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On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation

  • Yinxia Wang , Yuzhu Wang EMAIL logo and Feiyan Zhao
Published/Copyright: August 29, 2025
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Advances in Nonlinear Analysis
From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

In this article, we investigate the initial value problem for the hyperbolic geometric flow equation, which is derived from hyperbolic geometric flow for Riemannian metric. We find that some of the quadratic nonlinear terms satisfy the null condition by rearranging the equation. Based on the ghost weight and standard energy estimate, the L 1 − L ∞ estimate of solutions, and the bootstrap argument, lifespan of smooth solutions is established under the assumption of small data.

Keywords: hyperbolic geometric flow equation; lifespan; smooth solutions; null condition
MSC 2010: 35L72; 35L15

1 Introduction

The hyperbolic geometric flow equation takes the following form:

(1.1) v t t − Δ ln v = 0 ,

which comes from hyperbolic geometric flow introduced by Kong and Liu [18]. Let M be an n -dimensional complete Riemannian manifold with Riemannian metric g i j . The hyperbolic geometric flow is characterized by the following evolutionary equation for the metric g i j :

(1.2) ∂ 2 g i j ∂ t 2 = − 2 R i j ,

where R i j stands for the Ricci curvature tensor of g i j . For the study on the hyperbolic geometric flow, we refer to the recent articles [5,6,17–23].

On Riemann surface, all of the information about curvature is contained in the scalar curvature function R ; then, the hyperbolic geometric flow equation (1.2) can be simplified. Let R = 2 K and K be the Gauss curvature. The Ricci curvature is given by

(1.3) R i j = 1 2 R g i j ,

and then, the hyperbolic geometric flow equation (1.2) simplifies the following equation for the special metric:

(1.4) ∂ 2 g i j ∂ t 2 = − R g i j .

The metric for a surface can always be written (at least locally) in the following form:

(1.5) g i j = v ( t , x , y ) δ i j ,

where v ( t , x , y ) > 0 . Therefore, we have

(1.6) R = − Δ ln v v .

(1.1) is immediately derived by combining (1.4)–(1.6).

Equation (1.1) is a two-dimensional quasilinear wave equation if there exists C 0 > 0 such that C 0 − 1 ≤ v ≤ C 0 . Since the 1990s, global existence and lifespan of smooth solutions to the Cauchy problem for two-dimensional quasilinear wave equations with small initial data have been attracted many researcher’s interests, we refer to [1,3,9,24–27,30]. If the nonlinear term on the right depends on the unknown function, lifespan of smooth solutions was established in [24–26]. The null conditions play very important roles in studying global existence and lifespan of smooth solutions to the Cauchy problem for two-dimensional quasilinear wave equations. Global smooth solutions were obtained by Li and Zhou [27] if the equation at least has cubic nonlinearity, which satisfies the null condition. Zha [30] proved global existence of smooth solutions if both quadratic and cubic nonlinearities satisfy the null condition. The compact support condition is necessary to obtain sharp lifespan in all the aforementioned results because the nonlinear term depends on the unknown function itself.

If the nonlinear term on the right is dependent on the unknown function, Alinhac [1] proved global existence of smooth solutions with compact support small initial data under the assumption that both quadratic and cubic nonlinearities satisfy the null condition. Cai et al. [3] made full use of the nonlinear structure to obtain global smooth solutions with non-compactly supported small initial data. Hou and Yin [9] proved that the global existence of smooth solutions to the general 2-D null-form wave equations with non-compactly supported initial data.

Let

u = ln v ,

then (1.1) can be transformed to

(1.7) u t t − e − u Δ u = − u t 2 .

In this article, we are concerned on the initial value problem for two-dimensional hyperbolic geometric flow equation (1.7) with

(1.8) t = 0 : u = u 0 ( x ) , u t = u 1 ( x ) , x ∈ R 2 ,

where ( t , x ) = ( t , x 1 , x 2 ) , and u = u ( t , x ) is the unknown function. u 0 ( x ) and u 1 ( x ) are given smooth functions.

To investigate the intrinsic structure of (1.7), it can be rearranged as

(1.9) u t t − Δ u = ∇ ⋅ [ ( e − u − 1 ) ∇ u ] + e − u ( ∣ ∇ u ∣ 2 − u t 2 ) + ( e − u − 1 ) u t 2 .

We find that e − u ( ∣ ∇ u ∣ 2 − u t 2 ) satisfies the null condition, which was introduced in [4,15]. In fact, let

(1.10) U = U ( s ) = U ( χ 0 t + χ 1 x 1 + χ 2 x 2 )

be any plane wave solution to two-dimensional linear homogeneous wave equation

(1.11) □ u = 0 .

Here, χ = ( χ 0 , χ 1 , χ 2 ) is a constant vector, and

(1.12) U ( 0 ) = U ′ ( 0 ) = 0 .

It follows from (1.10) and (1.11) that

( χ 0 2 − χ 1 2 − χ 2 2 ) U ″ ( s ) = 0 .

Noting that (1.12), then U is a nontrivial plane wave solution to linear wave equation, it suffices that the vector χ = ( χ 0 , χ 1 , χ 2 ) satisfies

χ 0 2 − χ 1 2 − χ 2 2 = 0 .

It is easy to know that

e − U ( χ 1 2 + χ 2 2 − χ 0 2 ) U ′ = 0 .

Therefore, the nonlinear term e − u ( ∣ ∇ u ∣ 2 − u t 2 ) in the quadratic nonlinearity term satisfies the null condition.

Some results on smooth solutions to two-dimensional hyperbolic geometric flow equation have been established in [20] and [21]. But the structure of (1.7) such as the null condition in partial quadratic nonlinearity not been fully excavated and used. Therefore, our main aim of this article is to excavate the structure of (1.7) and then obtain sharp estimate for lifespan of smooth solutions.

We have the following theorem concerning the lifespan of smooth solutions for problems (1.7) and (1.8).

Theorem 1.1

Suppose that u 0 and u 1 are the smooth functions such that u 0 ( x ) = u 1 ( x ) = 0 , when ∣ x ∣ ≥ 1 , and let K ≥ 8 . Then, there exist constants ε 0 > 0 and C > 0 , such that if

∑ ∣ a ∣ ≤ K ( ‖ ∂ ∂ a u 0 ‖ L 2 + ‖ ∂ a u 1 ‖ L 2 ) = ε ≤ ε 0 ,

then problems (1.7) and (1.8) admit a unique smooth solution u with the lifespan

(1.13) T ε ≥ C ε 2 .

Remark 1.2

There are also twofolds in this article. First, on the one hand, we find that the quadratic nonlinearity e − u ( ∣ ∇ u ∣ 2 − u t 2 ) satisfy the null condition by rearranging the equation; on the other hand, the quadratic nonlinearity ∇ ⋅ [ ( e − u − 1 ) ∇ u ] has divergence form, which is useful for the energy estimate and L ∞ estimate. Second, we may directly use Alinhac’s ghost weight energy method to obtain the L 2 estimate for ∂ u since the appearance of the null form.

Remark 1.3

To obtain the sharp estimate for smooth solutions, the compact support condition on the initial data that is precisely caused by the nonlinear term that depends on the unknown function u is necessary.

Remark 1.4

The reason the lifespan of the smooth solution obtained in Theorem 1.1 is sharp lies in that the time decay of solutions to the two-dimensional linear wave equation is t − 1 2 .

Key points We first observe that Alinhac’s ghost weight method can be applied to the hyperbolic geometric flow equation since one of the quadratic nonlinearities satisfies the null condition. Second, the L 2 -type norm of u itself cannot be bounded by the natural wave energy, and we make estimate for the corresponding linear equation using the energy method in the Fourier space. Third, in view of the appearance of u in the nonlinear term and its L 2 norm is growth with respect to time, we need to resort to the Hardy-type inequality to overcome the difficulty.

Main ideas In view of the null condition and divergence structure in the quadratic nonlinearity, we obtain the L 2 estimate for ∂ u = ( ∂ t u , ∇ u ) by combining Klainerman’s vector field and Alinhac’s ghost weight energy method. But the nonlinear term of the equation depends on the unknown function u itself, and we have to make estimate for the unknown function u itself. Unfortunately, we cannot obtain the L 2 estimate for u by applying the energy method to the equation itself, then we consider the corresponding linearized equation of the nonlinear equation by the energy method in the Fourier space and establish the L 2 estimate for u . The result is the logarithmic growth with respect to time, and we refer to Lemma 3.1 for the linear estimate and Lemma 4.2 for the nonlinear estimate. In additional, to close the energy, the L ∞ estimate for u and ∂ u is necessary. The L ∞ estimate for ∂ u is simple by the L 2 estimate for ∂ u and Klainerman Sobolev inequality. The same procedure for L ∞ estimate for u fails since the L 2 estimate for u is growth with respect to time. If we do, we cannot obtain the sharp lifespan. So we have to build the L ∞ estimate for u by making analysis for the corresponding linearized equation of the nonlinear equation. Based on the L ∞ estimate of solutions to the linearized equation, the L ∞ estimate for u follows. Then, the lifespan of smooth solutions to problems (1.7) and (1.8) is achieved by combining L 2 and L ∞ estimates for u and ∂ u , the local well-posedness, and the bootstrap argument.

