1 Introduction

In this note we study the weak and strong unique continuation properties for the heat equation on \({{\,\textrm{RCD}\,}}(K,N)\) spaces. Consider the equation

$$\begin{aligned} \Delta u-\partial _tu=0. \end{aligned}$$
(1.1)

Let \(T > 0\). On an \({{\,\textrm{RCD}\,}}(K,N)\) space \((X, \textrm{d}, \mathfrak {m})\), a solution to (1.1) on [0, T] is defined as follows.

Definition 1

A function \(u: X\rightarrow \mathbb {R}\) is a solution (resp. sub-solution, super-solution) of (1.1) if \(u\in W^{1,2}_{loc} (X\times [0,T])\) and that, for all \(\phi \in {\text {Lip}}_0(X\times [0,T])\) which is compactly supported with \(\phi \ge 0\), we have

$$\begin{aligned} \int _{X\times [0,T]} -\langle \nabla u, \nabla \phi \rangle -\phi \,\partial _tu\,\, d\mathfrak {m}dt=0\, (\text {resp. }\le 0, \ge 0). \end{aligned}$$
(1.2)

This paper is concerned with the validity of the strong and weak unique continuation property for the heat equation (1.1). To be precise, we say that

  • The heat equation satisfies the weak unique continuation property on an \({{\,\textrm{RCD}\,}}(K,N)\) space \((X, \textrm{d}, \mathfrak {m})\) if, for any solution \(u \in W^{1,2}(X \times [0, T])\) of (1.1), if u vanishes on any non-empty open subset of \(X \times [0,T]\), then \(u \equiv 0\).

  • A function \(u \in L^2(X \times [0,T])\) vanishes up to infinite order at some \((x_0, t_0) \in X \times [0,T]\) if there exists some \(R > 0\) so that for any integer \(N > 0\), there exists \(C(N) > 0\) so that

    $$\begin{aligned} \int _{B_{r}(x_0) \times ((t_0 - r^2, t_0 + r^2) \cap [0, T])} |u|^2\, \, d\mathfrak {m}dt \le Cr^N, 0<r<1 \end{aligned}$$

  • The heat equation satisfies the strong unique continuation property on an \({{\,\textrm{RCD}\,}}(K,N)\) space \((X, \textrm{d}, \mathfrak {m})\) if, for any solution \(u \in W^{1,2}(X \times [0,T])\) of (1.1), if u vanishes up to infinite order at any \((x_0, t_0) \in X \times (0,T]\), then \(u \equiv 0\).

Note that we restrict \(t_0\) to be away from 0 in the definition of strong unique continuation as it is always possible to solve the heat equation starting from some initial data \(u(\cdot , 0)\), which vanishes up to infinite order (spatially) at some \(x_0 \in X\). This would easily imply that the heat flow also vanishes up to infinite order at \((x_0,0)\) in the smooth case . We mention that unique continuation type results are related to giving an upper bound for the measure of the nodal set of non-trivial solutions and that, recently, there has also been work to establish lower bounds in the nonsmooth setting, see [12, 24].

Our first result in Section 1 gives the validity of the weak unique continuation property for compact \({{\,\textrm{RCD}\,}}(K,2)\) spaces:

Theorem 1

Let \((X, \textrm{d}, \mathfrak {m})\) be a compact \({{\,\textrm{RCD}\,}}(K,2)\) space. The heat equation on X satisfies the weak unique continuation property.

As in [26], where the same theorem was shown for harmonic functions, the main idea to handle the spatial direction is to leverage the \(C^{0}\)-Riemannian structure of non-collapsed \({{\,\textrm{RCD}\,}}(K,2)\) spaces from [37, 44]. To handle the time direction, we also show the time analyticity of solutions of (1.1) following the recent [51].

In [26], an \({{\,\textrm{RCD}\,}}(K,4)\) space was constructed on which there exists non-trivial harmonic functions which vanish up to infinite order at some point. The heat flow of any such harmonic function would immediately give a counterexample to strong unique continuation for the heat equation as well. As such, we will be primarily interested in the strong unique continuation property for non-stationary solutions of the heat equation in this paper. We extend our result from [26] as follows in Sections 3 and 5 respectively:

Theorem 2

There exists an \({{\,\textrm{RCD}\,}}(K,4)\) space and a non-trivial eigenfunction on it with eigenvalue \(\mu \ne 0\) which vanishes up to infinite order at one point.

Theorem 3

There exists an \({{\,\textrm{RCD}\,}}(K,4)\) space and a non-stationary solution of the heat equation on it which vanishes up to infinite order at one point.

Besides these results, we also give frequency estimate on metric horns in Sections 2 and 4 and establish a strong unique continuation type result for eigenfunctions and caloric functions on metric horns, where the classical strong unique continuation property fails by the results of [26].

We have to use the geometry setting of this equation to deal with the difficulty of coefficients which are non-Lipschitz. For discussions on smooth manifolds with Ricci curvarure lower bound and some related questions, see for example [9,10,11, 13,14,15, 19,20,21]. For references on the theory of \({{\,\textrm{RCD}\,}}\) spaces, see [1,2,3, 7, 8, 16,17,18, 23, 25, 27, 28, 34, 38, 43, 47, 48].

2 Weak unique continuation of caloric functions on \({{\,\textrm{RCD}\,}}(K,2)\) spaces

In this section we will consider the case of \({{\,\textrm{RCD}\,}}(K,2)\) spaces. In the smooth setting, the Ricci curvature in 2 dimensions completely determines sectional curvature and hence a Ricci lower bound translates to a sectional curvature lower bound. This is also true for any non-collapsed \({{\,\textrm{RCD}\,}}(K,2)\) space \((X,\textrm{d}, \mathfrak {m})\).

Theorem 2.1

([37]) If \(\mathfrak {m}= c\mathcal {H}^2\) with \(c \ge 0\) (i.e. \((X, \textrm{d}, \mathfrak {m})\) is non-collapsed), then \((X,\textrm{d})\) is an Alexandrov space with curvature bounded below by K.

Therefore, it suffices to consider 2-dimensional Alexandrov spaces of curvature at least K and collapsed \({{\,\textrm{RCD}\,}}(K,2)\) spaces. Alexandrov spaces are known to have a generalized Riemannian structure by [44]. We refer to [44] for relevant definitions.

Theorem 2.2

([44]) Let \((X,\textrm{d})\) be an n-dimensional Alexandrov space and denote \(S_X\) as the set of singular points. Then there exists a \(C^0\)-Riemannian structure on \(X\setminus S_X\subset X\) satisfying the following:

  1. (1)

    There exists an \(X_0\subset X\setminus S_X\) such that \(X\setminus X_0\) is of n-dimensional Hausdorff measure zero and that the Riemannian structure is \(C^{\frac{1}{2}}\)-continuous on \(X_0\subset X\);

  2. (2)

    The metric structure on \(X\setminus S_X\) induced from the Riemannian structure coincides with the original metric of X.

In a coordinate neighborhood U given by the \(C^0\)-Riemannian structure of X, with corresponding metric \(g_{ij}\), a solution u of (1.1) satisfies

$$\begin{aligned} \int _{U\times [0,T]} -g^{xy}\,u_x\, \phi _y-\phi \,\partial _tu\,\, d\mathfrak {m}\,dt=0 \end{aligned}$$
(2.1)

for all \(\phi \in Lip_0(X\times [0,T])\) supported in \(U\times [0,T]\) (Cf. [35]). Note that since \(g^{ij}\) may not be Lipschitz, the results of [36] do not apply. We will instead use techniques which are special in the 2-dimensional case. For more details, see for example [22].

