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We consider the energy critical heat equation in $\mathbb R^n$ for $n\ge 7$ $$\left\{ \begin{aligned} u_t & = \Delta u+ |u|^{\frac 4{n-2}}u \hbox{ in }\ \mathbb R^n \times (0, \infty), \\ u(\cdot,0) & = u_0 \ \hbox{ in }\ \mathbb R^n, \end{aligned}\right. $$ which corresponds to the $L^2$-gradient flow of the Sobolev-critical energy $$ J(u) = \int_{\mathbb R^n} e[u] , \quad e[u] := \frac 12 |\nabla u|^2 - \frac {n-2}{2n} |u|^{\frac {2n}{n-2} }. $$ Given any $k\ge 2$ we find an initial condition $u_0$ that leads to sign-changing solutions with {\em multiple blow-up at a single point} (tower of bubbles) as $t\to +\infty$. It has the form of a superposition with alternate signs of singularly scaled {\em Aubin-Talenti solitons}, $$ u(x,t) = \sum_{j=1}^k (-1)^{j-1} {\mu_j^{-\frac {n-2}2}} U \left( \frac {x}{\mu_j} \right)\, +\, o(1) \quad\hbox{as } t\to +\infty $$ where $U(y)$ is the standard soliton $ U(y) = % (n(n-2))^{\frac 1{n-2}} \alpha_n\left ( \frac 1{1+|y|^2}\right)^{\frac{n-2}2}$ and $$\mu_j(t) = \beta_j t^{- \alpha_j}, \quad \alpha_j = \frac 12 \Big ( \, \left( \frac{n-2}{n-6}\right)^{j-1} -1 \Big). $$ Letting $\delta_0$ the Dirac mass, we have energy concentration of the form $$ e[ u(\cdot, t)]- e[U] \rightharpoonup (k-1) S_n\,\delta_{0} \quad\hbox{as } t\to +\infty $$ where $S_n=J(U)$. The initial condition can be chosen radial and compactly supported. We establish the codimension $k+ n (k-1)$ stability of this phenomenon for perturbations of the initial condition that have space decay $u_0(x) =O( |x|^{-\alpha})$, $\alpha > \frac {n-2}2$, which yields finite energy of the solution.

We consider the Euler equations in $$\mathbb R^3$$ R 3 expressed in vorticity form $$\begin{aligned} \left\{ \begin{array}{l} \vec \omega _t + (\mathbf{u}\cdot {\nabla } ){\vec \omega } =( \vec \omega \cdot {\nabla } ) \mathbf{u} \\ \mathbf{u} = \mathrm{curl}\vec \psi ,\ -\Delta \vec \psi = \vec \omega . \end{array}\right. \end{aligned}$$ ω → t + ( u · ∇ ) ω → = ( ω → · ∇ ) u u = curl ψ → , - Δ ψ → = ω → . A classical question that goes back to Helmholtz is to describe the evolution of solutions with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called binormal curvature flow. Existence of true solutions concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings. We provide what appears to be the first rigorous construction of helical filaments, associated to a translating-rotating helix. The solution is defined at all times and does not change form with time. The result generalizes to multiple polygonal helical filaments travelling and rotating together.

We consider the Cauchy problem for the energy critical heat equation \begin{document}$ \begin{equation} \left\{ \begin{aligned} u_t & = \Delta u + u^3 {\quad\hbox{in } }\ \mathbb R^4 \times (0, T), \\ u(\cdot, 0) & = u_0 {\quad\hbox{in } } \mathbb R^4. \end{aligned}\right. ~~~~~~~~~~~~~~~~~~~~~~~(1)\end{equation} $\end{document} We find that for given points \begin{document}$ q_1, q_2, \ldots, q_k $\end{document} and any sufficiently small \begin{document}$ T>0 $\end{document} there is an initial condition \begin{document}$ u_0 $\end{document} such that the solution \begin{document}$ u(x, t) $\end{document} of (1) blows up at exactly those \begin{document}$ k $\end{document} points with a type Ⅱ rate, namely larger than \begin{document}$ (T-t)^{-\frac 12} $\end{document} . In fact \begin{document}$ \|u(\cdot, t)\|_\infty \sim (T-t)^{-1}\log^2(T-t) $\end{document} . The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin-Talenti bubbles.

