Center-stable manifold of the ground state in the energy space for the critical wave equation

Abstract

We construct a center-stable manifold of the ground state solitons in the energy space for the critical wave equation without imposing any symmetry, as the dynamical threshold between scattering and blow-up, and also as a collection of solutions which stay close to the ground states. Up to energy slightly above the ground state, this completes the 9-set classification of the global dynamics in our previous paper (DCDS 33:6, 2013). We can also extend the manifold to arbitrary energy size by adding large radiation. The manifold contains all the solutions scattering to the ground state solitons, and also some of those blowing up in finite time by concentration of the ground states.

Appendices

Appendix A: Concentration blowup in the interior of blowup region

Here we observe that type-II blow-up is not always on the dynamical boundary between the scattering to $0$ and blow-up. More precisely, we have

Proposition 6.1

Let $0<T<\infty$, $\vec {u}^0\in \mathrm{S }_{\mathrm{olution }}([0,T))\cap L^\infty ([0,T);\mathcal {H})$. Then for any $\delta >0$, there is $\vec {u}^1\in \mathrm{S }_{\mathrm{olution }}([0,T))$ with the following property: $\vec {u}^1(t)-\vec {u}^0(t)$ has a strong limit as $t\nearrow T$, and for any $t\in [0,T)$ and any $\psi \in \mathcal {H}$ with $\Vert \psi \Vert _\mathcal {H}<\delta$, the solution starting from $\vec {u}^1(t)+\psi$ blows up in positive finite time.

In other words, for any blow-up with bounded energy norm, there is another solution with the same blow-up profile, whose orbit is in the interior of the blow-up set of initial data, with arbitrarily large distance from the exterior.

Proof

Fix $R \ge 1+T$ such that $\Vert \vec {u}^0(0)\Vert _{\mathcal {H}(|x|>R)} \ll 1$. Let $u^2$ be the solution for the initial data $\vec {u}^2(0)=\Gamma (x/R)\vec {u}^0(0)$, where $\Gamma$ is a smooth radial function on $\mathbb {R}^d$ satisfying $\Gamma (x)=1$ for $|x|\le 3$ and $\Gamma (x)=0$ for $|x|\ge 4$. Then the finite speed of propagation implies that for $0<t<T$ and as long as $u^2$ exists,

• (1) $\vec {u}^2(t)=\vec {u}^0(t)$ on $|x|<3R-t$,

• (2) $\Vert \vec {u}^2(t)\Vert _{\mathcal {H}(|x|>R+t)}\ll 1$,

• (3) $\mathrm{supp }\vec {u}^2(t)\subset \{|x|<4R+t\}$.

Since the regions for (1) and for (2) cover $[0,T)\times \mathbb {R}^d$, we deduce that $u^2$ extends beyond $t<T$. Moreover, both $u^0$ and $u^2$ extend to $|x|>R+t$ for all $t>0$ by the smallness in the exterior cone. Hence $\vec {u}^2(t)-\vec {u}^0(t)$ has a strong limit in $\mathcal {H}$ as $t\nearrow T$.

Now fix $\delta >0$. Since $u^2$ is bounded in $\mathcal {H}$ for $0<t<T$,

$$\begin{aligned} \begin{aligned} M:=\sup \{E_{|x|<5R}(\vec {u}^2(t)+\psi )\mid t\in [0,T),\ \Vert \psi \Vert _\mathcal {H}<\delta \} \end{aligned}\end{aligned}$$

(7.1)

is finite. Then we can find a strong radial solution $u^3$ such that

• (1) $\mathrm{supp }\vec {u}^3(t)\subset \{|x|>6R-t\}$.

• (2) $\sup \{E_{|x|>5R}(\vec {u}^3(t)+\psi )\mid t\in [0,T),\ \Vert \psi \Vert _\mathcal {H}<\delta \}<-M-1$.

Indeed, it is easy to satisfy (1) and (2) at $t=0$ by using a very flat radial smooth function, since for any $\varphi ,\psi \in \mathcal {H}$ and any $0<\varepsilon \ll 1$,

$$\begin{aligned} \begin{aligned} E_{|x|>5R}(\varphi +\psi ) \le (1+\varepsilon )E_{|x|>5R}(\varphi )+C_\varepsilon (\Vert \psi \Vert _\mathcal {H}^2+\Vert \psi _1\Vert _{2^*}^{2^*}). \end{aligned}\end{aligned}$$

(7.2)

(1) is preserved for $t>0$ by the finite speed of propagation. For such initial data, the solution may blow up in finite time, but we can delay the blow-up time as much as we like by the rescaling $\vec S^\sigma$ with $\sigma \rightarrow -\infty$, which makes both (1) and (2) easier. This yields $u^3\in \mathrm{S }_{\mathrm{olution }}([0,2T])$ with the above properties.

