Main result: nonlinear stability
Let A source (x, t) represent a source solution defined by a specific form, while A(x, t) denotes the solution of a related equation with smooth initial data A in (x) We assume the initial data A in (x) is expressed as Rin(x)e iφ in (x) and is close to the source solution, measured by a norm that combines the differences in the radial and phase components This norm, defined in equation (1.4), indicates that the deviation from the source solution is sufficiently small, allowing for the construction of the solution A(x, t) in a specific format.
The equation A(x+p(x, t), t) = (r(x) + R(x, t))e^(i(ϕ(x)+φ(x,t)))e^(-iω₀t) defines a perturbation where the function p(x, t) is selected to eliminate non-decaying terms Initial values for p(x, 0), R(x, 0), and φ(x, 0) can be derived from the initial data A in (x) We will perform a linearization of this equation around the source and utilize this information to effectively determine p(x, t) The linearization and the dominant nonlinear terms suggest that as time approaches infinity, φ(x, t) converges to φₐ(x, t), with φₐ representing the phase modulation from zero eigenvalues This notation indicates that φₐ is an approximate solution to the governing dynamics of the perturbation φ, specifically a solution to a Burgers-type equation that encapsulates the primary dynamics of φ Consequently, the analysis indicates that the leading order dynamics of the perturbed source are characterized by the modulated source.
The functions p(x, t) and φ a (x, t) effectively eliminate non-decaying or slowly-decaying terms related to zero eigenvalues and quadratic nonlinearity, facilitating the closure of a nonlinear iteration scheme To further clarify these functions, we introduce the definition e(x, t) := errfn x+c g t.
−∞ e − x 2 dx (1.5) and the Gaussian-like term θ(x, t) := 1
, (1.6) whereM 0 is a fixed positive constant Now define φ a (x, t) :=−d
The constant q, defined in equation (5.10), along with the smooth functions δ ± = δ ± (t), play a crucial role in our findings Our primary conclusion demonstrates that the shifted solution A(x+p(x, t), t) approaches the modulated source, exhibiting a decay rate characteristic of a Gaussian function.
Theorem 1.1 states that for initial data expressed as Ain(x) = Rin(x)e iφ in (x), where Rin and φin are in C^3(R), there exists a positive constant ǫ0 If the condition ǫ := kA in (ã) − A source (ã, 0) kin ≤ ǫ0 is satisfied, then the solution A(x, t) to the qCGL equation exists globally in time Furthermore, there are constants η0, C0, M0 > 0, and δ ± ∞ ∈ R, with |δ ∞ ± | ≤ ǫC0, along with smooth functions δ ± (t).
∂x ℓ hA(x+p(x, t), t)−A mod (x, t)i≤ǫC 0 (1+t) κ [(1+t) − ℓ/2 +e − η 0 | x | ]θ(x, t), ∀x∈R, ∀t≥0, (1.9) for ℓ = 0,1,2 and for each fixed κ ∈ (0, 1 2 ) In particular, kA(ã+p(ã, t), t)−A mod (ã, t)kW 2,r → 0 as t→ ∞ for each fixedr > 1 − 1 2κ
Theorem 1.1 not only confirms the nonlinear stability of the source solutions of (1.1) but also offers a comprehensive analysis of the dynamics associated with small perturbations Specifically, the amplitude of the shifted solution A(x+p(x, t), t) approaches the source amplitude A source (x, t) with a decay rate resembling that of a Gaussian function: R(x, t)∼θ(x, t) Furthermore, the dynamics of the phase can be articulated by defining δ φ (t) as −d.
1 +δ − (t) i, (1.10) it then follows from (1.7) that φ a (x, t)−δ φ (t)e(x, t+ 1)+p(x, t)−δ p (t)e(x, t+ 1)≤ǫC 0 (1 +t) 1/2 θ(x, t) (1.11)
The functione(x, t) represents an expanding plateau that extends outward at a speed of ±c g, with its height stabilizing around one As time progresses, the interfaces broaden proportionally to √t, leading to the phase ϕ(x) + φ(x, t) converging towards ϕ(x) + φ a (x, t), where φ a (x, t) also resembles an expanding plateau.
Figure 1: Illustration of the graph of e(x, t), the difference of two error functions, for a fixed value oft.
As a direct consequence of Theorem 1.1, we obtain the following corollary.
Corollary 1.2 establishes that for any positive constant η, the space-time cone V is defined by the constraint −(c g − η)t ≤ x ≤ (c g − η)t Given the same conditions as in Theorem 1.1, there exist positive constants η1 and C1 such that the solution A(x, t) to the qCGL equation (1.1) adheres to these constraints.
|A(x, t)−A source (x−δ p (∞), t−δ φ (∞)/ω 0 )| ≤ǫC 1 e − η 1 t for all (x, t)∈V, in which δ p and δ φ are defined in (1.10).
Proof Indeed, within the coneV, we have
|e(x, t+ 1)−1|+θ(x, t)≤C 1 e − η 1 t for some constants η 1 , C 1 >0 The estimate (1.11) shows that p(x, t) andφ a (x, t) are constants up to an error of ordere − η 1 t The main theorem thus yields the corollary at once.
In the proof of Theorem 1.1, the functions δ ± (t) are developed through integral formulas designed to accurately represent the non-decaying component of the Green's function associated with the linearized operator The selections of p(x, t) and φ a (x, t) are informed by the understanding that the leading-order asymptotic behavior of the translation and phase variables is dictated by a nonlinear Burgers-type equation.
(φ a ±k 0 p) =q(∂ x φ a ±k 0 ∂ x p) 2 , (1.12) where q is defined in (5.10) See Section 5.4 The formulas (1.7) are related to an application of the Cole-Hopf transformation to the above equation.
In this proof, we will address two main challenges The first challenge involves the presence of embedded zero eigenvalues, which can be managed using a well-established yet complex technique introduced in [ZH98] This method entails incorporating an initially arbitrary function into the perturbation Ansatz, which is subsequently selected to neutralize the nondecaying components of the Green's function caused by the zero eigenvalues The second challenge pertains to the handling of quadratic order nonlinearity.
To illustrate this second difficulty, for the moment ignore the issue of the zero eigenvalues Suppose we were to linearize equation (1.1) in the standard way and set
A(x, t) =A source (x, t) + ˜A(x, t), with the hope of proving that the perturbation, ˜A(x, t), decays The function ˜A(x, t) would then satisfy an equation of the form
The equation (∂t - L) ˜A = Q(˜A) involves a linearized operator L characterized by the highest order derivatives (1 + iα)∂x², while Q(˜A) = O(|˜A|²) represents the nonlinearity containing quadratic terms The temporal Green's function for the heat operator is a Gaussian centered at x = y, but for the operator ∂t - L, it behaves like a Gaussian centered at x = y ± cgt, complicating the analysis due to embedded zero eigenvalues Quadratic terms can significantly impact the dynamics, as seen in the example u_t = u_xx - u², where the zero solution is stable for positive initial data but generally unstable Standard stability techniques become ineffective in the presence of these quadratic terms, particularly the nonlinear iteration method that relies on pointwise Green's function estimates This method fails because the convolution of a Gaussian Green's function with a quadratic function does not guarantee Gaussian behavior Consequently, to effectively analyze the equation, it is essential to adopt an Ansatz that eliminates the quadratic terms.