We proceed by taking local isothermal coordinates. To be precise, consider functions \(\sigma ,\rho :U \rightarrow \mathbb {R}^2\) satisfying

$$\begin{aligned} \sigma _x=\frac{g^{xy}\rho _x+g^{yy}\rho _y}{\sqrt{g^{xx}g^{yy}-(g^{xy})^2}}\end{aligned}$$
(2.2)
$$\begin{aligned} -\sigma _y=\frac{g^{xx}\rho _x+g^{xy}\rho _y}{\sqrt{g^{xx}g^{yy}-(g^{xy})^2}}, \end{aligned}$$
(2.3)

where derivatives are taken with respect to the coordinate chart for U. The existence of such functions is given by [41], see also [6, 22, 32]. This is equivalent to solving the complex equation

$$\begin{aligned} w_{\bar{z}}=\mu w_z, \end{aligned}$$
(2.4)

where \(\mu =\frac{g^{yy}-g^{xx}-2ig^{xy}}{g^{xx}+g^{yy}+2\sqrt{g^{xx}g^{yy}-(g^{xy})^2}}\). Note that

$$\begin{aligned} |w_z|^2=\frac{(\sigma _x\rho _y-\sigma _y\rho _x)}{4\sqrt{g^{xx}g^{yy}-(g^{xy})^2}}(g^{xx}+g^{yy}+2\sqrt{g^{xx}g^{yy}-(g^{xy})^2}) \end{aligned}$$
(2.5)

We have

$$\begin{aligned} u_x=u_\rho \rho _x+u_\sigma \sigma _x,\end{aligned}$$
(2.6)
$$\begin{aligned} u_y=u_\rho \rho _y+u_\sigma \sigma _y, \end{aligned}$$
(2.7)

and so

$$\begin{aligned} \int _{U\times [0,T]}&-(g^{xx}\,(u_\rho \rho _x+u_\sigma \sigma _x)\, (\phi _\rho \rho _x+\phi _\sigma \sigma _x))-(g^{xy}\,(u_\rho \rho _x+u_\sigma \sigma _x)\, (\phi _\rho \rho _y+\phi _\sigma \sigma _y))\nonumber \\&-(g^{xy}\,(u_\rho \rho _y+u_\sigma \sigma _y)\, (\phi _\rho \rho _x+\phi _\sigma \sigma _x))-(g^{yy}\,(u_\rho \rho _y+u_\sigma \sigma _y)\, (\phi _\rho \rho _y+\phi _\sigma \sigma _y))\nonumber \\&-\phi \,\partial _tu\,\, d\mathfrak {m}\,dt=0. \end{aligned}$$
(2.8)

After rearranging the terms,

$$\begin{aligned} \int _{U\times [0,T]} \sqrt{g^{xx}g^{yy}-(g^{xy})^2}(\sigma _x\rho _y-\sigma _y\rho _x)(u_\rho \phi _\rho +u_\sigma \phi _\sigma )-\phi \,\partial _tu\,\, d\mathfrak {m}\,dt=0. \end{aligned}$$
(2.9)

and so in the new coordinates, we have

$$\begin{aligned} \int _{\mathbb {R}^2\times [0,T]} (u_\rho \phi _\rho +u_\sigma \phi _\sigma )-\frac{\phi \,\partial _tu}{\sqrt{g^{xx}g^{yy}-(g^{xy})^2}(\sigma _x\rho _y-\sigma _y\rho _x)}\,\, dx\,dy\,dt=0. \end{aligned}$$
(2.10)

Thus in the coordinate \((\rho ,\sigma )\) the equation becomes

$$\begin{aligned} \Delta u-a\partial _t u=0, \end{aligned}$$
(2.11)

where a is measurable and Hölder on a full measure subset. Note that as now we have a possibly discontinuous coefficient in front of \(\partial _t\), the result in [36] cannot be used. We will instead use a geometrical argument to deal with this difficulty.

Now we are ready to state the main result of this section.

Theorem 1

Let \(u\in W^{1,2} (X\times [0,T])\) be a solution of (1.1) on an \({{\,\textrm{RCD}\,}}(K,2)\) space. If u vanishes on a non-empty open set \(\Gamma \subset X\times (0,T]\), then \(u\equiv 0\) on \(\text {proj}_X(\Gamma )\times [0,T]\). Moreover if X is compact, then \(u\equiv 0\) on \(X\times [0,T]\).

Proof

By a similar discussion as in [26], since all the collapsed cases are trivial we can only consider the non-collapsed case. We assume that u vanishes on a non-empty open set \(V\times (t_1,t_2)\).

Since u satisfies the heat equation, we have the following a priori estimates (cf.[29, Remark 5.2.11]):

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^2}\le \Vert u(\cdot ,0)\Vert _{L^2}, \end{aligned}$$
(2.12)
$$\begin{aligned} \Vert |\nabla u(\cdot ,t)|\Vert _{L^2}\le \frac{\Vert u(\cdot ,0)\Vert _{L^2}}{\sqrt{t}},\end{aligned}$$
(2.13)
$$\begin{aligned} \Vert \partial _t u(\cdot ,t)\Vert _{L^2} =\Vert \Delta u(\cdot ,t)\Vert _{L^2}\le \frac{\Vert u(\cdot ,0)\Vert _{L^2}}{t}, \end{aligned}$$
(2.14)

for any \(t \in (0,T]\).

We first prove that u vanishes on \(V\times [0,T]\) by showing that \(u(x,\cdot )\) is analytic with respect to t. For the corresponding arguments on Riemannian manifolds see for example [51]. For the reader’s convenience we sketch the argument here.

As in [39], for any \((x_0,t_0)\) with \(0< t_0 < T\), \(k>0\) and \(j\le k\), we consider

$$\begin{aligned} H_j^1&=B(x_0,\frac{j\sqrt{t_0}}{\sqrt{2k}})\times [t_0-\frac{jt_0}{2k},t_0+\frac{jt_0}{2k}],\end{aligned}$$
(2.15)
$$\begin{aligned} H_j^2&=B(x_0,\frac{(j+0.5)\sqrt{t_0}}{\sqrt{2k}})\times [t_0-\frac{(j+0.5)t_0}{2k},t_0+\frac{(j+0.5)t_0}{2k}]. \end{aligned}$$
(2.16)

So that u is defined on these sets, we extend u to be a solution of (1.1) on \(X \times [0, \infty )\).

