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,
{ut−F(∇2u)=0in Ω×[0,∞),u(x,0)=u0(x)on ¯Ω×{0},∂u∂β=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=cosθ≥c0>0, where ν is the inner normal vector to ∂Ω, ∂u∂β=<∇u,β>, where ∇u=(∂u∂x1,∂u∂x2,⋯∂u∂xn) and u0∈C∞(¯Ω) satisfies ∂u0∂β=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
{ut−F(∇2u)=0in Ω×[0,∞),u(x,0)=u0(x)on ¯Ω×{0},∂u∂ν=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)∀r∈Sn,λI≤Fr(r),|F(r)|≤μ0|r|,
(F2)∀r,X∈Sn,|FX(r)|≤μ1|X|,
(F3)∀r,X∈Sn,FXX(r)≤0,
where λ,μ0,μ1 are positive constsnts. Besides, we suppose
(F4) There exists a smooth function F∞, such that
s−1F(sr)→F∞(r)locallyuniformlyinC1(Sn),ass→+∞. |
First, we state our major results of this paper.
Theorem 1.1. Suppose Ω⊂Rn (n≥2) is a bounded domain with smooth boundary. If F satisfies (F1)–(F4),φ∈C∞(¯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,∂U∂β=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,∂u∂β=φ(x) on ∂Ω. |
{F(∇2v)=τ2 in Ω,∂v∂β=φ(x) on ∂Ω. |
Without loss of generality, we may assume τ1<τ2, then,
{∫10∂F∂uαβ[t∇2u+(1−t)∇2v]dt(u−v)αβ<0,∂(u−v)∂β=0. |
By maximal principle, the minimum of u−v can be achieved at the boundary, but ∂(u−v)∂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 Ω⊂Rn (n≥2) is a bounded domain with smooth boundary. If F satisfies (F1) and (F3), φ∈C∞(¯Ω). ∀ T>0, suppose u∈C4+α,4+α2(¯Ω×(0,T)) is a unique solution of the following nonlinear parabolic equation
{ut−F(∇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) |
n∑k=1Gpk(x,∇u)νk≥1C2, | (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 Ω⊂Rn (n≥2) is a bounded domain with smooth boundary. Assume F satisfies (F1)–(F4), φ∈C∞(¯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 Ω⊂Rn (n≥2) is a bounded domain, ∂Ω∈C3. Set
d(x)=dist(x,∂Ω), |
and
Ωμ={x∈Ω:d(x)<μ}. |
Then there exists a positive constant μ1>0 such that ∀ μ≤μ1,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,∑1≤i≤nνi∇iνj=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
{ ut−Δu=0, in Ω×[0,T), u(x,0)=u0(x), on ¯Ω×{0}, uβ=φ, on ∂Ω×[0,T), | (2.1) |
where Ω⊂Rn 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}n−1l=1 to be the unit tangent vector fields which joint with ν form a unit normal frame along ∂Ω. Assume β=βnν+Σn−1l=1βlTl, therefore, φ(x)=∂u∂β=<∇u,β>=∂u∂νβn+Σβlul, where ul=<∇u,Tl>.
Lemma 2.1. Suppose Ω⊂Rn (n≥2) 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 (Δ−∂∂t)(u2t)=Δu2t−∂∂t(u2t)=2utΔut+2|Dut|2−2ututt=2|Dut|2≥0, 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 x0∈Ω 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 Ω⊂Rn (n≥2) 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
‖v‖C0(Ω×[0,T))≤A0. | (2.2) |
Proof. Let A=‖v‖C0(Ω×[0,T)). Without loss of generality, we assume A≥δ=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
{∂w∂t−Δw=−ut(x0,t)Ain Ω×[0,T),w(x,0)=(u0(x)−u0(x0))Aon ¯Ω×{0},∂w∂β=1Aφ(x)on ∂Ω×[0,T). | (2.3) |
To finish the proof, we need the following propositions.
