A Fuchsian viewpoint on the weak null condition

In [35], Kerr showed that systems of wave equations of the form (1.1) with

$$\bar{a}{}_{IJ}^{K\alpha\beta}=\bar{I}{}^{KL}\bar{C}{}_{LIJ}\delta^{\alpha}_{0}\delta^{\beta}_{0},$$

(1.11)

where $\bar{I}{}^{KL}$ is a constant, positive definite, symmetric matrix and the $\bar{C}{}_{LIJ}$ are any constants satisfying

$$\bar{C}{}_{LIJ}=-\bar{C}{}_{ILJ},$$

(1.12)

have associated asymptotic equations that satisfy the bounded weak null condition. This is, in fact, easy to verify since the choice (1.11) leads, by (1.4)-(1.6), to the associated asymptotic equation

$$(2-t)\partial_{t}\xi^{K}=-\frac{2}{t}\chi(\rho)\rho^{m}\bar{I}{}^{KL}\bar{C}{}_{LIJ}\xi^{I}\xi^{J}.$$

(1.13)

Introducing the inner-product $(\xi|\eta)=\check{\bar{I}{}}{}_{IJ}\xi^{I}\eta^{J}$, where $(\check{\bar{I}{}}{}_{IJ})=(\bar{I}{}^{IJ})^{-1}$, and contracting (1.13) with $\check{\bar{I}{}}{}_{LK}\xi^{L}$, we get

$$(2-t)(\xi|\partial_{t}\xi)=-\frac{2}{t}\chi(\rho)\rho^{m}\check{\bar{I}{}}{}_{LK}\bar{I}{}^{KM}\bar{C}{}_{MIJ}\xi^{L}\xi^{I}\xi^{J}=-\frac{2}{t}\chi(\rho)\rho^{m}\bar{C}{}_{LIJ}\xi^{L}\xi^{I}\xi^{J}\overset{\eqref{Cbantisym}}{=}{}0.$$

(1.14)

But this implies $\partial_{t}((\xi|\xi))=0$, and so, we conclude that any solution of the asymptotic IVP (1.7) -(1.7) exists for all $t\in(0,1]$ and satisfies $(\xi(t)|\xi(t))=(\mathring{\xi}|\mathring{\xi})$. Letting $|\cdot|$ denote the Euclidean norm, we then have that $\frac{1}{\sqrt{C}}|\cdot|\leq\sqrt{(\cdot|\cdot)}\leq\sqrt{C}|\cdot|$ for some constant $C>0$, and consequently, by the above inequality, we arrive at the bound $\sup_{0<t\leq 1}|\xi(t)|\leq C|\mathring{\xi}|$, which verifies that the bounded weak null condition is fulfilled.

The calculation (1.14) also shows that this class of semilinear equations satisfies the structural condition from [33] called Condition H. Because of this, the global existence results established in from [33] apply and yield the existence of global solutions to (1.1) on the region $\bar{t}{}>0$ for suitably small initial data with compact support. We further note that due to the compact support of the initial data, the results of [33] can, in fact, be deduced as a special case of the global existence theory developed in [35], but do not apply to the situation we are considering in this article because we allow for non-compact initial data in addition to a less restrictive weak null condition.

Early global existence results for nonlinear wave equations in $3+1$ dimensions that violate the null condition but satisfy the weak null condition were established for quasilinear wave equations in [1, 41], systems of semilinear wave equations in [2], and the Einstein equations in wave coordinates in [43, 44]. More recent results can be found in [33], which we discussed above, for semilinear equations, and in the articles [12, 18, 28] and, as we discussed above, in [34, 35] for quasilinear equations.

Singular systems of hyperbolic equations that can be expressed in the form

$$B^{0}(t,u)\partial_{t}u+B^{i}(t,u)\nabla_{i}u=\frac{1}{t}\mathcal{B}{}(t,u)u+F(t,u)$$

(1.15)

are said to be Fuchsian. Traditionally, these systems have been viewed as singular initial value problems (SIVP) where asymptotic data is prescribed at the singular time $t=0$ and then (1.15) is used to evolve the asymptotic data away from the singular time to construct solutions on time intervals $t\in(0,T_{0}]$ for some, possibly small, $T_{0}>0$. The SIVP for Fuchsian equations has been studied by many mathematicians and has found a wide array of applications, for example, see [6, 13, 14, 17, 27, 31, 32, 37, 51, 52, 54] for applications in the analytic setting and [3, 4, 5, 8, 9, 10, 15, 36, 53, 55, 56] in the class of Sobolev regularity.

