Asymptotics for slowly converging evolution equations

Analyzing the behavior of solutions near singularities is essential to understand geometric equations such as minimal surfaces, harmonic maps and mean curvature flows. Many open problems reduce to questions on the singularity formation and its asymptotics. In pioneering works of Leon Simon [Sim83, Sim85], the idea of using Łojasiewicz gradient inequality [Łoj63] from real algebraic geometry was introduced for the first time, and the uniqueness of blow-ups was shown for a class of elliptic and parabolic equations. This uniqueness shows that the solution converges to the unique tangent cone or tangent flow as it approaches a singular point or infinity.

A natural subsequent question is to investigate the rate of convergence and the next order asymptotics that describes the difference between the solution and the limit. This often serves as a crucial starting point for further analysis. For example, recent progresses on the classification of ancient solutions to geometric flows [SAS20][DH21][BCS22][CM22] and complete non-compact solutions to minimal surface [SS86] are based on higher order asymptotics and its improvement. For the singularity formation in parabolic problem, the higher order asymptotics at a singularity gives structural results on the singularity set in a neighborhood [SX22][Gan21]. As the uniqueness of blow-ups implies the second differentiability of arrival time [CM15, CM18], the higher order asymptotics and the convergence rate have a strong relation to further regularity of arrival time [KS06][Ses08].

The convergence rate and direction (i.e., the secant at the limit) are mostly understood when the solution converges at an exponential rate. This is because when the solution decays exponentially, the equation is well-approximated by its linearization. However, without the integrability of the limit, it is possible to have solutions that converge algebraically slowly. In [AS88], Adams-Simon discovered a sufficient condition (later called the Adams-Simon positivity condition or simply $AS_{p}$ condition) on the limit so that they constructed a slowly converging solution to elliptic equations of the form (1.1). Carlotto-Chodosh-Rubinstein [CCR15] found explicit examples of critical points of normalized Yamabe functional with the $AS_{p}$ condition and constructed normalized Yamabe flows converging slowly. As questioned in [CCR15, p.1533], it is of great interest to understand the general behavior of slowly converging solutions. More precisely, one can ask whether such a solution must satisfy the Adams-Simon positivity condition and whether the higher order asymptotics follows the ansatz used in the construction of [AS88] and [CCR15].

We answer this question by showing that the Adams-Simon ‘non-negativity’ condition is necessary for slowly converging solutions to exist. Moreover, when a positivity is satisfied, the convergence rate and the higher asymptotics agree with those of previously constructed examples. (See Theorem 2.3 and 2.5 for detailed statements.) We concern with the elliptic and parabolic equations of forms

(1.1)

$$u^{\prime\prime}-mu^{\prime}+\mathcal{M}_{\Sigma}u=N_{1}(u),$$

and

(1.2)

$$u^{\prime}-\mathcal{M}_{\Sigma}u=N_{2}(u).$$

Here, $\mathcal{M}_{\Sigma}$ is the Euler-Lagrange operator of an analytic functional $\mathcal{F}_{\Sigma}$ as in (2.2)-(2.3), $m$ is a nonzero constant, and we assume $N_{1}(u)$ and $N_{2}(u)$ satisfy the structure (2.6). The presence of nonzero constant $m$ accounts that the equation (1.1) appears in the study of the isolated singularity or the asymptotics near the infinity for geometric variational problems. In particular, minimal surfaces, harmonic maps [Sim83, Sim85, AS88] and $G_{2}$ manifolds [Che18] fall into this category. Equation (1.2) models a geometric flow which converges to a stationary solution when $t$ goes to $+\infty$ or $-\infty.$ For instance, the (rescaled) mean curvature flow (see Appendix B) and the harmonic map heat flow [CM22] can be described by (1.2). Our results also apply to the normalized Yamabe flow. See Remark 2.9 (3) for more details.

