Philippe G Lefloch

Weiping Yan

Nonlinear stability of blow-up solutions to the Hyperbolic Mean Curvature flow

Keywords: Hyperbolic Mean Curvature flow, self-similar solution, quasilinear wave equation, nonlinear stability, weighted Sobolev space. Math. Subject Class. 35B44, 35L05, 35C06, 35B35

We study the so-called Hyperbolic Mean Curvature (HMC) flow introduced by LeFloch and Smoczyk in 2008 for the evolution of a closed hypersurface moving in the direction of its mean curvature vector. This flow stems from a geometrically natural action consisting of a kinetic energy and an internal energy. We study the initial value problem for this flow in the case of an entire graph (in arbitrary dimension) and we establish the existence of a (singular) self-similar solution and its nonlinear stability in a suitably weighted Sobolev space by relying on Nash-Moser iterations.

1 Introduction

Formulation of the problem The hyperbolic flow of interest

We are interested in the Hyperbolic Mean Curvature flow of a closed hypersurface moving in the direction of its mean curvature vector. This flow was introduced by LeFloch and Smoczyk in [START_REF] Lefloch | The hyperbolic mean curvature flow[END_REF] and can be described as follows. Let F : [0, T ] × M -→ R n+1 be a smooth family of immersions of an orientable manifold M of dimension n into R n+1 , and let ν be the unit normal vector defined along the hypersurface and chosen to be inward pointing (when the manifold is compact without boundary). In local coordinates (x i ) i=1,2,...,n , we can write

F i := ∇ i F := dF ∂ ∂x i = ∂F ∂x i , ν = F 1 ∧ . . . ∧ F n |F 1 ∧ . . . ∧ F n | .
For this hypersurfacen, let us introduce the total kinetic energy defined by

K(t) := 1 2 M | d dt F | 2 dµ t
and the total internal energy defined by

V (t) := 1 2 M |∇F | 2 dµ t = n 2 M dt.
Then, the action of interest reads

S(F ) := T 0 (K(t) -V (t)) dt = T 0 M 1 2 | d dt F | 2 - n 2 dµ t dt.
The Euler-Lagrange equation associated with this functional is as follows:

d 2 dt 2 F = eHν -∇e, (1.1)
d dt F t=0 = 0,
in which H denotes the mean curvature of the hypersurface, ν is the unit normal, and e is the local energy density, with

|∇F | 2 = g ij < ∇ i F, ∇ j F >= g ij g ij = n e := 1 2 | d dt F | 2 + |∇F | 2 = 1 2 | d dt F | 2 + n . (1.
2)

The case of entire graphs

For an entire graph, we assume that F has the form F (t, x) = x(t), u(t, x(t)) (1.3) for a time dependent family of height functions u : [0, T ) × M -→ R and a family of diffeomorphisms x(t) = (x 1 (t), . . . , x n (t)) defined on the hyperplane

M = Z ⊥ = {(x 1 , . . . , x n+1 ) ∈ R n+1 : x n+1 = 0}.
The hyperbolic mean curvature flow for graphs over a flat hypersurface Z ⊥ is governed by the quasilinear wave equation

-u tt + (e( u t w )g ij + u 2 t w 2 (g ij -δ ij ))u ij + 2 u t w 2 δ ij u i u tj = 0, (1.4)
where u tt := ∂ 2 u ∂t 2 , u tj := ∂ 2 u ∂t∂x j and

w := 1 + |Du| 2 = 1 + δ ij u i u j , g ij := δ ij -w -2 δ ik δ jl u k u l , e := 1 2 (| d dt F | 2 + n) = 1 2 u 2 t 1 + |Du| 2 + n .
(

1.5)

There exists a function σ and b = (u i ) i=1,...,n such that equation (1.4) recasting in a form of nonlinear hyperbolic system ∂σ ∂t -∂ ∂x j e(σ) b i w δ i,j = 0,

∂b i ∂t - ∂ ∂x i (σw) = 0,
which has a conservation form and is endowed with the entropy function

E(σ, b) = 1 2 (σ 2 + n) 1 + δ i,j b i b j .
Local-in-time well-posed is known for the graph equaton (1.4).

Theorem 1.1 (LeFloch and Smoczyk [START_REF] Lefloch | The hyperbolic mean curvature flow[END_REF]). Given any s > 1+ n 2 . and initial data (σ(0, x), b(0, x)) ∈ H s+1 (R n ) × H s (R n ), the graph equation (1.4) admits a classical solution u : [0,

T ) × R n → R in L ∞ ([0, T ), H s+1 (R n )) ∩ Lip([0, T ), H s (R n )).