While the SIVP approach for establishing the existence of solutions to (1.15) is useful for certain applications, there are many situations where the global initial value problem (GIVP) for the Fuchsian system (1.15) is the relevant problem to solve. In this case, initial data is prescribed at some finite time $t=T_{0}>0$, and the problem becomes to establish the existence of solutions to (1.15) all the way up to the singular time at $t=0$, that is, for $t\in(0,T_{0}]$. The flavour of this problem is that of a global existence problem and the study of such problems was initiated by the first author in [49]. In that article, existence results were established for the Fuchsian GIVP, which were then used to deduce the future nonlinear stability of perturbations of Friedmann-Lemaître-Robertson-Walker (FLRW) solutions to the Einstein-Euler equations with a positive cosmological constant and a linear equation of state $p=K\rho$ for $0<K\leq 1/3$. This result represents the first instance of a new method for establishing the global existence of solutions to systems of hyperbolic equations that we now refer to as the Fuchsian method. Specifically, the goal of the Fuchsian method is to transform a given system of hyperbolic equations into a GIVP for a suitable Fuchsian system, and then to deduce the existence of solutions to the Fuchsian GIVP from general existence theorems such as those established in [49].

The advantage of the Fuchsian method is that solving the Fuchsian GIVP is technically much simpler compared to establishing global existence results for the original system of hyperbolic equations. The most difficult part of applying the Fuchsian method is finding a suitable set of variables and coordinates needed to bring the original system of hyperbolic equations into the required Fuchsian form, and this problem is typically more geometric-algebraic than analytic. In recent years, the GIVP existence theory for Fuchsian systems has been further developed in the articles [46, 45, 11, 20], and these existence results and those from [49] have been used to deduce the global existence of solutions for a number of different systems of hyperbolic equations in the articles [11, 20, 40, 45, 46, 47, 50, 58].

The results of this article can be generalized and extended in a number of ways, which we describe briefly here. First, the hierarchical weak null condition defined in [34], which is general enough to encompass all known results with the exception of those from [33, 35] and this article, can be easily handled using the Fuchsian method. Indeed, by introducing a re-scaling of the form $\mathtt{V}{}^{K}=t^{\mu_{K}}V^{K}$ for a suitable choices of the constants $\mu_{K}$, it is straightforward, assuming the hierarchical weak null condition, to bring the system (3.34) into a Fuchsian form in terms of the variables $\mathtt{V}{}^{K}$ that satisfies all the hypotheses of Theorem 3.8 of [11]. An application of this theorem then yields the existence of solutions to the semilinear system of wave equations (1.1) on $\bar{M}{}_{r_{0}}$ for suitable initial data specified on $\bar{\Sigma}{}_{r_{0}}$. We further note that all of the results of this article also hold in a neighborhood of timelike infinity. This follows from replacing the cylinder at spatial infinity construction from Section 2.1 by an analogous cylinder at temporal infinity construction. With this change, the arguments in this article go through in a similar fashion. In particular, this type of result in conjunction with the results of the current article can be combined to establish a global existence result for (1.1) on the regions $\bar{t}{}>0$ for initial data specified on the constant time hyperspace $\bar{t}{}=0$ that, importantly, does not have to be compactly supported. We will report on these extensions in a separate article.

We further note that systems of quasilinear wave equations can also be handled via the Fuchsian method. However, this requires an extension of the Fuchsian GIVP existence theory from [11] to allow for spatial manifolds with boundaries and the use of boundary weighted Sobolev spaces. This work will be reported on in a separate article that is currently in preparation. Finally, we note these types of weighted results are also of interest in the semilinear setting because they allow for for more general choices of initial data.

To prove the main result of this article, we use the Fuchsian method. This involves a sequence of steps where we transform the wave equation (1.1) on the non-compact domain $\bar{M}{}_{r_{0}}$ into a Fuchsian equation on a compact domain that is in a form that allows us to directly apply the Fuchsian GIVP existence theory from [11], and thereby, establish the existence of solutions on $\bar{M}{}_{r_{0}}$ to semilinear wave equations satisfying the bounded weak null condition.