We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere $$S^2$$, $$\begin{aligned} u_t&= \Delta u + |\nabla u|^2 u \quad \text {in } \Omega \times (0,T)\\ u&= \varphi \quad \text {on } \partial \Omega \times (0,T)\\ u(\cdot ,0)&= u_0 \quad \text {in } \Omega , \end{aligned}$$where $$\Omega $$ is a bounded, smooth domain in $$\mathbb {R}^2$$, $$u: \Omega \times (0,T)\rightarrow S^2$$, $$u_0:\bar{\Omega } \rightarrow S^2$$ is smooth, and $$\varphi = u_0\big |_{\partial \Omega }$$. Given any k points $$q_1,\ldots , q_k$$ in the domain, we find initial and boundary data so that the solution blows-up precisely at those points. The profile around each point is close to an asymptotically singular scaling of a 1-corotational harmonic map. We build a continuation after blow-up as a $$H^1$$-weak solution with a finite number of discontinuities in space–time by “reverse bubbling”, which preserves the homotopy class of the solution after blow-up. Furthermore, we prove the codimension one stability of the one point blow-up phenomenon.

We consider the Cauchy problem for the energy critical heat equation $$\begin{cases}u_t ={\Delta}u+|u|^\frac{4}{n-2}u\;\;\;\text{in}\;\mathbb{R}^n\times(0,T)\\u(\centerdot,0)=u_0\;\;\;\text{in}\;\mathbb{R}^n\end{cases}$$ in dimension n = 5. More precisely we find that for given points q1,q2,...,qk and any sufficiently small T > 0 there is an initial condition u0 such that the solution u(x,t) of (0.1) blows-up at exactly those k points with rates type II, namely with absolute size ~(T-t)-α for α > $$\frac{3}{4}$$ . The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin–Talenti bubbles.

We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere \begin{document}$ S^2 $\end{document} , \begin{document}$ \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u & = u_b \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) & = u_0 \quad \text{in } \Omega , \end{align*} $\end{document} with \begin{document}$ u(x,t): \bar \Omega\times [0,T) \to S^2 $\end{document} . Here \begin{document}$ \Omega $\end{document} is a bounded, smooth axially symmetric domain in \begin{document}$ \mathbb{R}^3 $\end{document} . We prove that for any circle \begin{document}$ \Gamma \subset \Omega $\end{document} with the same axial symmetry, and any sufficiently small \begin{document}$ T>0 $\end{document} there exist initial and boundary conditions such that \begin{document}$ u(x,t) $\end{document} blows-up exactly at time \begin{document}$ T $\end{document} and precisely on the curve \begin{document}$ \Gamma $\end{document} , in fact \begin{document}$ | {\nabla} u(\cdot ,t)|^2 \rightharpoonup | {\nabla} u_*|^2 + 8\pi \delta_\Gamma \quad\mbox{as}\quad t\to T . $\end{document} for a regular function \begin{document}$ u_*(x) $\end{document} , where \begin{document}$ \delta_\Gamma $\end{document} denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a space-codimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng [ 5 , 6 ].

We consider the parabolic Allen–Cahn equation in Rn, n≥2, ut=Δu+(1−u2)u in Rn×(−∞,0]. We construct an ancient radially symmetric solution u(x,t) with any given number k of transition layers between −1 and +1. At main order they consist of k time-traveling copies of w with spherical interfaces distant O(log⁡|t|) one to each other as t→−∞. These interfaces are resemble at main order copies of the shrinking sphere ancient solution to mean the flow by mean curvature of surfaces: |x|=−2(n−1)t. More precisely, if w(s) denotes the heteroclinic 1-dimensional solution of w″+(1−w2)w=0 w(±∞)=±1 given by w(s)=tanh⁡(s2) we have u(x,t)≈∑j=1k(−1)j−1w(|x|−ρj(t))−12(1+(−1)k) as t→−∞ where ρj(t)=−2(n−1)t+12(j−k+12)log⁡(|t|log⁡|t|)+O(1),j=1,…,k.

