Ancient curve shortening flow in the disc with mixed boundary condition

Variational problems subject to boundary constraints are ubiquitous in pure and applied mathematics and physics. One of the simplest such problems is to find and study paths of critical (e.g. minimal) length amongst those joining a given point \(o\) in some domain \(\Omega \) to its boundary \(\partial \Omega \). When \(\Omega \) is a Euclidean domain, such paths are, of course, straight linear arcs from \(o\) to \(\partial \Omega \) which meet \(\partial \Omega \) orthogonally.

While characterizing all such curves is a non-trivial problem in general (even for convex Euclidean domains, say), the “Dirichlet–Neumann geodesics” in the unit disc in \(\R ^2\) are easily found: when \(o\) is the origin, they are the radii; when \(o\) is not the origin, there are exactly two, and their union is the diameter through \(o\).

One useful tool for analyzing such variational problems is the (formal) gradient flow (a.k.a. steepest descent flow), which in this case is the “Dirichlet–Neuman curve shortening flow”; this equation evolves each point of a given curve with velocity equal to the curvature vector at that point, subject to holding one endpoint fixed at \(o\) with the other constrained to \(\partial \Omega \), which is met orthogonally.

While curve shortening flow is now well-studied under other boundary conditions — particularly the “periodic” (i.e. no-boundary) [2] , [3] , [4] , [7] , [9] , [11] , [12] , [13] , [15] , “Neumann–Neumann” (a.k.a. free boundary) [5] , [6] , [8] , [10] , [14] , [16] , [17] , [19] , [20] and “Dirichlet–Dirichlet” [1] , [15] conditions — we are aware of no literature considering the mixed “Dirichlet–Neumann” condition.

Our main result (inspired by [6] ) is the following classification of the convex ancient solutions which arise in the simple setting of the unit disc.

En route to proving Theorem 1.1 , we establish the following convergence result (cf. [1] , [11] , [12] , [13] , [17] ), which is of independent interest (see the proof of Lemma 3.1 ).

Though the curvature monotonicity hypothesis appears unnaturally restrictive in Theorem 1.2 , we note that some such additional condition is required to prevent the development of self-intersections at the Dirichlet endpoint (resulting in subsequent cusplike singularities). Moreover, as Theorem 1.2 demonstrates in case the Dirichlet endpoint lies on the boundary, collapsing singularities may form at the Dirichlet endpoint even when the flow remains embedded. It is not hard to see that this can also occur when the Dirichlet endpoint lies to the interior (as a limiting case of the flow forming a cusp singularity just after losing embeddedness, say).

Fix a point \(o=(-d,0)\in D\) in the unit disc \(D\subset \R ^2\), with \(d\in (0,1]\). Denote by \(C_\theta \subset D\) the circular arc which passes through \(o\) and meets the boundary of \(D\) orthogonally at \((\sin \theta ,\cos \theta )\); that is,

\[ C_\theta :=\{(x,y)\in D:(x-\xi )^2+(y-\eta )^2=r^2\}\,, \]

where, defining \(a:=\frac {1}{2}(d^{-1}+d)\),

\[ (\xi ,\eta ):=(\cos \theta ,\sin \theta )+r(-\sin \theta ,\cos \theta )\;\; \text { and}\;\; r:=\frac {1+d^2+2d\cos \theta }{2d\sin \theta }=\frac {a+\cos \theta }{\sin \theta }\,. \]

Consider also the circular arc \(\check C_{\theta }\subset D\) which is symmetric about the \(y\)-axis and meets \(\partial D\) orthogonally at \((\cos \theta ,\sin \theta )\). That is,

\[ \check C_\theta :=\{x^2+(y-\check \eta )^2=\check r^2\}\,, \]

where

\[ \check \eta :=\csc \theta \;\;\text {and}\;\;\check r:=\cot \theta \,. \]

Proof of Proposition 2.1 .

Next consider \(\{\mathrm {H}_t\}_{t\in (-\infty ,\infty )}\), the fundamental domain of the horizontally oriented hairclip solution to curve shortening flow centred at \(o\); that is,

\[ \mathrm {H}_t:=\{(x,y)\in [0,\infty )\times [0,\tfrac {\pi }{2}]:\sin (y)=\mathrm {e}^{t}\sinh (x+d)\}\,. \]

Figure 1. Some timeslices of (one period of) the “hairclip” solution.