Now let $u^1$ be the strong solution for the initial data

$$\begin{aligned} \begin{aligned} \vec {u}^1(0)=\vec {u}^2(0)+\vec {u}^3(0). \end{aligned}\end{aligned}$$

(7.3)

Then the finite propagation property together with the disjoint supports of $u^2$ and $u^3$ implies that $\vec {u}^1=\vec {u}^2$ for $|x|<6R-t$, $\vec {u}^1=\vec {u}^3$ for $|x|>4R+t$, so $\vec {u}^1=\vec {u}^2+\vec {u}^3$ for $0<t<T$, and $\vec {u}^1(t)-\vec {u}^0(t)$ has a strong limit in $\mathcal {H}$ as $t\nearrow T$. Moreover, for any $t\in [0,T)$ and any $\psi \in \mathcal {H}$ satisfying $\Vert \psi \Vert <\delta$ we have

$$\begin{aligned} \begin{aligned} E(\vec {u}^1(t)+\psi ) =E_{|x|<5R}(\vec {u}^2(t)+\psi ) + E_{|x|>5R}(\vec {u}^3(t)+\psi )<-1, \end{aligned}\end{aligned}$$

(7.4)

hence the solution starting from $\vec {u}^1(t)+\psi$ has to blow up in finite time because of the negative energy, see [11, 18]. $\square$

Appendix B: Table of notation

${\triangleleft X^\triangleright }$	$=X^1-X^0$	(2.12)
$\vec {u}$	$=(u,\dot{u})$ vector in the phase space	(1.2)
$\varphi ^\dagger$	$=(\varphi _1,-\varphi _2)$ time inversion	(1.17)
${\langle \cdot |\cdot \rangle }$	$L^2$ inner product	(1.36)
$\fancyscript{B}_d$	Borel sets in $\mathbb {R}^d$
(CW)	The critical wave equation	(1.1)
$\mathcal {H}$, $\mathcal {H}_\perp$	Energy space, its subspace	(1.2),(1.35)
$\mathrm{S }_{\mathrm{olution }}(I)$	Solutions of (CW) on $I$	(1.27)
$E(\vec {u}),P(\vec {u})$	Total energy and momentum	(1.4),(1.5)
$E_B(\varphi ),K_B(\varphi )$	Restricted energy functionals	(3.15)
$U(t)$	Free propagator	(1.12)
$\mathcal {T}^c,\mathcal {S}^\sigma ,S_a^\sigma$	Invariant translation and scaling	(1.28)
${\mathrm{St }},{\mathrm{St }}_*^*,q_s,q_m$	Strichartz norms and exponents	(1.22)
$W$, $\mathrm{S }_{\mathrm{tatic }}(W)$	Ground states	(1.8),(1.9)
$\mathrm{S }_{\mathrm{oliton }}(W)$	Ground solitons	(1.10)
$\mathrm{dist }_W$, $d_W$	Distances to the ground states	(1.16),(1.59)
$L_+$, $\mathcal {L}$	Linearized operators around $W$	(1.30),(1.32)
$N(v),\underline{N}(v)$	Higher order terms	(1.31),(1.54)
$\rho$, $k$, $P_\perp$	Ground state of $L_+$	(1.34)
$g_\pm$, $\Lambda _\pm$	(un)stable modes of $J\mathcal {L}$	(1.39),(1.40)
$v,\lambda ,\gamma ,\lambda _\pm$	Components of $u$ around $W$	(1.38),(1.39)
$(\alpha ,\mu )$	Parameters to define the orthogonality	(1.42)
$(\widetilde{\sigma },\widetilde{c})$, $\fancyscript{T}_\varphi$, $\widetilde{\lambda }$	Local coordinates by the orthogonality	(1.52), Lemma 1.4
$Z=(Z_1,Z_2)$	Modulation operator in the equation	(1.54)
$\tau$	Rescaled time variable	(1.53)
$\Vert \varphi \Vert _E$, $\nu (\tau )$	Linearized energy norms	(1.43),(2.42)
$\mathcal {B}_\delta$, $\mathcal {B}_\delta ^+$, $\mathcal {B}_\delta '$	Small balls for different components	(1.45), (2.1)
$\mathcal {N}_\delta$, $\mathcal {N}_{\delta _1,\delta _2}$	Neighborhoods of $\mathrm{S }_{\mathrm{tatic }}(W)$	(1.45),(2.2)
$\Phi _{\sigma ,c}$, $\Psi _{\sigma ,c}$	Local coordinates around $\mathrm{S }_{\mathrm{tatic }}(W)$	(1.44)
$\mathcal {M}_0\sim \mathcal {M}_5$	Local manifold and its extensions	(2.8),(3.1),(3.33),(5.1)
$m_+,M_+$	Functions defining the local manifold	Theorem 2.1,(2.11)
$a_W,b_W$	Positive constants	(1.48),(1.49)
$\varepsilon _S$	Small Strichartz norm for scattering	(1.24)
$\delta _\Phi$, $\delta _m$	Small distances from $\mathrm{S }_{\mathrm{tatic }}(W)$	Lemma 1.4, Theorem 2.1
$\iota _I$	Smallness in the ignition lemma	Lemma 2.2
$\eta _l$	$\tau$-length for uniform Strichartz bound	Lemma 2.3
$\varsigma _m,\varsigma _*$	Smallness in radiative distance	(5.1), Theorems 6.3,6.5
$\varepsilon _*,\delta _*$	Smallness in the one-pass theorem	[14, Theorem 5.1]
$\kappa (\delta )$	Variational bound on $K$	[14, Lemma 4.1]
$B_{+a},B_{-a}$	Fattened and thinned sets by radius $a$	(3.3),(3.5)
${\mathcal {H}\!\downharpoonright \!B},\widetilde{\mathcal {H}}(B)$	Restrictions of $\mathcal {H}$ to $B$	(3.7),(3.10)
$X_B$	Extension operator from $B$ to $\mathbb {R}^d$	(3.14)
$\fancyscript{D}(u)$, $t_\pm (\varphi ,x)$	Maximal space-time domain of solution	(3.26)
$\fancyscript{R}_B^T$	Seminorm measuring radiation	(4.1)
$u^{\mathrm{\mathbf d }}$,$u^{\mathrm{\mathbf x }}$	Detached interior and exterior solutions	Lemma 4.4
$d_{\fancyscript{R}}(\varphi )$	Radiative distance to the ground states	(6.1)