The essential spectrum of L intersects the imaginary axis at the origin, indicating a zero eigenvalue with multiplicity two The corresponding eigenfunctions, ∂t A source and ∂x A source, represent time and space translations, respectively, but are not localized in space or in the (R, φ) coordinates This non-locality arises because the group velocities are directed outward from the defect's core, resulting in localized perturbations producing a non-local phase response Consequently, the perturbed phase φ(x, t) manifests as an outwardly expanding plateau, which must be considered in the analysis to successfully complete a nonlinear iteration scheme.
In the proof, we write the solutionA(x, t) in the form
The equation A(x+p(x, t), t) = [r(x) + R(x, t)]ei(ϕ(x)+φ(x,t))e − iω 0 t utilizes perturbation variables (R(x, t), φ(x, t)), offering several advantages when employing polar coordinates Firstly, these coordinates align with the phase invariance associated with the equation Secondly, the quadratic nonlinearity depends on R, φ_x, and their higher derivatives, excluding any zero-order term involving φ Lastly, based on the leading order terms, it is anticipated that the amplitude R will decay more rapidly over time than the phase φ, effectively transforming the equation into a form akin to ut = uxx - u².
The equation ˜ut = uxx - uux represents a significant relationship in the context of (R, φ) variables, though it lacks the structure of a conservation law This indicates that while the nonlinearity is crucial when considering ˜A, it becomes marginal in the (R, φ) framework, as noted in BK94.
In the context of marginal nonlinearity, the absence of an additional conservation law structure, such as that found in Burgers' equation, necessitates an alternative approach to address the marginal terms The findings from Section 5.4 indicate that the leading order dynamics of (R, φ) are primarily influenced by these marginal terms.
Existence of a family of sources for qCGL
In this subsection, we prove the following lemma concerning the existence and some qualitative proper- ties of the source solutions defined in (1.2).
Lemma 2.1 There exists a k0 ∈R with |k0|< 1 such that a source solution Asource(x, t) of (1.1) of the form (1.2)exists and satisfies the following properties.
1 The functions r(x) and ϕ(x) are C ∞ Let x 0 be a point at which r(x 0 ) = 0 Necessarily, r ′ (x0)6= 0 and rxx(x0) =ϕx(x0) = 0.
2 The functions r and ϕsatisfyr(x)→ ±r 0 (k 0 ) andϕ x (x)→ ±k 0 asx→ ±∞, respectively, where r 0 is defined in (2.1), below Furthermore, d ℓ dx ℓ r(x)∓r 0 (k 0 )+ d ℓ+1 dx ℓ+1 ϕ(x)∓k 0 x≤C 0 e − η 0 | x | , for integers ℓ≥0 and for some positive constantsC 0 and η 0
3 As x → ±∞, A source (x, t) converges to the wave trains A wt (x, t;±k 0 ) =±r 0 (k 0 )e i( ± k 0 x − ω nl (k 0 )t) , respectively, with r 2 0 = 1−k 0 2 +γ 1 r 4 0 , ω nl (k 0 ) =β+ (α−β)k 2 0 + (βγ 1 −γ 2 )r 4 0 (2.1)
4 If k 0 6= 0, the asymptotic group velocities c ± g := dω nl dk (k) |k= ± k 0 have opposite sign at ±∞ and satisfy c ± g =±cg, cg := 2k0(α−β ∗ ), (2.2) where β ∗ is defined (1.3) Without loss of generality, we assume thatc g >0.
In the context of the cubic CGL equation, where γ1 and γ2 are both equal to zero, an explicit formula for the traveling source, known as a Nozaki-Bekki hole, has been established in various studies [BN85, Leg01, PSAK95] These Nozaki-Bekki holes represent degenerate solutions of the CGL equation, characterized by their existence at a non-transverse intersection of stable and unstable manifolds Specifically, when γ1 and γ2 are zero, the formula for the standing Nozaki-Bekki holes is defined.
Asource(x, t) =r0tanh(κx)e − iω 0 t e − iδ log(2 cosh κx) , wherer 0 =p
1−k 2 0 ,k 0 =−δκ, and δ and κ are defined by κ 2 = (α−β) (α−β)δ 2 −3δ(1 +α 2 ), δ 2 +3(1 +αβ)
The equation (β−α) δ−2 = 0 defines δ as a root under the condition δ(α−β) 0, the difference is positive, resulting in a positive group velocity at +∞ (c + g) and a negative group velocity at −∞ (c − g), confirming that the solution acts as a source Conversely, when κ < 0, the signs of c ± g are reversed, yet the corresponding ends are also switched, maintaining the conclusion that the solution remains a source in all scenarios.
The qCGL equation represents a minor perturbation of the cubic CGL, with established solutions that persist as standing sources across a range of parameter values These solutions are derived from the transverse intersection of the two-dimensional center-stable manifold of the asymptotic wave train at infinity and the center-unstable manifold of the wave train at negative infinity, specifically unfolded concerning the wavenumbers of these wave trains Notably, the standing sources facilitate the connection between wave trains with a designated wavenumber, which is crucial for the validation of Lemma 2.1.
Proof of Lemma 2.1 As mentioned above, standing sources have been proven to exist in [Doe96] and [SS04a] Let A source (x, t) be that source, which we can write in the form
Plugging this into (1.1), we find that (r, ϕ) solves
It is shown in [Doe96] that there exists a locally unique wavenumberk 0 and a smooth solution (r, ϕ) of (2.3) so that Asource(x, t) converges to the wave trains of the form
A wt (x, t;±k 0 ) =±r 0 (k 0 )e i( ± k 0 x − ω nl (k 0 )t) , respectively asx→ ±∞ Putting this asymptotic Ansatz into (2.3) then yields γ 1 r 0 4 −r 0 2 + 1−k 2 0 = 0, ω nl (k 0 ) =αk 2 0 +βr 0 2 −γ 2 r 4 0
Rearranging terms then gives (2.1), and hence item 3 in the lemma Differentiating the above identities with respect tok and solving for dω dk nl (k) , we obtain item 4 as claimed.
For the first item, sincer(x)→ ±r 0 withr 0 ≈p
1−k 0 2 6= 0 (becauseγ 1 ≈0), there must be a point at which r(x) vanishes Without loss of generality, we assume that r(0) = 0 Since r(x) and ϕ(x) are smooth, evaluating the system (2.3) at x= 0 gives r xx (0)−2αr x (0)ϕ x (0) = 0, 2r x (0)ϕ x (0) +αr xx (0) = 0.
The equations suggest that \( r_{xx}(0) = r_x(0) \phi_x(0) = 0 \), but we assert that \( r_x(0) \) must be nonzero If it were zero, then the source function \( A_{\text{source}}(0) \) would equal its derivative, leading to \( A_{\text{source}}(x) \equiv 0 \), which contradicts our assumption Thus, we conclude that \( r_x(0) \) cannot be zero Furthermore, the exponential decay mentioned in item 2 arises directly from the solutions identified in [Doe96, SS04a], which exist at the intersection of stable and unstable manifolds of saddle equilibria.
Linearization
In order to linearize (1.1) aroundA source (x, t), we introduce the perturbation variables (R, φ) via
A(x, t) = [r(x) +R(x, t)]ei(ϕ(x)+φ(x,t))e − iω 0 t , (2.4) wherer and ϕare the amplitude and phase ofA source (x, t) Throughout the paper, we shall work with the vector variable
U :R rφ for the perturbation (R, φ) We have the following lemma.