We consider the parabolic one-dimensional Allen–Cahn equationThe steady state connects, as a ‘transition layer’, the stable phases –1 and +1. We construct a solution u with any given number k of transition layers between –1 and +1. Mainly they consist of k time-travelling copies of w, with each interface diverging as t → –∞. More precisely, we findwhere the functions ξj (t) satisfy a first-order Toda-type system. They are given byfor certain explicit constants γjk.

Finite topology self-translating surfaces for the mean curvature flow constitute a key element in the analysis of Type II singularities from a compact surface because they arise as limits after suitable blow-up scalings around the singularity. We prove the existence of such a surface M ⊂ R 3 that is orientable, embedded, complete, and with three ends asymptotically paraboloidal. The fact that M is self-translating means that the moving surface S ( t ) = M + t e z evolves by mean curvature flow, or equivalently, that M satisfies the equation H M = ν ⋅ e z where H M denotes mean curvature, ν is a choice of unit normal to M , and e z is a unit vector along the z -axis. This surface M is in correspondence with the classical three-end Costa–Hoffman–Meeks minimal surface with large genus, which has two asymptotically catenoidal ends and one planar end, and a long array of small tunnels in the intersection region resembling a periodic Scherk surface. This example is the first non-trivial one of its kind, and it suggests a strong connection between this problem and the theory of embedded complete minimal surfaces with finite total curvature.

For the generalized surface quasi-geostrophic equation $$\left\{ \begin{aligned} & \partial_t \theta+u\cdot \nabla \theta=0, \quad \text{in } \mathbb{R}^2 \times (0,T), \\ & u=\nabla^\perp \psi, \quad \psi = (-\Delta)^{-s}\theta \quad \text{in } \mathbb{R}^2 \times (0,T) , \end{aligned} \right. $$ $0, Comment: 26 pages; comments welcome

We construct globally defined in time, unbounded positive solutions to the energy-critical heat equation in dimension 3 ¶ u t = Δ u + u 5 in ℝ 3 × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) in ℝ 3 . ¶ For each [math] we find initial data (not necessarily radially symmetric) with [math] such that as [math] ¶ ∥ u ( ⋅ , t ) ∥ ∞ ∼ t γ − 1 2 if 1 < γ < 2 , ∥ u ( ⋅ , t ) ∥ ∞ ∼ t if γ > 2 , ∥ u ( ⋅ , t ) ∥ ∞ ∼ t ( ln t ) − 1 if γ = 2 . ¶ Furthermore we show that this infinite-time blow-up is codimensional-1 stable. The existence of such solutions was conjectured by Fila and King (Netw. Heterog. Media 7:4 (2012), 661–671).

We construct codimension 1 surfaces of any dimension that minimize a periodic nonlocal perimeter functional among surfaces that are periodic, cylindrically symmetric and decreasing. These surfaces may be seen as a nonlocal analogue of the classical Delaunay surfaces (onduloids). For small volume, most of their mass tends to be concentrated in a periodic array and the surfaces are close to a periodic array of balls (in fact, we give explicit quantitative bounds on these facts).

We consider the problem v_t & = ��v+ |v|^{p-1}v \quad\hbox{in }\ ��\times (0, T), v & =0 \quad\hbox{on } \partial ��\times (0, T ) , v& >0 \quad\hbox{in }\ ��\times (0, T) . In a domain $��\subset \mathbb R^d$, $d\ge 7$ enjoying special symmetries, we find the first example of a solution with type II blow-up for a power $p$ less than the Joseph-Lundgren exponent $$p_{JL}(d)=\infty, & \text{if $3\le d\le 10$}, 1+{4\over d-4-2\,\sqrt{d-1}}, & \text{if $d\ge11$}. $$ No type II radial blow-up is present for $p< p_{JL}(d)$. We take $p=\frac{d+1}{d-3}$, the Sobolev critical exponent in one dimension less. The solution blows up on circle contained in a negatively curved part of the boundary in the form of a sharply scaled Aubin-Talenti bubble, approaching its energy density a Dirac measure for the curve. This is a completely new phenomenon for a diffusion setting., 49 pages, no figures. arXiv admin note: text overlap with arXiv:1705.01672

