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Research article

Convergence of smooth solutions to parabolic equations with an oblique derivative boundary condition

  • Received: 24 October 2023 Revised: 10 December 2023 Accepted: 12 December 2023 Published: 02 January 2024
  • MSC : 35B45, 35G30

  • In this paper, the parabolic equation with oblique derivative boundary condition is considered. The long time behavior of the solution is derived by selecting the appropriate auxiliary functions and making priori estimates. Through blow up analysis, time-dependent gradient estimates are obtained, followed by second-order derivative estimates. Then, the convergence of smooth solution to parabolic equations with the oblique derivative boundary condition is obtained using standard theory.

    Citation: Hongmei Li. Convergence of smooth solutions to parabolic equations with an oblique derivative boundary condition[J]. AIMS Mathematics, 2024, 9(2): 2824-2853. doi: 10.3934/math.2024140

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  • In this paper, the parabolic equation with oblique derivative boundary condition is considered. The long time behavior of the solution is derived by selecting the appropriate auxiliary functions and making priori estimates. Through blow up analysis, time-dependent gradient estimates are obtained, followed by second-order derivative estimates. Then, the convergence of smooth solution to parabolic equations with the oblique derivative boundary condition is obtained using standard theory.



    In this paper, we consider the long time behavior of smooth solutions to the following parabolic equations with oblique derivative boundary value problems,

    {utF(2u)=0in   Ω×[0,),u(x,0)=u0(x)on   ¯Ω×{0},uβ=φ(x)  on   Ω×[0,), (1.1)

    where Ω is a bounded smooth domain in Rn, F is a smooth real function defined on Sn, Sn means n×n real symmetric matrix space. φ is a given function defined on ¯Ω, β is the inward unit vector along Ω, and satisfies the condition <ν,β>=βn=cosθc0>0, where ν is the inner normal vector to Ω, uβ=<u,β>, where u=(ux1,ux2,uxn) and u0C(¯Ω) satisfies u0β=φ(x).

    At present, there are many results on various boundary value problems of partial differential equations [1,2,3,4,5,6,7], and the oblique derivative boundary value problems of partial differential equations have been widely studied. The related problems of the oblique derivative boundary value problems of linear and quasilinear elliptic equations can be seen in the book [8,9,10]. The related results of nonlinear differential equations can be found in the literature [11,12,13,14,15,16]. In [13], Bao established the global Hölder gradient estimates for the W2,p solution of the nonlinear oblique derivative problems for the second-order fully nonlinear elliptic equations using the perturbation idea of Caffarelli. In [17], they studied the long time behavior of the solution in the classical senses through a blow up skill for the following parabolic equation

    {utF(2u)=0in   Ω×[0,),u(x,0)=u0(x)on   ¯Ω×{0},uν=φ(x)on   Ω×[0,),

    where ν is the inward unit normal vector. In this paper, we will consider the long-time behavior of the solution to the above problem when the boundary condition becomes the oblique cases.

    We need to make some structural assumptions about F :

    (F1)rSn,λIFr(r),|F(r)|μ0|r|,

    (F2)r,XSn,|FX(r)|μ1|X|,

    (F3)r,XSn,FXX(r)0,

    where λ,μ0,μ1 are positive constsnts. Besides, we suppose

    (F4) There exists a smooth function F, such that

    s1F(sr)F(r)locallyuniformlyinC1(Sn),ass+.

    First, we state our major results of this paper.

    Theorem 1.1. Suppose ΩRn (n2) is a bounded domain with smooth boundary. If F satisfies (F1)(F4),φC(¯Ω), then the smooth solution u(x,t) of (1.1) converges to U+τt, namely, D⊂⊂Ω, ζ<1 and 0<α<1,

    limt+u(,t)(U()+τt)C1+ζ(¯Ω)=0,limt+u(,t)(U()+τt)C4+α(¯D)=0, (1.2)

    where (U,τ) is a suitable solution to

    {F(2U)=τ  in Ω,Uβ=φ(x)  on Ω. (1.3)

    The constant τ depends only on Ω,φ and F. The solution to (1.3) is unique up to a constant.

    Remark. Note (1.3) that τ depends only on F,φ,Ω.

    Proof. Assume there exist two pairs (τ1,u) and (τ2,v) solving (1.3).

    Namely

    {F(2u)=τ1  in  Ω,uβ=φ(x)  on  Ω.
    {F(2v)=τ2  in  Ω,vβ=φ(x)  on  Ω.

    Without loss of generality, we may assume τ1<τ2, then,

    {10Fuαβ[t2u+(1t)2v]dt(uv)αβ<0,(uv)β=0.

    By maximal principle, the minimum of uv can be achieved at the boundary, but (uv)β=0 and strong maximal principle indicate that the minimum can only be reached internally, which is contradictory, thus τ1=τ2.

    The above proof indicates that τ here only depends on F,φ,Ω.

    In [18], Huang and Ye established a convergence result under assumptions of a priori estimate.

    Theorem 1.2. [18] Suppose ΩRn (n2) is a bounded domain with smooth boundary. If F satisfies (F1) and (F3), φC(¯Ω). T>0, suppose uC4+α,4+α2(¯Ω×(0,T)) is a unique solution of the following nonlinear parabolic equation

    {utF(2u)=0in  Ω×[0,T),u(x,0)=u0(x)on  ¯Ω×{0},G(x,u)=0  on  Ω×[0,T), (1.4)

    and u satisfies

    ||ut(,t)||C(¯Ω)+||u(,t)||C(¯Ω)+||2u(,t)||C(¯Ω)C1, (1.5)
    nk=1Gpk(x,u)νk1C2, (1.6)

    where C1 and C2 are positive constants independent of t>1. Then u(,t) converges to a function U+τt in C1+ξ(¯Ω)C4+α(¯D) as t+, D⊂⊂Ω,ξ<1 and α<α, that is (1.2) is satisfied.

    In the paper, we derive the estimate (1.5) for the problem (1.1).

    Theorem 1.3. Suppose ΩRn (n2) is a bounded domain with smooth boundary. Assume F satisfies (F1)(F4), φC(¯Ω), then we get the uniform (in t) estimate (1.5) for the solution to (1.1).

    Actually, in [19], a good proof of convergence result is provided, under the assumption of uniform (in t)ut(,t)C(¯Ω), u(,t)C(¯Ω) estimate of quasilinear equation. In this note, after we establish the estimate of ut(,t)C(¯Ω), u(,t)C(¯Ω), 2u(,t)C(¯Ω), we use Schauder method and the process in [17] to obtain the convergence result. We can also find more details in the work [18] of Huang and Ye.

    First of all, we give some notations.

    Suppose ΩRn (n2) is a bounded domain, ΩC3. Set

    d(x)=dist(x,Ω),

    and

    Ωμ={xΩ:d(x)<μ}.

    Then there exists a positive constant μ1>0 such that  μμ1,d(x)C3(¯Ωμ). As mentioned in Lieberman [8], we can prolong ν as Dd in Ωμ which is a C2 vector field. We also have the following expressions

    |ν|+|2ν|˜C(n,Ω)in  Ωμ,1inνiiνj=0in  Ωμ,|ν|=1in  Ωμ. (1.7)

    Furthermore, in this paper, to simplify the proof of the theorems, we use O(z) to represent an expression that there exists a uniform constant C>0 satisfying |O(z)|Cz.

    In the following part of the paper, we make the following arrangement. In the second section, we think about the special case of F(2u)=u, and use a blow-up technique to control u(,t)C(ˉΩ) and then derive the estimate of u(,t)C(ˉΩ) and 2u(,t)C(ˉΩ). In the third section, we study the general F(2u) and derive the priori estimates.

    In this part, we discuss the long time behavior of the following diffusion equation with oblique derivative boundary conditions

    { utΔu=0,  in   Ω×[0,T), u(x,0)=u0(x),  on   ¯Ω×{0}, uβ=φ,  on   Ω×[0,T), (2.1)

    where ΩRn is a bounded smooth domain, φ(x), u0(x)C(¯Ω), and u0,β=φ(x) on Ω.

    As before, we denote by ν the inner normal vector field along Ω. Set {Tl}n1l=1 to be the unit tangent vector fields which joint with ν form a unit normal frame along Ω. Assume β=βnν+Σn1l=1βlTl, therefore, φ(x)=uβ=<u,β>=uνβn+Σβlul, where ul=<u,Tl>.

    Lemma 2.1. Suppose ΩRn (n2) is a bounded domain with smooth boundary. If u(x,t) is a smooth solution to (2.1), then

    supΩ×[0,T)|ut|2=supΩ|ut(x,0)|2,

    so there exists a constant C=C(u0)>0, such that (x,t)Ω×[0,T),

    |ut|(x,t)C.

    Proof. Because (Δt)(u2t)=Δu2tt(u2t)=2utΔut+2|Dut|22ututt=2|Dut|20, from the weak maximum principle, we have

    supΩ×(0,T)|ut|2=supΩ×{0}Ω×(0,T)|ut|2.