Lemma 2.2 If the A(x, t), defined in (2.4), solves (1.1), then the linearized dynamics of U are
Proof Plugging (2.4) into (1.1) and using equation (2.3), we obtain the linearized system
R t =R xx −2αϕ x R x + (1−3r 2 −αϕ xx −ϕ 2 x + 5γ 1 r 4 )R−αrφ xx −2(αr x +rϕ x )φ x rφ t =rφ xx −(2αrϕ x −2r x )φ x +αR xx + 2ϕ x R x + (ω 0 +ϕ xx −αϕ 2 x −3βr 2 + 5γ 2 r 4 )R.
A rearrangement of terms yields the lemma.
Next, note that L is a bounded operator from ˙H 2 (R;C 2 )T
L ∞ (R;C 2 ) to L 2 (R;C 2 ), where the function space ˙H 2 (R;C 2 ) consists of functions U = (u 1 , u 2 ) so that u 1 ∈ H 2 (R;C) and ∂ x u 2 , ∂ 2 x u 2 ∈
L 2 (R;C) (Note that we do not require that u 2 ∈ L 2 , since rφ need not be localized.) We make the following assumption about the spectral stability ofL.
Hypothesis 2.1 The spectrum of the operator L in L 2 (R,C 2 ) satisfies the following two conditions:
• The spectrum does not intersect the closed right half plane, except at the origin.
• In the weighted space L 2 η (R,C 2 ), defined bykuk 2 η =R e − η | x | |u(x)| 2 dx withη >0 sufficiently small, there are exactly two eigenvalues in the closed right half plane, and they are both at the origin.
Lemma 2.3 states that the essential spectrum of the linearized operator L in L²(R, C²) is completely located in the left half-plane, specifically where Reλ ≤ 0, and intersects the imaginary axis solely at the origin, forming a parabolic curve as illustrated in Figure 2 Furthermore, the nullspace of L is spanned by the functions V₁(x) and V₂(x).
, (2.7) and the corresponding adjoint eigenfunctions, ψ 1,2 (x), are both exponentially localized: |ψ 1,2 (x)| ≤
Proof It follows directly from the translation and gauge invariance of qCGL that ∂xAsource(x, t) and
∂ t A source (x, t) are exact solutions of the linearized equation aboutA source (x, t) Consequently,V 1 (x), V 2 (x) belong to the kernel ofL By hypothesis 2.1 these are the only elements of the kernel.
Next, by standard spectral theory (see, for instance, Henry [Hen81]), the essential spectrum ofL in
L 2 (R,C 2 ) is the same as that of the limiting, constant-coefficient operatorL ± defined by
To find the spectrum ofL±, let us denote by Λ ± κ for each fixedκ∈R the constant matrix Λ ± κ :=−κ 2 D 2 ∓2iκk 0 D 1 −D ∞ 0 ,
The essential spectrum of the operator L is represented within the shaded area defined by the algebraic curves λ ± 1 (κ) and λ ± 2 (κ), which lie outside the set Ω ϑ and its boundary Γ = ∂Ω ϑ The eigenvalues λ ± 1 (κ) and λ ± 2 (κ) of Λ ± κ demonstrate that Reλ ± 1 (0) equals zero, while Reλ ± j (κ) remains negative for all κ ≠ 0 Notably, λ ± 1 (κ) approaches the origin as a parabolic curve, expressed as λ ± 1 (κ) = −ic ± g κ − dκ² + O(κ³) for sufficiently small κ, whereas λ ± 2 (κ) stays distanced from the imaginary axis Consequently, the spectrum of L ± is restricted to the shaded region to the left of the curves λ ± j (κ) for κ in R, as illustrated in Figure 2.
Finally, one can see from studying the asymptotic limits of (3.10) that the adjoint eigenfunctions are exponentially localized; alternatively, this property was shown more generally in [SS04a, Corollary 4.6].
3 Construction of the resolvent kernel
We develop the resolvent kernel and establish resolvent estimates for the linearized operator L, as defined in (2.5) These estimates will be utilized in §4 to derive pointwise estimates for the Green’s function.
LetG(x, y, λ) denote the resolvent kernel associated with the operatorL, which is defined to be the distributional solution of the system
The equation (λ− L)G(ã, y, λ) = δy(ã) illustrates the relationship between the operator and the Dirac delta function centered at y This section presents pointwise bounds on the resolvent kernel G(x, y, λ), categorized into three distinct frequency regions: low-frequency (as λ approaches 0), mid-frequency (where |λ| is between θ and M), and high-frequency (as λ approaches infinity) This classification is essential due to the behavior of the spatial eigenvalues linked to the four-dimensional first-order ODE associated with the equation, as outlined in (3.4) These eigenvalues play a crucial role in defining the characteristics of G and are influenced by the spectral parameter λ, allowing for a differentiated asymptotic analysis of G across the three frequency regions.
In this section, we establish key propositions, with real constants ϑ 1, ϑ 2, and ϑ 3 selected to ensure that the set Ω ϑ does not overlap with the spectrum of L, as illustrated in Figure 2 and detailed in Lemma 2.3 The proofs for these main results will be provided subsequently.
Proposition 3.1 (High-frequency bound) There exist positive constants ϑ 1,2 , M, C, and η so that
Proposition 3.2 (Mid-frequency bound) For any positive constants ϑ 3 and M, there exists a C C(M, ϑ 3 ) sufficiently large so that
Proposition 3.3 (Low-frequency bound) There exists an η 3 >0sufficiently small such that, for all λ with |λ|< η3, we have the expansion
Here cg >0is the group velocity defined in (2.2), V1, V2 are the eigenfunctions defined in (2.7), and the adjoint eigenfunctions satisfy ψ j (y) =O(e − η | y | ),for some fixed η >0.
Spatial eigenvalues
Let us first consider the linearized eigenvalue problem
(λ− L)U = 0 (3.3) and derive necessary estimates on behavior of solutions as x→ ±∞ We write the eigenvalue problem (3.3) as a four-dimensional first order ODE system For simplicity, let us denote
−1 0 whereI denotes the 2×2 identity matrix and D 0,1,2 are defined in (2.6) Let W = (U, U x ) be the new variable By (2.5), the eigenvalue problem (3.3) then becomes
LetA±(λ) be the asymptotic limits of A(x, λ) at x=±∞, and let
. so that, by Lemma 2.1,B 0 (x, λ)→B 0 (λ) as x→ ±∞ Thus, we have
The solutions to the limiting ordinary differential equation (ODE) system \( W x = A^{\pm}(\lambda)W \) are expressed as \( W_{\infty}(\lambda)e^{\nu^{\pm}(\lambda)x} \), where \( W_{\infty}(w, \nu^{\pm}(\lambda)w) \) and \( \nu^{\pm}(\lambda) \) represent the eigenvalues of \( A^{\pm}(\lambda) \) These eigenvalues, known as spatial eigenvalues, are distinct from the temporal eigenvalue parameter \( \lambda \) and are determined by the condition \( \text{det}(B_0(\lambda) \pm 2k_0 \nu^{\pm}(\lambda) C_0 - \nu^{\pm 2}(\lambda) I) = 0 \).
The behavior of spatial eigenvalues as functions of λ is crucial for understanding the resolvent kernel's properties This behavior can be analyzed by examining the limiting cases as λ approaches 0 and as |λ| approaches infinity, along with the intermediate regime The mid- and high-frequency regimes present the most straightforward scenarios for this analysis.
Mid- and high-frequency resolvent bounds
Proof of Proposition 3.1 The spatial eigenvalues and resolvent kernel can be analyzed in this regime using the following scaling argument Define ˜ x=|λ| 1/2 x, λ˜=|λ| − 1 λ, fW(˜x) =W(|λ| − 1/2 x).