The main results of the paper, Theorems 2.3-2.6, can be placed in line as a generalization of Thom’s gradient conjecture which we discuss in this paragraph. Let $x=x(t)\in\mathbb{R}^{n}$ be a solution to the gradient flow $x^{\prime}=-\nabla f(x)$ with an analytic potential $f:\mathbb{R}^{n}\to\mathbb{R}$ and suppose $x$ converges to the origin as $t\to\infty$. In [Tho89], Thom conjectured that the secant ${x}/{|x|}$ should converge to a direction $\theta_{0}\in\mathbb{S}^{n-1}$ as $t\to\infty$. Kurdyka-Mostowski-Parusiński [KKP00] settled this conjecture by showing a stronger result that the secant has a finite length in $\mathbb{S}^{n-1}$. However, the precise convergence rate is not yet completed revealed. If $\theta_{0}$ does not belong to $\ker\,(\nabla^{2}f(0))$, one obtains that $\theta_{0}$ has to be an eigenvector of the hessian for a positive eigenvalue $\lambda$, and the solution decays with higher order asymptotics $|x(t)-ce^{-\lambda t}\theta_{0}|=o(e^{-\lambda t})$ for some $c\in\mathbb{R}$. If $\theta_{0}$ belongs to the kernel of hessian, however, we merely know that the convergence takes place between algebraic rates $t^{-\alpha_{1}}\lesssim|x|\lesssim t^{-\alpha_{2}}$ for some $0<\alpha_{2}\leq\alpha_{1}\leq 1$. It is unknown if each solution has a specific convergence rate $|x|\sim t^{-\gamma}$ some $\gamma>0$.

Let us briefly summarize the main motive and results of the paper. In the slow decaying regime, the equation becomes well-approximated by a gradient flow on the kernel of $\mathcal{L}_{\Sigma}$, the linearization of $\mathcal{M}_{\Sigma}$ at $u\equiv 0$. The potential $f:\ker\mathcal{L}_{\Sigma}\approx\mathbb{R}^{J}\to\mathbb{R}$ is given by the pull-back of $\mathcal{F}_{\Sigma}$ through Lyapunov-Schmidt reduction (see Proposition 2.1). Near the origin, the flow dynamics shall be determined by the first non-constant homogeneous polynomial appearing in the expansion of potential $f$. Let us denote this polynomial by $f_{p}$. We reserve $p\geq 3$ to denote the degree of this polynomial and we call it the order of integrability. When the gradient flow $x^{\prime}=-\nabla f_{p}(x)$ is considered, one readily finds a radial solution converging to the origin whenever there exists a critical point of $\hat{f}_{p}:=f_{p}|_{\mathbb{S}^{J-1}}$ with positive critical value. The slowly decaying solutions of [AS88] and [CCR15] are obtained as perturbations of such radial ansatzes.

In Theorem 2.3 and 2.5, we prove slowly decaying solution $u(\omega,t)$ has a dichotomy on its convergence rate that either $t^{\frac{1}{p-2}}\|u(\cdot,t)\|_{L^{2}}$ converges to a positive number or diverges to infinity. In the first case, we further show $u(\cdot,t)/\|u(\cdot,t)\|_{L^{2}}$ converges smoothly to a critical point of $\hat{f}_{p}$, say $v\in\ker\mathcal{L}_{\Sigma}$, with positive critical value. This shows the secant $u(\cdot,t)/\|u(\cdot,t)\|_{L^{2}}$ has a limit and confirms Thom’s gradient conjecture for the concerned case. Note that one may interpret the theorem as the higher order asymptotics $u(t)=ct^{-\frac{1}{p-2}}v+o(t^{-\frac{1}{p-2}})$, where $c$ is a constant of $p$ and $f_{p}(v)$. Next, the second alternative shows a solution possibly decays even at a slower rate than $t^{-\frac{1}{p-2}}$. As seen in the classical gradient flow of potential $f=x_{1}^{4}+x_{2}^{8}$ on $\mathbb{R}^{2}$ (here, $p=4$ but a solution may decay at rate $t^{-\frac{1}{8-2}}$), the second alternative can actually take place. We show this can only occur if $\hat{f}_{p}$ admits critical point(s) of zero critical value and the secant $u(\cdot,t)/\|u(\cdot,t)\|_{L^{2}}$ accumulates on those critical point(s) as $t\to\infty$. It remains open whether the secant $u(\cdot,t)/\|u(\cdot,t)\|_{L^{2}}$ has a unique limit in the second case, and thereby, Thom’s conjecture holds for all slowly decaying solutions. We would like to point out that Theorem 2.5 is also novel for finite-dimensional gradient flows. For a finite-dimensional gradient flow given by $x^{\prime}=-\nabla f(x)$, the theorem states that there exists an integer $p\geq 3$ which depends only on $f$, such that any slowly converging solution satisfies either $t^{\frac{1}{p-2}}|x|=c(1+o(1))$ for some $c\in(0,\infty)$, or $t^{\frac{1}{p-2}}|x|\to\infty$.