    On the other hand, (u2t)β=2ututβ=2utφt=0.

    Hopf lemma shows that the maximum cannot appear on Ω×(0,T), then

    supΩ×(0,T)|ut|2=supΩ×{0}|ut|2=supΩ|Δu0|2.

    Take x0Ω and let v(x,t)=u(x,t)u(x0,t), in the following, we first give a time independent bound of |v| by using a blow-up method. With the C0 estimate of v, we then obtain the C2 estimate of v. Naturally, the estimates for |u| and |2u| follow. Finally, the convergence results are obtained by using [18].

    Lemma 2.2. Let ΩRn (n2) be a bounded domain with smooth boundary. If u(x,t) is a smooth solution to (2.1), v(x,t) as defined above, then there exists a constant A0>0, independent of T, so that

     vC0(Ω×[0,T))A0. (2.2)

    Proof. Let A=vC0(Ω×[0,T)). Without loss of generality, we assume Aδ=δ(u0)>0, (otherwise we get a constant solution to (2.1)). Assume A is unbounded, i.e., A, as T. Let

    w(x,t)=v(x,t)A.

    Then, w(x0,t)=0,t[0,T),|w|C0(¯Ω×[0,T))=1, and satisfies

    {wtΔw=ut(x0,t)Ain  Ω×[0,T),w(x,0)=(u0(x)u0(x0))Aon  ¯Ω×{0},wβ=1Aφ(x)on  Ω×[0,T). (2.3)

    To finish the proof, we need the following propositions.

    Proposition 2.3. Let wC3,2(Ω×[0,T)) and satisfy

    wtΔw=f(t), |w|1,in Ω×[0,T).

    Then,  Ω⊂⊂Ω,

    supΩ×[0,T)|w|C(dist(Ω,Ω),|f|L([0,T))).

    Remark. We can see the proof process of this proposition in [17], so we skip it here. Note that f=ut(x0,t)A, we have

    supΩ×[0,T)|w|C(dist(Ω,Ω),|f|L([0,T)))=C(dist(Ω,Ω),u0). (2.4)

    Proposition 2.4. Let ΩRn (n2) be a bounded domain with smooth boundary. Assume that wC3,2(¯Ω×[0,T) is a solution to (2.3), Then there is a constant C=C(Ω,n,u0,||φ||C3(¯Ω)) such that for σσ1,

    supΩσ×[0,T)|w|C. (2.5)

    Proof. For 0<T<T, We will prove that we can give |w| a bound independent of T on Ω×[0,T] and then take a limit.

    Let φ=φ(x)A=wβ=<w,β>=wνβn+n1l=1βlwl and ρ=wφdcosθ, then w=ρ+φdcosθ and φ(x)=(ρ+φdcosθ)νβn+n1l=1βl(ρ+φdcosθ)lρνβn+n1l=1βlρl=0ρν=n1l=1βlβnρl.

    Thus

    (ρν)2cos2θ=(n1l=1βlρl)2n1l=1β2ln1l=1ρ2l=(|ρ|2(ρν)2)sin2θ. (2.6)

    Therefore,

    (ρν)2|ρ|2sin2θ. (2.7)

    Let

    ϕ=|ρ|2(ni=1ρidi)2=ni,j=1(δijdidj)ρiρjni,j=1Cijρiρj,

    and

    Φ=logϕ+τd+μd2,

    where τ,μ are two constants to be determined later.

    Suppose that the maximum value of Φ on Ωσ×[0,T](σσ1) is obtained at (x0,t0). Let us discuss it in several cases:

    Case 1. t0=0. If this happens, it is easy to get the gradient estimate.

    Case 2. x0ΩσΩ. In this way, the estimate is transformed into interior gradient estimate.

    Case 3. x0Ω. Select a suitable coordinate at x0, so that xn=ν, and xi(i=1,,n1) are tangent along Ω. Then, we have

    dn=1,di=0,2dxnxα=0,2dxixj=κiδij,

    where 1i,j<n, 1αn1, and κi is the principal curvatures of Ω at x0.

    Because x0 is the maximum point of Φ, then we have,

    Φi=0,1i<n1, (2.8)

    and

    0Φn=ϕnϕ+τ. (2.9)

    By (2.8), for 1i<n1, we have

    0=Φi=(|ρ|2)i(nα=1ραdα)2i=2n1j=1ρjρji2n1j=1ρnρjdij=2n1j=1ρjρji+2ρnρiκi. (2.10)

    Using (2.10) to calculate ϕn, we obtain

    ϕn=(|ρ|2)n(nα=1ραdα)2n=2ni=1ρiρin2ρnρnn=2n1i=1ρiρin=2n1i=1ρiρin+2n1i,j=1ρiρjκij=2n1i,l=1ρi(βlβnρl)i+2n1i,j=1κijρiρj=2n1i,l=1ρiρliβlβn2n1i,l=1ρiρl(βlβn)i+2n1i,j=1κijρiρj=2ρnβnn1l=1ρlκlβl2n1i,l=1ρiρl(βlβn)i+2n1i,j=1κijρiρj, (2.11)

    where we denote by κij the Weingarten matrix.

    Thus,

    0Φn=2ρnβnn1l=1κlβlρl2n1i,l=1ρiρl(βlβn)i+2n1i,j=1κijρiρjϕ+τ. (2.12)

    From (2.7), we have

    c20|ρ|2|ρ|2cos2θϕ|ρ|2.

    If we make τ large enough determined by the geometry of Ω,c0 and |β|C1(Ω), this case can not happen.

    Case 4. x0Ωσ, and t0>0.

    First, we show that |w|2 gets the maximum value at the boundary.

    By simple calculation, we have Δ(|w|2)(|w|2)t0, then

    supΩ×[0,T]|w|2supΩ×[0,T]Ω×{0}|w|2. (2.13)

    Choose a special coordinate, such that ρ1=|ρ|,  ρl=0  (l=2,3,,n) and (ρij)  (2i,jn) is diagonal. We assume that |w| is large enough such that |ρ|,|w| are equivalent at this point.

    Under this coordinate and by the assumption that |w| at (x0,t0) is large enough, we first give a basic fact

    C11˜C(σ1,c0,|φ|C1(Ω),|u0|C1(Ω))>0. (2.14)

    In fact, the maximum point of |w| on Ω×[0,T] is denoted by (x1,t1), without loss of generality, we suppose that |w|(x1,t1)4supΩ|φcosθ|.

    We propose a precondition that

    μσ1. (2.15)

    Because of Φ(x1,t1)Φ(x0,t0), (2.7) and (2.13), then we obtain

    ϕ(x0,t0)e(τ+1)σ1ϕ(x1,t1)=C[|ρ|2(ρν)2](x1,t1)C[|ρ|2cos2θ](x1,t1)C|ρ|2(x1,t1)=C|wφcosθν|2(x1,t1)C|w|2(x1,t1)CsupΩ×[0,T]|w|2C|w|2(x0,t0)C|ρ|2(x0,t0). (2.16)

    Note that C may be different in each line of the above processes.

    Through an easy observation, it can be seen that

    C11˜C>0. (2.17)

    Since (x0,t0) is the maximum point, we have

    0=Φi=(Cklρkρl)iϕ+τdi+2μddi=ϕiϕ+τdi+2μddi. (2.18)

    Hence one can see that

    ϕiϕ=τdi2μddi,Cklρkiρl=ϕ2(τ+2μd)diCkl,i2ρkρl. (2.19)

    For i=1, we get

    C11ρ11+nδ=2Cδ1ρδ1=12C11,1ρ1ϕ2ρ1(τ+2μd)d1. (2.20)

    For δ>1, we have

    C11ρ1δ+C1δρδδ=12C11,δρ1ϕ2ρ1(τ+2μd)dδ. (2.21)

    Then

    ρ1δ=C1δC11ρδδC11,δ2C11ρ1(τ+2μd)dδ2ρ1=C1δC11ρδδ+O(|ρ|). (2.22)

    Replace (2.22) back to (2.20), we have

    ρ11=(C1δC11)2ρδδ+Cδ1C11,δ2(C11)2ρ1+Cδ1ρ1(τ+2μd)dδ2C11C11,12C11ρ1ρ1(τ+2μd)d12=(C1δC11)2ρδδ+O(|ρ|). (2.23)

    At this point we still have

    0Φt=ϕtϕ=2Cklρkρltϕ, (2.24)

    and

    0Φ=ϕϕ(ϕϕ)2+(τ+2μd)d+2μ|d|2. (2.25)

    Combining (2.19), (2.24) and (2.25), we gain

    0ΦΦt=ϕϕtϕ+(τ+2μd)d+2μ|d|2(τ+2μd)2|d|2ϕϕtϕ+[2μ(τ+2μd)2]|d|2(τ+2μd)k0. (2.26)

    Where dk0 on Ωσ1.