In these scaled variables, (3.4) becomes
For λ in the set Ω ϑ, the value ˜λ lies on the unit circle and remains distanced from the negative real axis Consequently, there exists a constant η greater than 0, ensuring that the real part of qλ(1˜ ±iα) exceeds η for all λ in Ω ϑ as |λ| approaches infinity Refer to Figure 3a for visual representation.
Figure 3 illustrates the spatial eigenvalues, which are conveniently represented on the real axis As |λ| approaches infinity, these eigenvalues become distinctly separated from the imaginary axis Conversely, as |λ| approaches zero, one spatial eigenvalue, influenced by the spatial limit as x approaches ±∞, nears the origin.
The asymptotic behavior allows us to gain insights into the resolvent kernel The eigenprojections linked to ˜A(˜λ), denoted as ˜P s,u (˜λ), are smooth functions of λ and correspond to the stable and unstable subspaces By defining fW s,u (˜x) as ˜P s,u (˜λ)fW(˜x), we can analyze the interactions within these subspaces more effectively.
! +O(|λ| − 1/2 fW) (3.7) where the matrices ˜A s,u (˜λ) satisfy
RehA˜ s (˜λ)W, Wi C 4 ≤ −η|W| 2 , RehA˜ u (˜λ)W, Wi C 4 ≥η|W| 2 , for all W ∈ C 4 Here, hã,ãi C 4 denotes the usual inner product in C 4 Taking the inner product of equation (3.7) with (fW s ,Wf u ), we get
For sufficiently large |λ|, stable solutions exhibit a decay rate faster than e^(-η|x˜|), while unstable solutions increase at a rate of at least e^(η|x˜|) Consequently, the resolvent kernel GeW(˜x,y,˜ λ) linked to the scaled equation (3.6) adheres to a uniform bound.
|Ge W (˜x,y,˜ λ)˜ | ≤Ce − η | ˜ x − y ˜ | Going back to the original variables, the resolvent kernel associated with (3.4) satisfies
The resolvent kernelG(x, y, λ) is by definition is the two-by-two upper-left block of the matrixG W (x, y, λ). This proves the proposition.
Proof of Proposition 3.2 The proof is immediate by the analyticity of G(x, y, λ) in λ.
Low frequency resolvent bounds via exponential dichotomies
As λ approaches the origin, it nears the boundary of the essential spectrum, leading to the loss of hyperbolicity in the asymptotic matrices A±(λ) This results in one spatial eigenvalue from both +∞ and −∞ approaching zero, with the eigenvalue satisfying ν ± (λ) = O(λ) More specifically, ν ± (λ) is expressed as −λ/c ± g + dλ² c³ g + O(λ²³), where c ± g is defined in (2.2) Notably, at +∞, there are three spatial eigenvalues that remain bounded away from the imaginary axis as λ approaches 0, with two having positive real parts and one a negative real part Conversely, at −∞, three spatial eigenvalues also stay bounded away from the imaginary axis, with two exhibiting negative real parts and one a positive real part, as illustrated in Figures 3b and 3c.
In the context of the positive half line x ≥ 0, we define ν + s (λ) as the spatial eigenvalue approaching +∞ with a negative real part that remains bounded away from zero for all λ > 0, indicating a strong-stable direction Conversely, ν + c (λ) represents the spatial eigenvalue from +∞ with a negative real part that behaves as O(λ) as λ approaches 0, signifying a center-stable direction Specifically, we have ν + s (λ) = −η + O(λ) for some η > 0, and ν + c (λ) = −λ/cg + O(λ²) as λ → 0 Utilizing the conjugation lemma, it can be shown that a basis of bounded solutions to equation (3.4) on the positive half line R + is formed by e ν + c (λ)x W + c (x, λ) and e ν s + (λ)x W + s (x, λ), where W + j (x, λ) converges exponentially fast to W ∞ j (λ) as x approaches +∞ for each j = c, s.
V 1,2 , defined in (2.7), correspond to solutionsW 1,2 of (3.4) for λ= 0 Since, x →±∞lim W 1 (x)
W 1,2 (x) =a c 1,2 e ν c + (0)x W + c (x,0) +a s 1,2 e ν s + (0)x W + s (x,0), x≥0, wherea c 1,2 6= 0 There are also corresponding solutions e − ν + j (λ)y Ψ j + (y, λ) of the adjoint equation associ- ated with (3.4),
For λ = 0, the adjoint eigenfunctions are denoted by Ψ 1,2 (x), and they are related to the adjoint eigenfunctions ψ 1,2 via (Ψ 1,2 ) 2 =D T 2 ψ 1,2 , where (Ψ 1,2 ) 2 denotes the second component of Ψ 1,2
The four-dimensional ordinary differential equation (ODE) associated with equation (3.1) incorporates a nonautonomous term linked to the Dirac delta function, resulting in the system described by (3.4) The exponential dichotomy, denoted as Φ s,u +, indicates that Φ s + (x, y, λ) and Φ u + (x, y, λ) decay exponentially to zero as x and y approach infinity, under specific conditions The resolvent kernel, which addresses (3.1) on the positive half-line, corresponds to the upper-left two-by-two block of Φ s,u + According to equation (3.9), Φ s + (x, y, λ) can be expressed as a combination of exponential terms and analytic functions in λ for the range 0 ≤ y ≤ x Similarly, it is established that Φ u + (x, y, λ) behaves as O(e − η | x − y | ) for 0 ≤ x ≤ y, with a corresponding construction for the negative half-line utilizing the unstable spatial eigenvalues ν − u,c (λ).
To extend the exponential dichotomy from the half line to the entire real line, we must acknowledge that this extension will not be analytic due to the presence of an eigenvalue at the origin, leading to a pole in the resolvent kernel However, it is possible to construct a meromorphic extension This approach will follow a strategy akin to those outlined in [BHSZ10,§4.4] and [BSZ10,§4.2].
And note that span{W 1 , W 2 }=E + s (0) Similarly, since span{W 1 , W 2 }=E − u (0) Next, set
E 0 pt := span{W1(0), W2(0)}, E 0 ψ := span{Ψ1(0),Ψ2(0)}, so that E 0 pt ⊕E 0 ψ = C 4 The following lemma is analogous to [BHSZ10, Lemma 4.10] and [BSZ10, Lemma 6], and more details of the proof can be found in those papers.
Lemma 3.4 establishes the existence of a small positive value ǫ, such that for each λ in the punctured ball B ǫ (0) excluding zero, there is a unique mapping h + (λ) from E + u (λ) to E + s (λ), satisfying E u − (λ) = graph h + (λ) This mapping can be expressed as h + (λ) = h + p (λ) + h + a (λ), where h + a is analytic for λ in B ǫ (0) excluding zero, and h + p (λ)W ψ = 1 λM W ψ, with bases {W1(0), W2(0)} and {Ψ1(0), Ψ2(0)}, where M is the inverse of M ψ (0) Similarly, for each λ in B ǫ (0) excluding zero, there exists a unique map h − (λ) from E − s (λ) to E u − (λ), ensuring that E + s (λ) = graph h − (λ) This mapping also has a meromorphic representation similar to h + (λ), with h − p (λ) = −h + p (λ).
In this proof, we utilize the coordinates (W pt, W ψ) within the space E 0 pt ⊕ E 0 ψ The exponential dichotomies Φ s ± (x, y, λ) and Φ u ± (x, y, λ) are analytic for all x ≥ y ≥ 0 and y ≥ x ≥ 0, respectively Consequently, there exist functions h ψ (λ) and g ψ,pt (λ) that are analytic for all λ close to zero.