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$ and denote the regular part of the Green's function on $\Omega$ with Dirichlet boundary condition as $H(x,y)$. Assume that $q \in \Omega$ and $n\geq 5$. We prove that there exists an integer $k_0$ such that for any integer $k\geq k_0$ there exist initial data $u_0$ and smooth parameter functions $\xi(t)\to q$, $0, Comment: 60 pages; comments welcome

We construct new type II ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as t → - ∞ {t\to{-}\infty} , to a tower of two spheres. Their curvature operator changes sign. We allow two time-dependent parameters in our ansatz. We use perturbation theory, via fixed point arguments, based on sharp estimates on ancient solutions of the approximated linear equation and careful estimation of the error terms which allow us to make the right choice of parameters. Our technique may be viewed as a parabolic analogue of gluing two exact solutions to the rescaled equation, that is the spheres, with narrow cylindrical necks to obtain a new ancient solution to the Yamabe flow. The result generalizes to the gluing of k spheres for any k ≥ 2 {k\geq 2} , in such a way the configuration of radii of the spheres glued is driven as t → - ∞ {t\to{-}\infty} by a First order Toda system.

We consider the wave equation $\varepsilon^2(-\partial_t^2 + \Delta)u + f(u) = 0$ for $0, Comment: 34 pages

We consider the Allen–Cahn equation Δ u + u ( 1 − u 2 ) = 0 in R 3 . We construct two classes of axially symmetric solutions u = u ( | x ′ | , x 3 ) such that the (multiple) components of the zero set look for large | x ′ | like catenoids, namely | x 3 | ∼ A log ⁡ | x ′ | . In Theorem 1, we find a solution which is even in x 3 , with Morse index one and a zero set with exactly two components, which are graphs. In Theorem 2, we construct a solution with a zero set with two or more nested catenoid-like components, whose Morse index become as large as we wish. While it is a common idea that nodal sets of the Allen–Cahn equation behave like minimal surfaces, these examples show that the nonlocal interaction between disjoint portions of the nodal set, governed in suitably asymptotic regimes by explicit systems of 2d PDE, induces richness and complex structure of the set of entire solutions, beyond the one in minimal surface theory.

The goal of this note is to study the asymptotic behavior of positive solutions for a class of semilinear elliptic equations which can be realized as minimizers of their energy functionals. This class includes the Fisher-KPP and Allen–Cahn nonlinearities. We consider the asymptotic behavior in domains becoming infinite in some directions. We are in particular able to establish an exponential rate of convergence for this kind of problems.

Let $\Omega$ be a smooth bounded domain in $\R^n$, $n\ge 5$. We consider the semilinear heat equation at the critical Sobolev exponent $$ u_t = \Delta u + u^{\frac{n+2}{n-2}} \inn \Omega\times (0,\infty), \quad u =0 \onn \pp\Omega\times (0,\infty). $$ Let $G(x,y)$ be the Dirichlet Green's function of $-\Delta$ in $\Omega$ and $H(x,y)$ its regular part. Let $q_j\in \Omega$, $j=1,\ldots,k$, be points such that the matrix $$ \left [ \begin{matrix} H(q_1, q_1) & -G(q_1,q_2) &\cdots & -G(q_1, q_k) -G(q_1,q_2) & H(q_2,q_2) & -G(q_2,q_3) \cdots & -G(q_3,q_k) \vdots & & \ddots& \vdots -G(q_1,q_k) &\cdots& -G(q_{k-1}, q_k) & H(q_k,q_k) \end{matrix} \right ] $$ is positive definite. For any $k\ge 1$ such points indeed exist. We prove the existence of a positive smooth solution $u(x,t)$ which blows-up by bubbling in infinite time near those points. More precisely, for large time $t$, $u$ takes the approximate form $$ u(x,t) \approx \sum_{j=1}^k \alpha_n \left ( \frac { \mu_j(t)} { \mu_j(t)^2 + |x-\xi_j(t)|^2 } \right )^{\frac {n-2}2} . $$ Here $\xi_j(t) \to q_j$ and $0