    Next, we calculate the term ϕϕtϕ. Note that

    ¯I=ϕϕt=(Cijρiρj)ϕt=2[Cij(ρ)iρjCijρiρtj]+2Cijρikρjk+4Cij,kρikρj+Cijρiρj=I+II+III+IV. (2.27)

    For the term I,

    I=2[Cij(ρ)iρjCijρiρtj]=2Cij{[(w)i(φdcosθ)i]ρjρiwtj}=2Cij{[wti(φdcosθ)i]ρjρiwtj}=2Cij(φdcosθ)iρj=O(|ρ|). (2.28)

    For the term IV,

    IV=Cij,kkρiρj=O(|ρ|2). (2.29)

    For the term III,

    III=4Cij,kρikρj=4ρ1Ci1,kρik=4ρ1C11,1ρ11+4ρ1Cδ1,1ρ1δ+4ρ1C11,δρ1δ+4ρ1Cδ1,δρδδ=III1+III2+III3+III4, (2.30)

    where

    III1=4ρ1C11,1ρ11=4ρ1C11,1[(C1δC11)2ρδδ+O(ρ1)]=O(|w|)ρδδ+O(|w|2),
    III2+III3=4ρ1(Cδ1,1+C11,δ)ρ1δ=4ρ1(Cδ1,1+C11,δ)(C1δC11ρδδ+O(|ρ|))=O(|w|)ρδδ+O(|w|2),

    then

    III=O(|w|)ρδδ+O(|w|2). (2.31)

    For the term II,

    II=2Cijρikρjk=2C1iρikρ1k+2Ciδρikρδk=II1+II2, (2.32)

    where

    II1=2C1iρikρ1k=(Ci1,kρiϕρ1(τ+2μd)dk)ρ1k=(C11,1ρ1ρ1(τ+2μd)d1)ρ11+(C11,δρ1ρ1(τ+2μd)dδ)ρ1δ=II11+II12,
    II11=(C11,1ρ1ρ1(τ+2μd)d1)ρ11=(C11,1ρ1ρ1(τ+2μd)d1)((C1δC11)2ρδδ+O(ρ1))=nδ=2O(|w|)ρδδ+O(|w|2),
    II12=(C11,δρ1ρ1(τ+2μd)dδ)ρ1δ=(C11,δρ1ρ1(τ+2μd)dδ)(C1δC11ρδδ+O(|ρ|))=nδ=2O(|w|)ρδδ+O(|w|2),

    then

    II1=nδ=2O(|w|)ρδδ+O(|w|2).

    Where

    II2=2Ciδρikρδk=2C1δρ1kρδk+2Cαδραkρδk=2C1δρ11ρδ1+2C1δρ1δρδδ+2Cαδρα1ρδ1+2Cδδρ2δδ=II21+II22+II23+II24,
    II21=2C1δρ11ρδ1=2[nδ=2(C1δ)2C11ρδδ+O(|ρ|)]×[nδ=2(C1δC11)2ρδδ+O(|ρ|)]=2(C11)3nα,β=2C1αC1β(C1αραα)(C1βρββ)+nδ=2O(|ρ|)ρδδ+O(|ρ|2),
    II22=2nδ=2C1δρ1δρδδ=nδ=22(C1δ)2C11ρ2δδ+nδ=2O(|ρ|)ρδδ,
    II23=2Cαδρα1ρδ1=2nα,β=2Cαβ[C1αC11ραα+O(|ρ|)]×[C1βC11ρββ+O(|ρ|)]=2(C11)2nα,β=2Cαβ(C1αραα)(C1βρββ)+nδ=2O(|ρ|)ρδδ+O(|ρ|2),

    hence

    II2=II21+II22+II23+II24=2(C11)2nα,β=2Cαβ(C1αραα)(C1βρββ)2(C11)3nα,β=2C1αC1β(C1αραα)(C1βρββ)+2nδ=2Cδδρ2δδnδ=22(C1δ)2C11ρ2δδ+nδ=2O(|ρ|)ρδδ+O(|ρ|2).

    Thus

    II=II1+II2=2(C11)2nα,β=2Cαβ(C1αραα)(C1βρββ)2(C11)3nα,β=2C1αC1β(C1αραα)(C1βρββ)+2nδ=2Cδδρ2δδnδ=22(C1δ)2C11ρ2δδ+nδ=2O(|ρ|)ρδδ+O(|ρ|2). (2.33)

    And

    ϕϕt=2(C11)2nα,β=2Cαβ(C1αραα)(C1βρββ)2(C11)3nα,β=2C1αC1β(C1αραα)(C1βρββ)+2nδ=2Cδδρ2δδnδ=22(C1δ)2C11ρ2δδ+nδ=2O(|ρ|)ρδδ+O(|ρ|2)=Π1+Π2. (2.34)

    For the term Π1,

    Π1=2(C11)2nα,β=2Cαβ(C1αραα)(C1βρββ)2(C11)3nα,β=2C1αC1β(C1αραα)(C1βρββ)=2(C11)3nα,β=2(C11CαβC1αC1β)(C1αραα)(C1βρββ)=2(C11)3nα,β=2[(1d21)δαβdαdβ](C1αραα)(C1βρββ)0.

    The above formula is nonnegative, because the matrix ((1d21)δαβdαdβ)α,β2 is semi positive definite, due to |d|2=1.

    Next we set out to deal with the term Π2,

    Π2=2nδ=2Cδδρ2δδnδ=22(C1δ)2C11ρ2δδ+nδ=2O(|ρ|)ρδδ+O(|ρ|2)=2C11nδ=2(C11Cδδ(C1δ)2)ρ2δδ+nδ=2O(|ρ|)ρδδ+O(|ρ|2)=2nδ=21d21d2δC11ρ2δδ+nδ=2O(|ρ|)ρδδ+O(|ρ|2)=2nδ=2eδρ2δδ+nδ=2O(|ρ|)ρδδ+O(|ρ|2),

    where eδ=1d21d2δC11.

    According to equation wwt=f(t), we can obtain by Lemma 2.1 that ρ=O(1). Joint with ρ11=(C1δC11)2ρδδ+O(ρ1), we get

    nδ=2(1+(C1δC11)2)ρδδ=O(|ρ|).

    Therefore

    ρ22=O(|ρ|)nδ=3(C11)2+(C1δ)2(C11)2+(C12)2ρδδ.

    Substituting ρ22 into Π2, we can get

    Π2=2nδ=2eδρ2δδ+nδ=2O(|ρ|)ρδδ+O(|ρ|2)=2e2(nδ=3(C11)2+(C1δ)2(C11)2+(C12)2ρδδ)2+2nδ=3eδρ2δδ+nδ=3O(|ρ|)ρδδ+O(|ρ|2).

    Now let us consider the quadratic form in Π2, which is a quadratic form of ρ33,ρ44,,ρnn.

    Let

    Λ=2e2(nδ=3(C11)2+(C1δ)2(C11)2+(C12)2ρδδ)2+2nδ=3eδρ2δδ=2e2(nδ=3εδρδδ)2+2nδ=3eδρ2δδ.

    Through observation, we know that

    0eδ1, (2δn), nδ=2eδ=n2,

    so at most one of e2,,en is zero, it is obvious that 0<C0εδC1,δ=3,,n, the quadratic form Λ is positive definite.

    Next, we give a positive controllable lower bound for the eigenvalues of this quadratic form.

    We can regard Λ as a 3n-5 variables function, and its definition domain is

    D={(e2,e3,,en,ε3,,εn,ρ33,,ρnn)|0eδ1,nδ=2eδ=n2,0<C0εδC1,δ=3,,n,nδ=3ρ2δδ=1}.

    It is easy to see that D is a compact set. The minimum value of Λ on D is denoted by λ0, then the positive number λ0 is a general positive lower bound of the eigenvalue of the quadratic form, that is

    Λλ0nδ=3ρ2δδ.

    Therefore, in light of ax2+bxb24a, if a>0 we can obtain

    ¯IΠ2λ0nδ=3ρ2δδ+nδ=3O(|ρ|)ρδδ+O(|ρ|2)k1|ρ|2. (2.35)

    Meanwhile, in consideration of ϕ=|ρ|2C11

    ΦΦt=¯Iϕ+(τ+2μd)d+2μ(τ+2μd)2k1˜C(τ+2μd)k0(τ+2μd)2+2μ. (2.36)

    First, select μ to make

    2μ=k1˜C+(τ+1)2+(τ+1)k0+1.

    Then, select σσ1 to make 2μσ1, then we have a contradiction 0ΦΦt>0, so |ρ| must be bounded.

    Namely,

    |w|C(Ω,n,u0,||φ||C3(¯Ω)), (x,t)Ωσ×[0,T]. (2.37)

    Since the bound is independent of T, Proposition 2.4 is proved. By the uniform estimate of ut, we can deduce the estimate of uniform bound of wt, Combining with Propositions 2.3 and 2.4, we then get the uniform Ck,α estimate for kZ+ and 0<α<1 by the Schauder theory.