Main result

In view of the above theorem, it is natural to investigate whether singular solutions do exist and whether they are stable. In this paper, we study the stability of self-similar singular solutions to the graph equation (1.4). We set |x| = r and u(t, x) = v(t, r) and the quasilinear wave equation (1.4) reduces to

-v tt + 1 2 ( v 2 t 1 + v 2 r + n)(v rr + n r v r - v rr v 2 r + r -1 v 3 r 1 + v 2 r ) - v 2 t (v rr v 2 r + r -1 v 3 r ) (1 + v 2 r ) 2 + 2 v t v tr v r 1 + v 2 r
= 0, t ∈ (0, T ).

(1.6)

We supplement equation (1.6) with an initial data, say v(0, r) = v 0 (r), v t (0, r) = v 1 (r), r ∈ (0, T ].

(1.7)

Let Ω = (0, T ) × (0, T -t) be the right-angled triangle domain.

If v(t, r) is a solution of (1.6)-(1.7), there exists an exact scaling invariance so that v(t, r) can be mapped to another solution given by v µ (t, r) = µv( t µ , r µ ), with µ > 0.

(1.8)

Under this acaling the conserved energy

E(u) = ∞ 0 ( 1 2 v 2 t + n 2 4 v 2 r + F (v t , v r ))rdr,
where

F (v t , v r ) = v 2 t 2(1 + v 2 r ) (v rr + n r v r - v rr v 2 + r -1 v 3 r 1 + v 2 r ) - n(v rr v 2 + r -1 v 3 r ) 2(1 + v 2 r ) - v 2 t (v rr v 2 r + r -1 v 3 r ) (1 + v 2 r ) 2 + 2 v t v tr v r 1 + v 2 r , (1.9)
can be transformed as E(u µ ) = µE(u).

In general, the scaling (1.8) invariant Sobolev spaces are H sn × H sn-1 with the index s n = 1 + n 2 corresponds to the critical regularity. So we have an energy-supercritical case for any n ≥ 3. One expects smooth finite energy initial data to lead to finite time blow-up, and the blow up behavior should be described by the self-similar blow-up solution.

Namely, we will study the local behavior of self-similar solutions in the backward light-cone L T := {(t, r), t ∈ (0, T ), r ∈ (0, T -t]}.

We are in a position to state the main result in the present paper.

Theorem 1.2.

• Let κ ∈ (0, 1]. For any dimension n ≥ 3, the radial graph equation (1.6) admits a family of self-similar solutions

v T (t, r) = κ(T -t)φ( r T -t ) (1.10)
defined in the backward light-cone L T with φ( r T -t ) ∈ C ∞ ((0, 1]) and (φ(0), φ (0)) = (0, φ 1 ) (with φ 1 = 0).

• Furthermore, let k ≥ 3 be a fixed positive constant, and be a small positive constant and consider any v 0 , v 1 ∈ C ∞ ((0, T )) in (1.7). If

(v 0 , v 1 ) -(v T (0, •), v T t (0, •)) H k ((0,T ))×H k-1 ((0,T )) < , (1.11)
then there exist small positive constants κ 0 and T such that for any fixed κ ∈ (0, κ 0 ], the problem (1.6)-(1.7) with the boundary condition v| ∂L T ≡ 0 and v r | ∂L T ≡ 0 admits a unique solution v = v(t, r) satisfying for all t ∈ (0, T )

(v(t, •), v t (t, •)) -(v T (t, •), v T t (t, •) H k ((0,T ))×H k-1 ((0,T )) |T -t| 2 .
(1.12)

Thoughout this paper, we denote the usual norm of L 2 ((0, T )) and H k ((0, T )) by • L 2 ((0,T )) and • H k ((0,T )) , respectively. For all real number k ≥ 0 and all γ ≥ 1, we introduce the weighted Sobolev space

H γ,k (Ω) := {v ∈ D (Ω)/(T -t) -γ v ∈ H k ((0, T ))}, (1.13)
equipped with the norm

v H γ,k (Ω) := (T -t) -γ ∂ k v H k ((0,T )) , which is equivalent to the norm (T -t) -γ v H k ((0,T )) . The space L p γ ((0, T )) is the set of mea- surable functions v such that (T -t) -2γp v belongs to L p ((0, T )). The space L 2 ((0, T ); H γ,k (Ω)) is equipped with the norm v 2 L 2 ((0,T );H γ,k (Ω)) := T 0 v(t, •) 2 H γ,k (Ω) dt.
Furthermore, we introduce the function space