In Theorem 2.4 and 2.6, we show the higher order asymptotic behavior of fast (exponentially) decaying solutions for elliptic and parabolic equations, respectively. This type of result has been expected among researchers and has been obtained for specific problems such as [SS86][CM22]. Nevertheless, we could not find proper literature covering the general forms (1.1) and (1.2), so we provide a proof in Section 4. The elliptic equation (1.1) can be viewed as a perturbation of the second order ODE $u^{\prime\prime}-mu^{\prime}+\mathcal{L}_{\Sigma}u=0$. Suppose $\mathcal{L}_{\Sigma}$ has an eigenvalue larger than $4^{-1}m^{2}$, a solution might oscillate while decaying exponentially. This type of solutions has been constructed for minimal graphs over Simons cones in $\mathbb{R}^{4}$ and $\mathbb{R}^{6}$. See [CP18, Remark 1.21] and [BDGG69] for more details. For this reason, the original form of Thom’s gradient conjecture is not true in the elliptic problem. Moreover, if $4^{-1}m^{2}$ is an eigenvalue of $\mathcal{L}_{\Sigma}$ and $m<0$, a resonance might occur and result in a solution that decays at a rate $te^{2^{-1}mt}$. Note [HS85] considered this possibility for minimal graphs over stable (but not strictly stable) minimal cones.

This paper is organized as follows. In Section 2, we introduce the notation, condition, spectral property of linearized operator $\mathcal{L}_{\Sigma}$, and main theorems. In Section 3, we set up the elliptic problem (1.1) as a first order ODE system on function spaces. In Section 4, Theorem 2.4 and Theorem 2.6 for exponentially decaying solutions to elliptic and parabolic equations are proved, respectively. In Section 5, we show in Proposition 5.1 that slowly converging solutions to (1.1) are governed by a finite-dimensional gradient flow with a small perturbation. The parabolic analogue, Proposition 6.1, is proved in Section 6. In Section 7, we complete the proofs for Theorems 2.3 and 2.5 by analyzing a finite-dimensional gradient flow with a small perturbation, using a version of the Łojasiewicz argument motivated by [KKP00]. Appendix A contains auxiliary tools we need in the paper. In Appendix B, we show that (rescaled) mean curvature flows can be written in the form (1.2).

Let us introduce our setting to study (1.1) and (1.2) and state the main results. Let $\Sigma$ be a closed $n$-dimensional Riemannian manifold and $d\mu$ be a smooth volume form on $\Sigma$ which is mutually absolutely continuous with respect to the volume form induced by the metric. Let $\mathbf{V}\to\Sigma$ be a smooth vector bundle equipped with a smooth inner product $\left\langle\cdot,\cdot\right\rangle$. For $u\in\mathbf{V}$, we write $|u|^{2}:=\left\langle u,u\right\rangle$. For $\omega\in\Sigma$, we denote by $\mathbf{V}_{\omega}$ the fiber over $\omega$. Let $\not{\nabla}$ be a connection on $\mathbf{V}$ which is compatible with the inner product.