    Proof of Lemma 2.2. We continue to prove Lemma 2.2. For nZ+, denoted by wn=w|¯Ω×[0,n], suppose An=sup¯Ω×[0,n]|wn| which is obtained at the point (xn,tn). For (x,s)¯Ω×[0,1], Let gn(x,s)=wn(x,s+tn1), Then gn(x,s) suits

    {gnsΔgn=f(s+tn1)Anin   Ω×[0,1],gn(x,0)=wn(x,tn1)on   ¯Ω×{0},gnβ=φ(x)Anon   Ω×[0,1]. (2.38)

    Since we have obtained the uniform C1 estimate of wn(x,t) independent of t[0,n], gn(x,s) also has the uniform estimate of gradient independent of n and s. Therefore, (for convenience, we set gn(x,0)=wn(x,tn1)Δ=an(x)), {an(x)} and its derivative sequence are uniformly bounded. Thus, from the Arzela-Ascoli theorem, gn(x,0) has convergent subsequences. Without losing generality, we suppose that gn(x,0) converges to a continuous function g0(x) defined on ¯Ω satisfying g0(x0)=0 and supxΩ|g0(x)|1.

    From the relationship between gn and wn, we can obtain the uniform Ck,α estimate of gn on ¯Ω×[0,1]. So we choose a subsequence of gn converges in the sense of Ck,α (kZ+ and 0<α<1) to g on ¯Ω×[0,1]. Clearly, we get

    {gsΔg=0in   Ω×[0,1],g(x,0)=g0(x)on   ¯Ω×{0},gβ=0on   Ω×[0,1]. (2.39)

    Because of gsΔg=0, g obtains the maximum value on Ω×{0} or Ω×[0,1], but gβ=0 shows that it can only be achieved at Ω×{0}, however g reaches the maximum at s=1. It is a contradiction by the maximum principle and Hopf Lemma for the parabolic differential equations. Thus, we complete the proof of Lemma 2.2.

    Theorem 2.5. T>0, supposing that u is a smooth solution to (2.1), then we have the estimate,

    ut(,t)C(¯Ω)+u(,t)C(¯Ω)+2u(,t)C(¯Ω)C, (2.40)

    where C is a constant independent of t and T.

    Proof. From the definition of v, v satisfies the following equation

    {vtΔv=ut(x0,t)in    Ω×(0,T),v(x,0)=u0(x)u0(x0)on   ¯Ω×{0},vβ=φ(x)on   Ω×(0,T). (2.41)

    From Lemma 2.2 we have |v|A0, the step similar to Propositions 2.3 and 2.4 deduces

    v(,t)C(¯Ω)C.

    Schauder theory then deduces

    2v(,t)C(¯Ω)C.

    Since v(x,t)=u(x,t)u(x0,t), we get

    u(,t)C(¯Ω)+2u(,t)C(¯Ω)C.

    Combining with Lemma 2.1, we finish the proof of Theorem 2.5.

    In this part, we consider

    { ut=F(uij)  in   Ω×[0,T), u(x,0)=u0(x)  on   Ω×{0}, uβ=φ(x)  on   Ω×[0,T), (3.1)

    where ΩRn is a smooth bounded domain, φ(x), u0(x)C(¯Ω), so that u0,β=φ(x) on Ω. Moreover, we assume that F satisfies (F1)(F4).

    Lemma 3.1. Let ΩRn (n2) be a bounded domain with smooth boundary. Assuming that u(x,t) is a smooth solution to (3.1), there is a constant C0=C0(u0)>0 such that (x,t)Ω×(0,),

    |ut|(x,t)C0.

    Proof. Let Fiju denote rij|r=2uF(r) and L=Fijuijt, take the derivative of t on both sides of ut=F(2u), we have

    utt=Fijuuijt,

    then L(u2t)=2ni,j=1Fijuutiutj+2Fijuututij2ututt2ni,j=1Fijuutiutj0, from the weak maximum principle, we get

    supΩ×(0,T)|ut|2=supΩ×{0}Ω×(0,T)|ut|2.

    Since, (u2t)β=2ututβ=2utuβt=0. Hopf lemma makes it impossible for the maximum to occur on Ω×(0,T), then

    supΩ×(0,T)|ut|2=supΩ×{0}|ut|2=supΩ|F(2u0)|2.

    Let v(x,t)=u(x,t)u(x0,t) where x0Ω. Similar to Section 2, we first give a time-independent bound for |v| by a blow-up technique. Then from the C0 estimate of v, we get the bound of vC2. Naturally, it follows the estimates of |u| and |2u|. Finally, we get the convergence result according to the method of [1].

    Lemma 3.2. Let ΩRn (n2) be a bounded domain with smooth boundary. If u(x,t) is a smooth solution to (3.1), v(x,t) is defined as above, then there is a constant A0>0, independent of T, such that

     vL(Ω×[0,T))A0. (3.2)

    Proof. Let A=vC0(Ω×[0,T)). Without loss of generality, we assume that Aδ=δ(u0)>0, (otherwise the solution to (3.1) is a constant). Assume A is unbounded, that is, A, as T. Let

    w(x,t)=v(x,t)A.

    Obviously w satisfies w(x0,t)=0,t[0,T),|w|C0(¯Ω×[0,T))=1, and

    {wt1AF(A2w)=ut(x0,t)Ain  Ω×[0,T),w(x,0)=(u0(x)u0(x0))Aon  ¯Ω×{0},wβ=1Aφ(x)on  Ω×[0,T). (3.3)

    In order to prove the above estimate, we need the following propositions.

    Proposition 3.3. If wC3,2(Ω×[0,T)) satisfies |w|M for a normal number M and

    {wt1AF(A2w)=f(t)in  Ω×[0,T),w(x,0)=(u0(x)u0(x0))A=w0(x)on  ¯Ω×{0}. (3.4)

    Then Ω⊂⊂Ω,

    supΩ×[0,T)|w|C(λ, μ0, μ1, M, w0, dist(Ω, Ω), |f|L([0,T))).

    Remark. One can refer to [17] for the proof of this proposition. Note that f=ut(x0,t)A,M=1 in problem (3.3), we get

    supΩ×[0,T)|Dw|C(λ,μ0,μ1,dist(Ω,Ω),u0). (3.5)

    Proposition 3.4. If ΩRn (n2) is a bounded domain with smooth boundary. Assuming that wC3,2(¯Ω×[0,T)) is a solution to (3.3), there exists a constant C=C(Ω,n,u0,φ,λ,μ0,μ1) such that for σσ1,

    supΩσ×[0,T)|w|C. (3.6)

    Proof. For 0<T<T, we will complete the proof on Ωσ×[0,T] and show that the bound is independent of T.

    Let

    Φ=logϕ+τd+μd2,ϕ=|ρ|2(ni=1ρidi)2=ni,j=1(δijdidj)ρiρjni,j=1Cijρiρj,

    where τ, μ are positive constants to be determined later, ρ=wφdcosθ, and φ=φ(x)A=wβ=wνβn+n1l=1βlwl.

    Assume Φ gets the maximum value at (x0,t0) on Ωσ×[0,T].

    Case 1. t0=0. we get

    |w|2(x0,0)C(Ω,n,u0).

    Case 2. x0ΩσΩ. In this case, the estimate follows from interior gradient estimate in Proposition 3.3.

    Case 3. x0Ω. Similar to the process of Proposition 2.4, we can choose the appropriate τ to guarantee this case does not occur.

    Case 4. x0Ωσ, and t0>0.

    Select a particular coordinate, so that ρ1=|ρ|, ρl=0 (l=2,3,,n) and (ρij)  (2i,jn) is diagonal. We assume that |w| is large enough at this point so that |ρ|, |w| are equivalent.

    Through a process similar to Proposition 2.4, we have

    C11˜C( σ1, c0, |φ|C1(Ω), |u0|C1(Ω))>0.

    At the maximum point (x0,t0), we have

    0=Φi=(Cklρkρl)iϕ+τdi+2μddi=ϕiϕ+τdi+2μddi,

    thus it can be seen

    ϕiϕ=τdi2μddi,Cklρkiρl=ϕ2(τ+2μd)diCkl,i2ρkρl.

    When i=1, it follows

    C11ρ11+nδ=2Cδ1ρδ1=12C11,1ρ1ϕ2ρ1(τ+2μd)d1.

    When δ>1, we obtain

    C11ρ1δ+C1δρδδ=12C11,δρ1ϕ2ρ1(τ+2μd)dδ.

    Thus,

    ρ1δ=C1δC11ρδδC11,δ2C11ρ1(τ+2μd)dδ2ρ1=C1δC11ρδδ+O(|ρ|),

    and

    ρ11=(C1δC11)2ρδδ+Cδ1C11,δ2(C11)2ρ1+Cδ1ρ1(τ+2μd)dδ2C11C11,12C11ρ1ρ1(τ+2μd)d12=(C1δC11)2ρδδ+O(|ρ|).

    At the same time, at this point we have

    0Φt=ϕtϕ=2Cklρkρltϕ,

    and

    0Φij=ϕijϕ(τ+2μd)2didj+(τ+2μd)dij+2μdidj.