The superscripts indicate the range of the associated function For example, g ψ (λ)W pt ∈ E 0 ψ for all
W pt ∈E pt 0 We want to writeE u − (λ) as the graph of a functionh + (λ) :E + u (λ)→E + s (λ) This requires
, where for each ˜W pt we need to write W pt in terms of W ψ so that the above equation holds In components,
W˜ pt =W pt +g pt (λ)W ψ , λh ψ (λ) ˜W pt =W ψ +λg ψ (λ)W pt , which implies λh ψ (λ)
=W ψ +λg ψ (λ)W pt Rearranging the terms in this equation, we find λ h ψ (λ)−g ψ (λ)
M ψ (λ) :=h ψ (λ)−g ψ (λ) :E pt 0 →E 0 ψ For the moment, assume thatM ψ (0) is invertible, with
The relationship between M pt (0) and M ψ (0) is defined by the equation M pt (0) := M ψ (0) − 1 : E 0 ψ → E 0 pt It is established that M ψ (λ) remains invertible for values of λ close to zero Consequently, we can express M pt (λ) as M pt (0) + λM˜ pt (λ), where M˜ pt (λ) is an analytic function of λ This formulation allows for the solution of the equation presented in (3.14).
= 1 λM pt (0)V ψ + [ ˜M pt (λ)(1−λh ψ (λ)g pt (λ))−M pt (0)h ψ (λ)g pt (λ)]W ψ +O(λ)W ψ
It only remains to justify (3.15) We have that M ψ (0) =h ψ (0)−g ψ (0), whereh ψ and g ψ represent the graphs ofE − u andE + s , respectively Following the same argument as in the proof of [BSZ10, Lemma
Recall that the first component of Wj(y) corresponds to the eigenfunctions Vj defined in (2.7), and (D − 2 1 ) T (Ψ j ) 2 corresponds to ψ j Thus, we find that
According to [SS04a, Corollary 4.6] and Hypothesis 2.1, the matrix M ψ (0) is proven to be invertible This invertibility indicates that the linearization around the source, when analyzed in a suitably weighted space, has zero as an eigenvalue with a multiplicity of two.
Following [BHSZ10, BSZ10], the meromorphic extension of the exponential dichotomy for x > y is then given by Φ(x, y, λ) :
We are now ready to derive pointwise bounds on the resolvent kernel forλ∈Bǫ(0).
Proof of Proposition 3.3 We give the details only for y ≤x; the analysis for x ≤y is similar Recall that the resolvent kernel Gis just the upper-left two-by-two block of Φ.
Case I: 0≤y≤x Due to (3.12) and (3.17), we have Φ(x, y, λ) =−Φ s + (x,0, λ)h + p (λ)Φ u + (0, y, λ) + e ν + c (λ)(x − y) W + c (x, λ)hΨ c + (y, λ),ãi+O(e − η | x − y | ).
By definition (3.13), we can expand the first term as Φ s + (x,0, λ)h + p (λ)Φ u + (0, y, λ) = Φ s + (x,0, λ)
By definition (3.11), we have Φ s + (x,0, λ)W + (0) = e ν c + (λ)x W + (x)(1 +O(λ)) +O(λe ν + s (λ)x ) = e ν + c (λ)x W + (x) +O(λe ν + c (λ)x ) for anyW+that is a linear combination ofW1andW2, where we have used the fact thathΨ s + (0,0), Wj(0)i0 Similarly, (3.18) implies that we also have Φ u + (0, y, λ) ∗ Ψ j (y) = Ψ j (y) +O(λe − η | y | ) Thus, we have Φ s + (x,0, λ)h + p (λ)Φ u + (0, y, λ) = 1 λ
We can also write e ν + c (λ)(x − y) W + c (x, λ)hΨ c + (y, λ),ãi= e ν + c (λ)(x − y) W + (x)hΨ + (y),ãi+O(λe ν + c (λ)(x − y) ), whereW + (x) =W + c (x,0) and Ψ + (y) = Ψ c + (y,0).
(3.19) whereV + (x) is a linear combination ofV 1 (x) andV 2 (x) This proves Proposition 3.3 in this case.
Case II: y ≤x≤0 A a result of (3.17), we obtain Φ(x, y, λ) = Φ u − (x,0, λ)h − p (λ)Φ s − (0, y, λ) +O(e ν − c (λ)x e − η | y | ) +O(e − η | x − y | ) with Φ u − (x,0, λ)h − p (λ)Φ s − (0, y, λ) = 1 λ
Using an argument similar to the previous case, by (3.18) we can write Φ u − (x,0, λ)W j (0) = e ν − c (λ)x W j (x)+
O(λe ν − c (λ)x ), and also Φ s − (0, y, λ) ∗ Ψ j (0) = Ψ j (y) +O(λe − η | y | ) Thus, we get
X2 j=1 e ν − c (λ)x V j (x)hψ j (y),ãi+O(e ν c − (λ)x e − η | y | + e − η | x − y | ), which yields the proposition fory≤x≤0.
Case III: y≤0≤x Again by (3.17), we write Φ(x, y, λ) = Φ s + (x,0, λ)Φ u − (0,0, λ)h − p (λ)Φ s − (0, y, λ) +O(e ν c + (λ)x e − η | y | )
Combining all the three cases yields Proposition 3.3.
Remark 3.5 If we denote byG R (x, y, λ) the first row in the matrixG(x, y, λ), thenG R (x, y, λ) has a better bound than that ofG(x, y, λ) byO(e − η | x | ) This is due to the structure ofV 1 (x) = (r x , rϕ x ) and
V2(x) = (0, r),sincerx=O(e − η | x | ) More precisely, we have
This last estimate means that the R-component of the Green’s function will decay faster than the φ-component by a factor of t − 1/2 , after subtracting the terms resulting from the eigenfunctions.
In this section, we derive pointwise bounds on the temporal Green’s function defined by
The integral representation of the resolvent kernel is given by Z Γ e λt G(x, y, λ)dλ, where Γ is a contour positioned outside the essential spectrum The kernel G(x, y, λ), as established in Section 3, is analytic in λ beyond the essential spectrum Consequently, the contour Γ can be defined as the boundary of the set Ω ϑ, as described in (3.2) and illustrated in Figure 2.
Lemma 4.1 If G(x, y, t) is defined in (4.1), then G(x, y, t) is the temporal Green’s function of ∂ t − L, where L is the linearized operator defined as in (2.5) In particular, the solution to the inhomogeneous system
(∂ t − L)U(x, t) =f(x, t) is given by the standard Duhamel formula
G(x, y, t−s)f(y, s) dsds, as long as the integrals on the right hand side are well-defined.
Proof By Propositions 3.1 and 3.2, G(x, y, λ) is uniformly bounded outside of the essential spectrum, and therefore e λt G(x, y, λ) is integrable by moving the contour Γ so that Γ =∂Ω ϑ for λlarge Hence,
G(x, y, t) is well-defined for x6=y and fort >0 A direct calculation then yields
This verifies thatG(x, y, t) is indeed the Green’s function.
The main result of this section is the following proposition, which contains pointwise bounds on the Green’s function This proposition will be proven in §4.1 and§4.2, below.