We consider the Yamabe equation Δ u + n ( n − 2 ) 4 | u | 4 n − 2 u = 0 in R n , n ⩾ 3 . Let k ⩾ 1 and ξ j k = ( e 2 j π i k , 0 ) ∈ R n = C × R n − 2 . For all large k we find a solution of the form u k ( x ) = U ( x ) − ∑ j = 1 k μ k − n − 2 2 U × ( μ k − 1 ( x − ξ j ) ) + o ( 1 ) , where U ( x ) = ( 2 1 + | x | 2 ) n − 2 2 , μ k = c n k 2 for n ⩾ 4 , μ k = c k 2 ( log k ) 2 for n = 3 and o ( 1 ) → 0 uniformly as k → + ∞ .

Let Ω be a bounded, smooth domain in \({\mathbb{R}^2}\). We consider the functional $$I(u) = \int_\Omega e^{u^2}\,dx$$ in the supercritical Trudinger-Moser regime, i.e. for \({\int_\Omega |\nabla u|^2dx > 4\pi}\). More precisely, we are looking for critical points of I(u) in the class of functions \({u \in H_0^1 (\Omega )}\) such that \({\int_\Omega |\nabla u|^2 \, dx = 4\, \pi \, k\, (1+\alpha)}\), for small α > 0. In particular, we prove the existence of 1-peak critical points of I(u) with \({\int_\Omega |\nabla u|^2dx = 4\pi(1 + \alpha)}\) for any bounded domain Ω, 2-peak critical points with \({\int_\Omega |\nabla u|^2dx = 8\pi(1 + \alpha)}\) for non-simply connected domains Ω, and k-peak critical points with \({\int_\Omega |\nabla u|^2 dx = 4k \pi(1 + \alpha)}\) if Ω is an annulus.

We establish nondegeneracy of the explicit family of finite mass solutions of the Liouvillle equation with a singular source of integer multiplicity, in the sense that all bounded elements in the kernel of the linearization correspond to variations along the parameters of the family.

In a bounded domain \({\Omega \subset \mathbb R^2}\) with smooth boundary we consider the problem $$\Delta u = 0 \quad {\rm{in }}\, \Omega, \qquad \frac{\partial u}{\partial \nu} = \frac1\varepsilon f(u) \quad {\rm{on }}\,\partial\Omega,$$ where ν is the unit normal exterior vector, e > 0 is a small parameter and f is a bistable nonlinearity such as f(u) = sin(πu) or f(u) = (1 − u2)u. We construct solutions that develop multiple transitions from −1 to 1 and vice-versa along a connected component of the boundary ∂Ω. We also construct an explicit solution when Ω is a disk and f(u) = sin(πu).

We construct a new class of entire solutions for the Allen–Cahn equation Δu+(1−u2)u=0, in R2(∼C). Given k⩾1, we find a family of solutions whose zero level sets are, away from a compact set, asymptotic to 2k straight lines (which we call the ends). These solutions have the property that there exist θ0

Let (M ,g) be an $N$-dimensional smooth (compact or noncompact) Riemannian manifold. We introduce the elliptic Jacobi-Toda system on (M ,g). We review various recent results on its role in the construction of solutions with multiple interfaces of the Allen-Cahn equation on compact manifolds and entire space, as well as multiple-front traveling waves for its parabolic counterpart.

We consider the Allen–Cahn equation Δu+u(1−u2)=0in RN. A celebrated conjecture by E. De Giorgi (1978) states that if u is a bounded solution to this problem such that ∂xNu>0, then the level sets {u=λ}, λ∈R, must be hyperplanes at least if N⩽8. We construct a family of solutions which shows that this statement does not hold true for N⩾9. To cite this article: M. del Pino et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).