    Then,

    0FijΦijΦt=Fijϕijϕtϕ+(τ+2μd)Fijdij+[2μ(τ+2μd)2]Fijdidj.

    First, we come to calculate Fklϕklϕt,

    Fklϕklϕt=2CijFklρiklρj2Cijρjρit+2CijFklρikρjl+4FklCij,kρilρj+FklCij,klρiρj=I+II+III+IV,

    where

    I=2CijFklρiklρj2Cijρjρit=2Cij[Fklρiklρit]ρj=2Cij[Fkl(φdcosθ)ikl]ρj=O(|w|),IV=FklCij,klρiρj=O(|w|2),III=4FklCij,kρilρj=4FklCi1,kρilρ1=4ρ1Fk1C11,kρ11+4ρ1Fk1Cδ1,kρ1δ+4ρ1FkδC11,kρ1δ+4ρ1FkδCδ1,kρδδ=III1+III2+III3+III4,

    and

    III1=4ρ1Fk1C11,kρ11=4ρ1Fk1C11,k[(C1δC11)2ρδδ+O(ρ1)]=O(|w|)ρδδ+O(|w|2),III2+III3=4ρ1Fk1Cδ1,kρ1δ+4ρ1FkδC11,kρ1δ=4ρ1(Fk1Cδ1,k+FkδC11,k)ρ1δ=4ρ1(Fk1Cδ1,k+FkδC11,k)(C1δC11ρδδ+O(|ρ|))=O(|w|)ρδδ+O(|w|2),

    thus,

    III=O(|w|)ρδδ+O(|w|2).

    For the second term

    II=2FklCijρikρjl=2FklC1iρikρ1l+2FklCiδρikρδl=II1+II2,II1=2FklC1iρikρ1l=Fkl(Ci1,kρiϕρ1(τ+2μd)dk)ρ1l=Fk1(C11,kρ1C11ρ1(τ+2μd)dk)ρ11+Fkδ(C11,kρ1C11ρ1(τ+2μd)dk)ρ1δ=II11+II12,

    and

    II11=Fk1(C11,kρ1+C11ρ1(τ+2μd)dk)ρ11=Fk1(C11,kρ1+C11ρ1(τ+2μd)dk)((C1δC11)2ρδδ+O(ρ1))=O(|w|)ρδδ+O(|w|2),II12=Fkδ(C11,kρ1C11ρ1(τ+2μd)dk)ρ1δ=Fkδ(C11,kρ1C11ρ1(τ+2μd)dk)(C1δC11ρδδ+O(|ρ|))=O(|w|)ρδδ+O(|w|2),

    therefore,

    II1=O(|w|)ρδδ+O(|w|2).

    For II2, we have

    II2=2FklCiδρikρδl=2FklC1δρ1kρδl+2FklCαδραkρδl=2F1lC1δρ11ρδl+2FαlC1δρ1αρδl+2F1lCαδρα1ρδl+2FαlCαδρααρδl=II21+II22+II23+II24,

    where

    II21=2F1lC1δρ11ρδl=2F11C1δρ11ρ1δ+2F1δC1δρ11ρδδ=2F11[(C1δ)2C11ρδδ+O(|ρ|)]×[(C1δC11)2ρδδ+O(|ρ|)]+2F1δC1δρδδ[(C1αC11)2ραα+O(|ρ|)]=2(C11)3F11nα,β=2(C1α)2(C1β)2ρααρββ+2(C11)2nα,β=2(C1α)2F1βC1βρααρββ+nδ=2O(|ρ|)ρδδ+O(|ρ|2),
    II22=2FαlC1δρ1αρδl=2Fα1C1δρ1αρ1δ+2FαδC1δρ1αρδδ=2Fα1C1δ[C1αC11ραα+O(|ρ|)][C1δC11ρδδ+O(|ρ|)]+2FαδC1δρδδ[C1αC11ραα+O(|ρ|)]=2(C11)2nα,β=2F1αC1α(C1β)2ρααρββ2C11nα,β=2FαβC1αC1βρααρββ+nδ=2O(|ρ|)ρδδ+O(|ρ|2),
    II23=2F1lCαδρα1ρδl=2F11Cαδρ1αρ1δ+2F1δCαδρα1ρδδ=2F11Cαδ[C1αC11ραα+O(|ρ|)][C1δC11ρδδ+O(|ρ|)]+2F1δCαδρδδ(C1αC11ραα+O(|ρ|))=2F11(C11)2nα,β=2CαβC1αC1βρααρββ2C11nα,β=2F1βCαβC1αρααρββ+nδ=2O(|ρ|)ρδδ+O(|ρ|2),II24=2FαlCαδρααρδl=2Fα1Cαδρααρδ1+2FαδCαδρααρδδ=2Fα1Cαδραα[C1δC11ρδδ+O(|ρ|)]+2FαδCαδρααρδδ=2C11nα,β=2F1αCαβC1βρααρββ+2nα,β=2FαβCαβρααρββ+nδ=2O(|ρ|)ρδδ,

    then,

    II2=2C11nα,β=2CαβF1αC1βρααρββ+2nα,β=2FαβCαβρααρββ+2F11(C11)2nα,β=2CαβC1αC1βρααρββ2C11nα,β=2F1βCαβC1αρααρββ+2(C11)2nα,β=2F1αC1α(C1β)2ρααρββ2C11nα,β=2FαβC1αC1βρααρββ2(C11)3F11nα,β=2(C1α)2(C1β)2ρααρββ+2(C11)2nα,β=2F1β(C1α)2C1βρααρββ+nδ=2O(|ρ|)ρδδ+O(|ρ|2),

    thus, we have

    II2=4C11nα,β=2F1αCαβC1βρααρββ+2nα,β=2FαβCαβρααρββ+2F11(C11)2nα,β=2CαβC1αC1βρααρββ2(C11)3F11nα,β=2(C1α)2(C1β)2ρααρββ+4(C11)2nα,β=2F1αC1α(C1β)2ρααρββ2C11nα,β=2FαβC1αC1βρααρββ+nδ=2O(|ρ|)ρδδ+O(|ρ|2).

    We mainly deal with the quadratic term in II2,

    Π=4C11nα,β=2F1αCαβC1βρααρββ+2nα,β=2FαβCαβρααρββ+2F11(C11)2nα,β=2CαβC1αC1βρααρββ2(C11)3F11nα,β=2(C1α)2(C1β)2ρααρββ+4(C11)2nα,β=2F1αC1α(C1β)2ρααρββ2C11nα,β=2FαβC1αC1βρααρββ.

    Simplify the above formula, we get

    Π=nα,β=22[Fαβ(C11)2+F11C1αC1βF1αC1βC11F1βC1αC11][C11CαβC1αC1β](C11)3ρααρββ,

    where C11CαβC1αC1β=(1d21)δαβdαdβ.

    To deal with the above quadratic form, let us make the following preparations:

    Definition 3.5. Suppose A, B are two m-order symmetric matrices, its Hadamard product is defined as AB=(aijbij)m×m, that is, the element product at the corresponding position is defined as the element at the corresponding position of the Hadamard product matrix.

    Theorem 3.6. If A and B are two m order positive semi-definite matrices, AB is also a m order positive semi-definite matrix; If A and B are two m order positive definite matrices, AB is also a m order positive definite matrix.

    Corollary 3.7. If AλE, B0, then ABλEB.

    With the above knowledge about matrices, let's look at the two matrices contained in Π, one is

    A=nα,β=2(Fαβ(C11)2+F11C1αC1βF1αC1βC11F1βC1αC11),

    and the other is

    B=nα,β=2((1d21)δαβdαdβ).

    Because |d|2=1, it's easy to see that matrix B is positive semi-definite.

    Let's consider symmetric matrix A.

    Remark that Fij is positive definite and by the assumption we know that λEFijΛE, for any X=(x2,x3,,xn), we have

    nα,β=2(Fαβ(C11)2+F11C1αC1βF1αC1βC11F1βC1αC11)xαxβ=(C11)2nα,β=2Fαβxαxβ+F11(nα=2C1αxα)22C11(nα=2F1αxα)(nα=2C1αxα)(C11)2[nα,β=2Fαβxαxβ1F11(nα=2F1αxα)2]=(C11)2nα,β=2(FαβF1αF1βF11)xαxβ.

    We want to show that the matrix (FαβF1αF1βF11)2α,βn is positive definite and its eigenvalues are bounded from below by λ. In fact, since λEFijΛE, we have that the matrix (Fij)diag{0,λ,λ,,λ} is positive semi-definite. However, according to a series of elementary transformations we can deduce that (Fij)diag{0,λ,λ,,λ} is congruent with (F1100FαβλδαβF1αF1βF11). Therefore, (FαβF1αF1βF11)2α,βn is positive definite and its eigenvalues are bounded from below by λ.