The derivation of the Fuchsian equation begins in Section 2.1 where we map the domain $\bar{M}{}$, see (2.1), onto the cylinder at spatial infinity $(0,2)\times(0,\infty)\times\mathbb{S}^{2}$, which compactifies the outgoing null-rays. We then use this mapping in Section 2.3 to push-forward the wave equation (1.1) on $\bar{M}{}_{r_{0}}$. This yields the conformal wave equation (2.19) (see also (2.22)) defined on the manifold with boundary $M_{r_{0}}$ (see (2.23)) whose closure is now compact. We then proceed, in Section 3.1, to write the conformal wave equation in the first order form (3.13) using the variables $V^{K}=(V^{K}_{\mathcal{I}}{})$ defined by (3.5) and (3.11). This choice of variables is motivated by the structure of the most singular terms appearing in the conformal wave equation (see (3.4)) and the requirement that the resulting first order system be symmetric hyperbolic. The first order equation (3.13) is then replaced, in Section 3.2, by the extended system (3.39). The point of the extended system is twofold. First, it is defined on an extended spacetime region of the form $(0,1)\times\mathcal{S}{}$ where $\mathcal{S}{}$ is now a closed manifold, which, unlike $M_{r_{0}}$ is needed to apply the existence theory developed in [11], and second, its solutions yield solutions to (3.13) on $M_{r_{0}}\subset(0,1)\times\mathcal{S}{}$ that are independent of the particular form of the initial data on the “unphysical” part of the initial hypersurface $\{1\}\times\mathcal{S}{}$ given by $\{1\}\times\mathcal{S}{}\setminus\Sigma_{r_{0}}$. Here, $\Sigma_{r_{0}}$ is the spatial hypersurface where the initial data is prescribed. The upshot of this is that we lose nothing by working with the extended system (3.39) rather than the first order form of the conformal wave equation given by (3.13). Next, we differentiate the extended system (3.39) to obtain the system (3.59), which can be viewed as an evolution equation for the variables $W^{K}_{j}=t^{\kappa}(\mathcal{D}{}_{j}V^{K})$, and we use the flow of the asymptotic equation (1.5) to change variables from $V_{0}$ to a new variable $Y$ defined by (3.67)-(3.66). The point of this change of variables is that it removes the most singular term from the “time” component of the extended system (3.39), which results in the evolution equation (3.68). In Section 3.5, we then apply the projector $\mathbb{P}{}$ to the extended system (3.39) to obtain an equation for $X^{K}=t^{-\nu}\mathbb{P}{}V^{K}$ given by (3.85), and we combine the three systems (3.39), (3.59) and (3.85) into the single Fuchsian system (3.88) to obtain an evolution system for $Z=(W^{K}_{j},X^{K},Y^{K})$. It is then shown in Section 3.6 that, under the flow assumptions from Section 3.4.1, the Fuchsian system (3.88) satisfies, for a suitable choice of the parameters $\kappa,\nu$, all the assumptions needed to apply the Fuchsian GIVP existence theory from [11]. Applying Theorem 3.8. from [11] then yields the GIVP result for the Fuchsian system (3.88) that is stated in Theorem 4.1. This, in turn, yields, by construction, a small initial data global existence result for the original system of wave equations (1.1). Finally, we easily derive Corollary 4.3 from Theorem 4.1 using Proposition 3.2, which is the main result of the article and is stated informally as Theorem 1.2 in the Introduction.

The first step in transforming the system of semilinear wave equations (1.1) into Fuchsian form is to compactify the neighborhoods of spatial infinity defined by (1.9). To this end, we let

$$\bar{M}{}=\{\,(\bar{t}{},\bar{r}{})\in(-\infty,\infty)\times(0,\infty)\>|\>-\bar{t}{}^{2}+\bar{r}{}^{2}>0\,\}\times\mathbb{S}^{2}$$

(2.1)

and follow [21], see also [11, §4.1.1.], by mapping $\bar{M}{}$ to

$$M=(0,2)\times(0,\infty)\times\mathbb{S}^{2}$$

using the diffeomorphism

$$\psi\>:\>\bar{M}{}\longrightarrow M\>:\>(\bar{x}{}^{\mu})=(\bar{t}{},\bar{r}{},\theta,\phi)\longmapsto(x^{\mu})=\biggl{(}1-\frac{\bar{t}{}}{\bar{r}{}},\frac{\bar{r}{}}{-\bar{t}{}^{2}+\bar{r}{}^{2}},\theta,\phi\biggr{)},$$

(2.2)

where we label the coordinates on $M$ by

$$(x^{\mu})=(x^{0},x^{1},x^{2},x^{3})=(t,r,\theta,\phi).$$

The inverse of the mapping (2.2) is readily obtained by solving

$$t=1-\frac{\bar{t}{}}{\bar{r}{}},\quad r=\frac{\bar{r}{}}{-\bar{t}{}^{2}+\bar{r}{}^{2}},$$

(2.3)

for $(\bar{t}{},\bar{r}{})$ to get

$$\bar{t}{}=\frac{1-t}{rt(2-t)},\quad\bar{r}{}=\frac{1}{tr(2-t)}.$$

(2.4)

