the right hand side function ϕ in Eq (1.2) depends on u, that is ϕ=ϕ(x,u), it is interesting to ask whether we can solve them. We intend to solve (1.2) by the parabolic flow method.

    Equation (1.1) covers some of the important geometric flows in complex geometry. If k=n and l=0, (1.1) is known as the complex Monge-Ampère flow

    u(x,t)t=logχnuωnlogϕ(x), (x,t)M×[0,T),

    which is equivalent to the Kähler-Ricci flow. The result of Yau [28] was reproduced by Cao [2] through Kähler-Ricci flow. Using the complex Monge-Ampère flow, similar results on a compact Hermitian manifold and a compact almost Hermitian manifold were proved by Gill [9] and Chu [5], respectively. To study the normalized twisted Chern-Ricci flow

    ωtt=Ric(ωt)ωt+η,

    which is equivalent to the following Mong-Ampère flow

    φt=log(θt+ddcφ)nΩφ,

    [25,26] considered the following complex Monge-Ampère flow

    φt=log(θt+ddcφ)nΩF(t,x,φ),

    where Ω is a smooth volume form on M. From this, we can see that the given function ϕ depends on u in some geometric flows. If l=0, (1.1) is called as the complex k-Hessian flow

    u(x,t)t=logCknχkuωnkωnlogϕ(x,u), (x,t)M×[0,T).

    The solvability of complex k-Hessian flow was showed by Sheng-Wang [21].

    In this paper, our research can be viewed as a generalization of Tô's work in [26] and Sheng-Wang's work in [21]. To solve the complex Hessian quotient flow, the condition of the parabolic C-subsolution is needed. According to Phong and Tô [14], we can give the definition of the parabolic C-subsolution to Eq (1.1).

    Definition 1.1. Let u_(x,t)C2,1(M×[0,T)) and χu_Γωk, if there exist constants δ, R>0, such that for any (x,t)M×[0,T),

    logσk(λ)σl(λ)tu_logϕ(x,u_),  λλ(u_)+δIΓn,

    implies that

    |λ|<R,

    then u_ is said to be a parabolic C-subsolution of (1.1), where λ(u_) denotes eigenvalue set of χu_.

    Obviously, we can give the equivalent definition of parabolic C-subsolution of (1.1).

    Definition 1.2. Let u_(x,t)C2,1(M×[0,T)) and χu_Γωk, if there exist constant ˜δ>0, for any (x,t)M×[0,T), such that

    limμlogσk(λ(u_)+μei)σl(λ(u_)+μei)>u_t+˜δ+logϕ(x,u_), 1in, (1.3)

    then u_ is said to be a parabolic C-subsolution of (1.1).

    Our main result is

    Theorem 1.3. Let (M,g) a compact Hermitian manifold and χ be a smooth real (1,1)-form on M. Assume there exists a parabolic C-subsolutionu_ for Eq (1.1) and

    tu_max{supM(logσk(λ(u0))σl(λ(u0))logϕ(x,u0)),0}, (1.4)
    ϕz(x,z)ϕ>cϕ>0, (1.5)

    where cϕ is a constant.Then there exits a unique smooth solution u(x,t) to (1.1) all timewith

    supxM(u0(x)u_(x,0))=0. (1.6)

    Moreover, u(x,t) isC convergent to a smooth function u, which solves Eq (1.2).

    The rest of this paper is organized as follows. In Section 2, we give some important lemmas and estimate on |ut(x,t)|. In Section 3, we prove C0 estimates of Eq (1.1) by the parabolic C-subsolution condition and the Alexandroff-Bakelman-Pucci maximum principle. In Section 4, using the parabolic C-subsolution condition, we establish the C2 estimate for Eq (1.1) by the method of Hou-Ma-Wu [10]. In Section 5, we adapt the blowup method of Dinew and Kolodziej [7] to obtain the gradient estimate. In Section 6, we give the proof of the long-time existence of the solution to the parabolic equation and its convergence, that is Theorem 1.3.

    In this section, we give some notations and lemmas. In holomorphic coordinates, we can set

    ω=1gi¯jdzid¯zj=1δijdzid¯zj,  χ=1χi¯jdzid¯zj,
    χu=1(χi¯j+ui¯j)dzid¯zj=1Xi¯jdzid¯zj,
    χu_=1(χi¯j+u_i¯j)dzid¯zj=1X_i¯jdzid¯zj.