This article is organized as follows. In Section 2, we introduce the vector fields and give some results. The L 2 and L 1 − L ∞ estimates of solutions to the corresponding linear wave equation are established in Section 3, which play very important roles in investigating hyperbolic geometric flow equation. In Section 4, we shall build the L 2 and L ∞ estimates of smooth solutions using Alinhac’s ghost weight energy method, and L 2 and L 1 − L ∞ estimates of solutions to the corresponding linear wave equation are obtained in Section 3. Then, lifespan of smooth solutions follows by combining L 2 and L ∞ estimate and the bootstrap argument in Section 5.

2 Vector fields and some useful lemmas

In this section, we give Klainerman’s vector field and list some useful lemmas. In order to apply Klainerman’s vector field method, we first introduce the vector fields:

Translations:

∇ = ( ∂ i ) i = 1 2 , ∂ = ( ∂ t , ∂ 1 , ∂ 2 ) = ( ∂ t , ∇ ) .

Lorentz boosts:

L i = x i ∂ t + t ∂ i , i = 1 , 2 ,

Scaling vector field:

S = t ∂ t + r ∂ r = t ∂ t + ∑ i = 1 2 x i ∂ i ,

Rotations:

Ω ≔ x 1 ∂ 2 − x 2 ∂ 1 .

We shall use Γ to denote a general vector field in { ∂ , L 1 , L 2 , S , Ω } . For the multi-index a = ( a 1 , … , a 7 ) , we set

Γ a = Γ 1 a 1 … Γ 7 a 7 , Γ ∈ { ∂ , L 1 , L 2 , S , Ω } .

From [27], it holds that

(2.1) ∣ ∂ ϕ ∣ ≤ C ⟨ ∣ x ∣ − t ⟩ − 1 ∣ Γ ϕ ∣ ,

with ⟨ ∣ x ∣ − t ⟩ = 1 + ∣ ∣ x ∣ − t ∣ .

The good derivatives are

T i = ∂ i + ω i ∂ t , i = 1 , 2 ,

where ω i = x i ∣ x ∣ with i = 1 , 2. The good derivatives will appear in Alinhac’s ghost weight method. Note that

T i = ω i ( ∂ t + ∂ r ) + ( ω ⊥ ) i ∣ x ∣ Ω ,

∂ t + ∂ r = 1 t + ∣ x ∣ ∑ i = 1 2 ω i L i + S ,

and

1 ∣ x ∣ Ω = 1 t + ∣ x ∣ ( ω Λ L + Ω ) ,

where ( ω ⊥ ) i denotes the ith component of ω ⊥ = { − ω 2 , ω 1 } .

Then, we have

(2.2) ∣ T ϕ ∣ ≤ C ⟨ ∣ x ∣ + t ⟩ − 1 ∣ Γ ϕ ∣ .

Klainerman’s vector field Γ has the following commutation relation between the wave operator □ and ∂ , which is critical in dealing with nonlinear wave equation. We may refer to [27] for the proof.

Lemma 2.1

For any given multi-index k = ( k 1 , … , k 7 ) , we achieve

(2.3) [ ∂ α , Γ k ] = ∑ ∣ l ∣ ≤ ∣ k ∣ − 1 C ˜ k l Γ l ∂ = ∑ ∣ l ∣ ≤ ∣ k ∣ − 1 C ¯ k l ∂ Γ l , α = 0 , 1 , 2 ,

where [ ⋅ , ⋅ ] stands for the Poissons bracket, i.e., [ A , B ] = A B − B A , and ∂ = ( ∂ t , ∂ 1 , ∂ 2 ) = ( ∂ 0 , ∂ 1 , ∂ 2 ) , and C ˜ k l and C ¯ k l are the constants.

Lemma 2.2

For any given multi-index k = ( k 1 , … , k 7 ) , it holds that

(2.4) [ □ , Γ k ] u = ∑ ∣ l ∣ ≤ ∣ k ∣ − 1 C k l Γ l □ u ,

where [ ⋅ , ⋅ ] stands for the Poissons bracket, i.e., [ A , B ] = A B − B A , and C k l are the constants.

Let

Q ( ϕ , ψ ) = Q α β ∂ α ϕ ∂ β ψ = ∇ ϕ ∇ ψ − ∂ t ϕ ∂ t ψ .

Obviously, Q ( ϕ , ψ ) = Q α β ∂ α ϕ ∂ β ψ satisfies the null condition, and we call Q is a null form. For null form Q , we have the following estimate.

Lemma 2.3

Assume that Q ( ϕ , ψ ) is a null form, then we have

(2.5) ∣ Q ( ϕ , ψ ) ∣ = ∣ Q α β ∂ α ϕ ∂ β ψ ∣ ≤ C ( ∣ T ϕ ∣ ∣ ∂ ψ ∣ + ∣ ∂ ϕ ∣ ∣ T ψ ∣ ) .

The Klainerman Sobolev inequality that is proved in [11] is to use the L 2 norms of products Γ a of Lorentz vector fields Γ to ϕ and obtain a weighted control of ∣ ϕ ( t , x ) ∣ .

Lemma 2.4

There exists a constant C > 0 such that

(2.6) ⟨ ∣ x ∣ + ∣ t ∣ ⟩ n − 1 2 ⟨ ∣ x ∣ − t ⟩ 1 2 ∣ ϕ ( t , x ) ∣ ≤ C ∑ ∣ a ∣ ≤ n + 2 2 ‖ Γ a ϕ ( t ) ‖ L 2 .

Since the appearance of u ( x , t ) in the nonlinear term and the L 2 norm of u ( x , t ) is growth with respect to t , to obtain sharp lifespan, we need the following lemma, which is called as Hardy-type inequality and comes from [28].

Lemma 2.5

Assume that Supp ϕ = { x ∣ ∣ x ∣ ≤ 1 + t } , then it holds that

(2.7) ‖ ⟨ ∣ x ∣ − t ⟩ − 1 ϕ ( t ) ‖ L 2 ≤ ‖ ∂ x ϕ ( t ) ‖ L 2 .

To deal with some nonlinear terms, we need the following lemma, which is found in [27].

Lemma 2.6

Suppose that G = G ( ϕ ) is a sufficiently smooth function of ϕ with

G ( 0 ) = 0 .

For any given integer N ≥ 0 , if a vector function ϕ = ϕ ( x , t ) satisfies

∑ ∣ a ∣ ≤ [ N 2 ] ‖ Γ a ϕ ( ⋅ , t ) ‖ L ∞ ≤ ν 0 , ∀ t ∈ [ 0 , T ] ,

where [ ⋅ ] stands for the integer part of a real number and ν 0 is a positive constant, then it holds that

(2.8) ∑ ∣ a ∣ ≤ N ‖ Γ a G ( ϕ ( ⋅ , t ) ) ‖ L P ≤ C ( ν 0 ) ∑ ∣ a ∣ ≤ N ‖ Γ a ϕ ( ⋅ , t ) ‖ L p , ∀ t ∈ [ 0 , T ] ,

provided that all norms appearing on the right-hand side of (2.8) are bounded, where C ( ν 0 ) is a positive constant depending on ν 0 , and p is a real number with 1 ≤ p ≤ ∞ .

3 Linear wave equation

In this section, our main aim is to build and give the estimate of solutions to two-dimensional linear wave, which will play very important roles in dealing with hyperbolic geometric flow equation.

Lemma 3.1

Assume that ϕ = ϕ ( t , x ) is the solution to the initial value problem for two-dimensional wave equation

(3.1) □ ϕ = Φ + ∑ j = 0 2 ∂ j Ψ ,

with the initial value

(3.2) t = 0 : ϕ = ϕ 0 ( x ) , ∂ t ϕ = ϕ 1 ( x ) ,

where Φ = Φ ( t , x ) and Ψ = Ψ ( t , x ) : R 1 + 2 ⟶ R are two sufficiently nice functions, then we have

(3.3) ‖ ϕ ( t ) ‖ L 2 ≤ C ( ‖ ϕ 0 ‖ L 2 + ‖ ϕ 1 ‖ L 2 + ‖ Ψ ( ξ , 0 ) ‖ L 2 ) + C log ( 2 + t ) 1 2 ( ‖ ϕ 1 ‖ L 1 + ‖ Ψ ( ξ , 0 ) ‖ L 1 ) + C ∫ 0 t ( ‖ Φ ( τ ) ‖ L 2 + ‖ Ψ ( τ ) ‖ L 2 ) d τ + C log ( 2 + t ) 1 2 ∫ 0 t ‖ Φ ( τ ) ‖ L 1 d τ .