The importance of the diffeomorphism (2.2) is that it defines a compactification of null-rays in $\bar{M}{}$. Decomposing the boundary of $M$ as

$$\partial M=\mathscr{I}{}^{+}\cup i^{0}\cup\mathscr{I}{}^{-},$$

where

$$\mathscr{I}{}^{+}=\{0\}\times(0,\infty)\times\mathbb{S}^{2},\quad\mathscr{I}{}^{-}=\{2\}\times(0,\infty)\times\mathbb{S}^{2}{\quad\text{and}\quad}i^{0}=[0,2]\times\{0\}\times\mathbb{S}^{2},$$

the boundary components $\mathscr{I}{}^{\pm}$ correspond to portions of ($+$) future and ($-$) past null-infinity, respectively, while $i^{0}$ corresponds to spatial infinity. Furthermore, the spacelike hypersurface $\{1\}\times(0,\infty)\times\mathbb{S}^{2}$ in $M$ corresponds to the constant time hypersurface $\bar{t}{}=0$ in Minkowski spacetime.

Before proceeding with the transformation to Fuchsian form, we first derive an expansion formula for the tensor components $\bar{a}{}_{IJ}^{K\alpha\beta}$ that will play an important role in the calculations that follow. To start, we compute the Jacobian matrix

$$(\bar{J}{}^{\alpha}_{\mu})=\begin{pmatrix}1&0&0&0\\ 0&\sin(\theta)\cos(\phi)&\sin(\theta)\sin(\phi)&\cos(\theta)\\ 0&\frac{\cos(\theta)\cos(\phi)}{\bar{r}{}}&\frac{\cos(\theta)\sin(\phi)}{\bar{r}{}}&-\frac{\sin(\theta)}{\bar{r}{}}\\ 0&-\frac{\csc(\theta)\sin(\phi)}{\bar{r}{}}&\frac{\csc(\theta)\cos(\phi)}{\bar{r}{}}&0\end{pmatrix}$$

(2.5)

from the change of variables $(\hat{x}{}^{\mu})=(\bar{t}{},\bar{r}{}\cos(\phi)\sin(\theta),\bar{r}{}\sin(\phi)\sin(\theta),\bar{r}{}\cos(\theta))$ from spherical to Cartesian coordinates. Using this and the tensorial transformation law

$$\bar{a}{}_{IJ}^{K\alpha\beta}=\bar{J}{}^{\alpha}_{\mu}\hat{a}{}_{IJ}^{K\mu\nu}\bar{J}{}^{\beta}_{\nu},$$

(2.6)

we can expand the components (2.6) in powers of $\bar{r}{}$ as

$$\bar{a}{}_{IJ}^{K\alpha\beta}=\frac{1}{\bar{r}{}^{2}}\bar{e}{}_{IJ}^{K\alpha\beta}+\frac{1}{\bar{r}{}}\bar{d}{}_{IJ}^{K\alpha\beta}+\bar{c}{}_{IJ}^{K\alpha\beta}$$

(2.7)

where the expansions coefficients can be used to define the following geometric objects (see Appendix A for our indexing conventions) on $\mathbb{S}^{2}$:

1. (a)

   smooth functions $\bar{e}{}_{IJ}^{K\mathpzc{p}{}\mathpzc{q}{}}$, $\bar{d}{}_{IJ}^{K\mathpzc{p}{}\mathpzc{q}{}}$ and $\bar{c}{}_{IJ}^{K\mathpzc{p}{}\mathpzc{q}{}}$,

2. (b)

   smooth vector fields $\bar{e}{}_{IJ}^{K\mathpzc{q}{}\Lambda}$, $\bar{e}{}_{IJ}^{K\Lambda\mathpzc{q}{}}$, $\bar{d}{}_{IJ}^{K\mathpzc{q}{}\Lambda}$, $\bar{d}{}_{IJ}^{K\Lambda\mathpzc{q}{}}$, $\bar{c}{}_{IJ}^{K\mathpzc{q}{}\Lambda}$, and $\bar{c}{}_{IJ}^{K\Lambda\mathpzc{q}{}}$,

3. (c)

   and smooth (2,0)-tensor fields $\bar{e}{}_{IJ}^{K\Lambda\Sigma}$, $\bar{d}{}_{IJ}^{K\Lambda\Sigma}$ and $\bar{c}{}_{IJ}^{K\Lambda\Sigma}$.