By [43], see also [21], we may choose a cutoff function \(\phi _1(x,t)\) supported in \(H^2_j\) such that \(\phi _1=1\) on \(H^1_j\), satisfying

$$\begin{aligned} |\nabla \phi _1(x,t)|^2+|\partial _t \phi _1(x,t)|\le \frac{Ck}{t_0}. \end{aligned}$$
(2.17)

Indeed, \(\phi _1\) may be chosen to be a product of two cutoff functions on the spatial and time coordinates respectively, so that the spatial and time derivatives are well-defined without further measure theoretic arguments. It follows that

$$\begin{aligned} \int _{H^2_j}|\partial _tu|^2\phi _1^2\,d\mathfrak {m}\,dt&=-\int _{H_j^2}(\langle \nabla \partial _t u,\nabla u\rangle \phi _1^2+\partial _t u\langle \nabla u,\nabla \phi _1^2\rangle )\,d\mathfrak {m}\,dt\\&=-\int _{H_j^2}(\frac{1}{2}\partial _t |\nabla u|^2\phi _1^2+\partial _t u\langle \nabla u,\nabla \phi _1^2\rangle )\,d\mathfrak {m}\,dt \\&\le \int _{H^2_j}(\frac{3}{4}|\partial _t u|^2\phi _1^2+\frac{Ck}{t_0}|\nabla u|^2)\,d\mathfrak {m}\,dt, \end{aligned}$$

where the last inequality follows from integration by parts, Young’s inequality and (2.17). This gives that

$$\begin{aligned} \int _{H_j^1}|\partial _tu|^2\,d\mathfrak {m}\,dt\le \frac{Ck}{t_0}\int _{H_j^2}|\nabla u|^2\,d\mathfrak {m}\,dt. \end{aligned}$$
(2.18)

Similarly, we may choose a cutoff function \(\phi _2(x,t)\) supported in \(H^1_{j+1}\) such that \(\phi _2=1\) on \(H^2_j\), satisfying

$$\begin{aligned} |\nabla \phi _2(x,t)|^2+|\partial _t \phi _2(x,t)|\le \frac{Ck}{t_0}. \end{aligned}$$
(2.19)

Arguing as before, we have

$$\begin{aligned} \int _{H_{j+1}^1}|\nabla u|^2\phi _2^2\,d\mathfrak {m}\,dt&=-\int _{H_{j+1}^1}u\,\partial _t u\,\phi _2^2 +u\langle \nabla u,\nabla \phi _2^2\rangle \,d\mathfrak {m}\,dt\nonumber \\&=-\int _{H_{j+1}^1}\frac{1}{2}\,\partial _t u^2\,\phi _2^2 +u\langle \nabla u,\nabla \phi _2^2\rangle \,d\mathfrak {m}\,dt\nonumber \\&\le \int _{H_{j+1}^1}(\frac{1}{4}|\nabla u|^2\phi ^2_2+\frac{Ck}{t_0}u^2)\,d\mathfrak {m}\,dt, \end{aligned}$$
(2.20)

and so

$$\begin{aligned} \int _{H_j^2}|\nabla u|^2\,dx\,dt\le \frac{Ck}{t_0}\int _{H_{j+1}^1}u^2\,dx\,dt. \end{aligned}$$
(2.21)

By using (2.18),(2.21) inductively on j, we obtain

$$\begin{aligned} \int _{H_1^1}(\partial _t^k u)^2\,dx\,dt\le \frac{(Ck)^{2k}}{t_0^{2k}}\int _{H^1_{k+1}}u^2\,dx\,dt\le \frac{(Ck)^{2k}}{t_0^{2k-1}}, \end{aligned}$$
(2.22)

where C at the end also depends on the \(L^2\)-norm of \(u_0\).

From [5, 31, 46, 49] we know that the (p, p)-Poincaré inequality holds on \({{\,\textrm{RCD}\,}}(K,N)\) space for any \(p \ge 1\) (see, for example, [33, Section 4] for a careful discussion of Poincaré inequalties in the \({{\,\textrm{RCD}\,}}(K,N)\) setting). Thus Moser iteration ([31, Lemma 3.10]) shows that for \(R<1\) there exist a mean value inequality

$$\begin{aligned} |\partial _t^ku(x_0,t_0)|^2\le C(\frac{R^{\nu _2}}{|B(x_0,R)|})\frac{1}{R^{\nu _2(1+\frac{2}{\nu _2})}}\int _{B(x,R)\times (t-R^2,t)}|\partial _t^ku|^2\,d\mathfrak {m}\,dt. \end{aligned}$$
(2.23)

Thus by taking \(R=\frac{\sqrt{t_0}}{\sqrt{2k}}\) and using (2.22), we obtain

$$\begin{aligned} |\partial _t^ku(x_0,t_0)|^2\le \frac{C}{|B(x_0,\frac{\sqrt{t_0}}{\sqrt{2k}})|(\frac{\sqrt{t_0}}{\sqrt{2k}})^2}\int _{H_1^1}|\partial _t^ku|^2\,d\mathfrak {m}\,dt\le \frac{C}{|B(x_0,\frac{\sqrt{t_0}}{\sqrt{2k}})|}\frac{(Ck)^{2k+1}}{t_0^{2k}},\,\,t_0<k. \end{aligned}$$
(2.24)

Using volume comparison, this implies that \(u(x_0,\cdot )\) is analytic with respect to t. In particular, since u vanishes on \(V \times (t_1,t_2)\) by assumption, u also vanishes on \(V\times [0,T]\).

We now prove the second assertion of the theorem by contradiction. Assume X is compact and \(u\not \equiv 0\).

Let \(\phi _k\) be eigenfunctions of \(-\Delta \) corresponding to eigenvalues \(\lambda _k\) with \(\Vert \phi _k\Vert _{L^2}=1\) and \(0 = \lambda _0\lambda _1 \le \lambda _2 \le ... \rightarrow \infty \), see [30] for a discussion on this. It follows from the estimates obtained in the appendix of [4] that u admits the representation

$$\begin{aligned} u(x,t)=\sum _{k=0}^\infty a_k e^{-\lambda _kt}\phi _k(x). \end{aligned}$$
(2.25)

Let j be the first index for which \(a_j \ne 0\). As it does not affect the argument, we assume for simplicity that the dimension of the eigenspace corresponding to \(\lambda _j\) is 1. This gives that for any \(x_0\in V\),

$$\begin{aligned} 0=\mathop {lim}\limits _{t\rightarrow \infty }e^{\lambda _j t}u(x_0,t)=a_j \phi _j(x_0), \end{aligned}$$
(2.26)

where we have used [4, Proposition 7.1] to bound the values of \(\phi _k(x_0)\) for \(k > j\) to obtain the second equality. As we assumed that \(a_j\ne 0\), we conclude that \(\phi _j(x_0)=0\). This shows that \(\phi _j(x_0)=0\) for all k and \(x_0\in V\). From [26], this implies that \(\phi _j\equiv 0\), which is a contradiction. It follows that \(u \equiv 0\). Note that similar arguments also work for Dirichlet problem on non-compact space.

Remark 1

The previous argument actually shows the time analyticity of caloric functions with respect to time on any \({{\,\textrm{RCD}\,}}(K,N)\) space, since that part of the argument does not require any assumptions on dimension.

Remark 2

In [36], Fourier transform was used to reduce the problem to a solution of an elliptic equation on \(X \times \mathbb {R}\). In this case, one does not have weak unique continuation for elliptic equations on \({{\,\textrm{RCD}\,}}(K,3)\) spaces, so a different argument had to be used.

Finally, we recall a well-known counterexample given by Miller [40] which indicates that in general we cannot expect weak unique continuation for parabolic operators with time-dependent coefficients even if the time-slices of the corresponding metric have a uniform Ricci curvature bound.

Proposition 1

([40]) There exists a smooth function \(u:\mathbb {R}^2\times [0,\infty )\rightarrow \mathbb {R}\) such that:

$$\begin{aligned} u_t=((1+A(t)+a)u_x)_x+(bu_y)_x+(bu_x)_y+((1+C(t)+c)u_y)_y, \end{aligned}$$
(2.27)

where \(A=C=a=b=c=u=0\) on \(t\ge T\), a, b, c, u are smooth on \(\mathbb {R}^2\times [0,\infty )\), A(t), C(t) are smooth on (0, T) and Hölder on \([0,\infty )\). Moreover, u, a, b, c are period in x and y with period \(2\pi \).