Given any \(\lambda >0\), define \(\{\mathrm {H}^\lambda _t\}_{t\in (-\infty ,\infty )}\) by parabolically rescaling the hairclip by \(\lambda \). That is,

\[ \mathrm {H}^\lambda _t:=\lambda ^{-1}\mathrm {H}_{\lambda ^{2}t}=\{(x,y)\in [0,\infty )\times [0,\tfrac {\pi }{2\lambda }]:\sin (\lambda y)=\mathrm {e}^{\lambda ^2t}\sinh (\lambda (x+d))\}\,. \]

Observe that \(\{\mathrm {H}^\lambda _t\}_{t\in (-\infty ,\infty )}\) satisfies

\[ \frac {\kappa }{\cos \theta }=\lambda \tan (\lambda y)\;\;\text {and}\;\;\frac {\kappa }{\sin \theta }=\lambda \tanh (\lambda (x+d))\,, \]

where \(\theta \in [0,\frac {\pi }{2}]\) is the angle the tangent vector makes with the \(x\)-axis. From this we see, in particular, that \(\kappa \) is positive and monotone increasing with respect to arclength from \(o\).

Proof.

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Proposition 2.3 ( A priori estimates) . A priori estimates'); rotateIcon(this);">

Let \(\Gamma \subset D_+\) be a smooth, convex embedding of a closed interval with left endpoint \(o=(-d,0)\), \(d\in (0,1]\), and right endpoint meeting \(\partial D\) orthogonally, and suppose that the curvature \(\kappa \) of \(\Gamma \) increases monotonically with respect to arclength from \(o\). Denote by \(\underline \theta \) resp. \(\overline \theta \) the least resp. greatest value taken by the turning angle along \(\Gamma \) and by \(\overline \kappa =\kappa (\overline \theta )\) the greatest value taken by \(\kappa \).

The circle \(C_{\overline \theta }\) lies below \(\Gamma \). Thus,

\(\seteqnumber{0}{}{1}\)

\begin{equation} \label {eq:curvature lower bound} \overline \kappa \ge \frac {\sin \overline \theta }{a+\cos \overline \theta } \end{equation}

and

\(\seteqnumber{0}{}{2}\)

\begin{equation} \label {eq:gradient lower bound} \underline \theta \ge \mathrm {arccot}\left (\frac {1+a\cos \overline \theta }{b\sin \overline \theta }\right )\,, \end{equation}

where \(b:=\frac {1}{2}(d^{-1}-d)\) and we recall that \(a:=\frac {1}{2}(d^{-1}+d)\), with the right hand side taken to be zero in case \(d=1\).

Proof.

Suppose, to the contrary, that \(C_{\overline \theta }\) does not lie below \(\Gamma \). Then some point of \(\Gamma \) must lie strictly below \(C_{\overline \theta }\), and hence (since the endpoints of the two curves agree) upon translating \(C_{\overline \theta }\) downwards, the two curves will continue to intersect until some final moment, at which they must make first order contact at some interior point \(q\in \Gamma \). At this point, the curvature \(\kappa \) of \(\Gamma \) must be no less than \(1/r(\overline \theta )\) (the curvature of \(C_{\overline \theta }\)). But then, by the monotonicity of \(\kappa \), \(\kappa \) must exceed \(1/r(\overline \theta )\) on the whole segment of \(\Gamma \) joining \(q\) to \(\partial D\), in which case (since \(\Gamma \) and \(C_{\overline \theta }\) make first order contact at \(\partial D\)) the point \(q\) must lie strictly above \(C_{\overline \theta }\), which is absurd. So \(C_{\overline \theta }\) must indeed lie below \(\Gamma \). The first inequality is then immediate and the second is straightforward.

Figure 2. Scaled hairclip timeslice and circular arcs through the prescribed boundary points \(o\) and \((\cos \theta ,\sin \theta )\).

For each \(d\in (0,1]\) and \(\rho \in (0,\frac {\pi }{2})\), let \(\Gamma ^\rho \subset D_+\) be a smooth oriented arc satisfying the following properties.

• The left endpoint of \(\Gamma ^\rho \) is \(o=(-d,0)\), where its curvature vanishes, and its right endpoint meets \(\partial D\) orthogonally at \((\cos \rho ,\sin \rho )\).
• \(\Gamma ^\rho \) is convex.
• The curvature of \(\Gamma ^\rho \) is monotone increasing with respect to arclength from \(o\).