    λ(u) and λ(u_) denote the eigenvalue set of {Xi¯j} and {X_i¯j} with respect to {gi¯j}, respectively. In local coordinates, (1.1) can be written as

    tu=logσk(λ(u))σl(λ(u))logϕ(x,u). (2.1)

    For simplicity, set

    F(λ(u))=logσk(λ(u))σl(λ(u)),

    then (2.1) is abbreviated as

    tu=F(λ(u))logϕ(x,u). (2.2)

    We use the following notation

    Fi¯j=FXi¯j,  F=iFi¯i,  Fi¯j,p¯q=2FXi¯jXp¯q.

    For any x0M, we can choose a local holomorphic coordinates such that the matrix {Xi¯j} is diagonal and X1¯1Xn¯n, then we have, at x0M,

    λ(u)=(λ1,,λn)=(X1¯1,,Xn¯n),
    Fi¯j=Fi¯iδij=(σk1(λ|i)σkσl1(λ|i)σl)δij,  F1¯1Fn¯n.

    To prove a priori C0-estimate for solution to Eq (1.1), we need the following variant of the Alexandroff-Bakelman-Pucci maximum principle, which is Proposition 10 in [20].

    Lemma 2.1. [20] Let v:B(1)R be a smooth function, which meets the conditionv(0)+ϵinfB(1)v, whereB(1) denotes the unit ball in Rn.Define the set

    Ω={xB(1):|Dv(x)|<ϵ2,   and  v(y)v(x)+Dv(x)(yx),yB(1)}.

    Then there exists a costant c0>0 such that

    c0ϵnΩdet(D2v).

    Next, we give an estimate on |ut(x,t)|.

    Lemma 2.2. Under the assumption of Theorem 1.3, let u(x,t) be a solution to (1.1). Then for any (x,t)M×[0,T), we have

    min{infMut(x,0),0}ut(x,t)max{supMut(x,0),0}. (2.3)

    Furthermore, there is a constant C>0 such that

    supM×[0,T)|tu(x,t)|supM|tu(x,0)|C,

    where C depends on H=|u0|C2(M) and |ϕ|C0(M×[H,H]).

    Proof. Differentiating (2.2) on both sides simultaneously at t, we obtain

    (ut)t=Fi¯jXi¯jtϕzϕut=Fi¯j(ut)i¯jϕzϕut. (2.4)

    Set uεt=utεt, ε>0. For any T(0,T), suppose uεt achieves its maximum Mt at (x0,t0)M×[0,T]. Without loss of generality, we may suppose Mt0. If t0>0, From the parabolic maximum principle and (2.4), we get

    0(uεt)tFi¯j(uεt)i¯j+ϕzϕuεt(ut)tεFi¯j(ut)i¯j+ϕzϕutεϕzϕt0εεϕzϕt0.

    This is obviously a contradiction, so t0=0 and

    supM×[0,T]uεt(x,t)=supMut(x,0),

    that is

    supM×[0,T]ut(x,t)=supM×[0,T](uεt(x,t)+εt)supMut(x,0)+εT.

    Letting ε0, we obtain

    supM×[0,T]ut(x,t)supMut(x,0).

    Since T(0,T) is arbitrary, we have

    supM×[0,T)ut(x,t)supMut(x,0). (2.5)

    Similarly, setting uεt=ut+εt, ε>0, we obtain

    infM×[0,T)ut(x,t)infMut(x,0). (2.6)

    (2.1) yields

    |ut(x,0)|=|logσk(λ(u0))σl(λ(u0))logϕ(x,u0)|C. (2.7)

    Combining (2.5)–(2.7), we complete the proof of Proposition 2.2.

    From the concavity of F(λ(u)) and the condition of the parabolic C-subsolution, we give the following lemma, which plays an important role in the estimation of C2.

    Lemma 2.3. Under the assumption of Theorem 1.3 and assuming that X1¯1Xn¯n, there exists two positive constantsN and θ such that we have either

    Fi¯i(u_i¯iui¯i)t(u_u)θ(1+F) (2.8)

    or

    F1¯1θN(1+F). (2.9)

    Proof. Since u_ is a parabolic C-subsolution to Eq (1.1), from Definition 1.2, there are uniform constants ˜δ>0 and N>0, such that

    logσk(λ(u_)+Ne1)σl(λ(u_)+Nue1)>u_t+˜δ+logϕ(x,u_). (2.10)

    If ϵ>0 is sufficiently small, it can be obtained from (2.10)

    logσk(λ(u_)ϵI+Ne1)σl(λ(u_)ϵI+Nue1)u_t+˜δ+logϕ(x,u_).