3 Elliptic frequency estimate on Metric horn

In this section we will give a frequency estimate on the metric horn, which allows us to prove a form of unique continuation. Recall from [26] that the standard formulation of strong unique continuation does not hold at the horn tip. The form of unique continuation we will prove in this section will therefore assume a higher order of decay at the horn tip, see Remark 1 at the end of the section for more details.

On a weighted warped product \((X,dr^2+f^2(r)g_{S^{n-1}},e^{-\psi (r)}dvol)\), given function \(\varphi \), we have that

$$\begin{aligned} \Delta \varphi =\partial _r^2\varphi +(n-1)\partial _r\varphi (log\,f)_r+ f^{-2}\Delta _{S^{n-1}}\varphi -\partial _r\varphi \partial _r\psi , \end{aligned}$$
(3.1)

away from \(r=0\). In the case of the standard metric horn,

$$\begin{aligned} f(r)&=\frac{1}{2}r^{1+\epsilon },\end{aligned}$$
(3.2)
$$\begin{aligned} \psi (r)&=-(N-n)(1-\eta )log(r), \end{aligned}$$
(3.3)

and so,

$$\begin{aligned} \Delta \varphi =\partial _r^2\varphi +\partial _r\varphi (\frac{(n-1)(1+\epsilon )+(N-n)(1-\eta )}{r})+\frac{4}{r^{2+2\epsilon }}\Delta _{S^{n-1}}\varphi . \end{aligned}$$
(3.4)

From the equation of Laplacian on metric horn, in particular, for \(\varphi = r^\alpha \), we have that

$$\begin{aligned} \Delta \varphi =r^{\alpha -2}\alpha (\alpha +N-2+(n-1)\epsilon -(N-n)\eta ) \end{aligned}$$
(3.5)

and

$$\begin{aligned} {\text{ Hess }}\,\varphi =\alpha (\alpha -1)r^{\alpha -2}dr\otimes dr+(\frac{r^{1+\epsilon }}{2})^2\alpha (1+\epsilon )r^{\alpha -2}g_{S^{n-1}}. \end{aligned}$$
(3.6)

If we take \(\alpha =2\), then

$$\begin{aligned} {\text{ Hess }}\,\varphi = 2(dr^2+(1+\epsilon )(\frac{r^{1+\epsilon }}{2})^2g_{S^{n-1}}). \end{aligned}$$
(3.7)

Given an eigenfunction u with \(\Delta u=\lambda u\), define scale-invariant quantities I(r), E(r) and the frequency function U(r) with respect to the level sets of the distance function \(d_p\), where p is the horn tip. For \(r>0\), we denote

$$\begin{aligned} B_r=\{x\,|\,d_p(x)<r\},\end{aligned}$$
(3.8)
$$\begin{aligned} \partial B_r=\{x\,|\,d_p(x)=r\}, \end{aligned}$$
(3.9)

and define

$$\begin{aligned}&I(r)=r^{1-n}\int _{\partial B_r}u^2 d\mathfrak {m}_r, \end{aligned}$$
(3.10)
$$\begin{aligned}&E(r)=r^{2-n}\int _{B_r}|\nabla u|^2+\lambda u^2 d\mathfrak {m}=r^{2-n}\int _{\partial B_r}\frac{u}{|\nabla d_p|} \,\langle \nabla u,\nabla d_p\rangle d\mathfrak {m}_r, \end{aligned}$$
(3.11)
$$\begin{aligned}&U(r)=\frac{E(r)}{I(r)}, \end{aligned}$$
(3.12)

where \(\mathfrak {m}= e^{-\psi }(r)dvol\) is the weighted volume measure of the metric horn and \(\mathfrak {m}_r = e^{-\psi }(r)dvol_r\) is the corresponding weighted area measure on \(\partial B_r\).

We first compute the derivative of I. We remark that all computations are done away from the cone tip p, so no regularity issues will arise from p itself. Let \(\phi \) be any smooth function compactly supported on \((0,\infty )\), we have

$$\begin{aligned} -\int I(t)\phi '(t)dt&=-\int t^{1-n}\int _{\partial B_t}u^2\phi '(t)dt=-\int u^2 d^{1-n}_p\phi '(d_p)\\&=-\int u^2 d^{1-n}_p\langle \nabla d_p,\nabla \phi (d_p)\rangle =\int {\text {div}}(u^2 d^{1-n}_p\nabla d_p)\phi (d_p)\\&=\int \frac{1}{2-n}(2u\langle \nabla u, \nabla d_p^{2-n}\rangle +u^2 \Delta d_p^{2-n})\phi (d_p)\\&=\int _t\int _{\partial B_t} \frac{1}{(2-n)|\nabla d_p|}(2u\langle \nabla u, \nabla d^{2-n}_p\rangle +u^2 \Delta d^{2-n}_p)\phi (d_p)dt. \end{aligned}$$

This shows that

$$\begin{aligned} I'(r)&=\int _{\partial B_r} \frac{1}{(2-n)|\nabla d_p|}(2u\langle \nabla u, \nabla d^{2-n}_p\rangle +u^2 \Delta d^{2-n}_p)\\&=\frac{2E(r)}{r} +\int _{\partial B_r} \frac{1}{(2-n)|\nabla d_p|}u^2 \Delta d^{2-n}_p,\quad a.e.\,\, t\in (0,1),\nonumber \end{aligned}$$
(3.13)

and so by using (3.5),

$$\begin{aligned} (\log I)'(r)-\frac{2U(r)}{r}=\frac{N-n+(n-1)\epsilon -(N-n)\eta }{r}. \end{aligned}$$
(3.14)

For E, from the definition (3.11) and the coarea formula, we have that

$$\begin{aligned} E'(r)=(2-n)\frac{E(r)}{r}+r^{2-n}\int _{\partial B_r}({|\nabla u|^2}+\lambda u^2). \end{aligned}$$
(3.15)

Now consider the vector field \(X=\frac{\langle \nabla (d_p^2),\nabla u\rangle \nabla u-\frac{1}{2}|\nabla u|^2\nabla (d_p^2)}{|\nabla d_p|^3}\), we have that

$$\begin{aligned} {\text {div}}(X)={{\text{ Hess }}(d_p^2)(\nabla u,\,\nabla u)-\frac{1}{2}|\nabla u|^2\Delta d_p^2}+\lambda u \langle \nabla (d_p^2),\,\nabla u\rangle \end{aligned}$$
(3.16)

By Integrating both sides in \(B_r\) we have that

$$\begin{aligned} \int _{\partial B_r}\frac{2r}{|\nabla d_p|^3}|\langle \nabla d_p,\,\nabla u\rangle |^2-r\frac{|\nabla u|^2}{|\nabla d_p|}&=\int _{\partial B_r}\langle X,\,\nabla d_p\rangle \\&=\int _{B_r}{{\text{ Hess }}(d_p^2)(\nabla u,\,\nabla u)-\frac{1}{2}|\nabla u|^2\Delta d_p^2}\nonumber \\&\quad +\lambda u \langle \nabla (d_p^2),\,\nabla u\rangle . \nonumber \end{aligned}$$
(3.17)

Combining (3.15) and (3.17) we have that

$$\begin{aligned} E'(r)=&(2-n)\frac{E(r)}{r}+r^{2-n}\int _{\partial B_r}\frac{2}{|\nabla d_p|^3}|\langle \nabla d_p,\,\nabla u\rangle |^2+\lambda u^2\\&-r^{1-n}\int _{B_r}({{\text{ Hess }}(d_p^2)(\nabla u,\,\nabla u)-\frac{1}{2}|\nabla u|^2\Delta d_p^2}+\lambda u \langle \nabla (d_p^2),\,\nabla u\rangle ). \nonumber \end{aligned}$$
(3.18)