We denote by $L^{2}(\Sigma;\mathbf{V})$ the space of $L^{2}$-sections of $\mathbf{V}$ with respect to $d\mu$. Namely, a section $u$ belongs to $L^{2}(\Sigma;\mathbf{V})$ provided

$\displaystyle\|u\|^{2}_{L^{2}}:=\int_{\Sigma}|u|^{2}\,d\mu<\infty.$

For $\ell\in\mathbb{N}_{0}$, we denote by $H^{\ell}(\Sigma;\mathbf{V})$ the collection of sections that satisfies

$\displaystyle\|u\|^{2}_{H^{\ell}}:=\sum_{i=0}^{\ell}\int_{\Sigma}\left|\not{\nabla}^{i}u\right|^{2}\,d\mu<\infty.$

For $-\infty<a<b\leq\infty$, we define $Q_{a,b}:=\Sigma\times[a,b)$ and equip $Q_{a,b}$ with the product metric. We denote by $\widetilde{\mathbf{V}}$ the pull back bundle of $\mathbf{V}$ through the projection $Q_{a,b}\to\Sigma$. For $u\in C^{s}(Q_{a,b};\widetilde{\mathbf{V}})$ and $t\in[a,b)$, $u(t)\in C^{s}(\Sigma;\mathbf{V})$ is defined through $u(t)(\omega):=u(\omega,t)$. Its $H^{\ell}$-norm is denoted by

$$\|u\|_{H^{\ell}}(t):=\|u(t)\|_{H^{\ell}}.$$

Also,

(2.1)

$\displaystyle\|u\|_{C^{s}}(t):=\sup_{\omega\in\Sigma}\sup_{k+\ell\leq s}\left|\frac{\partial^{k}}{\partial t^{k}}\not{\nabla}^{\ell}u(\omega,t)\right|.$

Here, note that the norms of time derivatives are included in the definition of $\|\cdot\|_{C^{s}}(t)$. We often use $u^{\prime}$ as an abbreviation of $\frac{\partial u}{\partial t}$ .

Our assumption on $\mathcal{M}_{\Sigma}$ is almost identical to the one in [Sim83]. The only difference is that the volume form, $d\mu$, is not necessarily the one induced from the Riemannian metric. Let $\mathcal{F}_{\Sigma}$ be a functional defined for $u\in C^{1}(\Sigma;\mathbf{V})$ by

(2.2)

$\displaystyle\mathcal{F}_{\Sigma}(u):=\int_{\Sigma}F(\omega,u,\not{\nabla}u)\,d\mu.$

Here the integrand $F$ satisfies the following:

1. (1)

   $F=F(\omega,z,p)$ is a smooth function defined on an open set of $\mathbf{V}\times(T\Sigma\otimes\mathbf{V})$ that contains the zero section.

2. (2)

   For each $\omega\in\Sigma$, $F(\omega,\cdot,\cdot)$ is analytic on $\mathbf{V}_{\omega}\times(T_{\omega}\Sigma\otimes\mathbf{V}_{\omega})$.

3. (3)

   $F$ satisfies the Legendre-Hadamard ellipticity condition

   $\displaystyle D^{2}_{p}F(\omega,0,0)[\eta\otimes\xi,\eta\otimes\xi]\geq c|\eta|^{2}|\xi|^{2},$

   for $c>0$ independent of $\omega\in\Sigma$, $\eta\in T_{\omega}\Sigma$ and $\xi\in\mathbf{V}_{\omega}$.

$\mathcal{M}_{\Sigma}$ is defined to be the negative Euler-Lagrange operator of $\mathcal{F}_{\Sigma}$. Namely, for any $\zeta\in C^{\infty}(\Sigma;\mathbf{V})$,

(2.3)

$\displaystyle\int_{\Sigma}\left\langle\mathcal{M}_{\Sigma}(u),\zeta\right\rangle\,d\mu=-\frac{d}{ds}\mathcal{F}_{\Sigma}(u+s\zeta)|_{s=0}.$

We further assume $0$ is a critical point of $\mathcal{F}_{\Sigma}$. Namely, we assume $\mathcal{M}_{\Sigma}(0)=0$. We denote by $\mathcal{L}_{\Sigma}$ the linearization of $\mathcal{M}_{\Sigma}$ at $0$. It is clear that the difference between $\mathcal{M}_{\Sigma}(u)$ and $\mathcal{L}_{\Sigma}u$ is a quadratic term of the form