For example, we could take \(\Gamma ^\rho :=\mathrm {H}^{\lambda _\rho }_{t_\rho }\cap D\), where \((\lambda _\rho ,t_\rho )\) are the unique choice of \((\lambda ,t)\) which ensure that \(\mathrm {H}_t^\lambda \) meets \(\partial D\) orthogonally at \((\cos \rho ,\sin \rho )\).

  

Lemma 3.1 (Very old (but not ancient) solutions) .

For each \(d\in (0,1]\) and \(\rho \in (0,\frac {\pi }{2})\) there exist \(\alpha _\rho <0\) such that \(\alpha _\rho \to -\infty \) as \(\rho \to 0\) and a smooth More precisely,\(\{\Gamma ^\rho _t\}_{t\in [\alpha _\rho ,\omega _d)}\) is given by a family of immersions of the interval \([0,1]\) which is of class \(C^\infty ([0,1]\times (\alpha _\rho ,\omega _d))\cap C^{3+\beta ,1+\frac {\beta }{2}}([0,1)\times [\alpha _\rho ,\omega _d))\cap C^{2+\beta ,1+\frac {\beta }{2}}((0,1]\times [\alpha _\rho ,\omega _d))\) for any \(\beta \in (0,1)\). Without additional compatibility conditions at the boundary points, higher regularity at the initial time may fail. However, if the curvature of \(\Gamma ^\rho \) is odd resp. even at its left resp. right boundary point, then the solution will be smooth up to the left resp. right boundary point at the initial time.× ¹ curve shortening flow \(\{\Gamma ^\rho _t\}_{t\in [\alpha _\rho ,\omega _d)}\) in \(D\) exhibiting the following properties.

• \(\Gamma ^\rho _{\alpha _\rho }=\Gamma ^\rho \).
• For each \(t\in [\alpha _\rho ,\omega _d)\), \(\Gamma ^\rho _t\) is an oriented embedding of a closed interval, with left endpoint \(o=(-d,0)\) and right endpoint meeting \(\partial D\) orthogonally.
• For each \(t\in [\alpha _\rho ,\omega _d)\), \(\Gamma ^\rho _t\) is convex.
• For each \(t\in [\alpha _\rho ,\omega _d)\), the curvature of \(\Gamma ^\rho _t\) is monotone increasing with respect to arclength from \(o\).
• If \(d<1\), then \(\omega _d=\infty \) and \(\Gamma ^\rho _t\) converges uniformly in the smooth topology as \(t\to \infty \) to the minimizing arc \(\{(x,0):x\in [-1,-d]\}\).
• If \(d=1\), then \(\omega _d\in (0,\infty )\) and \(\Gamma ^\rho _t\) converges uniformly as \(t\to \omega _d\) to the point \(o\), and there are sequences of times \(t_j\nearrow \omega _d\), points \(p_j\in \Gamma ^\rho _{t_j}\), and scales \(\lambda _j\nearrow \infty \) such that the sequence \(\{\lambda _j(\Gamma ^\rho _{\lambda _j^{-2}t+t_j}-p_j)\}_{t\in [\lambda _j^2(\alpha _\rho -t_j),\lambda _j^2(\omega _\rho -j^{-1}-t_j))}\) converges locally uniformly in the smooth topology as \(j\to \infty \) to the right half of the downwards translating Grim Reaper.

Proof.

Form the “odd doubling” \(\check \Gamma ^\rho \) of \(\Gamma ^\rho \) by taking the union of \(\Gamma ^\rho \) with its rotation through angle \(\pi \) about \(o\). Since \(\check \Gamma ^\rho \) is a regular curve of class \(C^2\) and there exists a ball \(B\) about \((\cos \rho ,\sin \rho )\) (of radius \(1/10\), say) which is disjoint from the rotation of \(\Gamma ^\rho \) through angle \(\pi \) about \(o\), where \(\Gamma ^\rho \) meets \(\partial D\) orthogonally, Stahl's short-time existence theorem for free boundary mean curvature flow [20], Theorem 2.1 yields a solution \(\{\check \Gamma ^\rho _t\}_{t\in [0,\delta )}\) to Neumann–Neumann curve shortening flow with boundary on the odd doubling of \(\partial D\cap B\) for a short time \(\delta >0\). Since this solution is uniquely determined by its initial condition, it must be invariant under rotation through angle \(\pi \) about \(o\), and hence descend to a solution \(\{\hat \Gamma ^\rho _t\}_{t\in [0,\delta )}\) to Dirichlet–Neumann curve shortening flow in \(D\) with Dirichlet condition \(o\) and initial condition \(\Gamma ^\rho \). Denote by \(T\) the maximal time of existence of the latter.