Proposition 2.3. Let w∈C3,2(Ω×[0,T)) and satisfy
∂w∂t−Δw=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 Ω⊂Rn (n≥2) be a bounded domain with smooth boundary. Assume that w∈C3,2(¯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 σ≤σ1,
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=∂w∂β=<∇w,β>=∂w∂νβn+n−1∑l=1βlwl and ρ=w−φ′dcosθ, then w=ρ+φ′dcosθ and φ′(x)=∂(ρ+φ′dcosθ)∂νβn+n−1∑l=1βl(ρ+φ′dcosθ)l⇒∂ρ∂νβn+n−1∑l=1βlρl=0⇒∂ρ∂ν=−n−1∑l=1βlβnρl.
Thus
(∂ρ∂ν)2cos2θ=(−n−1∑l=1βlρl)2≤n−1∑l=1β2ln−1∑l=1ρ2l=(|∇ρ|2−(∂ρ∂ν)2)sin2θ. | (2.6) |
Therefore,
(∂ρ∂ν)2≤|∇ρ|2sin2θ. | (2.7) |
Let
ϕ=|∇ρ|2−(n∑i=1ρidi)2=n∑i,j=1(δij−didj)ρiρj≜n∑i,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′](σ≤σ1) 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. x0∈∂Ωσ⋂Ω. In this way, the estimate is transformed into interior gradient estimate.
Case 3. x0∈∂Ω. Select a suitable coordinate at x0, so that ∂∂xn=ν, and ∂∂xi(i=1,⋯,n−1) are tangent along ∂Ω. Then, we have
dn=1,di=0,∂2d∂xn∂xα=0,∂2d∂xi∂xj=−κiδij, |
where 1≤i,j<n, 1≤α≤n−1, and κi is the principal curvatures of ∂Ω at x0.
Because x0 is the maximum point of Φ, then we have,
Φi=0,1≤i<n−1, | (2.8) |
and
0≥Φn=ϕnϕ+τ. | (2.9) |
By (2.8), for 1≤i<n−1, we have
0=Φi=(|∇ρ|2)i−(n∑α=1ραdα)2i=2n−1∑j=1ρjρji−2n−1∑j=1ρnρjdij=2n−1∑j=1ρjρji+2ρnρiκi. | (2.10) |
Using (2.10) to calculate ϕn, we obtain
ϕn=(|∇ρ|2)n−(n∑α=1ραdα)2n=2n∑i=1ρiρin−2ρnρnn=2n−1∑i=1ρiρin=2n−1∑i=1ρiρin+2n−1∑i,j=1ρiρjκij=2n−1∑i,l=1ρi(−βlβnρl)i+2n−1∑i,j=1κijρiρj=−2n−1∑i,l=1ρiρliβlβn−2n−1∑i,l=1ρiρl(βlβn)i+2n−1∑i,j=1κijρiρj=2ρnβnn−1∑l=1ρlκlβl−2n−1∑i,l=1ρiρl(βlβn)i+2n−1∑i,j=1κijρiρj, | (2.11) |
where we denote by κij the Weingarten matrix.
Thus,
0≥Φn=2ρnβnn−1∑l=1κlβlρl−2n−1∑i,l=1ρiρl(βlβn)i+2n−1∑i,j=1κijρiρjϕ+τ. | (2.12) |
From (2.7), we have
c20|∇ρ|2≤|∇ρ|2cos2θ≤ϕ≤|∇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. x0∈Ωσ, 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)t≥0, then
supΩ×[0,T′]|∇w|2≤sup∂Ω×[0,T′]⋃Ω×{0}|∇w|2. | (2.13) |
Choose a special coordinate, such that ρ1=|∇ρ|, ρl=0 (l=2,3,⋯,n) and (ρij) (2≤i,j≤n) 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)≥4sup∂Ω|φ′cosθ|.