Now consider the frequency function U(r). We have that

$$\begin{aligned} \frac{d}{dr}log\,U(r)=&\frac{E'(r)}{E(r)}-\frac{I'(r)}{I(r)}\\ =&\frac{1}{E(r)}(-\frac{2E^2(r)}{rI(r)}+r^{2-n}\int _{\partial B_r}|2\langle \nabla d_p,\,\nabla u\rangle |^2\nonumber \\&+\lambda u^2)-\frac{1}{I(r)}\int _{\partial B_r} \frac{1}{2-n}u^2 \Delta d^{2-n}_p+\frac{2-n}{r}\nonumber \\&-\frac{1}{E(r)}(r^{1-n}\int _{B_r}{\text{ Hess }}(d_p^2)(\nabla u,\,\nabla u)-\frac{1}{2}|\nabla u|^2\Delta d_p^2+\lambda u \langle \nabla (d_p^2),\,\nabla u\rangle ).\nonumber \end{aligned}$$
(3.19)

From Cauchy-Schwarz inequality we know that

$$\begin{aligned} -\frac{2E^2(r)}{rI(r)}+2r^{2-n}\int _{\partial B_r}|\langle \nabla d_p,\,\nabla u\rangle |^2\ge 0. \end{aligned}$$
(3.20)

Combining (3.19) and (3.20) we have that

$$\begin{aligned} \frac{d}{dr}log\,U(r)\ge&\frac{\lambda r}{U(r)}-\frac{1}{I(r)}\int _{\partial B_r} \frac{1}{2-n}u^2 \Delta d^{2-n}_p+\frac{2-n}{r}\\&-\frac{1}{E(r)}(r^{1-n}\int _{B_r}{\text{ Hess }}(d_p^2)(\nabla u,\,\nabla u)-\frac{1}{2}|\nabla u|^2\Delta d_p^2-\frac{\lambda }{2} u^2 \Delta d_p^2)\nonumber . \end{aligned}$$
(3.21)

As we have that

$$\begin{aligned} -\frac{1}{2-n}\int _{\partial B_r} u^2 \Delta d^{2-n}_p=-\frac{1}{2}\int _{\partial B_r} u^2 {\text {div}}( d^{-n}_p \nabla d_p^2)=-\frac{r^{-n}}{2}\int _{\partial B_r} u^2 \Delta d_p^2+nr^{-n}\int _{\partial B_r} u^2, \end{aligned}$$
(3.22)

it follows that

$$\begin{aligned} \begin{aligned} \frac{d}{dr}log\,U(r)&\ge \frac{\lambda r}{U(r)}+\frac{2}{r}-\frac{1}{E(r)}(r^{1-n}\int _{B_r}{\text{ Hess }}(d_p^2)(\nabla u,\,\nabla u))\\&\ge \frac{\lambda r}{U(r)}-\frac{r^{1-n}\int _{B_r}2\epsilon r^{2+2\epsilon } g_{S^{n-1}}(\nabla u,\nabla u)}{r^{2-n}\int _{B_r}|\nabla u|^2}\\&\ge \frac{\lambda r}{U(r)}- \frac{2\epsilon }{r}. \end{aligned} \end{aligned}$$
(3.23)

This shows that

$$\begin{aligned} (r^{2\epsilon } U(r))'\ge \lambda r^{1+2\epsilon }, \end{aligned}$$
(3.24)

and so

$$\begin{aligned} r^{2\epsilon }U(r)-U(1)\le \frac{\lambda }{2+2\epsilon }(r^{2+2\epsilon }-1), \end{aligned}$$
(3.25)

which is equivalent to

$$\begin{aligned} U(r)\le r^{-2\epsilon }U(1)+\frac{\lambda }{2+2\epsilon }(r^{2}-r^{-2\epsilon })\le Cr^{-2\epsilon }. \end{aligned}$$
(3.26)

Combining with (3.14) we obtain

$$\begin{aligned} (\log I)'(r)\le Cr^{-1-2\epsilon }. \end{aligned}$$
(3.27)

This gives

$$\begin{aligned} \log I(r)\ge \log I(1)+C (1-r^{-2\epsilon }), \end{aligned}$$
(3.28)

and so in all

$$\begin{aligned} I(r)\ge Ce^{-Cr^{-2\epsilon }}. \end{aligned}$$
(3.29)

Remark 1

From [26], we know that strong unique continuation fails on metric horn. In particular, any harmonic function with \(u(0)=0\) satisfies \(u(x)=O(e^{-C(\log r)^2})\). However the discussion above shows that if an eigenfunction, in particular a harmonic function, vanishes at the tip up to order \(u(x)=O(e^{-Cr^{-2\epsilon }})\), then \(u\equiv 0\).

4 Failure of strong unique continuation of eigenfuctions on metric horn

In this section we will prove the following theorem:

Theorem 4.1

There exists an \({{\,\textrm{RCD}\,}}(K,4)\) space and a non-trivial eigenfunction on it with eigenvalue \(\mu \ne 0\), which vanishes up to infinite order at one point.

Proof

Consider any modified metric horn X constructed in [26, Section 6], which uses techniques in [50]. Let us denote the eigenfunctions on \(S^{n-1}\) as \(\{\varphi _i\}_{i=1}^\infty \) such that

$$\begin{aligned}&\Delta _{S^{n-1}}\varphi _i=-\mu _i\varphi _i,\,\,\mu _i>0 \end{aligned}$$
(4.1)
$$\begin{aligned}&\int _{S^{n-1}}\varphi _i^2dS=1. \end{aligned}$$
(4.2)

Assume \(\varphi \) is an \(L^2\) function which is smooth away from the tip, then \(\varphi \) may be decomposed as

$$\begin{aligned} \varphi (r,\theta )=f_0(r)+\sum _{i=1}^\infty f_i(r)\varphi _i(\theta ). \end{aligned}$$
(4.3)

Therefore, for any eigenfunction on metric horn with eigenvalue \(-\mu \) we have, for the decomposition (4.3) and each i,

$$\begin{aligned} f_i''(r)+f_i'(r)\left( \frac{(n-1)(1+\epsilon )+(N-n)(1-\eta )}{r}\right) +\frac{4}{r^{2+2\epsilon }}f_i(r)(-\mu _i)+\mu f_i(r)=0, \end{aligned}$$
(4.4)

for r sufficiently close to 0. We note that we do not have this formula for arbitrary r since the modified metric horn was constructed using a gluing procedure.