Since the curvature of \(\{\hat \Gamma ^\rho _t\}_{t\in [0,T)}\) satisfies

\[ \left \{\begin {aligned}(\partial _t-\Delta )\kappa ={}&\kappa ^3\\ \kappa ={}&0\;\;\text {at $o$, and}\\ \kappa _s={}&\kappa \;\;\text {at}\;\;\partial D\,, \end {aligned}\right . \]

where \(s\) denotes arclength from \(o\), the maximum principle (and Hopf boundary point lemma) ensure that \(\kappa \) remains positive on \(\hat \Gamma ^\rho _t\setminus \{o\}\) for \(t>0\).

For similar reasons, positivity of \(\kappa _s\) is also preserved. Indeed, using the commutator relation

\[ [\partial _t,\partial _s]=\kappa ^2\partial _s\,, \]

the identity \(0=\kappa _t=\Delta \kappa \) at \(o\), and the positivity of \(\kappa \) away from \(o\), we find that

\[ \left \{\begin {aligned}(\partial _t-\Delta )\kappa _s={}&4\kappa ^2\kappa _s\\ (\kappa _s)_s={}&0\;\;\text {at $o$, and}\\ \kappa _s>{}&0\;\;\text {at}\;\;\partial D\,, \end {aligned}\right . \]

so the claim once again follows from the maximum principle.

Since \(\overline \theta _t=\overline \kappa >0\) and \(\overline \theta <\pi \) (when \(d<1\), the maximum principle prevents \(\hat \Gamma ^\rho _t\) from ever reaching the minimizing arc — a stationary solution to the flow) we find that \(\overline \theta \) must attain a limit as \(t\to T\). We claim that this limit is \(\pi \). Indeed, if \(\overline \theta \le \theta _0<\pi \) for all \(t\in [0,T)\), then, representing the solution as a graph over the line \(\{(-d,y):y\in \R \}\), the “gradient estimate” ( 3 ) yields a uniform bound for the gradient, at least when \(d<1\). But then, by applying parabolic regularity theory (see, for instance, [18] ) to the graphical Dirichlet–Neumann curve shortening flow equation

\[ \left \{\begin {aligned}x_t={}&\frac {x_{yy}}{1+x_y^2}\;\;\text {in}\;\; [0,\overline y(t)]\\ x(0,t)={}&0\\ x_y(\overline y(t),t)={}&\cot \overline \theta (t)\,, \end {aligned}\right . \]

where \(\overline y(t):=\sin \overline \theta (t)\), we obtain uniform estimates for all derivatives of the graph functions \(x(\cdot ,t)\) (cf. [20] ). To obtain corresponding estimates when \(d=1\), we instead represent the solution as a graph over the “tilted” line through \((-1,0)\) and \((\cos (\overline \theta (T)),\sin (\overline \theta (T)))\) and use the “gradient estimate” \(\underline \theta \ge 0\). The Arzelà–Ascoli theorem and monotonicity of the flow now ensure that \(x(\cdot ,t)\) takes a smooth limit as \(t\to T\), at which point the flow can be smoothly continued by the above short time existence argument, violating the maximality of \(T\). We conclude that \(\overline \theta (t)\to \pi \) as \(t\to T\).

It now follows from ( 3 ) that \(\underline \theta (t)\to \pi \) as \(t\to T\). When \(d=1\), we conclude that \(\hat \Gamma ^\rho _t\) contracts to \(o\) as \(t\to T\). Note that in this case \(T<\infty \) since the lower barriers \(C_{\theta ^-(t)}\) contract to \(o\) in finite time. A more or less standard “type-I vs type-II" blow-up argument (cf. [17] ) then guarantees convergence to the half Grim Reaper after performing a standard type-II blow-up. (The flow must be type-II because the limit of a standard type-I blow-up — a shrinking semi-circle — violates the Dirichlet boundary condition.)