We propose a precondition that
μσ≤1. | (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−(∂ρ∂ν)2](x1,t1)≥C[|∇ρ|2cos2θ](x1,t1)≥C|∇ρ|2(x1,t1)=C|∇w−φ′cosθν|2(x1,t1)≥C|∇w|2(x1,t1)≥CsupΩ×[0,T′]|∇w|2≥C|∇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=−τdi−2μddi,Cklρkiρl=−ϕ2(τ+2μd)di−Ckl,i2ρkρl. | (2.19) |
For i=1, we get
C11ρ11+n∑δ=2Cδ1ρδ1=−12C11,1ρ1−ϕ2ρ1(τ+2μd)d1. | (2.20) |
For δ>1, we have
C11ρ1δ+C1δρδδ=−12C11,δρ1−ϕ2ρ1(τ+2μd)dδ. | (2.21) |
Then
ρ1δ=−C1δC11ρδδ−C11,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)dδ2C11−C11,12C11ρ1−ρ1(τ+2μd)d12=(C1δC11)2ρδδ+O(|∇ρ|). | (2.23) |
At this point we still have
0≤Φt=ϕtϕ=2Cklρkρltϕ, | (2.24) |
and
0≥△Φ=△ϕϕ−(∇ϕϕ)2+(τ+2μd)△d+2μ|∇d|2. | (2.25) |
Combining (2.19), (2.24) and (2.25), we gain
0≥△Φ−Φt=△ϕ−ϕtϕ+(τ+2μd)△d+2μ|∇d|2−(τ+2μd)2|∇d|2≥△ϕ−ϕtϕ+[2μ−(τ+2μd)2]|∇d|2−(τ+2μd)k0. | (2.26) |
Where △d≥−k0 on Ωσ1.
Next, we calculate the term △ϕ−ϕtϕ. Note that
¯I=△ϕ−ϕt=△(Cijρiρj)−ϕt=2[Cij(△ρ)iρj−Cijρiρtj]+2Cijρikρjk+4Cij,kρikρj+△Cijρiρj=I+II+III+IV. | (2.27) |
For the term I,
I=2[Cij(△ρ)iρj−Cijρiρtj]=2Cij{[(△w)i−(△φ′dcosθ)i]ρj−ρiwtj}=2Cij{[wti−(△φ′dcosθ)i]ρj−ρiwtj}=−2Cij(△φ′dcosθ)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,kρi−ϕρ1(τ+2μd)dk)ρ1k=(−C11,1ρ1−ρ1(τ+2μd)d1)ρ11+(−C11,δρ1−ρ1(τ+2μd)dδ)ρ1δ=II11+II12, |
II11=(−C11,1ρ1−ρ1(τ+2μd)d1)ρ11=(−C11,1ρ1−ρ1(τ+2μd)d1)((C1δC11)2ρδδ+O(ρ1))=n∑δ=2O(|∇w|)ρδδ+O(|∇w|2), |
II12=(−C11,δρ1−ρ1(τ+2μd)dδ)ρ1δ=(−C11,δρ1−ρ1(τ+2μd)dδ)(−C1δC11ρδδ+O(|∇ρ|))=n∑δ=2O(|∇w|)ρδδ+O(|∇w|2), |
then
II1=n∑δ=2O(|∇w|)ρδδ+O(|∇w|2). |