The only terms of the expansion (2.7) that we will need to consider in any detail are the $\bar{c}{}_{IJ}^{K\alpha\beta}$. Now, it can be easily verified that the non-vanishing $\bar{c}{}_{IJ}^{K\alpha\beta}$ are given by

	$\displaystyle\bar{c}{}_{IJ}^{K00}$	$\displaystyle=\hat{a}{}_{IJ}^{K00},$		(2.8)
	$\displaystyle\bar{c}{}_{IJ}^{K01}$	$\displaystyle=\sin(\theta)(\hat{a}{}_{IJ}^{K01}\cos(\phi)+\hat{a}{}_{IJ}^{K02}\sin(\phi))+\hat{a}{}_{IJ}^{K03}\cos(\theta),$		(2.9)
	$\displaystyle\bar{c}{}_{IJ}^{K10}$	$\displaystyle=\sin(\theta)(\hat{a}{}_{IJ}^{K10}\cos(\phi)+\hat{a}{}_{IJ}^{K20}\sin(\phi))+\hat{a}{}_{IJ}^{K30}\cos(\theta)$		(2.10)
and
	$\displaystyle\bar{c}{}_{IJ}^{K11}$	$\displaystyle=\sin^{2}(\theta)\left(\hat{a}{}_{IJ}^{K11}\cos^{2}(\phi)+(\hat{a}{}_{IJ}^{K12}+\hat{a}{}_{IJ}^{K21})\sin(\phi)\cos(\phi)+\hat{a}{}_{IJ}^{K22}\sin^{2}(\phi)\right)$
		$\displaystyle\qquad+\sin(\theta)\cos(\theta)((\hat{a}{}_{IJ}^{K13}+\hat{a}{}_{IJ}^{K31})\cos(\phi)+(\hat{a}{}_{IJ}^{K23}+\hat{a}{}_{IJ}^{K32})\sin(\phi))+\hat{a}{}_{IJ}^{K33}\cos^{2}(\theta).$		(2.11)

Furthermore, with the help of (2.5) and (2.6), we find via a straightforward calculation that the $\bar{b}{}^{K}_{IJ}$, which are defined by (1.4), can be expressed as

	$\displaystyle\bar{b}{}^{K}_{IJ}$	$\displaystyle=\hat{a}{}_{IJ}^{K00}-\sin(\theta)(\hat{a}{}_{IJ}^{K01}\cos(\phi)+\hat{a}{}_{IJ}^{K02}\sin(\phi))-\hat{a}{}_{IJ}^{K03}\cos(\theta)-\sin(\theta)(\hat{a}{}_{IJ}^{K10}\cos(\phi)+\hat{a}{}_{IJ}^{K20}\sin(\phi))$
		$\displaystyle\qquad+\sin^{2}(\theta)\left(\hat{a}{}_{IJ}^{K11}\cos^{2}(\phi)+(\hat{a}{}_{IJ}^{K12}+\hat{a}{}_{IJ}^{K21})\sin(\phi)\cos(\phi)+\hat{a}{}_{IJ}^{K22}\sin^{2}(\phi)\right)$
		$\displaystyle\qquad+\sin(\theta)\cos(\theta)((\hat{a}{}_{IJ}^{K13}+\hat{a}{}_{IJ}^{K31})\cos(\phi)+(\hat{a}{}_{IJ}^{K23}+\hat{a}{}_{IJ}^{K32})\sin(\phi))-\hat{a}{}_{IJ}^{K30}\cos(\theta)+\hat{a}{}_{IJ}^{K33}\cos^{2}(\theta).$		(2.12)

Letting

$$\tilde{g}{}=\psi_{*}\bar{g}{}$$

denote the push-forward of the Minkowski metric (1.2) from $\bar{M}{}$ to $M$ using the map (2.2), we find after a routine calculation that

$$\tilde{g}{}=\Omega^{2}g$$

(2.13)

where

$$\Omega=\frac{1}{r(2-t)t}$$

(2.14)

and

$$g=-dt\otimes dt+\frac{1-t}{r}(dt\otimes dr+dr\otimes dt)+\frac{(2-t)t}{r^{2}}dr\otimes dr+{\centernot{g}}.$$

(2.15)

Using the map (2.2) to push-forward the wave equations (1.1) yields the system of wave equations

$$\tilde{g}{}^{\alpha\beta}\tilde{\nabla}{}_{\alpha}\tilde{\nabla}{}_{\beta}\tilde{u}{}^{K}=\tilde{a}{}^{K\alpha\beta}_{IJ}\bar{\nabla}{}_{\alpha}\tilde{u}{}^{I}\tilde{\nabla}{}_{\beta}\tilde{u}{}^{J}$$