When \(d<1\), we conclude that \(\hat \Gamma ^\rho _t\) converges uniformly to the minimizing arc \(\{(x,0):x\in [-1,-d]\}\) as \(t\to T\). But then, for large enough \(t\), \(\hat \Gamma ^\rho _t\) may be represented as a graph over the \(x\)-axis with small gradient, at which point parabolic regularity, short-time existence and the Arzelà–Ascoli theorem guarantee that \(T=\infty \) and \(\hat \Gamma ^\rho _t\) converges uniformly in the smooth topology to the minimizing arc.

Finally, since \(\overline \theta \) is monotone, there is a unique time \(-\alpha _\rho >0\) such that \(\overline \theta (-\alpha _\rho )=\frac {\pi }{2}\); since the Neumann–Neumann circle \(\check C_{\theta _\rho }\), where \(\sin \theta _\rho =\frac {2\sin \rho }{1+\sin ^2\rho }\), lies above \(\Gamma ^\rho \), we find (by suitably time translating the upper barrier \(\{\check C_{\theta ^+(t)}\}_{t\in (-\infty ,0)}\), as in [6] ) that

\[ \alpha _\rho <\frac {1}{2}\log \left (\frac {2\sin \rho }{1+\sin ^2\rho }\right )\,. \]

Time-translating the solution \(\{\hat \Gamma ^\rho \}_{t\in [0,\infty )}\) by \(\alpha _\rho \) now yields the desired very old (but not ancient) solution \(\{\Gamma ^\rho _t\}_{t\in [\alpha _\rho ,\infty )}\).

Taking the limit as \(\rho \to 0\) of these very old (but not ancient) solutions yields our desired ancient solution.

Theorem 3.2.

Given any \(d\in (0,1]\), there exists a convex ancient Dirichlet–Neumann curve shortening flow \(\{\Gamma _t\}_{t\in (-\infty ,\omega _d)}\) in the upper half disc \(D_+\) which converges uniformly in the smooth topology to the unstable critical arc \([-d,1]\times \{0\}\) as \(t\to -\infty \). When \(d<1\), \(\omega _d=\infty \) and \(\{\Gamma _t\}_{t\in (-\infty ,\omega _d)}\) converges uniformly in the smooth topology as \(t\to +\infty \) to the minimizing arc \([-1,-d]\times \{0\}\). When \(d=1\), \(\omega _d<\infty \) and \(\Gamma ^\rho _t\) converges uniformly as \(t\to \omega _d\) to the point \(o\), and there are a sequence of times \(t_j\nearrow \omega _d\), right endpoints \(p_j\in \Gamma ^\rho _{t_j}\), and scales \(\lambda _j\nearrow \infty \) such that the sequence \(\{\lambda _j(\Gamma ^\rho _{\lambda _j^{-2}t+t_j}-p_j)\}_{t\in [\lambda _j^2(\alpha _\rho -t_j),\lambda _j^2(\omega _\rho -j^{-1}-t_j))}\) converges locally uniformly in the smooth topology as \(j\to \infty \) to the right half of the downwards translating Grim Reaper.

Proof.

Given any sequence of angles \(\rho _j\searrow 0\), consider the sequence of corresponding very old (but not ancient) solutions \(\{\Gamma ^j_t\}_{t\in [\alpha _j,\infty )}\) constructed in Lemma 3.1 . Differentiating the Neumann boundary condition and applying the estimate ( 2 ) yields the inequality

\[ \overline \theta _t=\overline \kappa \ge \frac {\sin \overline \theta }{a+\cos \overline \theta } \]

on each of these solutions. It follows, by the ODE comparison principle, that each \(\{\Gamma ^j_t\}_{t\in [\alpha _j,\infty )}\) satisfies