Proof

Fourier transform and integration by parts entail that the solution to problems (3.1) and (3.2) in Fourier space is given by

ϕ ( ξ , t ) = cos ( ∣ ξ ∣ t ) ϕ ^ 0 ( ξ ) + sin ( ∣ ξ ∣ t ) ∣ ξ ∣ ϕ ^ 1 ( ξ ) + ∫ 0 t sin ( ∣ ξ ∣ ( t − τ ) ) ∣ ξ ∣ ( Φ ^ + ∑ j = 0 2 i ξ j Ψ ^ + ∂ t Ψ ^ ) ( ξ , τ ) d τ = cos ( ∣ ξ ∣ t ) ϕ ^ 0 ( ξ ) + sin ( ∣ ξ ∣ t ) ∣ ξ ∣ ( ϕ ^ 1 ( ξ ) − Ψ ^ ( ξ , 0 ) ) + ∫ 0 t sin ( ∣ ξ ∣ ( t − τ ) ) ∣ ξ ∣ Φ ^ + ∑ j = 1 2 i ξ j Ψ ^ ( ξ , τ ) d τ + ∫ 0 t cos ∣ ξ ∣ ( t − τ ) Ψ ^ ( ξ , τ ) d τ .

Then, we have from Plancherel theorem

(3.4) ‖ ϕ ( t ) ‖ L 2 ≤ ‖ cos ( ∣ ξ ∣ t ) ϕ ^ 0 ( ξ ) ‖ L 2 + sin ( ∣ ξ ∣ t ) ∣ ξ ∣ ( ϕ ^ 1 ( ξ ) − Ψ ^ ( ξ , 0 ) ) L 2 + ∫ 0 t sin ( ∣ ξ ∣ ( t − τ ) ) ∣ ξ ∣ Φ ^ ( τ ) L 2 d τ + ∑ j = 1 2 ∫ 0 t sin ( ∣ ξ ∣ ( t − τ ) ) ∣ ξ ∣ i ξ j Ψ ^ ( τ ) L 2 d τ + ∫ 0 t ‖ cos ∣ ξ ∣ ( t − τ ) Ψ ^ ( τ ) ‖ L 2 d τ ≜ : A 1 + A 2 + A 3 + A 4 + A 5 .

In what follows, we estimate A j ( j = 1 , … , 4 ) . Obviously,

A 1 ≤ ‖ ϕ 0 ‖ L 2 .

By the detailed calculation, we arrive at

A 2 = ∫ R 2 sin ( ∣ ξ ∣ t ) ∣ ξ ∣ ( ϕ ^ 1 − Ψ ^ ( ξ , 0 ) ) ( ξ ) 2 d ξ 1 2 ≤ ∫ ∣ ξ ∣ ≤ 1 sin ( ∣ ξ ∣ t ) ∣ ξ ∣ ϕ ^ 1 ( ξ ) − Ψ ^ ( ξ , 0 ) 2 d ξ 1 2 + ∫ ∣ ξ ∣ ≥ 1 sin ( ∣ ξ ∣ t ) ∣ ξ ∣ ϕ ^ 1 ( ξ ) − Ψ ^ ( ξ , 0 ) 2 d ξ 1 2 ≤ ( ‖ ϕ ^ 1 ‖ L ∞ + ‖ Ψ ^ ( ξ , 0 ) ‖ L ∞ ) ∫ 0 t sin 2 ρ ρ d ρ 1 2 + ‖ ϕ 1 ( t ) ‖ L 2 + ‖ Ψ ^ ( ξ , 0 ) ‖ L 2 ≤ ( ‖ ϕ 1 ‖ L 1 + ‖ Ψ ( ξ , 0 ) ‖ L 1 ) ( log ( 2 + t ) ) 1 2 + ‖ ϕ 1 ( t ) ‖ L 2 + ‖ Ψ ( ξ , 0 ) ‖ L 2 .

The same procedure leads to

A 3 ≤ ∫ 0 t ( ‖ Φ ( τ ) ‖ L 1 ( log ( 2 + t − τ ) ) 1 2 + ‖ Φ ( τ ) ‖ L 2 ) d τ ≤ ∫ 0 t ( ( log ( 2 + t ) ) 1 2 ‖ Φ ( τ ) ‖ L 1 + ‖ Φ ( τ ) ‖ L 2 ) d τ .

Plancherel theorem implies that

A 4 + A 5 ≤ C ∫ 0 t ‖ Ψ ^ ( τ ) ‖ L 2 d τ ≤ C ∫ 0 t ‖ Ψ ( τ ) ‖ L 2 d τ .

Plugging the estimates for A i ( i = 1 , … , 5 ) into (3.4) immediately yields (3.3).□

Lemma 3.2

Assume that ϕ = ϕ ( t , x ) is the solution to the initial value problem for two-dimensional wave equation

□ ϕ = 0 ,

with the initial value

t = 0 : ϕ = ϕ 0 ( x ) , ∂ t ϕ = ϕ 1 ( x ) .

Assume that ϕ 0 ∈ W 3,1 , ϕ 1 ∈ W 2,1 , then we have

(3.5) ∣ ϕ ( x , t ) ∣ ≤ C ( 1 + t ) − 1 2 ( ‖ ϕ 0 ‖ W 3,1 + ‖ ϕ 1 ‖ W 2,1 ) .

Proof

The proof of (3.5) has been given in [27], and the proof process is complex. Hou and Yin [9] obtained weighted decay estimate under some strong condition, and since we only need the time-decay rate, the analysis given in this article is simpler. The result holds for non-compactly supported initial data.

Poisson formula gives

(3.6) u ( x , t ) = 1 2 π ∫ ∣ y − x ∣ ≤ t ϕ 1 ( y ) t 2 − ∣ y − x ∣ 2 d y + 1 2 π ∂ t ∫ ∣ y − x ∣ ≤ t ϕ 0 ( y ) t 2 − ∣ y − x ∣ 2 d y .

We first prove that

(3.7) ∫ ∣ y − x ∣ ≤ t ϕ 1 ( y ) t 2 − ∣ y − x ∣ 2 d y ≤ C ( 1 + t ) − 1 2 ‖ ϕ 1 ‖ W 2,1 .

In fact, when 0 ≤ t ≤ 1 , it holds that

(3.8) ∫ ∣ y − x ∣ ≤ t ϕ 1 ( y ) t 2 − ∣ y − x ∣ 2 d y ≤ ‖ ϕ 1 ‖ L σ ∫ ∣ y − x ∣ ≤ t ( t 2 − ∣ y − x ∣ 2 ) − σ ′ 2 d y 1 σ ′ ≤ C ‖ ϕ 1 ‖ L σ ∫ 0 t r ( t 2 − r 2 ) − σ ′ 2 d r 1 σ ′ ≤ C ‖ ϕ 1 ‖ W 2,1 ∫ 0 t r ( t 2 − r 2 ) − σ ′ 2 d r 1 σ ′ ≤ C t − 1 2 + 1 σ ′ ‖ ϕ 1 ‖ W 2,1 ≤ C ( 1 + t ) − 1 2 ‖ ϕ 1 ‖ W 2,1 ,

where σ satisfies 2 < σ ≤ ∞ .

When t ≥ 1 , it is suffice to prove

(3.9) ∫ ∣ y − x ∣ ≤ t ϕ 1 ( y ) t 2 − ∣ y − x ∣ 2 d y ≤ t − 1 2 ‖ ϕ 1 ‖ W 1,1 .

We rewrite ∫ ∣ y − x ∣ ≤ t ϕ 1 ( y ) t 2 − ∣ y − x ∣ 2 d y as

∫ ∣ y − x ∣ ≤ t ϕ 1 ( y ) t 2 − ∣ y − x ∣ 2 d y = ∫ ∣ y ∣ ≤ t ϕ 1 ( x − y ) t 2 − ∣ y ∣ 2 d y = ∫ ∣ y ∣ ≤ t − 1 2 ϕ 1 ( x − y ) t 2 − ∣ y ∣ 2 d y + ∫ t − 1 2 ≤ ∣ y ∣ ≤ t ϕ 1 ( x − y ) t 2 − ∣ y ∣ 2 d y .