    Set λ=λ(u_)ϵI+Ne1, then

    F(λ)u_t+˜δ+logϕ(x,u_). (2.11)

    Using the concavity of F(λ(u)) gives

    Fi¯i(u_i¯iui¯i)=Fi¯i({X_i¯iXi¯i)=Fi¯i({X_i¯iϵδii+Nδi1Xi¯i)+ϵFNF1¯1F(λ)F(λ(u))+ϵFNF1¯1. (2.12)

    From Lemma 2.2 and (1.4), we obtain

    u_t(x,t)ut(x,t), (x,t)M×[0,T). (2.13)

    In addition, it can be obtained from the condition (1.6)

    u_(x,0)u(x,0), xM×[0,T). (2.14)

    (2.13) and (2.14) deduce that

    u_(x,t)u(x,t), (x,t)M×[0,T). (2.15)

    It follows from this that

    ϕ(x,u_)ϕ(x,u). (2.16)

    Combining (2.2), (2.11) and (2.16) gives that

    F(λ)F(λ(u))u_t(x,t)ut(x,t)+˜δ+logϕ(x,u_)logϕ(x,u)u_t(x,t)ut(x,t)+˜δ. (2.17)

    Put (2.17) into (2.12)

    Fi¯i(u_i¯iui¯i)u_t(x,t)ut(x,t)+˜δ+ϵFNF1¯1˜δ+ϵFNF1¯1.

    Let

    θ=min{˜δ2, ϵ2}.

    If F1¯1Nθ(1+F), Inequality (2.8) is obtained, otherwise Inequality (2.9) must be true.

    In this section, we prove the C0 estimates by the existence of the parabolic C-subsolution and the Alexandroff-Bakelman-Pucci maximum principle.

    Proposition 3.1. Under the assumption of Theorem1.3, let u(x,t) be a solution to (1.1). Then there exists a constant C>0 such that

    |u(x,t)|C0(M×[0,T))C,

    where C depends on |u0|C2(M) and |u_|C2(M×[0,T)).

    Proof. Combining (2.13), (2.14) and ϕ(x,z)z0 yields

    u_t(x,t)+logϕ(x,u_)ut(x,t)+logϕ(x,u). (3.1)

    Let's rewrite Eq (2.2) as

    F(λ(u))=tu+logϕ(x,u). (3.2)

    when fix t[0,T), Eq (3.2) is elliptic. From (3.1), we see that the parabolic C-subsolution u_(x,t) is a C-subsolution to Eq (3.2) in the elliptic sense. From (2.15), we have

    supM×[0,T)(uu_)=0.

    Our goal is to obtain a lower bound for L=infM×t(uu_). Note that λ(u)Γk, which implies that λ(u)Γ1, then Δ(uu_)˜C, where Δ is the complex Laplacian with respect to ω. According to Tosatti-Weinkove's method [22], we can prove that uu_L1(M) is bounded uniformly. Let G:M×MR be the associated Green's function, then, by Yau [28], there is a uniform constant K such that

    G(x,y)+K0, (x,y)M×M, and yMG(x,y)ωn(y)=0.

    Since

    supM×[0,T)(uu_)=0,

    then for fixed t[0,T) there exists a point x0M such that (uu_)(x0,t)=0. Thus

    (uu_)(x0,t)=M(uu_)dμyMG(x0,y)Δ(uu_)(y)ωn(y)=M(uu_)dμyM(G(x0,y)+K)Δ(uu_)(y)ωn(y)M(uu_)dμ+˜CKMωn,

    that is

    M(u_u)dμ=M|(uu_)|dμ˜CKMωn.

    Let us work in local coordinates, for which the infimum L is achieved at the origin, that is L=u(0,t)u_(0,t). We write B(1)={z:|z|<1}. Let v=uu_+ϵ|z|2, for a small ϵ>0. We have infv=L=v(0), and v(z)L+ϵ for zB(1). From Lemma 2.1, we get

    c0ϵ2nΩdet(D2v). (3.3)

    At the same time, if xΩ, then D2v(x)0 implies that

    ui¯j(x)u_i¯j(x)+ϵδij0.

    If ϵ is sufficiently small, then

    λ(u)λ(u_)δI+Γn.

    Set μ=λ(u)λ(u_). Since λ(u) satisfies Eq (3.2), then

    F(λ(u_)+μ)=tu+logϕ(x,u),  μ+δIΓn. (3.4)

    u_ is a C-subsolution to Eq (3.2) in the elliptic sense, so there is a uniform constant R>0, such that

    |μ|R,

    which means |vi¯j|C, for any xΩ. As in Blocki [1], for xΩ, we have D2v(x)0 and so

    D2v(x)22ndet(vi¯j)2C.