(2.4)

$$\mathcal{M}_{\Sigma}u-\mathcal{L}_{\Sigma}u=\sum_{j=0}^{2}c_{j}\cdot\not{\nabla}^{j}u,$$

where $c_{j}=c_{j}(\omega,u,\not{\nabla}u)$ are smooth with $c_{j}(\omega,0,0)=0$. The Legendre-Hadamard condition implies $\mathcal{L}_{\Sigma}$ is elliptic. In particular, there exists a constant $C>0$ such that for any $u\in C^{2}(\Sigma;\mathbf{V})$,

(2.5)

$\displaystyle\int_{\Sigma}-\left\langle\mathcal{L}_{\Sigma}u,u\right\rangle+C|u|^{2}\,d\mu\approx\|u\|^{2}_{H^{1}}.$

Furthermore, $\mathcal{L}_{\Sigma}$ is self-adjoint with respect to $d\mu$. This can be seen from

$\displaystyle\int_{\Sigma}\left\langle\mathcal{L}_{\Sigma}u,v\right\rangle\,d\mu=-\frac{\partial^{2}}{\partial s_{1}\partial s_{2}}\mathcal{F}_{\Sigma}(s_{1}u+s_{2}v)\bigg{|}_{s_{1}=s_{2}=0}.$

We suppose $N_{1}(u)$ and $N_{2}(u)$ in (1.1) and (1.2) are of the form

(2.6)

$$\begin{split}N_{1}(u)=&a_{1}\cdot Du^{\prime}+a_{2}\cdot u^{\prime}+a_{3}\cdot\mathcal{M}_{\Sigma}(u),\\ N_{2}(u)=&b_{1}\cdot\mathcal{M}_{\Sigma}(u).\end{split}$$

Here $a_{i}=a_{i}(\omega,u,\not{\nabla}u,u^{\prime})$ and $b_{1}=b_{1}(\omega,u,\not{\nabla}u)$ are smooth with $a_{i}(\omega,0,0,0)=0$ and $b_{1}(\omega,0,0)=0$; $D=\left\{\frac{\partial}{\partial t},\not{\nabla}\right\}$.

Let $\lambda_{1}\geq\lambda_{2}\geq\dots$ be the eigenvalues of $\mathcal{L}_{\Sigma}$ and $\varphi_{1},\varphi_{2},\dots$ be the corresponding eigensections which form a complete orthonormal basis of $L^{2}(\Sigma;\mathbf{V})$. We separate $\mathbb{N}$ into four parts according to the eigenvalues.

(2.7)

$$\begin{split}I_{1}:=&\{i\in\mathbb{N}\,:\,\lambda_{i}>4^{-1}m^{2}\},\ I_{2}:=\{i\in\mathbb{N}\,:\,\lambda_{i}=4^{-1}m^{2}\},\\ I_{3}:=&\{i\in\mathbb{N}\,:\,\lambda_{i}=0\},\ I_{4}:=\mathbb{N}\setminus(I_{1}\cup I_{2}\cup I_{3}).\end{split}$$

Note that $I_{1}$, $I_{2}$ and $I_{3}$ are (possibly empty) finite sets and $\ker\mathcal{L}_{\Sigma}$ is spanned by $\{\varphi_{i}\}_{i\in I_{3}}$. Let $J$ be the cardinality of $I_{3}$, the dimension of $\ker\mathcal{L}_{\Sigma}$. This implies $I_{3}=\{\iota+1,\iota+2,\dots,\iota+J\}$ for some $\iota\in\mathbb{N}_{0}$.

We denote by $\Pi^{T}$ and $\Pi^{\perp}$ the orthogonal projection of $L^{2}(\Sigma;\mathbf{V})$ to $\ker\mathcal{L}_{\Sigma}$ and $\left(\ker\mathcal{L}_{\Sigma}\right)^{\perp}$ respectively. The following is a version of the implicit function theorem. See [Sim96, §3].