Obviously, we have

∫ ∣ y ∣ ≤ t − 1 2 ϕ 1 ( x − y ) t 2 − ∣ y ∣ 2 d y ≤ C t − 1 2 ‖ ϕ 1 ‖ L 1 .

Let ϖ be a smooth cutoff function satisfying

0 ≤ ϖ ( r ) ≤ 1 , ϖ ( r ) = 1 , t − 1 2 ≤ r ≤ t , 0 , 0 ≤ r ≤ t − 2 3 .

Then

∫ t − 1 2 ≤ ∣ y ∣ ≤ t ϕ 1 ( x − y ) t 2 − ∣ y ∣ 2 d y ≤ C t − 1 2 ∫ t − 1 2 ≤ ∣ y ∣ ≤ t ϕ 1 ( x − y ) t − ∣ y ∣ d y = C t − 1 2 ∫ t − 1 2 ≤ ∣ y ∣ ≤ t ϖ ( ∣ y ∣ ) ϕ 1 ( x − y ) − y ∣ y ∣ ∇ t − ∣ y ∣ d y ≤ C t − 1 2 ∫ 0 ≤ ∣ y ∣ ≤ t ϖ ( ∣ y ∣ ) ϕ 1 ( x − y ) − y ∣ y ∣ ∇ t − ∣ y ∣ d y ≤ C t − 1 2 ∫ 0 ≤ ∣ y ∣ ≤ t ϖ ′ ( ∣ y ∣ ) ∣ ϕ 1 ( x − y ) ∣ t − ∣ y ∣ d y + C t − 1 2 ∫ 0 ≤ ∣ y ∣ ≤ t ∣ ∇ ϕ 1 ( x − y ) ∣ + ∣ ϕ 1 ( x − y ) ∣ ∣ y ∣ ϖ ( ∣ y ∣ ) t − ∣ y ∣ d y ≤ C t − 1 2 ∫ R 2 ( ∣ ϕ 1 ( y ) ∣ + ∣ ∇ ϕ 1 ( y ) ∣ ) d y ≤ C t − 1 2 ‖ ϕ 1 ‖ W 1,1 .

The aforementioned two estimates entail that (3.9) holds.

Noting that

(3.10) ∂ t ∫ ∣ y − x ∣ ≤ t ϕ 0 ( y ) t 2 − ∣ y − x ∣ 2 d y = 1 t ∫ ∣ y − x ∣ ≤ t ϕ 0 ( y ) t 2 − ∣ y − x ∣ 2 d y + ∫ ∣ y − x ∣ ≤ t y − x t ∇ ϕ 0 ( y ) t 2 − ∣ y − x ∣ 2 d y .

Applying (3.7) to the second term on the right, we immediately obtain

(3.11) ∫ ∣ y − x ∣ ≤ t y − x t ∇ ϕ 0 ( y ) t 2 − ∣ y − x ∣ 2 d y ≤ C ( 1 + t ) − 1 2 ‖ ϕ 0 ‖ W 3,1 .

In what follows, we estimate 1 t ∫ ∣ y − x ∣ ≤ t ϕ 0 ( y ) t 2 − ∣ y − x ∣ 2 d y .

When t ≥ 1 , the aforementioned same procedure implies that

(3.12) 1 t ∫ ∣ y − x ∣ ≤ t ϕ 0 ( y ) t 2 − ∣ y − x ∣ 2 d y ≤ C ( 1 + t ) − 1 2 ‖ ϕ 0 ‖ W 3,1 .

When t ≤ 1 , we obtain from direct calculation and Sobolev embedding theorem that

(3.13) 1 t ∫ ∣ y − x ∣ ≤ t ϕ 0 ( y ) t 2 − ∣ y − x ∣ 2 d y = ∫ ∣ z ∣ ≤ 1 ϕ 0 ( x + t z ) 1 − z 2 d z ≤ C ‖ ϕ 0 ‖ L ∞ ≤ C ( 1 + t ) − 1 2 ‖ ϕ 0 ‖ W 2,1 .

Collecting (3.6) and (3.10)–(3.13) immediately yields (3.5). We complete the proof of Lemma 3.2.□

To estimate the L ∞ norm of u , we need the following two lemmas on L 1 − L ∞ estimate, which have been established in [10].

Lemma 3.3

Assume that ϕ = ϕ ( t , x ) is the solution to the initial value problem

□ ϕ = Φ ,

with the initial value

t = 0 : ϕ = 0 , ∂ t ϕ = 0 ,

where Φ = Φ ( t , x ) : R 1 + 2 ⟶ R is a sufficiently nice function, then we have

(3.14) ∣ ϕ ( x , t ) ∣ ≤ C ( 1 + t ) − 1 2 ∫ 0 t ∑ ∣ α ∣ ≤ 1 ‖ ( 1 + τ ) − 1 2 Γ α Φ ( τ ) ‖ L 1 d τ .

Lemma 3.4

Assume that ϕ = ϕ ( t , x ) is the solution to the initial value problem

□ ϕ = ∑ j = 0 2 ∂ j Ψ

with the initial value

t = 0 : ϕ = 0 , ∂ t ϕ = 0 ,

where Ψ = Ψ ( t , x ) : R 1 + 2 ⟶ R is a sufficiently nice function, then we have

(3.15) ∣ ϕ ( x , t ) ∣ ≤ C ( 1 + t ) − 1 2 ∫ 0 t ( 1 + τ ) 1 2 ‖ Ψ ( τ ) ‖ L ∞ + ( 1 + τ ) − 3 2 ∑ ∣ α ∣ ≤ n + 1 ‖ Γ α Ψ ( τ ) ‖ L 1 d τ .

4 A priori estimate

In this section, we make a priori estimate. To this end, for K ≥ 8 and 0 < σ < 1 24 , we define

(4.1) X ( t ) = sup 0 ≤ τ ≤ t ∑ ∣ a ∣ ≤ K ‖ ∂ Γ a u ( τ ) ‖ L 2 2 + sup 0 ≤ τ ≤ t ∑ ∣ a ∣ ≤ K ( 1 + τ ) − 2 σ ‖ Γ a u ( τ ) ‖ L 2 2 + sup 0 ≤ τ ≤ t ( 1 + τ ) ∑ ∣ a ∣ ≤ K 2 + 1 ‖ Γ a u ( τ ) ‖ L ∞ 2 .

By Lemmas 2.1 and 2.2, it follows from (1.7) that

(4.2) □ Γ a u = ∇ ⋅ [ ( e − u − 1 ) ∇ Γ a u ] + ∑ b ≤ a C 1 b a Γ b ∇ ⋅ [ ( e − u − 1 ) ∇ u ] − ∇ ⋅ [ ( e − u − 1 ) ∇ Γ a u ] + ∑ b ≤ a Γ b e − u ( ∣ ∇ u ∣ 2 − u t 2 ) + ∑ b ≤ a Γ b [ ( e − u − 1 ) u t 2 ] = ∇ ⋅ [ ( e − u − 1 ) ∇ Γ a u ] + ∑ c + d < a , 0 ≤ α , β ≤ 2 C 1 b a α β ∂ α [ Γ c ( e − u − 1 ) ∂ β Γ d u ] + ∑ b + c ≤ a C 2 b a Γ b e − u Γ c Q ( ∂ u , ∂ u ) + ∑ b ≤ a C 3 b a Γ b [ ( e − u − 1 ) u t 2 ] = ∇ ⋅ [ ( e − u − 1 ) ∇ Γ a u ] + ∑ c + d < a , 0 ≤ α , β ≤ 2 C 1 b a α β ∂ α [ Γ c ( e − u − 1 ) ∂ β Γ d u ] + ∑ b + c 1 + c 2 ≤ a C 2 b c 1 c 2 a Γ b e − u Q ′ ( ∂ Γ c 1 u , ∂ Γ c 2 u ) + ∑ b ≤ a C 3 b c d a Γ b ( e − u − 1 ) ∂ Γ c u ∂ Γ c u ≕ I 1 + I 2 + I 3 + I 4 .

Let

q ( τ ) = arctan τ , τ = ∣ x ∣ − t ,

then

q ′ ( τ ) = 1 ( 1 + ∣ ∣ x ∣ − t ∣ 2 )

and

e q ( τ ) ∈ ( e − π 2 , e π 2 ) .

Lemma 4.1

Under the assumption of Theorem 1.1, it holds that

(4.3) sup 0 ≤ τ ≤ t ∑ ∣ a ≤ K ‖ ∂ Γ a u ( τ ) ‖ L 2 2 ≤ C ε 2 + C ( 1 + t ) 1 2 ( X 3 2 ( t ) + X 2 ( t ) + X 3 ( t ) ) .