    From this and (3.3), we obtain

    c0ϵ2nΩdet(D2v)Cvol(Ω). (3.5)

    On the other hand, by the definition of Ω in Lemma 2.1, for xΩ, we get

    v(0)v(x)Dv(x)x>v(x)ϵ2,

    and so

    |v(x)|>|L+ϵ2|.

    It follows that

    M|v(x)|Ω|v(x)||L+ϵ2|vol(Ω). (3.6)

    Since uu_L1(M) is bounded uniformly, M|v(x)| is also bounded uniformly. If L is very large, Inequality (3.6) contradicts (3.5), which means that L has a lower bound. For any t[0,T), Inequality (3.1) holds, thus

    |u(x,t)|C0(M×[0,T))|L|+supM×[0,T)|u_|C.

    In this section, we prove that the second-order estimates are controlled by the square of the gradient estimate linearly. Our calculation is a parabolic version of that in Hou-Ma-Wu [10].

    Proposition 4.1. Under the assumption of Theorem1.3, let u(x,t) be a solution to (1.1). Then there exists a constant ˜C such that

    supM×[0,T)|1¯u|˜C(supM×[0,T)|u|2+1),

    where ˜C depends χ, ω, |ϕ|C2(M×[C,C]), |u_|C2(M×[0,T)), |tu_|C0(M×[0,T)) and |u0|C2(M).

    Proof. Let λ(u)=(λ1,,λn) and λ1 is the maximum eigenvalue. For any T<T, we consider the following function

    W(x,t)=logλ1+φ(|u(x,t)|2)+ψ(u(x,t)u_(x,t)), (x,t)M×[0,T], (4.1)

    where φ and ψ are determined later. We want to apply the maximum principle to the function W. Since the eigenvalues of the matrix {Xi¯j} with respect to ω need not be distinct at the point where W achieves its maximum, we will perturb {Xi¯j} following the technique of [20]. Let W achieve its maximum at (x0,t0)M×[0,T]. Near (x0,t0), we can choose local coordinates such that {Xi¯j} is diagonal with X1¯1Xn¯n, and λ(u)=(X1¯1,,Xn¯n). Let D be a diagonal matrix such that D11=0 and 0<D22<<Dnn are small, satisfying Dnn<2D22. Define the matrix ˜X=XD. At (x0,t0), ˜X has eigenvalues

    ˜λ1=λ1, ˜λi=λiDii, ni2.

    Since all the eigenvalues of ˜X are distinct, we can define near (x0,t0) the following smooth function

    ˜W=log˜λ1+φ(|u|2)+ψ(uu_), (4.2)

    where

    φ(s)=12log(1s2K), 0sK1,
    ψ(s)=Elog(1+s2L), L+1sL1,
    K=supM×[0,T]|u|2+1,
    L=supM×[0,T]|u|+supM×[0,T]|u_|+1,
    E=2L(C1+1),

    and C1>0 is to be chosen later. Direct calculation yields

    0<14Kφ12K,  φ=2(φ)2>0, (4.3)

    and

    C1+1ψ2(C1+1),  ψ4ϵ1ϵ(ψ)2, ϵ14E+1. (4.4)

    Without loss of generality, we can assume that λ1>1. From here on, all calculations are done at (x0,t0). From the maximum principle, calculating the first and second derivatives of the function ˜W gives

    0=˜Wi=˜λ1,iλ1+φ(|u|2)i+ψ(uu_)i,1in, (4.5)
    0˜Wi¯i=˜λ1,i¯iλ1˜λ1,i˜λ1,¯iλ21+φ(|u|2)i¯i+φ|(|u|2)i|2+ψ(uu_)i¯i+ψ|(uu_)i|2. (4.6)
    0˜Wt=˜λ1,tλ1+φ(|u|2)t+ψ(uu_)t. (4.7)

    Define

    L:=Fi¯j¯zjzit.

    Obviously,

    0L˜W=Llog˜λ1+Lφ(|u|2)+Lψ(uu_). (4.8)

    Next, we will estimate the terms in (4.8). Direct calculation shows that

    Llog˜λ1=Fi¯i˜λ1,i¯iλ1Fi¯i|˜λ1,i|2λ21˜λ1,tλ1. (4.9)