C l γ := 2 i=0 C i ((0, T ); H γ,l-i (Ω)) with the norm v 2 C l γ := sup t∈(0,T ) 2 i=0 ∂ i t v 2 H γ,l-i (Ω) ,
For convenience, we denote the weighted space H γ,k (Ω) by H γ,k . In Theorem 1.2, the solution v(t, r) is a regular solution of (1.6) in Ω. More precisely, the solution takes the form

v(t, r) = κ(T -t)φ( r T -t ) + w(t, r),
where w(t, r) satisfies one dimensional quasilinear wave equation with singular coefficient The equation for w is a quasilinear wave equation, and there is no special structure in nonlinear term. Although the linearized equation at self-similar solution κ(T -t)φ( r T -t ) is no "loss of derivative", there is "loss of derivative" in solving nonlinear equation. To overcome this difficulty, we have to rely on Nash-Moser iteration technique to construct an approximation solution w(t, r) in weighted Sobolev space H γ,k (Ω). Moreover, there should be a boundary condition on the right-angled triangle domain Ω. Here we supplement the boundary condition v| ∂L T ≡ 0 and v r | ∂L T ≡ 0. This is an important condition for getting suitable energy estimates to linearized equation.

, ν(T -ν -1 t)φ( ν -1 r T -ν -1 t
) is also a solution of the radial graph equation (1.6). This implies that the blow up time can advance the time T . Remark 1.4. In fact, w(t, r) H 2,k (Ω) < ∞, that is, w(t, r) H k ((0,T )) |T -t| 2 . The order 2 of T -t is determined by energy estimates of linearized equation when we deal with the singular coefficient 1 T -t . Hence, the self-similar solution (T -t)φ( r T -t ) is stable in the sense that there exists an open set of radial initial data that leads to a solution which converges to φ as t → T - in the backward light-cone of the blow up point (t, r) = (T, 0). Remark 1.5. In Theorem 1.2, we impose a condition on the existence time T of solution to (1.4), i.e. T is required as a small parameter. It is a natural condition. Because the existence of solution is short time smooth as pointed out in [START_REF] Lefloch | The hyperbolic mean curvature flow[END_REF].

Here we deal with the case of dimensional n ≥ 3. When n = 2, there is degenerate in the coefficient n 2 -y 2 of linear operator, where y = r T -t ∈ [0, 1]. Obviously, y = 1 is the degenerate point. Hence it is still open problem on the case of two dimension to (1.4).

Further background

Hyperbolic mean curvature flow is a geometric and physical model describing the motion of closed hypersurface in Riemann manifold. One of simple application is an idealized mathematical model for a moving soap bubble. This flow is different from classical mean curvature flow, the motion of equation of it is hyperbolic equations. LeFloch and Smoczyk [START_REF] Lefloch | The hyperbolic mean curvature flow[END_REF] first established a hyperbolic mean curvature flow (1.1), which is a strickly hyperbolic equation when the tangential part vanishing. Local well-posed of classical solution and global existence of weak solution were given. Meanwhile, the solution is short time smooth. There exist many results on the local existence of smooth solutions and other properties of solutions; see [START_REF] He | Hyperbolic mean curvature flow[END_REF][START_REF] Nguyen | On smoothness of timelike maximal cylinders in three-dimensional vacuum spacetimes[END_REF][START_REF] Notz | Closed hypersurfaces driven by mean curvature and inner pressure[END_REF][START_REF] Yan | The motion of closed hypersurfaces in the central force fields[END_REF]. The analysis of singularities is an important topic in the study of nonlinear hyperbolic PDEs. There are many works on the analysis of singularities for the mean curvature flow [START_REF] Chen | Two-dimensional graphs moving by mean curvature flow[END_REF][START_REF] Colding | Rigidity of generic singularities of mean curvature flow[END_REF][START_REF] Colding | Uniqueness of blowups and Lojasiewicz inequalities[END_REF][START_REF] Colding | The singular set of mean curvature flow with generic singularities[END_REF][START_REF] Li | The blow-up locus of heat flows for harmonic maps[END_REF]. In the present paper, we study the existence of self-similar blow up solutions to the graph equation (1.4) of hyperbolic mean curvature flow (1.1). In dealing with the nonlinear stability of self-similar to quasilinear wave equation (1.4), we can not use the semigroup method [START_REF] Donninger | On stable self-similar blow up for equivariant wave maps[END_REF][START_REF] Donninger | On stable self-similar blow up for equivariant wave maps: the linearized problem[END_REF]. Since there is a loss of derivative in dealing with nonlinear terms, the fixed point theory cannot be applied. This means that it does not work when we transform this problem into asymptotic stability problem in self-similar coordinates. So we have to use Nash-Moser iteration scheme to overcome this difficulty, as was used also in [START_REF] Yan | The motion of closed hypersurfaces in the central force fields[END_REF].