Proposition 4.1 The Green’s function G(x, y, t) defined as in (4.1)may be decomposed as
G(x, y, t) =e(x, t)V 1 (x)hψ 1 (y),ãi+e(x, t)V 2 (x)hψ 2 (y),ãi+Gb(x, y, t), where V1,2(x) are defined in (2.7), the adjoint functions satisfy ψ1,2(y) = O(e − η | y | ), e(x, t) is the sum of error functions defined in (1.5), and Gb(x, y, t) satisfies
In addition, the first row of Gb(x, y, t), denoted by GbR(x, y, t), satisfies the better bound
The proof, primarily detailed in [ZH98], involves key steps that are essential for completeness The central concept is to manipulate the contour Γ to achieve a minimization of the integral defined within the context.
The selection of the minimizing contour is determined by techniques such as the saddle point method, the method of stationary phase, or the method of steepest descents Since the integration is conducted solely with respect to the spectral parameter λ, we have the flexibility to choose the contour based on the variables (x, y, t).
We first split the contour Γ into Γ 1 S Γ 2 , with Γ 1 :=∂B(0, M)∩Ω ϑ , Γ 2 :=∂Ω ϑ \B(0, M), (4.2) for some appropriate M to be determined below We consider two cases: when |x−y|/tis sufficiently large and when it is bounded.
Figure 4: Illustration of the contoursΓ 1 and Γ 2
Large | x − y | /t
Setz:=η 2 |x−y|/(2t) andM := (z/η) 2 Since M is sufficiently large, we can apply Proposition 3.1 so that
Letλ 0 andλ ∗ 0 be the two points of intersection of Γ 1 and Γ 2 We observe that, after possibly makingη smaller, for all λ∈Γ 1 we have
Also, on Γ2 we have Reλ = −ϑ(1 +|Imλ|) (choosing ϑ = min{ϑ1, ϑ2} and deforming the contour slightly) and so
2 t η 2 Combining these last two estimates and noting thatz= η 2 | x 2t − y | is large and η is small, we have
2 ≤Ct − 1/2 e − η 2 ( | x − y | +t) forη >0 independent of the amplitude of | x − t y | , as long as it is sufficiently large.
Bounded | x − y | /t
We now turn to the critical case where |x−y|/t≤S for some fixed S We first deform Γ into Γ 1 S Γ 2 as in (4.2) with M now being sufficiently small.
The integral over Γ 2 is relatively straightforward Again letλ 0 ,λ ∗ 0 be the points where Γ 1 meets Γ 2
We have Reλ 0 =−ϑ(1+ ˜ϑ) and Imλ 0 = ˜ϑfor some ˜ϑ >0 Moreover, on Γ 2 we have|G(x, y, λ)| ≤C|λ| − 1 2 by the mid- and high-frequency resolvent bounds Thus, we can estimate
Z ∞ θ |Imλ| − 1 2 e − ϑ[ | Im λ |−| Im λ 0 | ]t |d Imλ| ≤Ct − 1 2 e − ϑt
Noting that |x−y| ≤St, we have
To estimate the integral over Γ1, we ensure that M and ϑ are sufficiently small to keep Γ2 outside the essential spectrum, allowing the λ-expansion for the low-frequency resolvent kernel G(x, y, λ) to be applicable We analyze various scenarios based on the locations of x and y.
In the case where \(0 \leq y \leq x\), we analyze the expansion given by equation (3.19) According to equation (3.8), we find that \(\nu + c(\lambda) = -\lambda c_g + d\lambda^2 c_3 g + O(\lambda^3)\), where \(d\) and \(c_g\) are positive constants defined in equations (1.3) and (2.2), respectively To estimate the contribution from the term \(O(\lambda^q e^{\nu + c(\lambda)}(x - y))\), we define \(z_1 := x - y - c_g t\).
Then λ = z 1 /z 2 minimizes ν + c (λ)(x−y) when λ is real Define Γ 1 to be the portion contained in
−1 c g Re(λ−λ 2 d/c 2 g )≡ −1 c g (λ min −λ 2 min d/c 2 g ) whereλ min is defined by z 1 /z 2 if|z 1 /z 2 | ≤ǫand by ±ǫifz 1 /z 2 ≷ǫ, forǫsmall.
With these definitions, we readily obtain that
For λ ∈ Γ1, the inequality (z1^2 t/2z2) - ηIm(λ)²t ≤ -z1²t/C0 - ηIm(λ)²t holds, indicating that z2 is bounded above It is essential to recognize that z1 effectively controls the term (|x| + |y|)/t, which is crucial in limiting the error term O(λ³)(|x| + |y|)/t that arises from the expansion Consequently, this leads to significant results for any q.
Thus, the contribution fromO(λ q e ν + c (λ)(x − y) ) to the Green function bounds is
Clearly, the termO(e − η(x − y) ) in Φ s (x, y, λ) contributes a time- and space-exponential decay: O(e − η( | x − y | +t) ). Thus, we are left with the term involving λ − 1 Precisely, consider the term
Figure 5: The contour Γ 1 is deformed into three straight lines.
Thus, by the Cauchy’s theorem, we can move the contour Γ 1 (as shown in Figure 5) to obtain α(x, t) = 1
2Residue λ=0 e λt e ( − λ/c g +dλ 2 /c 3 g )x , for someη >0 Rearranging and evaluating the residue term, one then has α(x, t) 1 2πP.V.
Note that the first term in α(x, t) can be explicitly evaluated, again by the Cauchy’s theorem and the standard dominated convergence theorem, as
, which is conveniently simplified to
, (4.3) plus a time-exponentially small error The second and third terms are clearly bounded by Ce − η | x | for η sufficiently small relative to r, and thus time-exponentially small for t≤C|x| In the case t≥C|x|,
For sufficiently large \( C > 0 \), we can adjust the contour to the interval \([-η - ir, -η + ir]\), yielding a complete residue of 1 along with a negligible time-exponentially small error from the shifted contour integral This result can again be represented using the error function (4.3) plus a minor time-exponentially small error Specifically, for \( t \geq C|x| \) and large \( C \), one can derive an estimation.
Thus, we have obtained for 0≤y≤x:
(4.4) in which we recall that V + (x) belongs to the span ofV 1 (x) and V 2 (x) We note that the errfn in (4.4) may be rewritten as errfn
√4dt plus error errfn −x+c g t p4d|x/cg|
=O(t − 1 e (x − c g t) 2 /M t ), forM >0 sufficiently large Note also that errfn
√ 4dt is time-exponentially small sincex, t, c g are all positive Thus, an equivalent expression forG(x, y, t) is
The first row of the Green function matrix G(x, y, t) exhibits a superior bound at the leading term due to the unique structure of V j (x), as highlighted in equation (2.7).
Case II: y ≤x≤0 In this case we recall that
X2 j=1 hW j (y),ãie ν − c (λ)x V j (x), where ν − c (λ) =− λ c − g + dλ 2 (c − g ) 3 +O(λ 3 ), withc − g =−c g 0 as in Lemma 2.1 The main result of this section is the following proposition, whose proof will be given in Section 5.4.
Proposition 5.2 Let δ ± =δ ± (t) be arbitrary smooth functions The functionU a = (R a , rφ a ) defined in (5.14) solves
Here L, T(p) and Q(R, φ, p) are defined in (5.4), (5.6), and (5.7), respectively, Υ( ˆφ a x , p x ) is defined in (5.15), andΣ ± (x, t, δ ± ) are defined by Σ ± (x, t, δ ± ) : = ∂U a
We now chose the function B(x, t) used in (5.9) to be
It is important to note that while L B B does not equal zero, the equation is asymptotically satisfied in a relevant manner This indicates that the function B remains adequate for our Ansatz, ensuring the validity of Proposition 5.3, which will be demonstrated in §5.5.