For the radial part, that is for \(\mu _0=0\),

$$\begin{aligned} f_0''(r)+f_0'(r)\left( \frac{(n-1)(1+\epsilon )+(N-n)(1-\eta )}{r}\right) +\mu f_0(r)=0, \end{aligned}$$
(4.5)

Denoting \(c=(n-1)(1+\epsilon )+(N-n)(1-\eta )\), we have

$$\begin{aligned} f_0(r) = k_1 r^{\frac{1 - c}{2}} J_{\frac{c - 1}{2}}(r \sqrt{\mu }) + k_2 r^{\frac{1 - c}{2}} Y_{\frac{c - 1}{2}}(r \sqrt{\mu }), \end{aligned}$$
(4.6)

where \(J_\nu ,\,Y_\nu \) are Bessel functions. For a discussion of the properties of Bessel functions, see [42, Appendix B]. We will use the fact that \(J_{\frac{c-1}{2}}(r) \approx r^{\frac{c-1}{2}}\) and \(Y_{\frac{c-1}{2}}(r) \approx r^{\frac{1-c}{2}}\) as \(r \rightarrow 0\). Since \(\varphi \) is assumed to be in \(L^2\), we have that \(f_0\) is in a weighted \(L^2\) space where the weight for small r is of the form \(r^{c}\). Taking this into consideration with the asymptotics of J and Y, we conclude that

$$\begin{aligned} f_0(r)=Cr^{\frac{1 - c}{2}} J_{\frac{c - 1}{2}}(r \sqrt{\mu })\approx C\mu ^{\frac{c-1}{4}}. \end{aligned}$$
(4.7)

For \(\mu _i=i(n+i-2)\) and denoting \(c=(n-1)(1+\epsilon )+(N-n)(1-\eta )\), we obtain

$$\begin{aligned} f_i''(r)+f_i'(r)\left( \frac{c}{r}\right) +\frac{4}{r^{2+2\epsilon }}f_i(r)(-\mu _i)+\mu f_i(r)=0. \end{aligned}$$
(4.8)

Define

$$\begin{aligned} f_i(r)=g_i(r^{-\epsilon }). \end{aligned}$$
(4.9)

Then

$$\begin{aligned} f_i'(r)&=-\epsilon r^{-\epsilon -1}g_i'(r^{-\epsilon })\\ f_i''(r)&=\epsilon (\epsilon +1) r^{-\epsilon -2}g_i'(r^{-\epsilon })+\epsilon ^2 r^{-2\epsilon -2}g_i''(r^{-\epsilon }). \end{aligned}$$

Thus

$$\begin{aligned} g_i''(r)+\frac{\epsilon +1-c}{\epsilon } r^{-1}g_i'(r)+\frac{-4\mu _i}{\epsilon ^2}g_i(r)+\frac{\mu }{\epsilon ^2} r^{-\frac{2}{\epsilon }-2} g_i(r)=0. \end{aligned}$$
(4.10)

If we further define

$$\begin{aligned} g_i(r)=k_i(r)r^{\frac{c-1-\epsilon }{2\epsilon }}, \end{aligned}$$
(4.11)

then

$$\begin{aligned} g_i'(r)&=k_i'(r)r^{\frac{c-1-\epsilon }{2\epsilon }}+k_i(r)\frac{c-1-\epsilon }{2\epsilon }r^{\frac{c-1-\epsilon }{2\epsilon }-1},\\ g_i''(r)&=k_i''(r)r^{\frac{c-1-\epsilon }{2\epsilon }}+2k_i'(r)\frac{c-1-\epsilon }{2\epsilon }r^{\frac{c-1-\epsilon }{2\epsilon }-1}\\&+k_i(r)\frac{c-1-\epsilon }{2\epsilon }(\frac{c-1-\epsilon }{2\epsilon }-1)r^{\frac{c-1-\epsilon }{2\epsilon }-2} \end{aligned}$$

and so

$$\begin{aligned} k_i''(r)=(\frac{c-1-\epsilon }{2\epsilon }(\frac{c-1-\epsilon }{2\epsilon }+1)r^{-2}+\frac{4\mu _i}{\epsilon ^2}-\frac{\mu }{\epsilon ^2} r^{-\frac{2}{\epsilon }-2})k_i(r). \end{aligned}$$
(4.12)

If we consider the solution with data \(k_i^{(1)}(r_\mu )=1=(k_i^{(1)})'(r_\mu )\), where \(r_\mu \) is chosen such that \(1\ge \frac{c-1-\epsilon }{2\epsilon }(\frac{c-1-\epsilon }{2\epsilon }+1)r^{-2}-\frac{\mu }{\epsilon ^2} r^{-\frac{2}{\epsilon }-2}\ge 0\) for \(r\ge r_\mu \), that is

$$\begin{aligned} r_\mu =\max \{\sqrt{\frac{c-1-\epsilon }{2\epsilon }(\frac{c-1-\epsilon }{2\epsilon }+1)},(\frac{c-1-\epsilon }{2\mu }(\frac{c-1-\epsilon }{2}+\epsilon ))^{\frac{-\epsilon }{2}}\}. \end{aligned}$$

So for \(r>r_\mu \) from \((\frac{4\mu _i}{\epsilon ^2}+1)k^{(1)}_i(r)\ge (k^{(1)}_i)''(r)\ge \frac{4\mu _i}{\epsilon ^2}k^{(1)}_i(r)\), we have

$$\begin{aligned} e^{(\sqrt{\frac{4\mu _i}{\epsilon ^2}}+1)(r-r_\mu )}\ge k_i^{(1)}(r)\ge \frac{1}{\sqrt{\frac{4\mu _i}{\epsilon ^2}}}e^{\sqrt{\frac{4\mu _i}{\epsilon ^2}}(r-r_\mu )}. \end{aligned}$$
(4.13)

Then we have that

$$\begin{aligned} k_i^{(2)}(r):=k_i^{(1)}(r)\int _{r}^\infty \frac{1}{(k_i^{(1)}(s))^2}ds. \end{aligned}$$
(4.14)

is also a solution of (4.12). So we get

$$\begin{aligned} \frac{1}{2({\frac{4\mu _i}{\epsilon ^2}}+\sqrt{\frac{4\mu _i}{\epsilon ^2}})}e^{(-\sqrt{\frac{4\mu _i}{\epsilon ^2}}-2)(r-r_\mu )}\le k_i^{(2)}(r)\le \frac{\sqrt{\frac{4\mu _i}{\epsilon ^2}}}{2}e^{(-\sqrt{\frac{4\mu _i}{\epsilon ^2}}+1)(r-r_\mu )},\,\, \text {for}\, r>r_\mu . \end{aligned}$$
(4.15)

And also

$$\begin{aligned} 0<C(\epsilon ,i)^{-1}\le (k_2^{(2)})^2(r_\mu )+((k_2^{(2)})')^2(r_\mu )\le C(\epsilon ,i). \end{aligned}$$
(4.16)

From ODE theory we have that \(k_i^{(1)},\,k_i^{(2)}\) form a basis of the solutions of (4.12). This tells us that

$$\begin{aligned} f_i(r)=(ak_i^{(1)}+bk_i^{(2)})(r^{-\epsilon })r^{-\frac{c-1-\epsilon }{2}}. \end{aligned}$$
(4.17)

As before, we have that \(f_i\) is in a weighted \(L^2\) space from assumption, this tells us that

$$\begin{aligned} f_i(r)=bk_i^{(2)}(r^{-\epsilon })r^{-\frac{c-1-\epsilon }{2}}. \end{aligned}$$
(4.18)

From (4.15) we have the decay rate \(f_i(r)=O(e^{-Cr^{-\epsilon }})\). This tells us that there exists eigenfunctions on \({{\,\textrm{RCD}\,}}(K,4)\) spaces which are not zero but vanish up to infinite order at one point.

Remark 1

In fact this argument tells us that any eigenfunction on this metric horn with zero integral on each link \(\{r=r_0\}\) vanishes up to infinite order.