Where
II2=2Ciδρikρδk=2C1δρ1kρδk+2Cαδραkρδk=2C1δρ11ρδ1+2C1δρ1δρδδ+2Cαδρα1ρδ1+2Cδδρ2δδ=II21+II22+II23+II24, |
II21=2C1δρ11ρδ1=2[−n∑δ=2(C1δ)2C11ρδδ+O(|∇ρ|)]×[n∑δ=2(C1δC11)2ρδδ+O(|∇ρ|)]=−2(C11)3n∑α,β=2C1αC1β(C1αραα)(C1βρββ)+n∑δ=2O(|∇ρ|)ρδδ+O(|∇ρ|2), |
II22=2n∑δ=2C1δρ1δρδδ=−n∑δ=22(C1δ)2C11ρ2δδ+n∑δ=2O(|∇ρ|)ρδδ, |
II23=2Cαδρα1ρδ1=2n∑α,β=2Cαβ[−C1αC11ραα+O(|∇ρ|)]×[−C1βC11ρββ+O(|∇ρ|)]=2(C11)2n∑α,β=2Cαβ(C1αραα)(C1βρββ)+n∑δ=2O(|∇ρ|)ρδδ+O(|∇ρ|2), |
hence
II2=II21+II22+II23+II24=2(C11)2n∑α,β=2Cαβ(C1αραα)(C1βρββ)−2(C11)3n∑α,β=2C1αC1β(C1αραα)(C1βρββ)+2n∑δ=2Cδδρ2δδ−n∑δ=22(C1δ)2C11ρ2δδ+n∑δ=2O(|∇ρ|)ρδδ+O(|∇ρ|2). |
Thus
II=II1+II2=2(C11)2n∑α,β=2Cαβ(C1αραα)(C1βρββ)−2(C11)3n∑α,β=2C1αC1β(C1αραα)(C1βρββ)+2n∑δ=2Cδδρ2δδ−n∑δ=22(C1δ)2C11ρ2δδ+n∑δ=2O(|∇ρ|)ρδδ+O(|∇ρ|2). | (2.33) |
And
△ϕ−ϕt=2(C11)2n∑α,β=2Cαβ(C1αραα)(C1βρββ)−2(C11)3n∑α,β=2C1αC1β(C1αραα)(C1βρββ)+2n∑δ=2Cδδρ2δδ−n∑δ=22(C1δ)2C11ρ2δδ+n∑δ=2O(|∇ρ|)ρδδ+O(|∇ρ|2)=Π1+Π2. | (2.34) |
For the term Π1,
Π1=2(C11)2n∑α,β=2Cαβ(C1αραα)(C1βρββ)−2(C11)3n∑α,β=2C1αC1β(C1αραα)(C1βρββ)=2(C11)3n∑α,β=2(C11Cαβ−C1αC1β)(C1αραα)(C1βρββ)=2(C11)3n∑α,β=2[(1−d21)δαβ−dαdβ](C1αραα)(C1βρββ)≥0. |
The above formula is nonnegative, because the matrix ((1−d21)δαβ−dαdβ)α,β≥2 is semi positive definite, due to |∇d|2=1.
Next we set out to deal with the term Π2,
Π2=2n∑δ=2Cδδρ2δδ−n∑δ=22(C1δ)2C11ρ2δδ+n∑δ=2O(|∇ρ|)ρδδ+O(|∇ρ|2)=2C11n∑δ=2(C11Cδδ−(C1δ)2)ρ2δδ+n∑δ=2O(|∇ρ|)ρδδ+O(|∇ρ|2)=2n∑δ=21−d21−d2δC11ρ2δδ+n∑δ=2O(|∇ρ|)ρδδ+O(|∇ρ|2)=2n∑δ=2eδρ2δδ+n∑δ=2O(|∇ρ|)ρδδ+O(|∇ρ|2), |
where eδ=1−d21−d2δC11.