\(\seteqnumber{0}{}{3}\)

\begin{equation} \label {eq:gradient upper bound} \overline \theta \le \theta ^- \end{equation}

for \(t\in [\alpha _j,0]\), where we recall that \(\theta ^-\) is the solution to ( 1 ). Since \(\theta ^-\) is independent of \(j\), this implies uniform estimates for the gradient on any time interval of the form \([-\infty ,-T]\), \(T>0\), when we represent \(\{\Gamma ^j_t\}_{t\in [\alpha _j,-T]}\) graphically over the \(x\)-axis. By parabolic regularity theory and the Arzelà–Ascoli theorem, we may then extract a smooth limit of the very old solutions \(\{\Gamma ^{j}_t\}_{t\in (-\infty ,0)}\) after passing to a subsequence. This limit is ancient, since \(\alpha _\rho \to -\infty \) as \(\rho \to 0\), reaches the point \((0,1)\) at time zero (since each \(\Gamma ^{j}_t\) intersects the convex domain bounded by \(\check C_{\theta ^+(t)}\) for each \(t\in (-\infty ,0)\)), and converges uniformly in the smooth topology as \(t\to -\infty \) to the unstable Dirichlet–Neumann geodesic from \(o\) due to the estimate ( 4 ) (and parabolic regularity theory). The longtime behaviour follows from the argument presented above.

3.1. Asymptotics.

We now prove precise asymptotics for the height of the ancient solution constructed in Theorem 3.2 , assuming the initial conditions for the old-but-not-ancient solutions \(\{\Gamma ^\rho _t\}_{t\in [\alpha _\rho ,\omega _d)}\) are given by the hairclip timeslices \(\Gamma ^\rho =\mathrm {H}^{\lambda _\rho }_{t_\rho }\cap D\).

Lemma 3.3.

On each old-but-not-ancient solution \(\{\Gamma ^\rho _t\}_{t\in [\alpha _\rho ,\omega _d)}\),

\(\seteqnumber{0}{}{4}\)

\begin{equation*} \frac {\kappa }{\cos \theta }\ge \lambda _\rho \tan (\lambda _\rho y)\,. \end{equation*}

Proof.

Note that equality holds on the initial curve \(\Gamma ^\rho =\mathrm {H}^{\lambda _\rho }_{t_\rho }\cap D\). Thus, given any \(\mu <\lambda _\rho \), the function

\[ w:=\frac {\kappa }{\cos \theta }-\mu \tan (\mu y) \]

is strictly positive on the initial curve \(\Gamma ^\rho \), except at the left endpoint, where it vanishes. Observe that

\[ w_s= \frac {\kappa _s}{\cos \theta }+\sin \theta \left (\frac {\kappa ^2}{\cos ^2\theta }-\mu ^2\sec ^2(\mu y)\right )\,. \]

In particular, at the left endpoint on the initial curve,

\(\seteqnumber{0}{}{4}\)

\begin{align*} w_s={}&\frac {\kappa _s}{\cos \theta }-\mu ^2\sin \theta =(\lambda _\rho ^2-\mu ^2)\sin \theta >0\,. \end{align*} Thus (since \(w_s\) is continuous at \(o\) at time zero), if \(w\) fails to remain non-negative at positive times, then this failure must occur immediately following some interior time \(t_\ast >0\). There are three possibilities: 1. \(w_s(\cdot ,t_\ast )=0\) at the left endpoint, 2. \(w(\cdot ,t_\ast )=0\) at the right endpoint; or 3. \(w(\cdot ,t_\ast )=0\) at some interior point, \(p_\ast \).

The first of the three possibilities is immediately ruled out by the Hopf boundary point lemma.

In the second case, the Hopf boundary point lemma and the Neumann boundary condition yield, at the right endpoint,

\(\seteqnumber{0}{}{4}\)

\begin{align*} 0>w_s ={}&\frac {\kappa }{\cos \theta }+\sin \theta \left (\frac {\kappa ^2}{\cos ^2\theta }-\mu ^2(1+\tan ^2(\mu y))\right )=\mu \tan (\mu y)-\mu ^2y\ge 0\,, \end{align*} which is absurd.

In the final case (having ruled out the first two), \(w\) must attain a negative interior minumum just following time \(t_\ast \). But at such a point, \(w<0\), \(w_s=0\) and

\(\seteqnumber{0}{}{4}\)

\begin{align*} 0\ge {}& (\partial _t-\Delta )w\\ ={}&\frac {(\partial _t-\Delta )\kappa }{\cos \theta }-\frac {\kappa (\partial _t-\Delta )\cos \theta }{\cos ^2\theta }+2\left (\frac {\kappa }{\cos \theta }\right )_s\frac {(\cos \theta )_s}{\cos \theta }-(\partial _t-\Delta )(\mu \tan (\mu y))\,. \end{align*} Since

\[ (\partial _t-\Delta )\kappa =\kappa ^3\,,\;\;(\partial _t-\Delta )\cos \theta =\kappa ^2\cos \theta \;\;\text {and}\;\;(\partial _t-\Delta )y=0\,, \]