The function $f$ plays a crucial role in this paper and we will call it reduced functional. The reduced functional $f$ is real analytic with $\nabla f(0)=0$ and $\nabla^{2}f(0)=0$. We may view $f$ as an analytic function defined on an open ball in $\mathbb{R}^{J}$ through the identification

(2.8)

$\displaystyle(x_{1},x_{2},\dots,x^{J})\mapsto\sum_{j=1}^{J}x_{j}\varphi_{\iota+j}.$

Let us review the integrable condition. The kernel $\ker\mathcal{L}_{\Sigma}$ is called integrable if for any $v\in\ker\mathcal{L}_{\Sigma}$, there exists a family $\{v_{s}\}_{s\in(0,1)}\subset C^{2}(\Sigma;\mathbf{V})$ such that $v_{s}\to 0$ in $C^{2}(\Sigma;\mathbf{V})$, $\mathcal{M}_{\Sigma}(v_{s})\equiv 0$ and $\lim_{s\to 0}v_{s}/s=v$ in $L^{2}(\Sigma;\mathbf{V})$. It is well-known [AS88, Lemma 1] that $\ker\mathcal{L}_{\Sigma}$ is integrable if and only if the reduced functional $f$ is a constant. Moreover, if integrable condition is satisfied, any decaying solution to (1.1) or (1.2) decays exponentially [AS88, CCR15]. For this reason, whenever non-exponentially decaying solution is considered, the integrable condition should necessarily fails. Namely, the reduced functional $f$ is not a constant function. In particular, there exists an integer $p\geq 3$ such that

(2.9)

$$f=f(0)+\sum_{j\geq p}f_{j},$$

where $f_{j}$ are homogeneous polynomials with degree $j$ and $f_{p}\not\equiv 0$. This integer $p$ is called the order of integrability [CCR15].

As explained in Introduction, the gradient flow of $f_{p}$ has a dominant role in the asymptotic behavior. Let $\hat{f}_{p}$ be the restriction of $f_{p}$ on $\{w\in\ker\mathcal{L}_{\Sigma}\,:\,\|w\|_{L^{2}}=1\}\approx\mathbb{S}^{J-1}$. Consider the critical points of $\hat{f}_{p}$:

(2.10)

$$\mathbf{C}:=\left\{w\in\ker\mathcal{L}_{\Sigma}\,:\,\|w\|_{L^{2}}=1\ \textup{and}\ w\ \textup{is a critical point of}\ \hat{f}_{p}\right\}.$$

If $w\in\mathbf{C}$ satisfies $f_{p}(w)>0$, then one checks that

$$x(t)=[p(p-2)f_{p}(w)(t+c)]^{-\frac{1}{p-2}}w$$

becomes a radial solution to the flow $x^{\prime}=-\nabla f_{p}(x)$. The higher order asymptotics in Theorem 2.3 and 2.5 will be modeled on such solutions.

Let us state main results concerning the asymptotic behavior and the convergence rate of decaying solutions. We begin with the elliptic equation (1.1).

Next, we consider decaying solutions to the parabolic equation (1.2).

In fact, it is possible that no slowly converging solution exists even in the presence of non-integrable kernel. e.g., we refer [CCK21][CS20]. Theorem 2.3 and 2.5 provide a criterion for this non-existence result.

We finish this section with two lemmas related to the Lyapunov-Schmidt reduction map $H$ given in Proposition 2.1. We decompose $u$ as follows: let $u^{T}=\Pi^{T}u$ and define $\tilde{u}^{\perp}$ by

(2.11)

$\displaystyle u=u^{T}+H(u^{T})+\tilde{u}^{\perp}.$

Since $u^{T}\in\ker\mathcal{L}_{\Sigma}$, $u^{T}$ can be written as a linear combination of $\{\varphi_{\iota+j}\}_{1\leq j\leq J}$ as

(2.12)

$$u^{T}=\sum_{j=1}^{J}x_{j}\varphi_{\iota+j}.$$