According to equation △w−wt=f(t), we can obtain by Lemma 2.1 that △ρ=O(1). Joint with ρ11=(C1δC11)2ρδδ+O(ρ1), we get
n∑δ=2(1+(C1δC11)2)ρδδ=O(|∇ρ|). |
Therefore
ρ22=O(|∇ρ|)−n∑δ=3(C11)2+(C1δ)2(C11)2+(C12)2ρδδ. |
Substituting ρ22 into Π2, we can get
Π2=2n∑δ=2eδρ2δδ+n∑δ=2O(|∇ρ|)ρδδ+O(|∇ρ|2)=2e2(n∑δ=3(C11)2+(C1δ)2(C11)2+(C12)2ρδδ)2+2n∑δ=3eδρ2δδ+n∑δ=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(n∑δ=3(C11)2+(C1δ)2(C11)2+(C12)2ρδδ)2+2n∑δ=3eδρ2δδ=2e2(n∑δ=3εδρδδ)2+2n∑δ=3eδρ2δδ. |
Through observation, we know that
0≤eδ≤1, (2≤δ≤n), n∑δ=2eδ=n−2, |
so at most one of e2,⋯,en is zero, it is obvious that 0<C0≤εδ≤C1,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)|0≤eδ≤1,n∑δ=2eδ=n−2,0<C0≤εδ≤C1,δ=3,⋯,n,n∑δ=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
Λ≥λ0n∑δ=3ρ2δδ. |
Therefore, in light of ax2+bx≥−b24a, if a>0 we can obtain
¯I≥Π2≥λ0n∑δ=3ρ2δδ+n∑δ=3O(|∇ρ|)ρδδ+O(|∇ρ|2)≥−k1|∇ρ|2. | (2.35) |
Meanwhile, in consideration of ϕ=|∇ρ|2C11
△Φ−Φt=¯Iϕ+(τ+2μd)△d+2μ−(τ+2μd)2≥−k1˜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 σ≤σ1 to make 2μσ≤1, then we have a contradiction 0≥△Φ−Φt>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 k∈Z+ and 0<α<1 by the Schauder theory.
Proof of Lemma 2.2. We continue to prove Lemma 2.2. For n∈Z+, 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+tn−1), Then gn(x,s) suits
{∂gn∂s−Δgn=−f(s+tn−1)Anin Ω×[0,1],gn(x,0)=wn(x,tn−1)on ¯Ω×{0},∂gn∂β=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,tn−1)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 supx∈Ω|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 (k∈Z+ and 0<α<1) to g on ¯Ω×[0,1]. Clearly, we get
{∂g∂s−Δg=0in Ω×[0,1],g(x,0)=g0(x)on ¯Ω×{0},∂g∂β=0on ∂Ω×[0,1]. | (2.39) |
Because of ∂g∂s−Δg=0, g obtains the maximum value on Ω×{0} or ∂Ω×[0,1], but ∂g∂β=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
{∂v∂t−Δv=−ut(x0,t)in Ω×(0,T),v(x,0)=u0(x)−u0(x0)on ¯Ω×{0},∂v∂β=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}, ∂u∂β=φ(x) on ∂Ω×[0,T), | (3.1) |
where Ω⊂Rn 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 Ω⊂Rn (n≥2) 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=Fiju∂ij−∂t, take the derivative of t on both sides of ut=F(∇2u), we have
utt=Fijuuijt, |
then L(u2t)=2n∑i,j=1Fijuutiutj+2Fijuututij−2ututt≥2n∑i,j=1Fijuutiutj≥0, 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 x0∈Ω. 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 ‖v‖C2. 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 Ω⊂Rn (n≥2) 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
‖v‖L∞(Ω×[0,T))≤A0. | (3.2) |
Proof. Let A=‖v‖C0(Ω×[0,T)). Without loss of generality, we assume that A≥δ=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
{∂w∂t−1AF(A∇2w)=−ut(x0,t)Ain Ω×[0,T),w(x,0)=(u0(x)−u0(x0))Aon ¯Ω×{0},∂w∂β=1Aφ(x)on ∂Ω×[0,T). | (3.3) |
In order to prove the above estimate, we need the following propositions.
Proposition 3.3. If w∈C3,2(Ω×[0,T)) satisfies |w|≤M for a normal number M and
{∂w∂t−1AF(A∇2w)=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 Ω⊂Rn (n≥2) is a bounded domain with smooth boundary. Assuming that w∈C3,2(¯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 σ≤σ1,
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−(n∑i=1ρidi)2=n∑i,j=1(δij−didj)ρiρj≜n∑i,j=1Cijρiρj, |
where τ, μ are positive constants to be determined later, ρ=w−φ′dcosθ, and φ′=φ(x)A=∂w∂β=∂w∂νβn+n−1∑l=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. x0∈∂Ωσ⋂Ω. In this case, the estimate follows from interior gradient estimate in Proposition 3.3.