we conclude that

\(\seteqnumber{0}{}{4}\)

\begin{align*} 0\ge {}&-2\left (\frac {\kappa }{\cos \theta }\right )_s\kappa \tan \theta +2\mu \tan (\mu y)(\mu \tan (\mu y))_s\sin \theta \\ ={}&2(\mu \tan (\mu y))_s\sin \theta \left (\mu \tan (\mu y)-\frac {\kappa }{\cos \theta }\right )\\ ={}&2\mu ^2\sec ^2(\mu y)\left (\mu \tan (\mu y)-\frac {\kappa }{\cos \theta }\right )\\ >{}&0\,, \end{align*} which is absurd.

Having ruled out each of the three possibilities, we conclude that \(w\ge 0\) for any \(\mu <\lambda _\rho \). The claim follows.

In the limit as \(\rho \to 0\), we then obtain

\(\seteqnumber{0}{}{4}\)

\begin{equation} \label {eq:sharp speed lower bound} \frac {\kappa }{\cos \theta }\ge \lambda _0\tan (\lambda _0y) \end{equation}

on the ancient solution, where \(\lambda _0=\lim _{\rho \to 0}\lambda _\rho \). We now find that, as a graph over the \(x\)-axis (for \(t\) sufficiently negative),

\[ (\sin (\lambda _0y))_t=\lambda \cos (\lambda _0 y)y_t=\lambda _0\cos (\lambda _0 y)\sqrt {1+y_x^2}\kappa =\lambda _0\cos (\lambda _0 y)\frac {\kappa }{\cos \theta }\ge \lambda _0^2\sin (\lambda y)\,, \]

and hence

\[ \left (\mathrm {e}^{-\lambda _0^2t}\sin (\lambda _0 y)\right )_t\ge 0\,, \]

which implies that the limit

\[ A(x):=\lim _{t\to -\infty }\mathrm {e}^{-\lambda _0^2t}y(x,t) \]

exists in \([0,\infty )\) for each \(x\in (-d,1)\).

Recall that \(\theta ^-(t)\sim \mathrm {e}^{\frac {t}{a+1}}\) for \(t\sim -\infty \). In particular, \(\overline \theta (t)\le \theta ^-(t)\) is integrable. We will exploit this fact to show that the limit \(A(x)\) is positive (at least near \(x=1\)). First, we shall show that \(\overline \kappa \) is integrable.

Lemma 3.4.

There exist \(\rho _0>0\), \(T>-\infty \), \(C<\infty \) and \(\delta >0\) such that

\[ \kappa \le C\mathrm {e}^{\delta t}\;\;\text {for}\;\; t\le T \]

on each old-but-not-ancient solution \(\{\Gamma ^\rho _t\}_{t\in [\alpha _\rho ,\omega _d)}\) with \(\rho <\rho _0\).

Proof.

Since

\[ (\partial _t-\Delta )\sin \theta =\kappa ^2\sin \theta \,, \]

we find that

\[ (\partial _t-\Delta )\frac {\kappa }{\sin \theta }=2\nabla \frac {\kappa }{\sin \theta }\cdot \frac {\nabla \sin \theta }{\sin \theta }\,. \]

So the maximum principle guarantees that the maximum of \(\frac {\kappa }{\sin \theta }\) occurs at the parabolic boundary. Now, at the left boundary point, \(\frac {\kappa }{\sin \theta }=0\), while at the right,

\(\seteqnumber{0}{}{5}\)

\begin{align*} \left (\frac {\kappa }{\sin \theta }\right )_s={}&\frac {\kappa }{\sin \theta }\left (\frac {\kappa _s}{\kappa }-\frac {\cos \theta \kappa }{\sin \theta }\right )=\frac {\overline \kappa }{\sin \overline \theta }\left (1-\frac {\overline \kappa }{\tan \overline \theta }\right )\,. \end{align*} By ( 4 ), we can find \(T>-\infty \) (independent of \(\rho \)) so that \(\cos \overline \theta (t)\ge \frac {1}{2}\) for all \(t\le T\). We thereby conclude that

\[ \frac {\kappa }{\sin \theta }\le \max \left \{2,\max _{t=\alpha _\rho }\frac {\kappa }{\sin \theta }\right \} \]

for all \(t\le T\). Since \(\max _{t=\alpha _\rho }\frac {\kappa }{\sin \theta }\le \lambda _\rho \tanh (\lambda _\rho (1+d))\to \lambda _0^2<1\) as \(\rho \to 0\), we find that