(2.16)

where $\tilde{\nabla}{}_{\alpha}$ is the Levi-Civita connection of the metric $\tilde{g}{}_{\alpha\beta}$,

	$\displaystyle\tilde{u}{}^{K}$	$\displaystyle=\psi_{*}\bar{u}{}^{K}$		(2.17)
and
	$\displaystyle\tilde{a}{}^{K\alpha\beta}_{IJ}$	$\displaystyle=\psi_{*}(\bar{a}{}^{K}_{IJ})^{\alpha\beta}.$		(2.18)

Since $M=\psi(\bar{M}{})$, it is clear the original system of semilinear wave equations (1.1) on $\bar{M}{}$ are completely equivalent to (2.16) on $M$.

Next, we observe that the Ricci scalar curvature of $\tilde{g}{}_{\alpha\beta}$ vanishes by virtue of $\tilde{g}{}_{\alpha\beta}$ being the push-forward of the Minkowski metric. Furthermore, a straightforward calculation using (2.15) shows that the Ricci scalar of the metric $g_{\alpha\beta}$ also vanishes. Consequently, it follows from the formulas (B.5)-(B.6) and (B.8)-(B.9), with $n=4$, from Appendix B that the system of wave equations (2.16) transform under the conformal transformation (2.13) into

$$g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}u^{K}=f^{K}$$

(2.19)

where $\nabla$ is the Levi-Civita connection of $g$,

$$\tilde{u}{}^{K}=rt(2-t)u^{K}$$

(2.20)

and

	$\displaystyle f^{K}=\tilde{a}{}_{IJ}^{K\mu\nu}\biggl{(}\frac{1}{rt(2-t)}\nabla_{\mu}u^{I}\nabla_{\nu}u^{J}$	$\displaystyle+\frac{1}{(rt(2-t))^{2}}\bigl{(}\nabla_{\mu}(rt(2-t))u^{I}\nabla_{\nu}u^{J}+\nabla_{\mu}u^{I}\nabla_{\nu}(rt(2-t))u^{J}\bigr{)}$
		$\displaystyle+\frac{1}{(rt(2-t))^{3}}\nabla_{\mu}\bigl{(}rt(2-t)\bigr{)}\nabla_{\nu}(rt(2-t))u^{I}u^{J}\biggr{)}.$		(2.21)

We will refer to this system as the conformal wave equations.

A routine computation involving (2.15) then shows that conformal wave equations (2.19) can be expressed as

$$(-2+t)t\partial_{t}^{2}u^{K}+r^{2}\partial_{r}^{2}u^{K}+2r(1-t)\partial_{r}\partial_{t}u^{K}+{\centernot{g}}^{\Lambda\Sigma}{\centernot{\nabla}}{}_{\Lambda}{\centernot{\nabla}}{}_{\Sigma}u^{K}+2(t-1)\partial_{t}u^{K}=f^{K}$$

(2.22)

where ${\centernot{\nabla}}{}_{\Lambda}$ is the Levi-Civita connection of the metric (1.3) on $\mathbb{S}^{2}$.

For the remainder of the article, we will focus on solving the conformal wave equations (2.22) on neighborhoods of spatial infinity of the form

$$M_{r_{0}}=\bigl{\{}(t,r)\in(1,0)\times(0,r_{0})\,\bigl{|}t>2-r_{0}/r\bigr{\}}\times\mathbb{S}^{2}\subset M,\qquad r_{0}>0,$$

(2.23)

where initial data is prescribed on the spacelike hypersurface

$$\Sigma_{r_{0}}=\{1\}\times(0,r_{0})\times\mathbb{S}^{2}$$

(2.24)

that forms the “top” of the domain $M_{r_{0}}$. Noting that

$$\psi(\bar{M}{}_{r_{0}})=M_{r_{0}}{\quad\text{and}\quad}\psi(\bar{\Sigma}{}_{r_{0}})=\Sigma_{r_{0}}$$

by (1.9) and (1.10), we conclude that any solution of the conformal wave equations on $M_{r_{0}}$ with initial data prescribed on $\Sigma_{r_{0}}$ corresponds uniquely to a solution of the semilinear wave equations (1.1) on $\bar{M}{}_{r_{0}}$ with initial data prescribed on $\bar{\Sigma}{}_{r_{0}}$.

We now turn to deriving expansion formulas for the tensor components $\tilde{a}{}^{K\alpha\beta}_{IJ}$, defined by (2.18), that will determine their behavior in the limit $t\searrow 0$. These results will be essential for transforming the conformal wave equations (2.22) into Fuchsian form, which will be carried out in the following section.