Proof

Multiplying (4.2) by e q Γ a u and integrating it with respect to x and t , we have

(4.4) ∫ 0 t ∫ R 2 e q □ Γ a u ∂ t Γ a u d x d τ = ∫ 0 t ∫ R 2 e q ( I 1 a + I 2 a + I 3 a + I 4 a ) ∂ t Γ a u d x d τ .

Using integration by parts and Stokes formula, we have

∫ R 2 e q □ Γ a u ∂ t Γ a u d x = 1 2 ∫ R 2 ∂ t [ e q ( ∣ ∂ t Γ a u ∣ 2 + ∣ ∇ Γ a u ∣ 2 ) ] d x − ∫ R 2 ∇ ⋅ ( e q ∂ t Γ a u ∇ Γ a u ) d x − ∫ R 2 1 2 ( ∣ ∂ t Γ a u ∣ 2 + ∣ ∇ Γ a u ∣ 2 ) ∂ t e q d x + ∫ R 2 ∂ t Γ a u ∇ Γ a u ∇ e q d x = 1 2 d d t ∫ R 2 [ e q ( ∣ ∂ t Γ a u ∣ 2 + ∣ ∇ Γ a u ∣ 2 ) ] d x + 1 2 ∑ i = 1 2 ∫ R 2 e q ∣ T i Γ a u ∣ 2 1 + ∣ ∣ x ∣ − t ∣ 2 d x .

Then, we have

(4.5) ∫ 0 t ∫ R 2 e q □ Γ a u ∂ t Γ a u d x d τ = 1 2 e q 2 ∂ t Γ a u L 2 2 + e q 2 ∇ Γ a u L 2 2 − 1 2 e q 2 ∂ t Γ a u ( x , 0 ) L 2 2 + e q 2 ∇ Γ a u ( x , 0 ) L 2 2 + 1 2 ∑ i = 1 2 ∫ 0 t ∫ R 2 e q ∣ T i Γ a u ∣ 2 1 + ∣ ∣ x ∣ − τ ∣ 2 d x d τ .

It follows from integration by parts, Stokes formula, and (4.1) that

(4.6) ∫ 0 t ∫ R 2 e q I 1 a ∂ t Γ a u d x d τ = ∫ 0 t ∫ R 2 e q ∇ ⋅ [ ( e − u − 1 ) ∇ Γ a u ] ∂ t Γ a u d x d τ = ∫ 0 t ∫ R 2 ∇ ⋅ [ e q ( e − u − 1 ) ∇ Γ a u ∂ t Γ a u ] d x − ∫ 0 t ∫ R 2 e q ( e − u − 1 ) ∇ Γ a u ∂ t ∇ Γ a u d x − ∫ 0 t ∫ R 2 ∇ e q ( e − u − 1 ) ∇ Γ a u ∂ t Γ a u d x = − 1 2 ∫ 0 t d d t ∫ R 2 [ e q ( e − u − 1 ) ∣ ∇ Γ a u ∣ 2 ] d x + ∫ 0 t ∫ R 2 ∂ t ( e q ) ( e − u − 1 ) ∣ ∇ Γ a u ∣ 2 d x d τ − 1 2 ∫ 0 t ∫ R 2 e q ∂ t u ∣ ∇ Γ a u ∣ 2 d x d τ − ∫ 0 t ∫ R 2 ∇ ( e q ) ( e − u − 1 ) ∇ Γ a u ∂ t Γ a u d x d τ ≤ − 1 2 ∫ 0 t ∫ R 2 [ e q ( e − u − 1 ) ∣ ∇ Γ a u ∣ 2 ] d x + 1 2 ∫ 0 t ∫ R 2 [ e q ( e − u ( x , 0 ) − 1 ) ∣ ∇ Γ a u ( x , 0 ) ∣ 2 ] d x + C X 3 2 ( t ) ∫ 0 t ( 1 + τ ) − 1 2 d τ ≤ − 1 2 ∫ 0 t ∫ R 2 [ e q ( e − u − 1 ) ∣ ∇ Γ a u ∣ 2 ] d x + 1 2 ∫ 0 t ∫ R 2 [ e q ( e − u ( x , 0 ) − 1 ) ∣ ∇ Γ a u ( x , 0 ) ∣ 2 ] d x + C X 3 2 ( t ) ( 1 + t ) 1 2 .

Thanks to Hölder’s inequality, Lemmas 2.5 and 2.6, and (4.1), we arrive at

(4.7) ∫ 0 t ∫ R 2 e q I 2 a ∂ t Γ a u d x = ∑ c + d < a , 0 ≤ α , β ≤ 2 ∫ R 2 e q ∂ α [ Γ c ( e − u − 1 ) ∂ β Γ c u ] ∂ t Γ a u d x d τ = ∑ c + d < a , 0 ≤ α , β ≤ 2 , ∣ c ∣ ≤ ∣ a ∣ 2 ∫ 0 t ∫ R 2 e q ∂ α [ Γ c ( e − u − 1 ) ∂ β Γ d u ] ∂ t Γ a u d x d τ + ∑ c + d < a , 0 ≤ α , β ≤ 2 , ∣ c ∣ ≥ ∣ a ∣ 2 ∫ 0 t ∫ R 2 e q ∂ α [ Γ c ( e − u − 1 ) ∂ β Γ d u ] ∂ t Γ a u d x d τ ≤ C ∑ c + d < a , ∣ c ∣ ≤ ∣ a ∣ 2 ∫ 0 t ‖ ∂ Γ c ( e − u − 1 ) ‖ L ∞ ‖ ∂ Γ d u ‖ L 2 ‖ ∂ t Γ a u ‖ L 2 d τ + C ∑ c + d < a , ∣ c ∣ ≤ ∣ a ∣ 2 ∫ 0 t ‖ Γ c ( e − u − 1 ) ‖ L ∞ ‖ ∂ 2 Γ d u ‖ L 2 ‖ ∂ t Γ a u ‖ L 2 d τ + C ∑ c + d < a , ∣ c ∣ ≥ ∣ a ∣ 2 ∫ 0 t ‖ ∂ Γ c ( e − u − 1 ) ‖ L 2 ‖ ∂ Γ d u ‖ L ∞ ‖ ∂ t Γ a u ‖ L 2 d τ + C ∑ c + d < a , ∣ c ∣ ≥ ∣ a ∣ 2 ∫ 0 t ‖ Γ c ( e − u − 1 ) ⟨ ∣ x ∣ − t ⟩ ≪ ‖ L 2 ‖ ∂ Γ Γ d u ‖ L ∞ ‖ ∂ t Γ a u ‖ L 2 d τ ≤ C X 3 2 ( t ) ∫ 0 t ( 1 + τ ) − 1 2 d τ ≤ C ( 1 + t ) 1 2 X 3 2 ( t ) .

The same procedure leads to

∫ 0 t ∫ R 2 e q I 3 a ∂ t Γ a u d x d τ = ∑ c 1 + c 2 ≤ a ∫ 0 t ∫ R 2 e q e − u Q ( Γ c 1 u , Γ c 2 u ) ∂ t Γ a u d x d τ + ∑ b + c 1 + c 2 ≤ a , 0 < ∣ b ∣ ≤ ∣ a ∣ 2 ∫ 0 t ∫ R 2 e q Γ b e − u Q ( Γ c 1 u , Γ c 2 u ) ∂ t Γ a u d x d τ + ∑ b + c 1 + c 2 ≤ a , ∣ b ∣ ≥ ∣ a ∣ 2 ∫ 0 t ∫ R 2 e q Γ b e − u Q ( Γ c 1 u , Γ c 2 u ) ∂ t Γ a u d x d τ ≤ C ∑ c 1 + c 2 ≤ a , ∣ c 1 ∣ ≤ ∣ a ∣ 2 ∫ 0 t ∫ R 2 e q ∣ T Γ c 1 u ∣ ∣ ∂ Γ c 2 u ∣ ∣ ∂ t Γ a u ∣ d x d τ + C ∑ c 1 + c 2 ≤ a , ∣ c 1 ∣ ≥ ∣ a ∣ 2 ∫ 0 t ∫ R 2 e q ∣ T Γ c 1 u ∣ ∣ ∂ Γ c 2 u ∣ ∣ ∂ t Γ a u ∣ d x d τ + ∑ b + c 1 + c 2 ≤ a , 0 < ∣ b ∣ ≤ ∣ a ∣ 2 ∫ 0 t ∫ R 2 e q ∣ Γ b ( e − u − 1 ) ∣ ( ∣ T Γ c 1 u ∣ ∣ ∂ Γ c 2 u ∣ + ∣ ∂ Γ c 1 u ∣ ∣ T Γ c 2 u ∣ ) ∣ ∂ t Γ a u ∣ d x d τ