Proposition 5.3 Let δ ± = δ ± (t) be arbitrary smooth functions and let B(x, t) defined as in (5.18). The approximate solution U a constructed as in (5.14)satisfies
(1 +δ ± )E ± (x, t) +R app 1 (x, t, δ ± ), where E ± (x, t) is defined by
(5.19) and the remainder R app 1 (x, t, δ ± ) satisfies
+C(|δ + |+|δ − |)(|δ˙ + |+|δ˙ − |)(1 +t) 1/2 θ(x, t), (5.20) for η 0 >0 as in Lemma 2.1 Here θ(x, t) denotes the Gaussian-like behavior (as in (1.6)): θ(x, t) = 1
The following lemma is relatively straightforward, but crucial to our analysis later on.
Lemma 5.2 Let E ± (x, t) be defined as in (5.19) Then
Proof of Lemma 5.1
In this subsection, we shall prove Lemma 5.1 First, we obtain the following simple lemma.
Lemma 5.3 Let p(x, t) be a given smooth function so that the map (x, t) 7→ (ξ(x, t), τ(x, t)), defined by ξ =x+p(x, t) and τ =t, is invertible If A(x, t) solves the qCGL equation (5.1), then the function
B(x, t) :=A(x+p(x, t), t) solves the modified qCGL equation
(1 +px) 2 (1 +iα)B xx (5.22) Proof Write B(x, t) =A(ξ(x, t), t) Then
(1 +p x ) 3 Inserting these expressions into the equation
Next, we write the solution to the new qCGL equation (5.21) in the amplitude and phase variables
B(x, t) = [r(x) +R(x, t)]ei(ϕ(x)+φ(x,t))e − iω 0 t , with (R, φ) denoting the perturbation variables As in Section 2.2, but now keeping all nonlinear terms, we collect the real and imaginary parts of the equations for R and φto find
The functionTj(p),j =R, φ, denotes the terms resulting fromT(p, B) that are linear inp The function
Q j (R, φ, p) collects terms that are quadratic in (R, φ, p), andN j (R, φ, p) denotes the remaining terms.
We now calculate these functions in detail.
First, note that (5.22) can be written
The expression T(p, B) is defined as T(p, B) = p t (1−p x )B x −(1 +iα)p xx B x −(1 +iα)p x (2−3p x )B xx +O(p 3 x +p 2 x p t +p x p xx ), valid under the condition |p x | 0, ensuring that for any initial data A_in with ǫ defined as kA_in(ã) - A_source(ã)kin ≤ ǫ₀, and for any κ in the range (0, 1/2), there are positive constants η, C₀, and M₀ This leads to the inequalities h₁(t) ≤ C₀(ǫ + h(t)²) and h₂(t) ≤ C₀(ǫ + h₁(t) + h(t)²) holding true for all t ≥ 0.
Using this proposition, we can add the inequalities in (6.18) and eliminateh 1 on the right-hand side to obtain h(t)≤C 0 (C 0 + 2)(ǫ+h(t) 2 ).
By leveraging the established inequality and the continuity of h(t), we can conclude that h(t) is bounded above by 2C₀(C₀ + 2)ǫ, given that 0 ≤ ǫ ≤ ǫ₀ is sufficiently small The proof of the main theorem hinges on the validation of Proposition 6.4, which will be the focus of the subsequent sections.
Bounds on the nonlinear terms
We recall that the nonlinear remainder N2( ˜R,φ, δ˜ ± )(x, t) is defined by
N2( ˜R,φ, δ˜ ± )(x, t) =N1( ˜R,φ, δ˜ ± )(x, t)−X ± Θ ± (t)R ± (x, t). with Θ ± (s) defined in (6.14), R ± (x, t) defined in (6.8), and N1( ˜R,φ, δ˜ ± ) defined in (6.3) We first note that
Z t 0 e − ηs h 1 (s)ds≤C(ǫ+h 1 (t)), where we recall that Lemma 6.3 implies thatδ ± 0 ∼ǫ By (5.20) and the definition of h1(t)
Here we leave the linear term inδ ± (t) in the above estimate for a different treatment, below Similarly, by definitions (6.14) and (6.8), we get
Next, from (6.4) and the definition of h 2 (t), we have
Recalling the estimate (6.3) forN1, we find that the nonlinear termN2( ˜R,φ, δ˜ ± )(x, t) satisfies
|N2( ˜R,φ, δ˜ ± )(x, t)| ≤Ch e − η | x | + (1 +t) − 1 i θ(x, t)(ǫ+|δ + (t)|+|δ − (t)|) +C(1 +t) − 1+κ θ(x, t)h 2 (t), (6.19) where we have used θ(x, t)≤2(1 +t) − κ and (1 +t) κ e − ηt ≤C(1 +t) − 1
Estimates for h 1 (t)
To establish the claimed estimate forh 1 (t), we differentiate the expression (6.16) to get δ˙ ± (t) = 2q d(1 +δ ± (t))
RhΨ ± (y),N2( ˜R,φ˜ y , δ ± )(y, t)idy (6.20) for each +/−case We first recall that|Ψ ± (y)| ≤2e − η 0 | y | and that e − η 0 2 |y| e −
M ≤C 1 e − η 0 4 |y| e − ηt , which holds for eachM ≥8cg/η0 and η sufficiently small This, together with the bound (6.19) on the nonlinear termN2( ˜R,φ˜ y , δ ± ), implies
Now, multiplying the equation (6.20) byδ ± and using the above estimate on the integral, we get d dt
, where we have used Young’s inequality and the fact that, as long as δ ± is bounded, higher powers of δ ± can be bounded by C|δ ± | 2 Applying the standard Gronwall’s inequality, we get
0 e − ηs dsis bounded, |δ ± (0)| ≤Cǫ, andh(t) is an increasing function, the above estimate yields
Using this into (6.22) and in (6.20), we immediately obtain
, which yields the first inequality in (6.18).
Finally, not that if we combine the bound (6.23) with the nonlinear estimate (6.19), we obtain
Pointwise estimates for ˜ R and r φ ˜
Let ˜U = ( ˜R, rφ) satisfy the integral formulation (6.17) We shall establish the following pointwise˜ bounds
(1 +t) − 1/2+κ θ(x, t) (6.26) fork+ℓ≤3 with ℓ= 0,1 andk= 1,2,3 We recall the integral formulation (6.17):
We give estimates for each term in this expression First, we consider the integral term in (6.27) that involves the initial data We recall that
Using this bound, together with equation (6.11), we see that
Mdy (6.29) Using the fact that, fort≥1, e − (x−y±c 4t g t)2 e − y
R t − 1/2 e − (x−y+cgt)2 8t + e − (x−y−cg 8t t)2 dy ≤ C 1 , we conclude that the integral in (6.29) is again bounded by ǫC 1 θ(x, t).
Now, we note that if we project the Green’s function on the R-component, say GeR(x, y, t), we get a better bound than that of (6.28); see (4.1) More precisely, we find
Using this better bound on the R-component, the above argument shows that
Next, for the second term in (6.27), we write
By the nonlinear estimates in (6.24), we have
This lemma is specifically designed to provide convolution estimates for the right-hand sides of the aforementioned inequalities, leading to the desired bound for the second term in equation (6.27) The proof of this lemma can be found in the Appendix.