Now, from (2.2) and (2.18), we find, after a routine calculation, that

	$\displaystyle\tilde{a}{}^{K00}_{IJ}$	$\displaystyle=4t^{2}r^{2}\tilde{b}{}^{K}_{IJ}+t^{3}r^{2}\tilde{c}{}^{K00}_{IJ},$		(2.25)
	$\displaystyle\tilde{a}{}^{K01}_{IJ}$	$\displaystyle=-4tr^{3}\tilde{b}{}^{K}_{IJ}+t^{2}r^{3}\tilde{c}{}^{K01}_{IJ},$		(2.26)
	$\displaystyle\tilde{a}{}^{K10}_{IJ}$	$\displaystyle=-4tr^{3}\tilde{b}{}^{K}_{IJ}+t^{2}r^{3}\tilde{c}{}^{K10}_{IJ},$		(2.27)
	$\displaystyle\tilde{a}{}^{K11}_{IJ}$	$\displaystyle=4r^{4}(1-2t)\tilde{b}{}^{K}_{IJ}+t^{2}r^{4}\tilde{c}{}^{K11}_{IJ},$		(2.28)
	$\displaystyle\tilde{a}{}^{K0\Lambda}_{IJ}$	$\displaystyle=-2tr(\bar{a}{}^{K0\Lambda}_{IJ}-\bar{a}{}^{K1\Lambda}_{IJ})\circ\psi^{-1}+t^{2}r(\bar{a}{}^{K0\Lambda}_{IJ}-3\bar{a}{}^{K1\Lambda}_{IJ})\circ\psi^{-1}+t^{3}r\bar{a}{}^{K1\Lambda}_{IJ}\circ\psi^{-1},$		(2.29)
	$\displaystyle\tilde{a}{}^{K\Sigma 0}_{IJ}$	$\displaystyle=-2tr(\bar{a}{}^{K\Sigma 0}_{IJ}-\bar{a}{}^{K\Lambda 1}_{IJ})\circ\psi^{-1}+t^{2}r(\bar{a}{}^{K\Sigma 0}_{IJ}-3\bar{a}{}^{K\Sigma 1}_{IJ})\circ\psi^{-1}+t^{3}r\bar{a}{}^{K\Sigma 1}_{IJ}\circ\psi^{-1},$		(2.30)
	$\displaystyle\tilde{a}{}^{K1\Lambda}_{IJ}$	$\displaystyle=2r^{2}(\bar{a}{}^{K0\Lambda}_{IJ}-\bar{a}{}^{K1\Lambda}_{IJ})\circ\psi^{-1}-2tr^{2}(\bar{a}{}^{K0\Lambda}_{IJ}-\bar{a}{}^{K1\Lambda}_{IJ})\circ\psi^{-1}-t^{2}r^{2}\bar{a}{}^{K1\Lambda}_{IJ}\circ\psi^{-1},$		(2.31)
	$\displaystyle\tilde{a}{}^{K\Sigma 1}_{IJ}$	$\displaystyle=2r^{2}(\bar{a}{}^{K\Sigma 0}_{IJ}-\bar{a}{}^{K\Sigma 1}_{IJ})\circ\psi^{-1}-2tr^{2}(\bar{a}{}^{K\Sigma 0}_{IJ}-\bar{a}{}^{K\Sigma 1}_{IJ})\circ\psi^{-1}-t^{2}r^{2}\bar{a}{}^{K\Sigma 1}_{IJ}\circ\psi^{-1}$		(2.32)
and
	$\displaystyle\tilde{a}{}^{K\Sigma\Lambda}_{IJ}$	$\displaystyle=\bar{a}{}^{K\Sigma\Lambda}_{IJ}\circ\psi^{-1},$		(2.33)