(4.8) + ∑ b + c 1 + c 2 ≤ a , ∣ b ∣ ≥ ∣ a ∣ 2 ∫ 0 t ∫ R 2 e q ∣ Γ b ( e − u − 1 ) ∣ ( ∣ T Γ c 1 u ∣ ∣ ∂ Γ c 2 u ∣ + ∣ ∂ Γ c 1 u ∣ ∣ T Γ c 2 u ∣ ) ∣ ∂ t Γ a u ∣ d x d τ ≤ C ∑ c 1 + c 2 ≤ a , ∣ c 1 ∣ ≤ ∣ a ∣ 2 ∫ 0 t ‖ T Γ c 1 u ‖ L ∞ ‖ ∂ Γ c 2 u ‖ L 2 ‖ ∂ t Γ a u ‖ L 2 d τ + 1 24 ∑ c 1 + c 2 ≤ a , ∣ c 1 ∣ ≥ ∣ a ∣ 2 ∫ 0 t ∫ R 2 e q ∣ T Γ c 1 u ∣ 2 1 + ∣ ∣ x ∣ − τ ∣ 2 d x + ‖ ⟨ ∣ x ∣ − τ ⟩ ∂ Γ c 2 u ∂ t Γ a u ‖ L 2 2 d τ + 1 24 ∑ b + c 1 + c 2 ≤ a , ∣ b ∣ ≤ ∣ a ∣ 2 , c 1 ≤ c 2 ∫ 0 t ∫ R 2 e q ∣ T Γ c 2 u ∣ 2 1 + ∣ ∣ x ∣ − τ ∣ 2 d x + ‖ ⟨ ∣ x ∣ − τ ⟩ Γ b ( e − u − 1 ) ∂ Γ c 1 u ∂ t Γ a u ‖ L 2 2 d τ + 1 24 ∑ b + c 1 + c 2 ≤ a , ∣ b ∣ ≤ ∣ a ∣ 2 , c 1 ≥ c 2 ∫ 0 t ∫ R 2 e q ∣ T Γ c 1 u ∣ 2 1 + ∣ ∣ x ∣ − τ ∣ 2 d x + ‖ ⟨ ∣ x ∣ − τ ⟩ Γ b ( e − u − 1 ) ∂ Γ c 2 u ∂ t Γ a u ‖ L 2 2 d τ + ∑ b + c 1 + c 2 ≤ a , ∣ b ∣ ≥ ∣ a ∣ 2 ∫ 0 t ‖ T Γ c 1 u ‖ L ∞ ‖ ⟨ ∣ x ∣ − τ ⟩ ∂ Γ c 2 u ‖ L ∞ Γ b ( e − u − 1 ) ⟨ ∣ x ∣ − τ ⟩ L 2 ‖ ∂ t Γ a u ‖ L 2 d τ ≤ C ( X 3 2 ( t ) + X 2 ( t ) + X 3 ( t ) ) ( 1 + t ) 1 2 + 1 8 ∑ c 1 ≤ a ∫ 0 t ∫ R 2 e q ∣ T Γ c 1 u ∣ 2 1 + ∣ ∣ x ∣ − τ ∣ 2 d x d τ

and

(4.9) ∫ 0 t ∫ R 2 e q I 4 a ∂ t Γ a u d x d τ = ∫ 0 t ∫ R 2 e q Γ a [ ( e − u − 1 ) ( ∂ t u ) 2 ] ∂ t Γ α u d x d t = ∑ b + c + d = a , ∣ b ∣ ≤ ∣ a ∣ 2 ∫ 0 t ∫ R 2 e q Γ b ( e − u − 1 ) ∂ Γ c u ∂ Γ d u ∂ t Γ a u d x d τ + ∑ b + c + d = a , ∣ b ∣ ≥ ∣ a ∣ 2 ∫ 0 t ∫ R 2 e q Γ b ( e − u − 1 ) ∂ Γ c u ∂ Γ d u ∂ t Γ a u d x d τ ≤ C ∑ b + c + d = a , ∣ b ∣ ≤ ∣ a ∣ 2 , c ≤ d ∫ 0 t ‖ Γ b ( e − u − 1 ) ‖ L ∞ ‖ ∂ Γ c u ‖ L ∞ ‖ ∂ Γ d u ‖ L 2 ‖ ∂ t Γ a u ‖ L 2 d τ + C ∑ b + c + d = a , ∣ b ∣ ≤ ∣ a ∣ 2 , c ≥ d ∫ 0 t ‖ Γ b ( e − u − 1 ) ‖ L ∞ ‖ ∂ Γ c u ‖ L 2 ‖ ∂ Γ d u ‖ L ∞ ‖ ∂ t Γ a u ‖ L 2 d τ

+ C ∑ b + c + d = a , ∣ b ∣ ≥ ∣ a ∣ 2 , c ≤ d ∫ 0 t Γ b ( e − u − 1 ) ⟨ ∣ x ∣ − τ ⟩ L 2 ‖ Γ Γ c u ‖ L ∞ ‖ ∂ Γ d u ‖ L ∞ ‖ ∂ t Γ a u ‖ L 2 d τ + C ∑ b + c + d = a , ∣ b ∣ ≥ ∣ a ∣ 2 , c ≥ d ∫ 0 t Γ b ( e − u − 1 ) ⟨ ∣ x ∣ − τ ⟩ L 2 ‖ ∂ Γ c u ‖ L ∞ ‖ Γ Γ d u ‖ L ∞ ‖ ∂ t Γ a u ‖ L 2 d τ ≤ C X 2 ( t ) ∫ 0 t ( 1 + τ ) − 1 d τ ≤ C X 2 ( t ) log ( 2 + t ) .

We institute (4.5)–(4.9) into (4.4) and obtain

sup 0 ≤ τ ≤ t ∑ ∣ a ∣ ≤ K ‖ ∂ Γ a u ( τ ) ‖ L 2 2 ≤ C ε 2 + C ( 1 + t ) 1 2 ( X 3 2 ( t ) + X 2 ( t ) + X 3 ( t ) ) .

Then, Lemma 3.1 follows.□

Lemma 4.2

Under the assumption of Theorem 1.1, it holds that

(4.10) sup 0 ≤ τ ≤ t ∑ ∣ a ∣ ≤ K ( 1 + τ ) − 2 σ ‖ Γ a u ( τ ) ‖ L 2 2 ≤ C ε 2 + C ( 1 + t ) ( X 2 ( t ) + X 3 ( t ) ) .

Proof

We apply Lemma 3.1 to (4.2) and obtain

(4.11) ‖ Γ α u ‖ L 2 ≤ C ( ‖ Γ α u ( 0 ) ‖ L 2 + ‖ ∂ t Γ α u ( 0 ) ‖ L 2 + ‖ I 2 ( 0 ) ‖ L 2 ) + C log 1 2 ( 2 + t ) ‖ ( ∂ t Γ α u ( 0 ) ‖ L 1 + ‖ I 2 ( 0 ) ‖ L 1 ) + C log 1 2 ( 2 + t ) ∫ 0 t ( ‖ I 3 ‖ L 1 + ‖ I 4 ‖ L 1 ) d τ + C ∫ 0 t ( ‖ I 1 + I 2 ‖ L 2 + ‖ I 3 ‖ L 2 + ‖ I 4 ‖ L 2 ) d τ .

Noting that Γ α u ( 0 ) , ∂ t Γ α u ( 0 ) , and I 2 ( 0 ) are the initial data u 0 and u 1 , and Supp u 0 , Supp u 1 = { x ∣ ∣ x ∣ ≤ 1 } , then we have

(4.12) ‖ Γ α u ( 0 ) ‖ L 2 + ‖ ∂ t Γ α u ( 0 ) ‖ L 2 + ‖ I 2 ( 0 ) ‖ L 2 ≤ C ε

and

(4.13) ‖ ∂ t Γ α u ( 0 ) ‖ L 1 + ‖ I 2 ( 0 ) ‖ L 1 ≤ C ε .