    So, (Fαβ(C11)2+F11C1αC1βF1αC1βC11F1βC1αC11)((C11)2λδαβ) and then by the corollary we have that

    Π2λC11nα,β=2((1d21)δαβdαdβ)δαβρααρββ=2λnα=2(1d21)d2αC11ρ2αα2λnα=2eαρ2αα.

    According to the first equation in (3.3), we can get aijρij=O(1), where λδijaijΛδij. Reuse ρ11=(C1δC11)2ρδδ+O(ρ1), there is

    nδ=2(aδδ+a11(C1δC11)22a1δC1δC11)ρδδ=O(|ρ|).

    Write

    γδ=aδδ+a11(C1δC11)22a1δC1δC11.

    Thus

    0<λ2ΛγδΛ(1+(1˜C)2+2˜C).

    Therefore

    ρ22=O(|ρ|)nδ=3γδγ2ρδδ.

    Then

    Π2λnα=2eαρ2αα=2λ[e2ρ222+nα=3eαρ2αα]=2λ[e2(nδ=3γδγ2ρδδ)2+nα=3eαρ2αα]+nδ=3O(|ρ|)ρδδ+O(|ρ|2).

    Consider the quadratic form in brackets in the above formula, which is about the quadratic form of ρ33,ρ44,,ρnn,

    Θ=e2(nδ=3γδγ2ρδδ)2+nδ=3eδρ2δδ.

    Since the coefficients e2,e3,,en satisfy

    0eδ1, δ=2,3,n, nδ=2eδ=n2,

    so, at most one of e2,,en is zero, and considering the condition about γδ, so this quadratic form is positive definite.

    Next, we give a positive controllable lower bound for the eigenvalues of this quadratic form.

    We can regard Θ as a 3n-4 variables function, and its definition domain is

    D={(e2,e3,,en,γ2,,γn,ρ33,,ρnn)|0eδ1,nδ=2eδ=n2,0<λ2ΛγδΛ(1+(1˜C)2+2˜C),nδ=3ρ2δδ=1}.

    It is easy to see that D is a compact set, so, the minimum value of Θ on D is written as λ0, then the positive mumber λ0 is a general positive lower bound of the eigenvalue of the quadratic form, that is

    Θ=e2(nδ=3γδγ2ρδδ)2+nδ=3eδρ2δδλ0nδ=3ρ2δδ.

    Therefore, on the basis of ax2+bxb24a, if a>0 we can obtain

    II2λλ0nδ=3ρ2δδ+nδ=3O(|ρ|)ρδδ+O(|ρ|2)k1|ρ|2.

    In consideration of ϕ=|ρ|2C11, and supposing dijk2δij we have

    0FijΦijΦt=Fijϕijϕtϕ+(τ+2μd)Fijdij+[2μ(τ+2μd)2]Fijdidjk1C11(τ+2μd)k2nΛ(τ+2μd)2Λ+2μλ.

    First, select μ to make

    2μλ=k1˜C+Λ(τ+1)2+(τ+1)k2nΛ+1.

    Then, select σσ1 to make 2μσ1, hence we have a contradiction 0FijΦΦt>0, so then |ρ| must be bounded.

    Then

    |Dw|2(x,t)C(λ,μ0,μ1,u0,||φ||C3(¯Ω),n,Ω), (x,t)¯Ωσ×[0,T]. (3.7)

    Since the bound is independent of T, the proof of Proposition 3.4 is completed.

    Proposition 3.8. If wC4,2(Ω×[0,T)) satisfies ||w||C1(Ω×[0,T))M1 ( M1>0) and

    {wt1AF(A2w)=f(t)in  Ω×[0,T),w(x,0)=w0(x)in  Ω. (3.8)

    Then  Ω⊂⊂Ω,

    supΩ×[0,T)|2w|C(λ, μ0, μ1, M1, w0, dist(Ω, Ω), ||f||L([0,T))).

    Remark. One can refer to [17] for the proof of this proposition.

    Proposition 3.9. If ΩRn (n2) is a bounded domain with smooth boundary. Assuming that wC4,2(¯Ω×[0,T)) is a solution to (3.3), there is a constant C=C(Ω,n,u0,φ,λ,μ0,μ1), such that for σσ1,

    supΩσ×[0,T)|2w|C(1+supΩ×[0,T)|wββ|). (3.9)

    Proof. For 0<T<T, we will give the bound of |2w| on Ωσ×[0,T] independent of T.

    Let

    H(x,t,ξ)=eαd(wξξ+Bw2ξ),

    where α,B (>0) to be determined later, and ξSn1 is a fixed unit vector, we can assume that |wξξ|1, otherwise, there is nothing to do. We first set the following differential inequality.

    ni,j=1FijHijHt0modHonΩσ×(0,T]. (3.10)

    In fact,

    0=Hi=αdiH+eαd(wξξi+B(w2ξ)i),Ht=eαd(wξξt+B(w2ξ)t),Hij=(αdijα2didj)H+eαd(wξξij+B(w2ξ)ij).

    Therefore

    ni,j=1FijHijHt=ni,j=1Fij(αdijα2didj)H+eαd(ni,j=1Fijwξξijwξξt)+Beαd(ni,j=1Fij(w2ξ)ij(w2ξ)t)=I+II+III,

    where

    \begin{equation*} \begin{aligned} |I|\leq& \mu_1(\alpha\widetilde{C}^2+\alpha^2)e^{\alpha d}|w_{\xi\xi}|+C_0(\alpha, \mu_1, n, \Omega), \\ II\geq& 0, \\ III = &2Be^{\alpha d}\sum\limits_{i, j = 1}^{n}F^{ij}w_{\xi i}w_{\xi j}+2Be^{\alpha d}w_\xi(\sum\limits_{i, j = 1}^{n}F^{ij}w_{\xi i j}-w_{\xi t})\\ \geq&2Be^{\alpha d}\lambda\sum\limits_{i = 1}^{n}|w_{\xi i}|^2.\end{aligned} \end{equation*}

    From Cauchy inequality, we have |w_{\xi\xi}|^2 = |\sum\limits_{i = 1}^{n}w_{\xi i}\xi^{i}|^2\leq \sum\limits_{i = 1}^{n}w_{\xi i}^2, and then according to the hypothesis |w_{\xi\xi}|\geq1 , we get

    \begin{equation*} \begin{aligned} III\geq2Be^{\alpha d}\lambda|w_{\xi \xi}|.\end{aligned} \end{equation*}

    Then if we take B = \frac{1}{2\lambda}(\mu_{1}(\alpha\widetilde{C}^2+\alpha^2)+C_0) , so (3.10) is proved.

    Suppose that the maximum point of H is (x_0, t_0, \xi_0), according to the maximum principle, it must occur on \Omega_\sigma\times\{0\}\times S^{n-1}, \ (\partial\Omega_\sigma\bigcap\Omega)\times[0, T']\times S^{n-1} \ \mbox{or}\ \partial\Omega\times[0, T']\times S^{n-1}. Let's discuss it one by one in the following situations.

    Case 1. (x_0, t_0, \xi_0)\in \Omega_\sigma\times\{0\}\times S^{n-1} . Then

    \begin{equation*} \begin{aligned} w_{\xi_0\xi_0}(x_0, t_0)\leq \max \{H(x_0, 0, \xi_0), 0\}\leq C(u_0, \Omega). \end{aligned} \end{equation*}

    Case 2. (x_0, t_0, \xi_0)\in (\partial\Omega_\sigma\bigcap\Omega)\times[0, T']\times S^{n-1} . In this case, it is transformed into the interior estimate, and Proposition 3.8 guarantees the conclusion.

    Case 3. (x_0, t_0, \xi_0)\in \partial\Omega\times[0, T']\times S^{n-1} . Under this condition, we have

    \begin{equation} \begin{aligned} 0\geq H_\beta = \alpha \beta_n(w_{\xi_0\xi_0}+Bw^2_{\xi_0})+w_{\xi_0\xi_0\beta}+2Bw_{\xi_0}w_{\xi_0\beta}.\end{aligned} \end{equation} (3.11)

    First, we suppose that \xi_0\cdot\nu = 0 .