    According to Inequality (78) in [20], we have

    ˜λ1,i¯iXi¯i1¯12Re(Xi¯11¯T1i1)C0λ1, (4.10)

    where C0 depending χ, ω, |ϕ|C2(M×[C,C]), |u_|C2(M×[0,T)), |tu_|C0(M×[0,T)) and |u0|C2(M)). From here on, C0 can always absorb the constant it represents before, and can change from one line to the next, but it does not depend on the parameter we choose later. By calculating the covariant derivatives of (4.7) in the direction z1 and ¯z1, we obtain

    ut1=Fi¯iXi¯i1(logϕ)1(logϕ)uu1, (4.11)

    and

    ut1¯1=Fi¯j,p¯qXi¯j1Xp¯q¯1+Fi¯iXi¯i1¯1(logϕ)1¯1(logϕ)1uu¯1(logϕ)u¯1u1(logϕ)uu|u1|2(logϕ)uu1¯1. (4.12)

    Notice that

    X1¯1i=χ1¯1i+u1¯1i=(χ11iχi11+Tpi1χp¯1)+Xi¯11T1i1λ1, (4.13)

    therefore

    |X1¯1i|2|Xi¯11|22λ1Re(Xi¯11¯T1i1)+C0(λ21+|X1¯1i|). (4.14)

    Combining (4.14) with

    ˜λ1,i=X1¯1i(D11)i,

    gives

    Fi¯i|˜λ1,i|2λ21=Fi¯i|X1¯1i|2λ21+2λ21Fi¯iRe(X1¯1i(D11)¯i)Fi¯i|(D11)i|2λ2iFi¯i|X1¯1i|2λ21C0λ21Fi¯i|X1¯1i|C0FFi¯i|Xi¯11|2λ21+2λ1Fi¯iRe(Xi¯11¯T1i1)C0λ21Fi¯i|X1¯1i|C0F. (4.15)

    Let's set λ1K, otherwise the proof is completed. Putting 4.10–4.12 and (4.15) into (4.9) yields

    Llog˜λ1Fi¯j,p¯qXi¯j1Xp¯q¯1λ1Fi¯i|Xi¯11|2λ21C0λ1Fi¯i|X1¯1i|λ1C0F+(logϕ)1¯1+(logϕ)1uu¯1+(logϕ)u¯1u1+(logϕ)uu|u1|2(logϕ)uχ1¯1λ1Fi¯j,p¯qXi¯j1Xp¯q¯1λ1Fi¯i|Xi¯11|2λ21C0λ1Fi¯i|X1¯1i|λ1C0FC0 (4.16)

    A simple computation gives

    Lφ(|u|2)=φFi¯i(|u|2)i¯i+φFi¯i|(|u|2)i|2φ(|u|2)t. (4.17)

    Next, we estimate the formula (4.17). Differentiating Eq (2.2), we have

    (ut)p=Fi¯iXi¯ip(logϕ)p(logϕ)uup, (4.18)

    and

    (ut)¯p=Fi¯iXi¯i¯p(logϕ)¯p(logϕ)uu¯p. (4.19)

    It follows from (4.18) and (4.19) that

    t|u|2=putpu¯p+pupu¯t¯p=Fi¯ip(Xi¯ipu¯p+Xi¯i¯pup)p(logϕ)pu¯pp(logϕ)¯pup2p(logϕ)u|u|2. (4.20)

    By commuting derivatives, we have the identity

    Fi¯iXi¯ip=Fi¯iui¯ip+Fi¯iχi¯ip=Fi¯iupi¯iFi¯iTqpiuq¯iFi¯iuqR   qi¯ip+Fi¯iχi¯ip. (4.21)

    Direct calculation gives

    Fi¯i(|u|2)i¯i=pFi¯i(upi¯iu¯p+u¯pi¯iup)+pFi¯i(upiu¯p¯i+up¯iu¯pi). (4.22)

    It follows from (4.21) that

    Fi¯iupi¯iu¯pFi¯iXi¯ipu¯p=Fi¯iTqpiuq¯iu¯p+Fi¯iu¯puqR   qi¯ipFi¯iχi¯ipu¯pC0K12Fi¯iXi¯iC0K12FC0KF. (4.23)

    Noticing that

    Fi¯iXi¯i=σk1(λ|i)σkλiσl1(λ|i)σlλi=kl, (4.24)

    from this and (4.23), we obtain

    Fi¯iupi¯iu¯pFi¯iXi¯ipu¯pC0K12C0K12FC0KF. (4.25)

    In the same way, we can get

    Fi¯iu¯pi¯iupFi¯iXi¯i¯pupC0K12C0K12FC0KF. (4.26)

    Using (4.20)–(4.26) in (4.17), we have

    Lφ(|u|2)φFi¯i|(|u|2)i|2+φ(C0K12C0K12FC0KF)+φp((logϕ)pu¯p+(logϕ)¯pup+2(logϕ)u|u|2)+pFi¯i(|upi|2+|u¯pi|2)φFi¯i|(|u|2)i|2+pFi¯i(|upi|2+|u¯pi|2)C0C0F. (4.27)