We conclude from Lemma 2.3, Hölder’s inequality, Lemma 2.5, and (4.1) that

(4.14) ∫ 0 t ‖ I 3 ‖ L 1 d τ ≤ C ∑ b + c + d ≤ a ∫ 0 t ‖ ∣ Γ b e − u ∣ ( ∣ T Γ c 1 u ∣ ∣ ∂ Γ c 2 u ∣ + ∣ ∂ Γ c 1 u ∣ ∣ T Γ c 2 u ∣ ) ‖ L 1 d τ ≤ C ∑ c 1 + c 2 ≤ a ∫ 0 t ‖ T Γ c 1 u ‖ L 2 ‖ ∂ Γ c 2 u ‖ L 2 + ‖ ∂ Γ c 1 u ‖ L 2 ‖ T Γ c 2 u ‖ L 2 d τ + C ∑ b + c + d ≤ a , ∣ b ∣ ≤ ∣ a ∣ 2 , ∣ b ∣ ≠ 0 ∫ 0 t ‖ Γ b ( e − u − 1 ) ‖ L ∞ ‖ T Γ c 1 u ‖ L 2 ‖ ∂ Γ c 2 u ‖ L 2 d τ + C ∑ b + c + d ≤ a , ∣ b ∣ ≥ ∣ a ∣ 2 ∫ 0 t ‖ Γ b ( e − u − 1 ) ⟨ ∣ x ∣ − τ ⟩ ‖ L 2 ‖ T Γ c 1 u ‖ L 2 ‖ ⟨ ∣ x ∣ − τ ⟩ ∂ Γ c 2 u ‖ L ∞ d τ ≤ C X ( t ) ∫ 0 t ( 1 + τ ) − 1 + σ d τ + X 3 2 ( t ) ∫ 0 t ( 1 + τ ) − 1 2 d τ ≤ C X ( t ) + X 3 2 ( t ) ( 1 + t ) 1 2 .

Using Hölder’s inequality, Lemmas 2.5, 2.6, and (2.1), (4.1), we achieve

(4.15) ∫ 0 t ‖ I 4 ‖ L 1 d τ ≤ C ∑ b + c + d ≤ a , ∣ b ∣ ≤ ∣ a ∣ 2 ∫ 0 t ‖ Γ b ( e − u − 1 ) ‖ L ∞ ‖ ∂ Γ c u ‖ L 2 ‖ ∂ Γ d u ‖ L 2 d τ + C ∑ b + c + d ≤ a , ∣ b ∣ ≥ ∣ a ∣ 2 ∫ 0 t ‖ Γ b ( e − u − 1 ) ⟨ ∣ x ∣ − τ ⟩ ‖ L 2 ‖ ⟨ ∣ x ∣ − τ ⟩ ∂ Γ c u ‖ L ∞ ‖ ∂ Γ d u ‖ L 2 d τ ≤ C X 3 2 ( t ) ∫ 0 t ( 1 + τ ) − 1 2 d τ ≤ C X 3 2 ( t ) ( 1 + t ) 1 2 .

Hölder’s inequality, Lemmas 2.5, 2.6, and (2.1) and (4.1) entail that

(4.16) ∫ 0 t ‖ I 1 + I 2 ‖ L 2 d τ ≤ C ∑ c + d ≤ a , ∣ c ∣ ≤ ∣ a ∣ 2 ∫ 0 t ‖ Γ c ( e − u − 1 ) ‖ L ∞ ‖ ∂ Γ d u ‖ L 2 d τ + C ∑ c + d ≤ a , ∣ c ∣ ≥ ∣ a ∣ 2 ∫ 0 t ‖ Γ c ( e − u − 1 ) ⟨ ∣ x ∣ − τ ⟩ ‖ L 2 ‖ ⟨ ∣ x ∣ − τ ⟩ ∂ Γ d u ‖ L ∞ d τ ≤ C X ( t ) ∫ 0 t ( 1 + τ ) − 1 2 d τ ≤ C X ( t ) ( 1 + t ) 1 2 .

Thanks to Lemma 2.3, Hölder’s inequality, Lemmas 2.5 and 2.6, and (2.1) and (4.1), we arrive at

(4.17) ∫ 0 t ‖ I 3 ‖ L 2 d τ ≤ C ∑ b + c 1 + c 2 ≤ a ∫ 0 t ‖ ∣ Γ b e − u ∣ ( ∣ T Γ c 1 u ∣ ∣ ∂ Γ c 2 u ∣ + ∣ ∂ Γ c 1 u ∣ ∣ T Γ c 2 u ∣ ) ‖ L 2 d τ ≤ C ∑ c 1 + c 2 ≤ a ∫ 0 t ( ‖ ∣ T Γ c 1 u ‖ L ∞ ‖ ∂ Γ c 2 u ‖ L 2 + ‖ ∂ Γ c 1 u ‖ L ∞ ‖ T Γ c 2 u ‖ L 2 ) d τ + C ∑ b + c 1 + c 2 ≤ a , ∣ b ∣ ≤ ∣ a ∣ 2 ∫ 0 t ‖ Γ b ( e − u − 1 ) ‖ L ∞ ( ‖ T Γ c 1 u ‖ L ∞ ‖ ∂ Γ c 2 u ‖ L 2 + ‖ ∂ Γ c 1 u ‖ L ∞ ‖ T Γ c 2 u ‖ L 2 ) d τ + C ∑ b + c 1 + c 2 ≤ a , ∣ b ∣ ≥ ∣ a ∣ 2 ∫ 0 t ‖ Γ b ( e − u − 1 ) ⟨ ∣ x ∣ − τ ⟩ ‖ L 2 ‖ T Γ c 1 u ‖ L ∞ ‖ ⟨ ∣ x ∣ − τ ⟩ ∂ Γ c 2 u ‖ L ∞ d τ ≤ C X ( t ) ∫ 0 t ( 1 + τ ) − 1 d τ + X 3 2 ( t ) ∫ 0 t ( 1 + τ ) − 3 2 d τ ≤ C X ( t ) log ( 2 + t ) + X 3 2 ( t ) .

It follows from Hölder’s inequality, Lemmas 2.5 and 2.6 and (2.1) and (4.1) that

∫ 0 t ‖ I 4 ‖ L 2 d τ ≤ C ∑ b + c + d ≤ a , ∣ b ∣ ≤ ∣ a ∣ 2 ∫ 0 t ‖ Γ b ( e − u − 1 ) ‖ L ∞ ‖ ∂ Γ c u ‖ L ∞ ‖ ∂ Γ d u ‖ L 2 d τ

(4.18) + C ∑ b + c + d ≤ a , ∣ b ∣ ≥ ∣ a ∣ 2 ∫ 0 t Γ b ( e − u − 1 ) ⟨ ∣ x ∣ − τ ⟩ L 2 ‖ ⟨ ∣ x ∣ − τ ⟩ ∂ Γ c u ‖ L ∞ ‖ ∂ Γ d u ‖ L ∞ d τ ≤ C X 3 2 ( t ) ∫ 0 t ( 1 + τ ) − 1 d τ ≤ C X 3 2 ( t ) log ( 2 + t ) .

We institute (4.12)–(4.18) into (4.11) immediately, which yields (4.10). Lemma 4.2 is proved.□

Lemma 4.3

Under the assumption of Theorem 1.1, it holds that

(4.19) sup 0 ≤ τ ≤ t ∑ ∣ a ∣ ≤ K 2 + 1 ( 1 + τ ) ‖ Γ a u ( τ ) ‖ L ∞ 2 ≤ C ε 2 + C ( ( 1 + t ) 2 σ X 2 ( t ) + ( 1 + t ) X 2 ( t ) + ( 1 + t ) X 3 ( t ) ) .

Proof

Let

(4.20) Γ a u = u ℓ + u ˜ + u ¯ ,

where u ℓ , u ˜ , and u ¯ satisfy

(4.21) □ u ℓ = 0 , u ℓ ( x , 0 ) = Γ a u ( x , 0 ) , ∂ t u ℓ ( x , 0 ) = ∂ t Γ a u ( x , 0 ) ,

(4.22) □ u ˜ = ∑ b + c ≤ a C 1 b c a ∂ ( Γ b ( e − u − 1 ) ∂ Γ c u ) , u ˜ ( x , 0 ) = 0 , ∂ t u ˜ ( x , 0 ) = 0 ,

and

(4.23) □ u ¯ = ∑ b + c 1 + c 2 ≤ a C 2 b c 1 c 2 a Γ b e − u Q ′ ( ∂ Γ c 1 u , ∂ Γ c 2 u ) + ∑ b + c + d ≤ a C 2 b c d a Γ b ( e − u − 1 ) ∂ Γ c u ∂ Γ d u , u ¯ ( x , 0 ) = 0 , ∂ t u ¯ ( x , 0 ) = 0 ,

respectively.

Applying Lemma 3.2 to (4.21) yields

(4.24) ∣ u ℓ ( x , t ) ∣ ≤ C ( 1 + t ) − 1 2 ( ‖ Γ a u ( x , 0 ) ‖ W 3,1 + ‖ ∂ t Γ a u ( x , 0 ) ‖ W 2,1 ) ≤ C ( 1 + t ) − 1 2 ε .