    Let's write w_{ij}\tau^i\mu^j with w_{\tau\mu}, take the tangential derivatives on both sides of w_\beta = \varphi' = \frac{\varphi}{A}, and we have

    \begin{equation*} \begin{aligned} \sum\limits_{p, q = 1}^{n}\sum\limits_{k = 1}^{n}C^{pq}(w_k \beta^k)_p\xi_0^q = \sum\limits_{p = 1}^{n}\sum\limits_{q = 1}^{n}C^{pq}(\varphi')_p\xi_0^q, \end{aligned} \end{equation*}

    where C^{pq} = \delta_{pq}-\nu^p\nu^q = \delta_{pq}-d_p d_q in \Omega_\sigma . Thus

    \begin{equation*} \begin{aligned} w_{\xi_0 \beta} = (\varphi')_{\xi_0}-\sum\limits_{k = 1}^{n}w_{k}{\beta^k}_{, q}\xi_0^q.\end{aligned} \end{equation*}

    It can be seen that there is a constant \Lambda = \Lambda(\varphi, \widetilde{C}, ||\nabla w||_{C^0(\overline{\Omega}\times[0, T))}) such that

    \begin{equation} \begin{aligned} |w_{\xi_0 \beta}|\leq\Lambda.\end{aligned} \end{equation} (3.12)

    Taking double tangential derivative on both sides of w_\beta = \varphi' = \frac{\varphi}{A}, we get

    \begin{equation*} \begin{aligned}\sum\limits_{i, j, k, p, q = 1}^{n}C^{jq}(C^{ip}(w_k\beta^k)_p)_q\xi_0^i\xi_0^j = \sum\limits_{i, j, p, q = 1}^{n}C^{jq}(C^{ip}\varphi'_p)_q\xi_0^i\xi_0^j, \end{aligned} \end{equation*}

    thus

    \begin{equation*} \begin{aligned} w_{\xi_0\xi_0\beta} = &\sum\limits_{i, j, p, q = 1}^{n}C^{jq}C^{ip}_{, q}\varphi'_p \xi_0^i\xi_0^j+\varphi'_{\xi_0\xi_0}-\sum\limits_{k, p, q = 1}^{n}\xi_0^p\xi_0^q(w_{kp}\beta^k_q+w_{kq}\beta^k_p+w_k \beta_{pq}^k)\\ -&\sum\limits_{i, p, q, k = 1}^{n}\xi_0^q C^{ip}_{, q}\xi_0^i(w_k\beta^k)_p.\end{aligned} \end{equation*}

    Therefore,

    \begin{equation*} \begin{aligned}|w_{\xi_0 \xi_0\beta}+2Bw_{\xi_0}w_{\xi_0 \beta}|\leq2\widetilde{C}|\nabla^{2}w|+C(||\varphi||_{C^2(\overline{\Omega})}, \widetilde{C}, ||\nabla w||_{C^0(\overline{\Omega}\times[0, T))}, B).\end{aligned} \end{equation*}

    Because w_t is bounded, operator F is uniformly elliptic, by classical theory of uniform elliptic differential equations, \forall (x, t)\in \Omega_\sigma \times[0, T'], we have

    \begin{equation*} \begin{aligned} |\nabla^2 w|\leq C_0(\lambda, \mu_1, u_0)(1+\sup\limits_{\gamma\in S^{n-1}}w^+_{\gamma\gamma}).\end{aligned} \end{equation*}

    Without loss of generality, we assume that \sup\limits_{\gamma\in S^{n-1}}w^+_{\gamma\gamma} = w_{\zeta\zeta} > 0.

    Choose a proper coordinate at x_0: \overrightarrow{e_1}, \cdots, \overrightarrow{e_{n-1}}, \overrightarrow{\beta}, such that \zeta = \sum\limits_{i = 1}^{n-1}a_i\overrightarrow{e_i}+a_n\overrightarrow{\beta}, let \zeta^{\top} = \sum\limits_{i = 1}^{n-1}a_i\overrightarrow{e_i}, then \zeta = \zeta^{\top}+a_n\overrightarrow{\beta} , we then have by (3.12)

    \begin{equation*} \begin{aligned}|\nabla^2 w|&\leq C_0(1+w_{\zeta\zeta})\\ &\leq C_0(1+w_{\zeta^\top\zeta^\top}+2a_{n}w_{\zeta^\top\beta}+a^2_{n}w_{\beta\beta})\\ & \leq C_1(1+2\Lambda+H(x_0, t_0, \xi_0)+|w_{\beta\beta}|)\\ &\leq C_1(1+2\Lambda+w_{\xi_0 \xi_0}+B||\nabla w||^2_{C^0(\overline{\Omega}\times[0, T))}+|w_{\beta\beta}|). \end{aligned} \end{equation*}

    Then,

    \begin{equation*} \begin{aligned}|w_{\xi_0\xi_0\beta}+2Bw_{\xi_0}w_{\xi_0\beta}|\leq 2C_1\widetilde{C}(1+w_{\xi_0 \xi_0}+|w_{\beta\beta}|)+C(||\varphi||_{C^2(\overline{\Omega})}, \widetilde{C}, B, ||\nabla w||_{C^0(\overline{\Omega}\times[0, T))}).\end{aligned} \end{equation*}

    Substitute the above inequality into (3.11), take \alpha = 2C_1\widetilde{C}+1 , and then we deduce

    \begin{equation*} \begin{aligned} w_{\xi_0 \xi_0}(x_0, t_0)\leq C(1+\sup\limits_{\partial\Omega\times[0, T)}|w_{\beta\beta}|), \end{aligned} \end{equation*}

    where C = C(\lambda, \mu_1, u_0, ||\varphi||_{C^2(\overline{\Omega})}, \widetilde{C}, B, ||\nabla w||_{C^0(\overline{\Omega}\times[0, T))}).

    If \xi_0\cdot\nu\neq0, similar to the above discussion process, let \xi_0 = \sum\limits_{i = 1}^{n-1}b_i\overrightarrow{e_i}+b_n\overrightarrow{\beta}, and \xi_0^{\top} = \sum\limits_{i = 1}^{n-1}b_i\overrightarrow{e_i}, then \xi_0 = \xi_0^{\top}+b_n\overrightarrow{\beta} ,

    then we obtain

    \begin{equation*} \begin{aligned} w_{\xi_0 \xi_0} = &w_{\xi_0^\top \xi_0^\top}+2b_nw_{\xi_0^\top \beta}+b_n^2w_{\beta\beta}\\ \leq& C(1+|w_{\beta\beta}|). \end{aligned} \end{equation*}

    Combined with all the above, we come to the conclusion that

    \begin{equation*} \begin{aligned} \sup\limits_{\Omega_{\sigma}\times[0, T']}|\nabla^2 w|\leq C(1+\sup\limits_{\partial\Omega\times[0, T)}| w_{\beta\beta}|), \end{aligned} \end{equation*}

    where C = C(\lambda, \mu_1, \Omega, n, \varphi, u_0, ||\nabla w||_{C^0(\overline{\Omega}\times[0, T))}) which is independent of T' , so we finish the proof of Proposition 3.9.

    Proposition 3.10. If \Omega \subset R^n (n\geq2) is a bounded domain with smooth boundary. Assume that w \in C^{4, 2}(\overline\Omega\times[0, T)) is a solution to (3.3), Then there is a constant C = C(\Omega, n, u_0, \varphi, \lambda, \mu_0, \mu_1), such that

    \begin{equation} \begin{aligned} \mathop {\sup }\limits_{{\partial\Omega}\times[0, T)}| w_{\beta\beta} | \le C. \end{aligned} \end{equation} (3.13)

    Proof. For any 0 < T' < T , we use the barrier function to give |w_{\beta\beta}| a bound independent of T' on \partial\Omega\times[0, T'] , and then take a limit.

    Let

    \begin{equation*} \begin{aligned} M_2 = \sup\limits_{\Omega\times[0, T)}|\nabla^2 w|.\end{aligned} \end{equation*}

    As before, we think about a function G(x, t) = \sum\limits_{i = 1}^n w_i\beta^i-\frac{\varphi}{A} defined on \Omega_\sigma \times[0, T'], and we have

    \begin{equation*} \begin{aligned} |G| < C(||\nabla w||_{C^0(\Omega\times[0, T))}, u_0, ||\varphi||_{C^0(\Omega)}): = \widehat{C}.\end{aligned} \end{equation*}

    Suppose the barrier function is

    \begin{equation*} \begin{aligned} H(x, t) = 4\widehat{C}K(d-Kd^2)\pm G, \end{aligned} \end{equation*}

    where

    \begin{equation} \begin{aligned} K\geq\frac{1}{2\sigma_1}, \end{aligned} \end{equation} (3.14)

    is a positive number to be determined. Clearly,

    \begin{equation} \begin{aligned} H = 0 \ \ \mbox{on} \ \partial\Omega\times[0, T'].\end{aligned} \end{equation} (3.15)

    Notice that if K\sigma = \frac{1}{2}, we get

    \begin{equation} \begin{aligned} H > 0\ \ \mbox{on} \ \ (\partial\Omega_\sigma\cap\Omega)\times[0, T'].\end{aligned} \end{equation} (3.16)

    On \Omega_\sigma\times\{0\}, note that G(x, 0) is a function related only to u_0(x) and we can suppose that

    \begin{equation} \begin{aligned} K\geq \widetilde{C}+\sqrt{\frac{\max\limits_{\overline{\Omega}}|\Delta G(x, 0)|}{4\widehat{C}}}, \end{aligned} \end{equation} (3.17)

    where \widetilde{C} is from (1.6).