    A simple calculation gives

    Lψ(uu_)=ψFi¯i|(uu_)i|2+ψ[Fi¯i(uu_)i¯it(uu_)]. (4.28)

    Substituting (4.27), (4.28) and (4.16) into (4.8),

    0Fi¯j,p¯qXi¯j1Xp¯q¯1λ1Fi¯i|Xi¯11|2λ21C0λ1Fi¯i|X1¯1i|λ1+φFi¯i|(|u|2)i|2+φpFi¯i(|upi|2+|u¯pi|2)C0C0F+ψFi¯i|(uu_)i|2+ψ[Fi¯i(uu_)i¯it(uu_)]. (4.29)

    Let δ>0 be a sufficiently small constant to be chosen later and satisfy

    δmin{11+4E,12}. (4.30)

    We separate the rest of the calculations into two cases.

    Case 1: λn<δλ1.

    Using(4.5), we find that

    Fi¯i|X1¯1i|2λ21=Fi¯i|φ(|u|2)i+ψ(uu_)i(D11)iλ1|22(φ)2Fi¯i|(|u|2)i|22Fi¯i|ψ(uu_)i(D11)iλ1|22(φ)2Fi¯i|(|u|2)i|2C0|ψ|2KFC0F. (4.31)

    From (4.13), we have

    |Xi¯11|2λ21|X1¯1i|2λ21+C0(1+|X1¯1i|λ1). (4.32)

    Combining (4.31) with (4.32), we conclude that

    Fi¯i|Xi¯11|2λ212(φ)2Fi¯i|(|u|2)i|2C0|ψ|2KFC0FC0Fi¯i|X1¯1i|λ1. (4.33)

    Note that the operator F is concave, which implies that

    Fi¯j,p¯qXi¯j1Xp¯q¯1λ10. (4.34)

    Applying (4.33) and (4.34) to (4.29) and using φ=2(φ)2 yield that

    0C0Fi¯i|X1¯1i|λ1C0λ1Fi¯i|X1¯1i|λ1+φpFi¯i(|upi|2+|u¯pi|2)C0|ψ|2KFC0FC0+ψFi¯i|(uu_)i|2+ψ[Fi¯i(uu_)i¯it(uu_)]. (4.35)

    Note that the fact

    |X1¯1i|λ1=|φ(upiu¯p+upu¯pi)ψ(uu_)i+(D11)iλ1|,

    It follows that

    C0Fi¯i|X1¯1i|λ1C0φK12Fi¯i(|upi|+|u¯pi|)+C0ψK12FC0F. (4.36)

    Using the following inequality

    K12(|upi|+|u¯pi|)14C0(|upi|2+|u¯pi|2)+C0K.

    deduces

    C0Fi¯i|X1¯1i|λ114φFi¯i(|upi|2+|u¯pi|2)+C0ψK12FC0F. (4.37)

    Note that λ1>1, we have

    C0λ1Fi¯i|X1¯1i|λ114φFi¯i(|upi|2+|u¯pi|2)+C0ψK12FC0F. (4.38)

    Since ψ>0, which implies that

    ψFi¯i|(uu_)i|20. (4.39)

    According to Lemma 2.3, there are at most two possibilities:

    (1) If (2.8) holds true, then

    ψ[Fi¯i(uu_)i¯it(uu_)]θ(1+F)|ψ|. (4.40)

    Substituting (4.37)—(4.40) into (4.35) and using φ14K yield that

    018KpFi¯i(|upi|2+|u¯pi|2)C0|ψ|2KFC0(F+1)+θ(1+F)|ψ|18KFi¯iλ2iC0(C1+1)2KF+θ(C1+1)(F+1)C0(F+1)δ2λ218nKFC0(C1+1)2KF+θ(C1+1)(F+1)C0(F+1). (4.41)

    We may set θC1C0. It follows from (4.41) that λ1˜CK.

    (2) If (2.9) holds true,

    F1¯1>θN(1+F). (4.42)

    According to ψ<0 and the concavity of the operator F, we have

    ψ[Fi¯i(uu_)i¯it(uu_)]=ψ[Fi¯i(Xi¯iX_i¯i)t(uu_)]ψ[F(χu)F(χu_)tu+tu_]=ψ[ϕ(x,u)+tu_F(χu_)]C0ψ. (4.43)

    Using (4.37)—(4.39) and (4.43) in (4.35), together with (4.42), we find that

    018KFi¯iλ2iC0|ψ|2KF+C0ψC0(F+1)θλ218NK(1+F)+δ2λ218nKFC0(C1+1)2KFC0(C1+1)C0(F+1). (4.44)

    Let λ1 be sufficiently large, so that

    θλ218NK(1+F)C0(C1+1)C0(F+1)0,

    It follows from (4.44)that λ1˜CK.