We consider the boundary value problem { − Δ u + u = λ e u , in Ω ∂ ν u = 0 on ∂ Ω where Ω is a bounded smooth domain in R 2 , λ > 0 and ν is the inner normal derivative at ∂Ω. This problem is equivalent to the stationary Keller–Segel system from chemotaxis. We establish the existence of a solution u λ which exhibits a sharp boundary layer along the entire boundary ∂Ω as λ → 0 . These solutions have large mass in the sense that ∫ Ω λ e u λ ∼ | log ⁡ λ | .

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions., Comment: arXiv admin note: substantial text overlap with arXiv:1509.08803

These lecture notes are devoted to the analysis of blow-up of solutions for some parabolic equations that involve bubbling phenomena. The term bubbling refers to the presence of families of solutions which at main order look like scalings of a single stationary solution which in the limit become singular but at the same time have an approximately constant energy level. This arise in various problems where critical loss of compactness for the underlying energy appears. Three main equations are studied, namely: the Sobolev critical semilinear heat equation in \(\mathbb{R}^{n}\), the harmonic map flow from \(\mathbb{R}^{2}\) into S2, the Patlak-Keller-Segel system in \(\mathbb{R}^{2}\).

We consider the initial value problem u t = Δ log u , u ( x , 0 ) = u 0 ( x ) ⩾ 0 in R 2 , corresponding to the Ricci flow, namely conformal evolution of the metric u ( d x 1 2 + d x 2 2 ) by Ricci curvature. It is well known that the maximal solution u vanishes identically after time T = 1 4 π ∫ R 2 u 0 . Assuming that u 0 is radially symmetric and satisfies some additional constraints, we describe precisely the Type II collapsing of u at time T: we show the existence of an inner region with exponentially fast collapsing and profile, up to proper scaling, a soliton cigar solution, and the existence of an outer region of persistence of a logarithmic cusp. This is the only Type II singularity which has been shown to exist, so far, in the Ricci Flow in any dimension. It recovers rigorously formal asymptotics derived by J.R. King [J.R. King, Self-similar behavior for the equation of fast nonlinear diffusion, Philos. Trans. R. Soc. London Ser. A 343 (1993) 337–375].

We study the existence of radial ground state solutions for the problem−div(∇u1+|∇u|2)=uq,u>0inRN,u(x)→0as|x|→∞, N⩾3, q>1. It is known that this problem has infinitely many ground states when q⩾N+2N−2, while no solutions exist if q⩽NN−2. A question raised by Ni and Serrin in [W.-M. Ni, J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations, Atti Convegni Lincei 77 (1985) 231–257] is whether or not ground state solutions exist for NN−2

Let D be a bounded, smooth domain in R N , N ⩾ 3 , P ∈ D . We consider the boundary value problem in Ω = D ∖ B δ ( P ) , Δ u + u p = 0 , u > 0 in Ω , u = 0 on ∂ Ω , with p supercritical, namely p > N + 2 N − 2 . We find a sequence p 1 p 2 p 3 ⋯ , with lim k → + ∞ p k = + ∞ , such that if p is given, with p ≠ p j for all j, then for all δ > 0 sufficiently small, this problem is solvable.

For all $N \geq 9$, we find smooth entire epigraphs in $\R^N$, namely smooth domains of the form $\Omega : = \{x\in \R^N\ / \ x_N > F (x_1,\ldots, x_{N-1})\}$, which are not half-spaces and in which a problem of the form $\Delta u + f(u) = 0 $ in $\Omega$ has a positive, bounded solution with 0 Dirichlet boundary data and constant Neumann boundary data on $\partial \Omega$. This answers negatively for large dimensions a question by Berestycki, Caffarelli and Nirenberg \cite{bcn2}. In 1971, Serrin \cite{serrin} proved that a bounded domain where such an overdetermined problem is solvable must be a ball, in analogy to a famous result by Alexandrov that states that an embedded compact surface with constant mean curvature (CMC) in Euclidean space must be a sphere. In lower dimensions we succeed in providing examples for domains whose boundary is close to large dilations of a given CMC surface where Serrin's overdetermined problem is solvable., Comment: 59 pages

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.