Remark 2

Since

$$\begin{aligned}&\int _{r=0}^{r_\mu ^{-\frac{1}{\epsilon }}}e^{2(-\sqrt{\frac{4\mu _i}{\epsilon ^2}}-2)r^{-\epsilon }}r^{-(c-1-\epsilon )} 2^{1-n}r^c\,dr\nonumber \\&=2^{1-n}\frac{(2(\sqrt{\frac{4\mu _i}{\epsilon ^2}}+2))^{\frac{2(1+\epsilon )}{\epsilon }-1}}{\epsilon }\int _{2(\sqrt{\frac{4\mu _i}{\epsilon ^2}}+2)r_\mu }^\infty e^{-r}r^{-\frac{2(1+\epsilon )}{\epsilon }}dr\nonumber \\ {}&\ge c(n,\epsilon ,\mu _i) e^{2(-\sqrt{\frac{4\mu _i}{\epsilon ^2}}-2)r_\mu }r_\mu ^{-\frac{2(1+\epsilon )}{\epsilon }}, \end{aligned}$$
(4.19)

the normalized coefficient \(c_{\mu ,i}\) for the eigenfunction \(f_i(r)\varphi _i(\theta )\) with \(L^2\)-norm 1 corresponding to eigenvalue \(\mu \) satisfies

$$\begin{aligned} c_{\mu ,i}\le c(n,\epsilon ,\mu _i)e^{(\sqrt{\frac{4\mu _i}{\epsilon ^2}}+2)r_\mu }r_\mu ^{\frac{1+\epsilon }{\epsilon }}. \end{aligned}$$
(4.20)

Remark 3

Let us use a more careful argument on \(\varphi \). That is, as we know that

$$\begin{aligned} |((k_i^{(2)})^2+((k_i^{(2)})')^2)|&=|2(k_i^{(2)})'(k_i^{(2)}+(k_i^{(2)})'')|\nonumber \\&\le 2|k_i^{(2)}(k_i^{(2)})'|C(\mu +1)\nonumber \\&\le C(\mu +1)|(k_i^{(2)})^2+((k_i^{(2)})')^2|. \end{aligned}$$
(4.21)

Thus if we denote the pasting radius as \(\tilde{r}\), by the discussion we have

$$\begin{aligned} C^{-1}e^{-C(\mu +1)(r_\mu -\tilde{r}^{-\frac{1}{\epsilon }})} \le |((k_i^{(2)})^2+((k_i^{(2)})')^2)'|(\tilde{r}^{-\frac{1}{\epsilon }})\le Ce^{C(\mu +1)(r_\mu -\tilde{r}^{-\frac{1}{\epsilon }})}. \end{aligned}$$
(4.22)

As we know that when transform back

$$\begin{aligned} C^{-1}e^{-C(\mu +1)(r_\mu -\tilde{r}^{-\frac{1}{\epsilon }})}\le |f_i^2+(f_i')^2|(\tilde{r})\le Ce^{C(\mu +1)(r_\mu -\tilde{r}^{-\frac{1}{\epsilon }})}. \end{aligned}$$
(4.23)

As outside \(\tilde{r}\) the space is a cone, the function is of the form \(f(r) = k_1 r^{(1 - b)/2} J_{1/2 \sqrt{(b - 1)^2 + 4 a}}(\sqrt{c} r) + k_2 r^{(1 - b)/2} Y_{1/2 \sqrt{(b - 1)^2 + 4 a}}(\sqrt{c} r)\). From the asymptotics of Bessel functions we can see that if we consider the eigenfunction on \(B_R\), the normalized coefficient satisfies

$$\begin{aligned} c_{\mu ,i,R}\le Ce^{C(\mu +1)(\mu ^{\frac{\epsilon }{2}}-\tilde{r}^{-\frac{1}{\epsilon }})}R^{-\frac{n-1}{2}}. \end{aligned}$$
(4.24)

5 Parabolic frequency estimate on metric horn

In this section we give a parabolic frequency estimate and the corresponding unique continuation type result on the metric horn. For simplicity, we consider the metric horn which is not modified. For previous discussions on parabolic frequency see for example [20, 45].

Consider the metric horn \((\mathbb {R}^n,\,\,g_{\epsilon ,n},\,\,e^{(N-n)(1-\eta )log(r)}dvol)\) with \( g_{\epsilon ,n}=dr^2+(\frac{1}{2}r^{1+\epsilon })^2g_{S^{n-1}}\). Note that we will use X to denote the metric horn in the following. Let p(x, y, t) be the heat kernel. Following [45], for \(x_0 \in X\) and \(t_0 \ge 0\), we define the backward heat kernel \(G_{x_0, t_0}: X \times (-\infty , t_0) \rightarrow \mathbb {R}\) as \(G_{x_0,t_0}(x,t):= p(x,x_0,t_0-t)\). For simplicity of notation, we denote \(G(x,t) = G_{o,0}(x,t)\). Let u be a solution of (1.1) on \(X \times [-R_0^2, 0]\) for some \(R_0 > 0\).

For \(R_0 \ge R > 0\), define

$$\begin{aligned} I(R)&=R^2\int _{t=-R^2}|\nabla u|^2 G \, d\mathfrak {m}, \end{aligned}$$
(5.1)
$$\begin{aligned} D(R)&=\int _{t=-R^2}u^2G \, d\mathfrak {m},\end{aligned}$$
(5.2)
$$\begin{aligned} N(R)&=\frac{I(R)}{D(R)}. \end{aligned}$$
(5.3)

From the definition of G, we have that

$$\begin{aligned} \frac{\partial \log G}{\partial t}=-\Delta \log G-|\nabla \log G|^2, \end{aligned}$$
(5.4)

and so

$$\begin{aligned} \log G(x,t)=-\frac{(N+(n-1)\epsilon -(N-n)\eta )}{2}\log (-t)+\frac{|x|^2}{4t}. \end{aligned}$$
(5.5)

Therefore,

$$\begin{aligned} I'(R)=&2R\int _{t=-R^2}|\nabla u|^2 Gd\mathfrak {m}-2R^3\int _{t=-R^2}(2\langle \nabla u,\,\nabla u_t\rangle G+|\nabla u|^2\partial _tG)d\mathfrak {m}\\ =&2R\int _{t=-R^2}|\nabla u|^2 Gd\mathfrak {m}+4R^3\int _{t=-R^2}u_t^2 G+u_t\langle \nabla u,\,\nabla G \rangle d\mathfrak {m}\nonumber \\&-2R^3\int _{t=-R^2}|\nabla u|^2\partial _tGd\mathfrak {m}.\nonumber \end{aligned}$$
(5.6)

Consider the vector field \(X=\langle \nabla G,\,\nabla u\rangle \nabla u\), then

$$\begin{aligned} div(X)={\text{ Hess }}_G(\nabla u,\,\nabla u)+\frac{1}{2}\langle \nabla G,\,\nabla |\nabla u|^2\rangle +\langle \nabla G,\,\nabla u\rangle \Delta u. \end{aligned}$$
(5.7)

On the metric horn,

$$\begin{aligned} \frac{{\text{ Hess }}_G}{G}(x,t)=\left( \frac{|x|^2}{4t^2}+\frac{1}{2t}\right) dr\otimes dr+\frac{\left( \frac{|x|^{1+\epsilon }}{2}\right) ^2}{2t}(1+\epsilon )g_{S^{n-1}}, \end{aligned}$$
(5.8)

and so

$$\begin{aligned} \int {\text{ Hess }}_G(\nabla u,\,\nabla u) +\langle \nabla G,\,\nabla u\rangle u_td\mathfrak {m}=\int \frac{1}{2}\Delta G|\nabla u|^2d\mathfrak {m}. \end{aligned}$$
(5.9)