    Case 2: λnδλ1.

    Let

    I={i{1,,n}|Fi¯i>δ1F1¯1}.

    Let us first treat those indices which are not in I. Similar to (4.31), we obtain

    iIFi¯i|X1¯1i|2λ212(φ)2iIFi¯i|(|u|2)i|2C0Kδ|ψ|2F1¯1C0F. (4.45)

    Using (4.32) yields that

    iIFi¯i|Xi¯11|2λ212(φ)2iIFi¯i|(|u|2)i|2C0Kδ|ψ|2F1¯1C0iIFi¯i|X1¯1i|λ1C0F. (4.46)

    Since

    Fi¯1,1¯i=Fi¯iF1¯1X1¯1Xi¯i andλiλnδλ1,

    which implies that

    iIFi¯1,1¯i1δ1+δ1λ1iIFi¯i,

    It follows that

    Fi¯1,1¯i|Xi¯11|2λ11δ1+δiIFi¯i|Xi¯11|2λ21. (4.47)

    Recalling φ=2(φ)2 and 0<δ12, we obtain from (4.5) that

    iIφFi¯i|(|u|2)i|2=2iIFi¯i|Xi¯11λ1+ψ(uu_)i+χ11iχi11+Tpi1χp¯1(D11)iλ1|22iIFi¯i(δ|Xi¯11λ1|22δ1δ(ψ)2|(uu_)i|2C0)2δiIFi¯i|Xi¯11λ1|24δ1δ(ψ)2Fi¯i|(uu_)i|2C0F. (4.48)

    Notice that ψ4ϵ1ϵ(ψ)2 if ϵ=14E+1. Since 14E+1δ, we get that

    ψFi¯i|(uu_)i|24δ1δ(ψ)2Fi¯i|(uu_)i|20. (4.49)

    Take (4.46)–(4.49) into (4.29),

    0C0iIFi¯i|X1¯1i|λ1C0λ1Fi¯i|X1¯1i|λ1+φpFi¯i(|upi|2+|u¯pi|2)C0FC0C0Kδ|ψ|2F1¯1+ψ[Fi¯i(uu_)i¯it(uu_)]. (4.50)

    Similar to (4.37) and (4.38), by using the third term of (4.50) to absorb the first two terms of it, We get that

    018KpFi¯i(|upi|2+|u¯pi|2)C0Kδ|ψ|2F1¯1C0FC0+ψ[Fi¯i(uu_)i¯it(uu_)]18KpFi¯iλ2iC0Kδ|ψ|2F1¯1C0(F+1)+ψ[Fi¯i(uu_)i¯it(uu_)]. (4.51)

    According to Lemma2.3, there are at most two possibilities:

    (1) If (2.8) holds true, then

    ψ[Fi¯i(uu_)i¯it(uu_)]θ(1+F)|ψ|. (4.52)

    Put (4.52) into (4.51)

    018KpFi¯iλ2iC0Kδ|ψ|2F1¯1C0(F+1)+θ(1+F)|ψ|18KpF1¯1λ21C0Kδ(1+C1)2F1¯1C0(F+1)+θ(1+F)(1+C1). (4.53)

    Here, C1 is determined finally, such that

    θC1C0.

    It follows from (4.53) that

    λ1˜CK.

    (2) If (2.9) holds true,

    F1¯1>θN(1+F). (4.54)

    Substituting (4.43) into (4.51) and using (4.54) give that

    018Kλ21C0Kδ|(1+C1)|2NθC0(1+C1)NθC0 (4.55)

    It follows that

    λ1˜CK.

    To obtain the gradient estimates, we adapt the blow-up method of Dinew and Kolodziej [7] and reduce the problem to a Liouville type theorem which is proved in [7].

    Proposition 5.1. Under the assumption of Theorem 1.3, let u(x,t) be a solution to (1.1). Then there exists a uniform constant ˜C such that

    supM×[0,T)|u|˜C. (5.1)

    Proof. Suppose that the gradient estimate (5.1) does not hold. Then there exists a sequence (xm,tm)M×[0,T) with tmT such that

    supM×[0,tm]|u(x,t)|=|u(xm,tm)| andlimm|u(xm,tm)|=.