Case 3. x0∈∂Ω. Similar to the process of Proposition 2.4, we can choose the appropriate τ to guarantee this case does not occur.
Case 4. x0∈Ωσ, and t0>0.
Select a particular coordinate, so that ρ1=|∇ρ|, ρl=0 (l=2,3,⋯,n) and (ρij) (2≤i,j≤n) 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=−τdi−2μddi,Cklρkiρl=−ϕ2(τ+2μd)di−Ckl,i2ρkρl. |
When i=1, it follows
C11ρ11+n∑δ=2Cδ1ρδ1=−12C11,1ρ1−ϕ2ρ1(τ+2μd)d1. |
When δ>1, we obtain
C11ρ1δ+C1δρδδ=−12C11,δρ1−ϕ2ρ1(τ+2μd)dδ. |
Thus,
ρ1δ=−C1δC11ρδδ−C11,δ2C11ρ1−(τ+2μd)dδ2ρ1=−C1δC11ρδδ+O(|∇ρ|), |
and
ρ11=(C1δC11)2ρδδ+Cδ1C11,δ2(C11)2ρ1+Cδ1ρ1(τ+2μd)dδ2C11−C11,12C11ρ1−ρ1(τ+2μd)d12=(C1δC11)2ρδδ+O(|∇ρ|). |
At the same time, at this point we have
0≤Φt=ϕtϕ=2Cklρkρltϕ, |
and
0≥Φij=ϕijϕ−(τ+2μd)2didj+(τ+2μd)dij+2μdidj. |
Then,
0≥FijΦij−Φt=Fijϕij−ϕtϕ+(τ+2μd)Fijdij+[2μ−(τ+2μd)2]Fijdidj. |
First, we come to calculate Fklϕkl−ϕt,
Fklϕkl−ϕt=2CijFklρiklρj−2Cijρjρit+2CijFklρikρjl+4FklCij,kρilρj+FklCij,klρiρj=I+II+III+IV, |
where
I=2CijFklρiklρj−2Cijρjρit=2Cij[Fklρikl−ρit]ρj=2Cij[−Fkl(φ′dcosθ)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,kρi−ϕρ1(τ+2μd)dk)ρ1l=Fk1(−C11,kρ1−C11ρ1(τ+2μd)dk)ρ11+Fkδ(−C11,kρ1−C11ρ1(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,kρ1−C11ρ1(τ+2μd)dk)ρ1δ=Fkδ(−C11,kρ1−C11ρ1(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)3F11n∑α,β=2(C1α)2(C1β)2ρααρββ+2(C11)2n∑α,β=2(C1α)2F1βC1βρααρββ+n∑δ=2O(|∇ρ|)ρδδ+O(|∇ρ|2), |
II22=2FαlC1δρ1αρδl=2Fα1C1δρ1αρ1δ+2FαδC1δρ1αρδδ=2Fα1C1δ[−C1αC11ραα+O(|∇ρ|)][−C1δC11ρδδ+O(|∇ρ|)]+2FαδC1δρδδ[−C1αC11ραα+O(|∇ρ|)]=2(C11)2n∑α,β=2F1αC1α(C1β)2ρααρββ−2C11n∑α,β=2FαβC1αC1βρααρββ+n∑δ=2O(|∇ρ|)ρδδ+O(|∇ρ|2), |
II23=2F1lCαδρα1ρδl=2F11Cαδρ1αρ1δ+2F1δCαδρα1ρδδ=2F11Cαδ[−C1αC11ραα+O(|∇ρ|)][−C1δC11ρδδ+O(|∇ρ|)]+2F1δCαδρδδ(−C1αC11ραα+O(|∇ρ|))=2F11(C11)2n∑α,β=2CαβC1αC1βρααρββ−2C11n∑α,β=2F1βCαβC1αρααρββ+n∑δ=2O(|∇ρ|)ρδδ+O(|∇ρ|2),II24=2FαlCαδρααρδl=2Fα1Cαδρααρδ1+2FαδCαδρααρδδ=2Fα1Cαδραα[−C1δC11ρδδ+O(|∇ρ|)]+2FαδCαδρααρδδ=−2C11n∑α,β=2F1αCαβC1βρααρββ+2n∑α,β=2FαβCαβρααρββ+n∑δ=2O(|∇ρ|)ρδδ, |
then,