So combining with (5.6) we have that

$$\begin{aligned} I'(R)&=4R^3\int _{t=-R^2}\left( \frac{x}{2t}\cdot \nabla u+u_t\right) ^2G(x,t)d\mathfrak {m}\nonumber \\&\quad +4R^3\int _{t=-R^2}\frac{\left( \frac{|x|^{1+\epsilon }}{2}\right) ^2}{2t}\epsilon g_{S^{n-1}}(\nabla u,\,\nabla u)G(x,t)d\mathfrak {m}\nonumber \\&\ge 4R^3\int _{t=-R^2}\left( \frac{x}{2t}\cdot \nabla u+u_t\right) ^2G(x,t)d\mathfrak {m}-\frac{2\epsilon }{R} I(R). \end{aligned}$$
(5.10)

Moreover,

$$\begin{aligned} D'(R)=-2R\int _{t=-R^2}(2uu_tG+u^2\partial _t G)d\mathfrak {m}=-4R\int _{t=-R^2}u(u_t+\frac{x}{2t}\cdot \nabla u)Gd\mathfrak {m}, \end{aligned}$$
(5.11)

so from integration by parts we have that

$$\begin{aligned} I(R)=-R^2\int _{t=-R^2}u(\partial _t u +\nabla u\cdot \frac{x}{2t}) G d\mathfrak {m}=\frac{R}{4}D'(R). \end{aligned}$$
(5.12)

From Cauchy–Schwarz inequality we obtain

$$\begin{aligned}&\left( \int _{t=-R^2}\left( \frac{x}{2t}\cdot \nabla u+u_t\right) ^2G(x,t)d\mathfrak {m}\right) \left( \int _{t=-R^2}u^2G(x,t)d\mathfrak {m}\right) \nonumber \\&\quad \ge \left( \int _{t=-R^2}u\left( \frac{x}{2t}\cdot \nabla u+u_t\right) G(x,t)d\mathfrak {m}\right) ^2, \end{aligned}$$
(5.13)

So we have that

$$\begin{aligned} (\log N)'(R)=\frac{I'}{I}(R)-\frac{D'}{D}(R)\ge -\frac{2\epsilon }{R}, \end{aligned}$$
(5.14)

and so

$$\begin{aligned} \log N(R)\le \log N(1)-2\epsilon \log R, \end{aligned}$$
(5.15)

which is equivalent to that

$$\begin{aligned} N(R)\le CR^{-2\epsilon }. \end{aligned}$$
(5.16)

Since

$$\begin{aligned} (\log D)'(R)=\frac{4I(R)}{RD(R)}=\frac{4}{R}N(R)\le CR^{-1-2\epsilon }, \end{aligned}$$
(5.17)

we obtain

$$\begin{aligned} \log D(R)\ge \log D(1)+C(1-R^{-2\epsilon }). \end{aligned}$$
(5.18)

So finally

$$\begin{aligned} D(R)\ge Ce^{-CR^{-2\epsilon }}. \end{aligned}$$
(5.19)

This gives a strong unique continuation type result for solution of heat equation on the metric horn.

6 Failure of strong unique continuation property of caloric on metric horn

In this section we will prove the following theorem:

Theorem 6.1

There exists an \({{\,\textrm{RCD}\,}}(K,4)\) space and a non-stationary solution of heat equation on it which vanishes up to infinite order at one point.

Similar as in the elliptic case, the heat equation becomes like

$$\begin{aligned} 0=\Delta \varphi -\partial _t \varphi =\partial _r^2\varphi +\partial _r\varphi \left( \frac{(n-1)(1+\epsilon )+(N-n)(1-\eta )}{r}\right) +\frac{4}{r^{2+2\epsilon }}\Delta _{S^{n-1}}\varphi -\partial _t \varphi . \end{aligned}$$
(5.2)

Denote the eigenfunctions on \(S^{n-1}\) as \(\{\varphi _i\}_{i=1}^\infty \) such that

$$\begin{aligned}&\Delta _{S^{n-1}}\varphi _i=-\mu _i\varphi _i,\,\,\mu _i>0\end{aligned}$$
(5.3)
$$\begin{aligned}&\int _{S^{n-1}}\varphi _i^2dS=1. \end{aligned}$$
(5.4)

We can decompose it as \(\varphi (r,\theta ,t)=f_0(r,t)+\sum _{i=1}^\infty f_i(r,t)\varphi _i(\theta )\). So we have

$$\begin{aligned}&\partial _r^2f_i(r,t)+\partial _rf_i(r,t)\left( \frac{(n-1)(1+\epsilon )+(N-n)(1-\eta )}{r}\right) \nonumber \\&+\frac{4}{r^{2+2\epsilon }}f_i(r,t)(-\mu _i)-\partial _t f_i(r,t)=0 \end{aligned}$$
(5.5)

For \(\mu _i=i(n+i-2)\) and denote \(c=(n-1)(1+\epsilon )+(N-n)(1-\eta )\)

$$\begin{aligned} \partial _r^2f_i(r,t)+\partial _rf_i(r,t)\left( \frac{c}{r}\right) +\frac{4}{r^{2+2\epsilon }}f_i(r,t)(-\mu _i)-\partial _t f_i(r,t)=0. \end{aligned}$$
(5.6)

We first consider the solution of heat equation on the compact modified metric horn constructed in [26, Section 6], where p is the tip of the metric horn. We denote the eigenfunctions with eigenvalue \(\nu _j\) with spherical components \(\varphi _i\) as \(q_{\nu _j}(r,\theta )=g_j(r)\varphi _i(\theta )\).

Then we have that

$$\begin{aligned} f_i(r,t)=\sum _j c_je^{-\nu _j t}g_j(r). \end{aligned}$$
(5.7)

Denote

$$\begin{aligned} r_\mu =\max \left\{ \sqrt{\frac{c-1-\epsilon }{2\epsilon }(\frac{c-1-\epsilon }{2\epsilon }+1)}, (\frac{c-1-\epsilon }{2\mu }(\frac{c-1-\epsilon }{2}+\epsilon ))^{\frac{-\epsilon }{2}}\right\} . \end{aligned}$$

By [52] we have \(C_1j^{\frac{2}{N}}\le \nu _j\le C_2j^{2}\), thus \(C_1j^{\frac{\epsilon }{N}}\le r_{\nu _j}\le C_2j^{\epsilon }.\)

From (4.15) we have that given any \(100r_{\nu _{k}}\le r^{-\epsilon } <100r_{\nu _{k+1}}\), which indicates that \(\frac{C_1}{k+1}<r<C_2k^{-\frac{1}{N}}\), we have

$$\begin{aligned} |f_i(r,t)|&\le \sum _{j=1}^k|c_j|e^{-\nu _jt}c(n,\epsilon ,\mu _i)e^{(\sqrt{\frac{4\mu _i}{\epsilon ^2}}+2)r_{\nu _j}}r_{\nu _j}^{\frac{1+\epsilon }{\epsilon }}e^{(-\sqrt{\frac{4\mu _i}{\epsilon ^2}}+1)r^{-\epsilon }}+\sum _{j=k+1}^\infty Ce^{-\nu _j t}\nu _j\nonumber \\&\le Ck\cdot k^{1+\epsilon }e^{-cr^{-\epsilon }}+\sum _{j=k+1}^\infty Ce^{-Ctj^{\frac{2}{N}}}j^2\nonumber \\&\le C r^{-(2+\epsilon )N}e^{-cr^{-\epsilon }}+Ce^{-Ctr^{-\frac{2}{N}}}r^{-2}\le e^{-cr^{-\epsilon }}. \end{aligned}$$
(5.8)

As the estimate is independent of k, we can see that \(f_i\) vanishes up to infinite order at tip.

Remark 5.9

The argument above actually shows that all caloric functions on the modified metric horn which does not contain the radial part \(f_0(r,t)\) vanishes up to infinite order at the tip.