    After passing to a subsequence, we may assume that limmxm=x0M. We choose a coordinate chart {U,(z1,,zn)} at x0, which we identify with an open set in Cn, and such that ω(0)=β=1idzid¯zi. We may assume that the open set contains ¯B1(0) and m is sufficiently large so that zm=z(xm)B1(0). Define

    |u(xm,tm)|=Cm,  ˜um(z)=u(zCm,tm).

    From this and Proposition 4.1, we have

    supM|˜um|=1, supM|1 ¯˜um|˜C.

    This yields that ˜um is contained in the Hölder space C1,γ(Cn) with a uniform. Along with a standard application of Azela-Ascoli theorem, we may suppose ˜um has a limit ˜uC1,γ(Cn) with

    |˜u|+|˜u|<C and |˜u(0)|0, (5.2)

    in particular ˜u is not constant. On the other hand, similar to the method of Dinew and Kolodziej [7], we have

    [χu(zCm)]k[ω(zCm)]nk=etuϕm(zCm,u)[χu(zCm)]l[ω(zCm)]nl.

    Fixing z, we obtain

    C2(kl)m[O(1C2m)β+1 ¯˜um(z)]k[(1+O(|z|2C2m))β]nk=etumϕm(zCm,um)[O(1C2m)β+1 ¯˜um(z)]l[(1+O(|z|2C2m))β]nl. (5.3)

    Lemma 2.2 gives tu is bounded uniformly. Since

    ϕm(zCm,um)supM×[C,C]ϕ,

    which implies that ϕm(zCm,um) is bounded uniformly. Taking the limits on both sides of 5.3 by m yields that

    (1¯˜u)kβnk=0. (5.4)

    which is in the pluripotential sense. Moreover, a similar reasoning tells us that for any 1pk,

    (1¯˜u)pβnp0. (5.5)

    Then, (5.4) and (5.5) imply ˜u is a k-subharmonic. By a result of Blocki [1], ˜u is a maximal k-subharmonic function in Cn. Applying the Liouville theorem in [7], we find that ˜u is a constant, which contradicts with (5.2).

    In this section, we shall give a proof of the long-time existence to the flow and its convergence, that is Theorem 1.3.

    From Lemma 2.2, Proposition 3.1, Propositions 4.1 and 5.1, we conclude that Eq 1.1 is uniformly parabolic. Therefore by Evans-Krylov's regularity theory [8,13,19,27] for uniformly parabolic equation, we obtain higher order derivative estimates. By the a priori estimates which don't depend on time, one can prove that the short time existence on [0,T) extends to [0,), that is the smooth solution exists at all time t>0. After proving C estimates on [0,), we are able to show the convergence of the solution flow.

    Let v=eγtut, where 0<γ<cϕ. Commuting derivative of v with respect to t and using (2.4), we obtain

    vt=eγtutt+γv=γv+eγt(Fi¯juti¯jϕzϕut)=Fi¯jvi¯j+(γϕzϕ)v.

    Using the condition (1.5) yields γϕzϕ<0, According to the parabolic maximum principle, it follows that

    supM×[0,)|v(x,t)|supM|ut(x,0)|supM|F(λ(u0))ϕ(x,u0)|C,

    which means that |ut| decreases exponentially, in particular

    t(u+Cγeγt)0.

    According to Proposition 3.1, it follows that u+Cγeγt is bounded uniformly and decreasing in t. Thus it converges to a smooth function u. From the higher order prior estimates, we can see that the function u(x,t) converges smoothly to u. Letting t in Eq (2.1),

    σk(λ(u))σl(λ(u))=ϕ(x,u).

    In this paper, we have considered the parabolic Hessian quotient equation (1.1), in which the right hand side function ϕ depends on u. Firstly, we prove C0 estimates of Eq (1.1) by the parabolic C-subsolution condition and the Alexandroff-Bakelman-Pucci maximum principle. Secondly, we establish the C2 estimate for Eq (1.1) by using the parabolic C-subsolution condition. Thirdly, we obtain the gradient estimate by adapting the blowup method. Finally we give the proof of the long-time existence of the solution to the parabolic equation and its convergence. As an application, we show the solvability of a class of complex Hessian quotient equations, which generalizes the relevant results.

    This work was supported by the Natural Science Foundation of Anhui Province Education Department (Nos. KJ2021A0659, gxgnfx2018017); Quality Enginering Project of Anhui Province Education Department (Nos. 2018jyxm0491, 2019mooc205, 2020szsfkc0686); Science Research Project of Fuyang Normal University (No. 2021KYQD0011)

    The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.



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