Lemma 6.5 For some C andM sufficiently large,
∂ x k Ge(x, y, t−s)h e − η | y | + (1 +s) − 1+κ i θ(y, s) dyds ≤ C(1 +t) − k 2 +κ θ(x, t), for k = 0,1,2,3 In addition, similar estimates hold for ΠRGe(x, y, t), with a gain of an extra factor
Next, we consider the last integral term
E ± (x, t−s)− E ± (x, t) hΨ ± (y),N2( ˜R,φ˜ y , δ ± )(y, s)idyds (6.31) in (6.27) Due to the bounds (6.21) and (6.23), we get
Also, due to (6.5) and the fact thatB(x, t) =e(x, t+ 1) andBx(x, t) =O(θ(x, t)), we have
Again note that the R-component of E ± (x, t) is bounded by
The estimate (6.32) is enhanced by either (1 + t) − 1/2 or e − η | x | when applied to the R-component It is important to note that e − η | x | θ(x, t) decays exponentially over time and space Therefore, it is sufficient to demonstrate that the integral on the right side of (6.32) is limited by Cθ(x, t) The proof of this lemma will be provided in the appendix.
Lemma 6.6 For each sufficiently large M, there is a constantC so that
In conclusion, by aggregating all the estimates into equation (6.27), we successfully derived the desired estimate (6.25) for k = 0 The process for obtaining estimates of the derivatives mirrors the method outlined previously, incorporating a time decay factor; we will forgo the detailed proof.
Estimates for h 2 (t)
In this section, we validate the estimate for h2(t) as stated in Proposition 6.4 While the estimates in (6.25) nearly confirm the desired inequality, there remains an issue near the core at x = 0, where r(0) = 0 According to Lemma 2.1, we can proceed under the assumption that positive constants a and b exist, specifically a, b > 1, 2.
Away from the core |x| ≥1 In this case, the second estimate in (6.25), (6.26), and the fact that
The term e − η | x | θ(x, t) can be bounded by Ce − η( | x | +t), which is negligible Additionally, we can express φ˜ t in terms of ( ˜R,φ) and their spatial derivatives, leading to the bound ˜ rφ˜ t ≤ C(ǫ+h(t)²)(1+t) − 1/2 θ(x, t) The factor (1+t) − 1/2 arises because the right-hand side of equation (6.2) does not include ˜φ, which has the slowest decay in the equation Consequently, the estimate for ˜φ t is established for |x| ≥ 1.
Near the core |x| ≤1 The second estimate in (6.26) with k= 2 gives
(r 2 φ˜ x ) x =r(2r x φ˜ x +rφ˜ xx ) =r((rφ)˜ xx −r xx φ),˜ by integration together with (6.33) we have
Here we note thatrφ(x, t) is finite, and so˜ r 2 φ˜vanishes atx= 0 sincer(0) = 0 Again by the estimate
|r(x)| ≥b 1 |x|from (6.33), the above estimate yields
In addition, if we write rxφ˜= (rφ)˜ x−rφ˜x and use the above estimate together with (6.25), we obtain the claimed estimate for ˜φat once thanks to the assumption that |r x | ≥b 1 >0.
Similarly, let us check the claimed estimate for ˜φ t As above, we write
(rφ)˜ xxt −r xx r (rφ)˜ t and note that, by (6.26), (rφ)˜ xxt and (rφ)˜ t are already bounded by C(ǫ+h(t) 2 )e − ηt It thus follows similarly to (6.34) that
To obtain the desired estimate for ˜φ t near the core, we utilize the equation r x φ˜ t = (rφ)˜ xt − rφ˜ xt, referencing equation (6.26) Next, we address the estimate for ˜φxx We note that we also possess an estimate for rφ˜xx for all x, expressed as rφ˜ xx = (rφ)˜ xx − 2r x φ˜ x − r xx φ, as shown in equation (6.35) To estimate ˜φ xx for x near zero, we can proceed with the derived expressions.
To analyze the expression (rφ)˜ xxx −3r xx φ˜ x −r xxx φ˜i, we integrate the identity from 0 to x According to equation (6.35), rφ˜ xx is finite at x=0, leading to the conclusion that 3 φ˜ xx vanishes at this point By applying the estimates from (6.26) with k=3, along with the estimates on ˜φ x and ˜φ, we derive significant results.
Again, since|r(x)| ≥b 1 |x|, we then obtain
This completes the proof of the claimed estimate h 2 (t) The key proposition (Proposition 6.4) is therefore proved, and so is the main theorem.
In this section, we prove the convolution estimates that we used in the previous sections These estimates can also be found in [BNSZ12].
Proof of Lemma 6.5 Let us recall that
Let us start with a proof of the first estimate in (A.2) We first note that there are constants
C˜ 1 e − y 2 /M ≤ |θ(y, s)| ≤ C 1 e − y 2 /M for all 0≤s≤1 Thus, for some constantC 1 that may change from line to line, we have
C˜ 1 θ(x, t) for all 0≤t≤1 Next, we write the first estimate in (A.2) as θ(x, t) − 1
R|Ge(x, y, t−s)|(1 +s) − 1+κ θ(y, s)dyds fort≥1 Combining only the exponentials in this expression, we obtain terms that can be bounded by exp
To estimate the expression in (A.3) where α j = ±c g, we follow the method outlined in [HZ06, Proof of Lemma 7] by completing the square for the last two exponents This approach allows us to present the results in a more generalized format.
2 and conclude that the exponent in (A.3) is of the form
The maximum value of the quadratic polynomial αx² + βx + γ can be determined using the formula -β²/(4α) + γ By applying this principle, we can conclude that the sum of the first two terms in equation (A.4), which depend solely on x and not on y, is less than or equal to zero Consequently, we can derive the estimate by omitting this term, leading to the expression exp.
2! forδ j =±1 Using this result, we can now estimate the integral (A.2) Indeed, we have θ(x, t) − 1
The first estimate in (A.2) is established, demonstrating that the expression ≤ C 1 (1 + t) − κ + C 1 is bounded due to κ being greater than zero The second estimate mirrors this process, utilizing the refined estimate (A.1) for GeR Additionally, derivative estimates can be derived in a similar manner, although further details are omitted.
Finally, it remains to show that
M s dyds ≤ Cθ(x, t), (A.6) where the Green function bounds read
, for |y| ≤1 The estimate (A.6) is clear when 0≤t≤1 Let us consider the case t≥1 The proof of this estimate uses the following bound: e
M , for fixed constantB and for largeM This is a simpler version of (A.5); see also (A.7), below We thus have θ(x, t) − 1
≤ C(1 +t) 1/2 t − 1/2 +C(1 +t)e − c 2 g 2M t, which is bounded fort≥1 This proves the estimate (A.6), and completes the proof of Lemma 6.5.
Proof of Lemma 6.6 We need to show that
The integral is expected to be small because the difference e(x, t−s)−e(x, t+1) approaches zero when s is within the interval [0, t/2] However, for values of s in the range [t/2, t], the exponential decay in s plays a significant role in the behavior of the integral.
√τ + 1 τ e − (x−c 8τ g τ)2 + e − (x+c 8τ g τ)2 dτ, where the last estimate follows by the fact thatze − z 2 is bounded for allz.
We shall give estimate for θ − 1 (x, t)(e(x, t−s+ 1)−e(x, t+ 1)) For instance, let us consider the single exponential term e
By combining these and completing the square inx, the terms in the exponential become
Since τ ≤t, if B is some fixed constant andM is sufficiently large we can neglect the exponential in x. That is, we have e
M , Using this and takingM large and η=c 2 g /M, we obtain θ(x, t) − 1
This proves the lemma for the case k = 0 The derivative estimates follow easily from the above proof.
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