where

	$\displaystyle\tilde{b}{}^{K}_{IJ}$	$\displaystyle=(\bar{a}{}^{K00}_{IJ}-\bar{a}{}^{K01}_{IJ}-\bar{a}{}^{K10}_{IJ}+\bar{a}{}^{K11}_{IJ})\circ\psi^{-1},$		(2.34)
	$\displaystyle\tilde{c}{}^{K00}_{IJ}$	$\displaystyle=-4\left((\bar{a}{}^{K00}_{IJ}-2\bar{a}{}^{K01}_{IJ}-2\bar{a}{}^{K10}_{IJ}+3\bar{a}{}^{K11}_{IJ})\right)\circ\psi^{-1}+t(\bar{a}{}^{K00}_{IJ}-5\bar{a}{}^{K01}_{IJ}-5\bar{a}{}^{K10}_{IJ}+13\bar{a}{}^{K11}_{IJ})\circ\psi^{-1}$
		$\displaystyle\hskip 113.81102pt+t^{2}(\bar{a}{}^{K01}_{IJ}+\bar{a}{}^{K10}_{IJ}-6\bar{a}{}^{K11}_{IJ})\circ\psi^{-1}+t^{3}\bar{a}{}^{K11}_{IJ}\circ\psi^{-1}$		(2.35)
	$\displaystyle\tilde{c}{}^{K01}_{IJ}$	$\displaystyle=2(3\bar{a}{}^{K00}_{IJ}-3\bar{a}{}^{K01}_{IJ}-5\bar{a}{}^{K10}_{IJ}+5\bar{a}{}^{K11}_{IJ})\circ\psi^{-1}-2t\left((\bar{a}{}^{K00}_{IJ}-2\bar{a}{}^{K01}_{IJ}-4\bar{a}{}^{K10}_{IJ}+5\bar{a}{}^{K11}_{IJ})\right)\circ\psi^{-1}$
		$\displaystyle\hskip 113.81102pt-t^{2}(\bar{a}{}^{K01}_{IJ}+2\bar{a}{}^{K10}_{IJ}-5\bar{a}{}^{K11}_{IJ})\circ\psi^{-1}-t^{3}\bar{a}{}^{K11}_{IJ}\circ\psi^{-1},$		(2.36)
	$\displaystyle\tilde{c}{}^{K10}_{IJ}$	$\displaystyle=2(3\bar{a}{}^{K00}_{IJ}-5\bar{a}{}^{K01}_{IJ}-3\bar{a}{}^{K10}_{IJ}+5\bar{a}{}^{K11}_{IJ})\circ\psi^{-1}-2t\left((\bar{a}{}^{K00}_{IJ}-4\bar{a}{}^{K01}_{IJ}-2\bar{a}{}^{K10}_{IJ}+5\bar{a}{}^{K11}_{IJ})\right)\circ\psi^{-1}$
		$\displaystyle\hskip 113.81102pt-t^{2}(2\bar{a}{}^{K01}_{IJ}+\bar{a}{}^{K10}_{IJ}-5\bar{a}{}^{K11}_{IJ})\circ\psi^{-1}-t^{3}\bar{a}{}^{K11}_{IJ}\circ\psi^{-1}$		(2.37)
and
	$\displaystyle\tilde{c}{}^{K11}_{IJ}$	$\displaystyle=2(2\bar{a}{}^{K00}_{IJ}-3\bar{a}{}^{K01}_{IJ}-3\bar{a}{}^{K10}_{IJ}+4\bar{a}{}^{K11}_{IJ})\circ\psi^{-1}$
		$\displaystyle\hskip 113.81102pt+2t(\bar{a}{}^{K01}_{IJ}+\bar{a}{}^{K10}_{IJ}-2\bar{a}{}^{K11}_{IJ})\circ\psi^{-1}+t^{2}\bar{a}{}^{K11}_{IJ}\circ\psi^{-1}.$		(2.38)

We further observe from (2.2), (2.4), (2.7)-(2.2) and (2.34) that

	$\displaystyle\bar{a}{}^{K\mathpzc{p}{}\mathpzc{q}{}}_{IJ}\circ\psi^{-1}$	$\displaystyle=\bar{c}{}_{IJ}^{K\mathpzc{p}{}\mathpzc{q}{}}+tr(2-t)\bar{d}{}_{IJ}^{K\mathpzc{p}{}\mathpzc{q}{}}+t^{2}r^{2}(2-t)^{2}\bar{e}{}_{IJ}^{K\mathpzc{p}{}\mathpzc{q}{}},$		(2.39)
	$\displaystyle\bar{a}{}^{K\alpha\Lambda}_{IJ}\circ\psi^{-1}$	$\displaystyle=tr(2-t)\bar{d}{}_{IJ}^{K\alpha\Lambda}+t^{2}r^{2}(2-t)^{2}\bar{e}{}_{IJ}^{K\alpha\Lambda},$		(2.40)
	$\displaystyle\bar{a}{}^{K\Sigma\beta}_{IJ}\circ\psi^{-1}$	$\displaystyle=tr(2-t)\bar{d}{}_{IJ}^{K\Sigma\beta}+t^{2}r^{2}(2-t)^{2}\bar{e}{}_{IJ}^{K\Sigma\beta}$		(2.41)
and
	$\displaystyle\tilde{b}{}^{K}_{IJ}$	$\displaystyle=\bar{b}{}^{K}_{IJ}.$		(2.42)