II2=−2C11n∑α,β=2CαβF1αC1βρααρββ+2n∑α,β=2FαβCαβρααρββ+2F11(C11)2n∑α,β=2CαβC1αC1βρααρββ−2C11n∑α,β=2F1βCαβC1αρααρββ+2(C11)2n∑α,β=2F1αC1α(C1β)2ρααρββ−2C11n∑α,β=2FαβC1αC1βρααρββ−2(C11)3F11n∑α,β=2(C1α)2(C1β)2ρααρββ+2(C11)2n∑α,β=2F1β(C1α)2C1βρααρββ+n∑δ=2O(|∇ρ|)ρδδ+O(|∇ρ|2), |
thus, we have
II2=−4C11n∑α,β=2F1αCαβC1βρααρββ+2n∑α,β=2FαβCαβρααρββ+2F11(C11)2n∑α,β=2CαβC1αC1βρααρββ−2(C11)3F11n∑α,β=2(C1α)2(C1β)2ρααρββ+4(C11)2n∑α,β=2F1αC1α(C1β)2ρααρββ−2C11n∑α,β=2FαβC1αC1βρααρββ+n∑δ=2O(|∇ρ|)ρδδ+O(|∇ρ|2). |
We mainly deal with the quadratic term in II2,
Π=−4C11n∑α,β=2F1αCαβC1βρααρββ+2n∑α,β=2FαβCαβρααρββ+2F11(C11)2n∑α,β=2CαβC1αC1βρααρββ−2(C11)3F11n∑α,β=2(C1α)2(C1β)2ρααρββ+4(C11)2n∑α,β=2F1αC1α(C1β)2ρααρββ−2C11n∑α,β=2FαβC1αC1βρααρββ. |
Simplify the above formula, we get
Π=n∑α,β=22[Fαβ(C11)2+F11C1αC1β−F1αC1βC11−F1βC1αC11][C11Cαβ−C1αC1β](C11)3ρααρββ, |
where C11Cαβ−C1αC1β=(1−d21)δαβ−dαdβ.
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 A∘B=(aij⋅bij)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, A∘B is also a m order positive semi-definite matrix; If A and B are two m order positive definite matrices, A∘B is also a m order positive definite matrix.
Corollary 3.7. If A≥λE, B≥0, then A∘B≥λE∘B.
With the above knowledge about matrices, let's look at the two matrices contained in Π, one is
A=n∑α,β=2(Fαβ(C11)2+F11C1αC1β−F1αC1βC11−F1βC1αC11), |
and the other is
B=n∑α,β=2((1−d21)δαβ−dαdβ). |
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 λE≤Fij≤ΛE, for any X=(x2,x3,⋯,xn), we have
n∑α,β=2(Fαβ(C11)2+F11C1αC1β−F1αC1βC11−F1βC1αC11)xαxβ=(C11)2n∑α,β=2Fαβxαxβ+F11(n∑α=2C1αxα)2−2C11(n∑α=2F1αxα)(n∑α=2C1αxα)≥(C11)2[n∑α,β=2Fαβxαxβ−1F11(n∑α=2F1αxα)2]=(C11)2n∑α,β=2(Fαβ−F1αF1βF11)xαxβ. |
We want to show that the matrix (Fαβ−F1αF1βF11)2≤α,β≤n is positive definite and its eigenvalues are bounded from below by λ. In fact, since λE≤Fij≤ΛE, 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 (F1100Fαβ−λδαβ−F1αF1βF11). Therefore, (Fαβ−F1αF1βF11)2≤α,β≤n is positive definite and its eigenvalues are bounded from below by λ.