We conclude from Lemma 3.3 and (4.22) that

(4.25) ∣ u ˜ ( x , t ) ∣ ≤ C ( 1 + t ) − 1 2 ∫ 0 t ∑ b + c ≤ a ( 1 + τ ) 1 2 ‖ Γ b ( e − u − 1 ) ∂ Γ c u ‖ L ∞ + ∑ ∣ b + c ∣ ≤ ∣ a ∣ + 3 ( 1 + τ ) − 3 2 ‖ Γ b ( e − u − 1 ) ∂ Γ c u ‖ L 1 d τ ≤ C ( 1 + t ) − 1 2 X ( t ) ∫ 0 t ( 1 + τ ) − 1 2 d τ + ∫ 0 t ( 1 + τ ) − 3 2 + σ d τ ≤ C X ( t ) .

Lemma 3.4, Hölder’s inequality, and 4.23 entail that

(4.26) ∣ u ¯ ( x , t ) ∣ ≤ C ( 1 + t ) − 1 2 ∫ 0 t ( 1 + τ ) − 1 2 ∑ ∣ b + c 1 + c 2 ∣ ≤ ∣ a ∣ + 1 ‖ Γ b Q ′ ( ∂ Γ c 1 u , ∂ Γ c 2 u ) ‖ L 1 + ∑ ∣ b + c + d ∣ ≤ ∣ a ∣ + 1 ‖ Γ b ( e − u − 1 ) ∂ Γ c 1 u ∂ Γ c 2 u ‖ L 1 d τ ≤ C ( 1 + t ) − 1 2 X ( t ) ∫ 0 t ( 1 + τ ) − 1 + σ d τ + X 3 2 ( t ) ∫ 0 t ( 1 + τ ) − 1 2 d τ

≤ C ( 1 + t ) − 1 2 ( 1 + τ ) σ X ( t ) + ( 1 + τ ) 1 2 X 3 2 ( t ) .

Substituting (4.24)–(4.26) into (4.20) immediately yields (4.19). Then, we complete the proof of Lemma 4.3.□

5 Lifespan of smooth solutions

In this section, we give the lifespan of smooth solutions to problems (1.7) and (1.8). In other words, we shall complete the proof of Theorem 1.1.

Proof

From (4.3), (4.10), (4.19), and (4.1), we conclude that

(5.1) X ( t ) ≤ C 0 ε 2 + C 1 ( 1 + t ) 1 2 X 3 2 ( t ) + C 2 ( 1 + t ) ( X 2 ( t ) + X 3 ( t ) ) .

Assume that

(5.2) X ( t ) ≤ M ε 2 ,

where M = 8 C 0 .

Then, (5.1) and (5.2) entail that

(5.3) X ( t ) ≤ C 0 ε 2 + C 1 ( 1 + t ) 1 2 M 3 2 ε 3 + C 2 ( 1 + t ) ( M 2 ε 4 + M 3 ε 6 ) .

Taking ε ≤ 1 M and t + 1 ≤ min { 1 64 C 1 2 M ε 2 , 1 8 C 2 M ε 2 } , then from (5.3), we arrive at

X ( t ) ≤ 1 2 M ε 2 .

Thus, by the local well-posedness of problems (1.7) and (1.8) and the continuous induction method, then problems (1.7) and (1.8) admit a global smooth solution u on [ 0 , T ε ] , with

T ε ≥ C ε 2 .

We complete the proof of Theorem 1.1.□

Acknowledgments

The authors wish to thank the referees for their very helpful comments and suggestions that improved the manuscript.

  1. Funding information: This work was supported in part by the Natural Science Foundation of Henan (Grant No. 252300421305) and Program for Innovative Research Team (in Science and Technology) in University of Henan Province (Grant No. 25IRTSTHN013).

  2. Author contributions: All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.

  3. Conflict of interest: The authors declare that they have no competing interests.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2024-09-21
Revised: 2025-06-16
Accepted: 2025-07-14
Published Online: 2025-08-29

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/anona-2025-0106
Keywords for this article
hyperbolic geometric flow equation; lifespan; smooth solutions; null condition
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. Incompressible limit for the compressible viscoelastic fluids in critical space
  3. Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
  4. Intervals of bifurcation points for semilinear elliptic problems
  5. On multiplicity of solutions to nonlinear Dirac equation with local super-quadratic growth
  6. Entire radial bounded solutions for Leray-Lions equations of (p, q)-type
  7. Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry
  8. Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces
  9. Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space
  10. Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations
  11. Existence results for non-coercive problems
  12. Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent
  13. Geometric characterization of generalized Hajłasz-Sobolev embedding domains
  14. Subharmonic solutions of first-order Hamiltonian systems
  15. Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights
  16. Stability of pyramidal traveling fronts in time-periodic reaction–diffusion equations with degenerate monostable and ignition nonlinearities
  17. Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth
  18. Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections
  19. Homoclinic solutions in periodic partial difference equations
  20. Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed
  21. Properties of minimizers for L2-subcritical Kirchhoff energy functionals
  22. Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces
  23. Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed
  24. Qualitative properties of solutions for dual fractional parabolic equations involving nonlocal Monge-Ampère operator
  25. Regularity for double-phase functionals with nearly linear growth and two modulating coefficients
  26. Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups
  27. Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians
  28. Existence of positive radial solutions of general quasilinear elliptic systems
  29. Low Mach number and non-resistive limit of magnetohydrodynamic equations with large temperature variations in general bounded domains
  30. Sharp viscous shock waves for relaxation model with degeneracy
  31. Bifurcation and multiplicity results for critical problems involving the p-Grushin operator
  32. Asymptotic behavior of solutions of a free boundary model with seasonal succession and impulsive harvesting
  33. Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds
  34. Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity
  35. Singularity for the macroscopic production model with Chaplygin gas
  36. Global strong solution of compressible flow with spherically symmetric data and density-dependent viscosities
  37. Global dynamics of population-toxicant models with nonlocal dispersals
  38. α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets
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  40. On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation
  41. Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping
  42. On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations
  43. Remark on the analyticity of the fractional Fokker-Planck equation
  44. Continuous dependence on initial data for damped fourth-order wave equation with strain term
  45. Unilateral problems for quasilinear operators with fractional Riesz gradients
  46. Boundedness of solutions to quasilinear elliptic systems
  47. Existence of positive solutions for critical p-Laplacian equation with critical Neumann boundary condition
  48. Non-local diffusion and pulse intervention in a faecal-oral model with moving infected fronts
  49. Nonsmooth analysis of doubly nonlinear second-order evolution equations with nonconvex energy functionals
  50. Qualitative properties of solutions to the viscoelastic beam equation with damping and logarithmic nonlinear source terms
  51. Shape of extremal functions for weighted Sobolev-type inequalities
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  56. Optimal large time behavior of the 3D rate type viscoelastic fluids
  57. Local well-posedness for the two-component Benjamin-Ono equation
  58. Self-similar blow-up solutions of the four-dimensional Schrödinger-Wave system
  59. Existence and stability of traveling waves in a Keller-Segel system with nonlinear stimulation
  60. Existence and multiplicity of solutions for a class of superlinear p-Laplacian equations
  61. On the global large regular solutions of the 1D degenerate compressible Navier-Stokes equations
  62. Normal forms of piecewise-smooth monodromic systems
  63. Fractional Dirichlet problems with singular and non-locally convective reaction
  64. Sharp forced waves of degenerate diffusion equations in shifting environments
  65. Global boundedness and stability of a quasilinear two-species chemotaxis-competition model with nonlinear sensitivities
  66. Non-existence, radial symmetry, monotonicity, and Liouville theorem of master equations with fractional p-Laplacian
  67. Global existence, asymptotic behavior, and finite time blow up of solutions for a class of generalized thermoelastic system with p-Laplacian
  68. Formation of singularities for a linearly degenerate hyperbolic system arising in magnetohydrodynamics
  69. Linear stability and bifurcation analysis for a free boundary problem arising in a double-layered tumor model
  70. Hopf's lemma, asymptotic radial symmetry, and monotonicity of solutions to the logarithmic Laplacian parabolic system
  71. Generalized quasi-linear fractional Wentzell problems
  72. Existence, symmetry, and regularity of ground states of a nonlinear Choquard equation in the hyperbolic space
  73. Normalized solutions for NLS equations with general nonlinearity on compact metric graphs
  74. Boundedness and global stability in a predator-prey chemotaxis system with indirect pursuit-evasion interaction and nonlocal kinetics
  75. Review Article
  76. Existence and stability of contact discontinuities to piston problems
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