    Now Let's compute \Delta H(x, 0) on \Omega_\sigma\times\{0\}. Combined with K\sigma = \frac{1}{2} , we get

    \begin{equation*} \begin{aligned} \Delta H(x, 0) = &4\widehat{C}K(\Delta d-2Kd\Delta d-2K)\pm\Delta G\\ \leq &4\widehat{C}K(\widetilde{C}-2K)\pm\Delta G\\ \leq &-4\widehat{C}K^2\pm\Delta G\leq0. \end{aligned} \end{equation*}

    From the fact H(x, 0)\geq0\ \ \mbox{on} \ \ \partial\Omega_\sigma derived from (3.15) and (3.16), we derive that

    \begin{equation} \begin{aligned} H > 0\quad \mbox{on} \quad \Omega_\sigma\times\{0\}.\end{aligned} \end{equation} (3.18)

    Now we start to think about the function H(x, t) on \Omega_\sigma\times(0, T'] .

    Set F^{ij} = \frac{\partial}{\partial r_{ij}}|_{r = A\nabla^2w }F(r) , thus on \Omega_\sigma\times(0, T'] ,

    \begin{equation*} \begin{aligned}\sum\limits_{i, j = 1}^{n}F^{ij}G_{ij}-G_t = &\sum\limits_{i, j, k = 1}^nF^{ij}w_{ijk}\beta^k-\sum\limits_{i = 1}^n w_{kt}\beta^k+\sum\limits_{i, j, k = 1}^{n}F^{ij}(w_{ik}\beta^k_j+w_{jk}\beta^k_i)-\sum\limits_{i, j = 1}^{n}\frac{1}{A}F^{ij}\varphi_{ij}\\ = &\sum\limits_{i, j, k = 1}^{n}F^{ij}(w_{ik}\beta^k_j+w_{jk}\beta^k_i)-\sum\limits_{i, j = 1}^{n}\frac{1}{A}F^{ij}\varphi_{ij}, \end{aligned} \end{equation*}

    consequently,

    \begin{equation*} \begin{aligned}|\sum\limits_{i, j = 1}^{n}F^{ij}G_{ij}-G_t|\leq C_2(\mu_1, \Omega, n, u_0, ||\varphi||_{C^2(\Omega)})[1+M_2].\end{aligned} \end{equation*}

    Hence, on \Omega_\sigma\times(0, T']

    \begin{equation*} \begin{aligned}\sum\limits_{i, j = 1}^{n}F^{ij}H_{ij}-H_t = &4\widehat{C}K\sum\limits_{i, j = 1}^{n}F^{ij}(d_{ij}-2Kd_{i}d_j-2Kdd_{ij})\pm(\sum\limits_{i, j = 1}^{n}F^{ij}G_{ij}-G_t)\\ \leq &4\widehat{C}K(\mu_1\widetilde{C}-2K\lambda)+C_2(1+M_2)\\ \leq &-4\widehat{C}\lambda K^2+C_2(1+M_2)\leq 0, \end{aligned} \end{equation*}

    if we take

    \begin{equation} \begin{aligned} K\geq \frac{\mu_1\widetilde{C}}{\lambda}+\sqrt{\frac{C_2(1+M_2)}{4\lambda\widehat{C}}}.\end{aligned} \end{equation} (3.19)

    Combined with (3.14), (3.17) and (3.19), let

    \begin{equation} \begin{aligned} K = \frac{1}{2\sigma_1}+\frac{\mu_1\widetilde{C}}{\lambda}+\sqrt{\frac{C_2(1+M_2)}{4\lambda\widehat{C}}}+\widetilde{C}+\sqrt{\frac{\max\limits_{\overline{\Omega}}|\Delta G(x, 0)|}{4\widehat{C}}}, \end{aligned} \end{equation} (3.20)

    and

    \begin{equation} \begin{aligned} \sigma = \frac{1}{2K}, \end{aligned} \end{equation} (3.21)

    then we get

    \begin{equation*} \begin{aligned} H_{\beta}\geq0\quad on \quad \partial\Omega\times[0, T'].\end{aligned} \end{equation*}

    On the other side, we obtain

    \begin{equation*} \begin{aligned} H_{\beta} = &4\widehat{C}K\beta_n\pm G_\beta\\ = &4\widehat{C}K\beta_n\pm(w_{kl}\beta^k\beta^l+w_{k}\beta^k_l\beta^l-\frac{1}{A}\varphi_l\beta^l). \end{aligned} \end{equation*}

    Therefore, from Proposition 3.9, \forall (x, t)\in \Omega_\sigma\times[0, T'], we gain

    \begin{equation*} \begin{aligned} |w_{\beta\beta}|\leq C\sqrt{1+M_2}\leq C\sqrt{1+|w_{\beta\beta}|}, \end{aligned} \end{equation*}

    therefore,

    \begin{equation*} \begin{aligned} |w_{\beta\beta}|\leq C. \end{aligned} \end{equation*}

    then the proof of Proposition 3.10 is completed.

    Proof of Lemma 3.2. We continue to prove Lemma 3.2. It is almost similar to the proof process in the last part of Lemma 2.2, From conditions (F_1), (F_2) and (F_4) , we can deduce the following uniformly parabolic differential equation

    \begin{equation} \begin{cases}\begin{aligned} &\frac{\partial g}{\partial s}-F_\infty(\nabla^2 g) = 0 &{\quad}\text{in }\ \ &\Omega\times[0, 1], \\ &g(x, 0) = g_0(x)&{\quad}\text{on }\ \ &\overline\Omega\times\{0\}, \\ &\frac{\partial g}{\partial \beta} = 0 &{\quad}\text{on }\ \ &\partial\Omega\times[0, 1), \end{aligned}\end{cases} \end{equation} (3.22)

    where g_0(x) is a continuous function defined on \overline{\Omega} and |g_0(x)| \le 1 .

    It can be inferred from F_\infty(0) = 0 that (3.22) can also be expressed as

    \begin{equation} \begin{cases}\begin{aligned} &\frac{\partial g}{\partial s}-\sum\limits_{i, j = 1}^n\int^1_0 F^{ij}_\infty(t\nabla^2 g)dt\cdot g_{ij} = 0 &{\quad}\text{in }\ \ &\Omega\times[0, 1], \\ &g(x, 0) = g_0(x)&{\quad}\text{on }\ \ &\overline\Omega\times\{0\}, \\ &\frac{\partial g}{\partial \beta} = 0 &{\quad}\text{on }\ \ &\partial\Omega\times[0, 1). \end{aligned}\end{cases} \end{equation} (3.23)

    However, similar to the proof of Lemma 2.2, for s\in[0, 1] , we have g(x_0, s) = 0 and for some \overline{x}\in \overline{\Omega}, \ |g(\overline{x}, 1)| = 1. This also runs counter to the maximum principle and Hopf Lemma of parabolic differential equations. Therefore, we receive (3.2) and finish the proof of Lemma 3.2.

    Theorem 3.11. For any T>0, if u is a smooth solution to (3.1), thus we get the estimate,

    \begin{equation} \begin{aligned} \| u_t(\cdot, t)\|_{C(\overline\Omega)}+\|\nabla u(\cdot, t)\|_{C(\overline\Omega)}+\|\nabla ^2u(\cdot, t)\|_{C(\overline\Omega)}\leq C, \end{aligned} \end{equation} (3.24)

    where C is a constant independent of t and T.

    Proof. The equation for v is

    \begin{equation} \begin{cases}\begin{aligned} &\frac{\partial v}{\partial t} - F(\Delta^2 v) = - u_t(x_0, t)&{\quad} \text{in }\ \ &\Omega \times (0, \infty ), \\ &v(x, 0) = u_0(x) - u_0(x_0)&{\quad} \text{on }\ \ &\Omega \times \left\{ 0 \right\}, \\ &\frac{\partial v}{\partial \beta } = \varphi &{\quad} \text{on }\ \ &\partial \Omega \times (0, \infty ). \end{aligned}\end{cases} \end{equation} (3.25)

    From Lemma 3.2 we gain |v|\leq A_0, A process similar to Propositions 3.3 and 3.4 deduces

    \|\nabla v(\cdot, t)\|_{C(\overline\Omega)}\leq C.

    Schauder theory derives

    \|\nabla^2 v(\cdot, t)\|_{C(\overline\Omega)}\leq C.

    Since v(x, t) = u(x, t)-u(x_0, t), combining with Lemma 3.1, we conclude that

    \| u_t(\cdot, t)\|_{C(\overline\Omega)}+\|\nabla u(\cdot, t)\|_{C(\overline\Omega)}+\|\nabla ^2u(\cdot, t)\|_{C(\overline\Omega)}\leq C.

    In this way, we have completed the proof of Theorem 3.11.

    Based on the conclusion of the above theorem, we have completed the proof of Theorem 1.3. On this basis, according to the Theorem 1.2, we ensure the validity of Theorem 1.1, thus obtaining the convergence conclusion of the equation solution discussed in this paper.

    The author declares she has not used Artificial Intelligence (AI) tools in the creation of this article.

    The author would like to thank Professor Peihe Wang for his guide and encouragement.

    The author is supported by Shandong Provincial Natural Science Foundation ZR2020MA018.



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