So, (Fαβ(C11)2+F11C1αC1β−F1αC1βC11−F1βC1αC11)≥((C11)2λδαβ) and then by the corollary we have that
Π≥2λC11n∑α,β=2((1−d21)δαβ−dαdβ)δαβρααρββ=2λn∑α=2(1−d21)−d2αC11ρ2αα≜2λn∑α=2eαρ2αα. |
According to the first equation in (3.3), we can get aijρij=O(1), where λδij≤aij≤Λδij. Reuse ρ11=(C1δC11)2ρδδ+O(ρ1), there is
n∑δ=2(aδδ+a11(C1δC11)2−2a1δC1δC11)ρδδ=O(|∇ρ|). |
Write
γδ=aδδ+a11(C1δC11)2−2a1δC1δC11. |
Thus
0<λ2Λ≤γδ≤Λ(1+(1˜C)2+2˜C). |
Therefore
ρ22=O(|∇ρ|)−n∑δ=3γδγ2ρδδ. |
Then
Π≥2λn∑α=2eαρ2αα=2λ[e2ρ222+n∑α=3eαρ2αα]=2λ[e2(n∑δ=3γδγ2ρδδ)2+n∑α=3eαρ2αα]+n∑δ=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(n∑δ=3γδγ2ρδδ)2+n∑δ=3eδρ2δδ. |
Since the coefficients e2,e3,⋯,en satisfy
0≤eδ≤1, δ=2,3⋯,n, n∑δ=2eδ=n−2, |
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)|0≤eδ≤1,n∑δ=2eδ=n−2,0<λ2Λ≤γδ≤Λ(1+(1˜C)2+2˜C),n∑δ=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(n∑δ=3γδγ2ρδδ)2+n∑δ=3eδρ2δδ≥λ0n∑δ=3ρ2δδ. |
Therefore, on the basis of ax2+bx≥−b24a, if a>0 we can obtain
II≥2λλ0n∑δ=3ρ2δδ+n∑δ=3O(|∇ρ|)ρδδ+O(|∇ρ|2)≥−k1|∇ρ|2. |
In consideration of ϕ=|∇ρ|2C11, and supposing dij≥−k2δij we have
0≥FijΦij−Φt=Fijϕij−ϕtϕ+(τ+2μd)Fijdij+[2μ−(τ+2μd)2]Fijdidj≥−k1C11−(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 σ≤σ1 to make 2μσ≤1, hence we have a contradiction 0≥FijΦ−Φt>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 w∈C4,2(Ω×[0,T)) satisfies ||w||C1(Ω×[0,T))≤M1 ( M1>0) and
{∂w∂t−1AF(A∇2w)=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 Ω⊂Rn (n≥2) is a bounded domain with smooth boundary. Assuming that w∈C4,2(¯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 σ≤σ1,
supΩσ×[0,T)|∇2w|≤C(1+sup∂Ω×[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 ξ∈Sn−1 is a fixed unit vector, we can assume that |wξξ|≥1, otherwise, there is nothing to do. We first set the following differential inequality.
n∑i,j=1FijHij−Ht≥0mod∇HonΩσ×(0,T′]. | (3.10) |
In fact,
0=Hi=αdiH+eαd(wξξi+B(w2ξ)i),Ht=eαd(wξξt+B(w2ξ)t),Hij=(αdij−α2didj)H+eαd(wξξij+B(w2ξ)ij). |
Therefore
n∑i,j=1FijHij−Ht=n∑i,j=1Fij(αdij−α2didj)H+eαd(n∑i,j=1Fijwξξij−wξξt)+Beαd(n∑i,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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