\(\seteqnumber{0}{}{5}\)

\begin{equation} \label {eq:C2 estimate constructed solution} \overline \kappa \le 2\sin \overline \theta \le 2\sin \theta ^- \end{equation}

for all \(\rho \) sufficiently small. The claim follows since \(\theta ^-\) is comparable to \(2^{1+\frac {a}{1+a}}\mathrm {e}^{\frac {t}{1+a}}\) as \(t\to -\infty \).

Corollary 3.5.

There exist \(T>-\infty \), \(C<\infty \) and \(\delta >0\) such that

\[ \frac {\kappa }{y}\le \lambda _0^2+C\mathrm {e}^{\delta t}\;\;\text {for}\;\; t<T \]

on the ancient solution.

Proof.

Given \(\rho <\rho _0\), set

\[ \eta _\rho (t):=\lambda _\rho ^2\left (\exp \big (\tfrac {C^2}{2\delta }\mathrm {e}^{2\delta t}\big )-1\right )\,, \]

where \(\rho _0\), \(C\) and \(\delta \) are the constants from Lemma 3.4 , so that

\[ \frac {\eta _\rho '}{\lambda _\rho ^2+\eta _\rho }=C^2\mathrm {e}^{2\delta t} \]

and hence, for \(t<T\),

\(\seteqnumber{0}{}{6}\)

\begin{align*} (\partial _t-\Delta )\big (\kappa -(\lambda _\rho ^2+\eta _\rho )y\big )={}&\kappa ^3-\eta _\rho 'y\\ \le {}&C^2\mathrm {e}^{2\delta t}\kappa -\frac {\eta _\rho '}{\lambda _\rho ^2+\eta _\rho }(\lambda _\rho ^2+\eta _\rho )y\\ ={}&C^2\mathrm {e}^{2\delta t}\big (\kappa -(\lambda _\rho ^2+\eta _\rho )y\big )\,. \end{align*} Since \(\kappa -(\lambda _\rho ^2+\eta _\rho )y=0\) at the left endpoint and \((\kappa -(\lambda _\rho ^2+\eta _\rho )y)_s=\kappa -(\lambda _\rho ^2+\eta _\rho )y\) at the right endpoint, we find that

\[ \kappa -(\lambda _\rho ^2+\eta _\rho )y\le \exp \left (\frac {C^2}{2\delta }\mathrm {e}^{2\delta t}\right )\big (\kappa -(\lambda _\rho ^2+\eta _\rho )y\big )\Big |_{t=\alpha _\rho } \]

on each of the old-but-not-ancient solutions with \(\rho <\rho _0\), and hence, taking \(\rho \to 0\),

\[ \kappa \le (\lambda _0^2+\eta _0)y \]

on the ancient solution. The claim follows since, by the mean value theorem, we may estimate \(\eta _0\le \frac {\lambda _0^2C^4}{4\delta ^2}\mathrm {e}^{2\delta t}\) for \(t<0\).

By the estimate ( 4 ) and Corollary 3.5 , we can find \(T>-\infty \), \(C<\infty \) and \(\delta >0\) such that our ancient solution satisfies

\[ (\log \overline y)_t=\frac {\overline \kappa }{\overline y\cos \overline \theta }\le \frac {1}{\sqrt {1-4C^2\mathrm {e}^{2\delta t}}}\frac {\overline \kappa }{\overline y}\le (1+8C^2\mathrm {e}^{2\delta t})\frac {\overline \kappa }{\overline y}\le \lambda _0^2+C^4\mathrm {e}^{2\delta t} \]

for \(t<T\). Integrating from time \(t<T\) to time \(T\) and rearranging then yields

\[ \overline y\ge B\mathrm {e}^{\lambda _0^2t}\,,\;\; B>0\,. \]

Since the gradient of the solution is bounded by \(\tan \overline \theta \le C\mathrm {e}^{\lambda _0^2t}\) for \(t\le T\), this guarantees that the limit \(A(x):=\mathrm {e}^{-\lambda _0^2t}y(x,t)\) is positive for all \(x>x_0\) where \(x_0<1\).