The organization of this paper is as follows. In Section 2, we show the existence of self-similar solutions to equation (1.4). In section 3, we first analyze the linearized graph equation (1.4) around the self-similar solution, which is a quasilinear wave equation with singular coefficients. Then the local existence of solution and some useful energy estimates of it are given. Then the nonlinear stability is proven by a Nash-Moser iteration scheme.

2 Existence of self-similar solutions

Objective

In this section, we prove there exists a self-similar solution to equation (1.4). It is natural to look for self-similar solutions of the form

v(t, r) = κ(T -t)φ( r T -t ), (2.1)
where T is a positive constant, and κ ∈ (0, 1]. We deal with the case of κ = 1. When κ ∈ (0, 1), it can be directly derived by using the same process of proof with the case of κ = 1. So we discuss it in the last of this section. Inserting the Ansatz (2.1) with κ = 1 into (1.6) by observing

∂ t v(t, r) = -φ(ρ) + ρφ (ρ), ∂ tt v(t, r) = (T -t) -1 ρ 2 φ (ρ), ∂ r v(t, r) = φ (ρ), ∂ rr v(t, r) = (T -t) -1 φ (ρ), ∂ tr v(t, r) = (T -t) -1 ρφ (ρ), the profile function φ satisfies a quasilinear ODE ( n 2 -y 2 )φ + n 2 2y φ + 1 2 (-φ + yφ ) 2 1 + φ 2 (φ + ny -1 φ - φ φ 2 + y -1 φ 3 1 + φ 2 ) - nφ φ 2 + ny -1 φ 3 2(1 + φ 2 ) - (-φ + yφ ) 2 (φ φ 2 + y -1 φ 3 ) (1 + φ 2 ) 2 + 2φ (-φ + yφ )((y -1)φ + y 2 φ ) 1 + φ 2 = 0, (2.2)
with initial data

φ(0) = φ 0 = 0, φ (0) = φ 1 = 0, (2.3)
where y = r T -t and = d dy . Obviously, equation (2.2) is singular at y = 0, this is due to the breakdown of spherical coordinates at the origin. We concern with the existence of smooth solutions in y ∈ [0, 1]. The mth derivative of solution to (2.2) at origin has the form

∂ m ∂r m φ( r T -t )| r=0 = (T -t) -m d m φ dy m (0).
Note that the closed interval y ∈ [0, 1] correponds to the interior of the past light cone of the point (t, r) = (T, 0). Hence every self-similar solution φ(y) ∈ C ∞ ([0, 1]) is an example of a singularity developing in finite time from smooth initial data.

A priori estimate

Obviously, (2.2) can be rewritten as

( n 2 -y 2 )φ + n 2 2y φ + [(-φ + yφ ) 2 (1 -2φ 2 ) + ((2y 3 -n)φ 2 -2yφ φ)(1 + φ 2 )]φ 2(1 + φ 2 ) 2 + (-φ + yφ ) 2 2y(1 + φ 2 ) (nφ - φ 3 1 + φ 2 ) - nφ 3 2y(1 + φ 2 ) + (-φ + yφ ) 2 φ 3 y(1 + φ 2 ) 2 + 2(y -1)φ 2 (-φ + yφ ) 1 + φ 2 = 0, (2.4)
which has a singularity at y = 0. This singularity correspond to the outgoing light cone {(t, r) : |r| = t, t > 0} and the incoming light cone {(t, r) : |r| = -t, t < 0} respectively. By the self-similar tranformation, the backward light-cone L T is transformed into LT := {(t, y), t ∈ (0, T ), y ∈ [0, 1]}.

We are now giving a priori estimate on φ (y) and φ (y) for y ∈ [0, 1].

Lemma 2.1. Assume that φ(y) is a local solution of (2.4) with y ∈ (0, 1). Then φ (y) and yφ (y) are bounded with y ∈ (0, 1). Furthermore, lim y→0 + φ(y) and lim y→0 + φ (y), lim y→1 -φ(y) and lim y→1 -φ (y) exist.

Proof. Let y 0 ∈ (0, 1) be near 1, and y 0 ∈ (0, 1) be near 0. To prove this result, we only need to prove φ (y) is bounded in (0, y 0 ), [y 0 , y 0 ] and (y 0 , 1), repectively. First, we show that |φ | is bounded in [y 0 , y 0 ]. For al ỹ ∈ [y 0 , y 0 ], we consider the interval B 1 = (ỹ -δ, ỹ + δ) with 0 < δ 1. In the interval B 1 , multiplying both sides of (2.4) with φ , we get

d dy (( n 2 -y 2 )|φ | 2 ) + (y + n 2 2y )|φ | 2 + [(-φ + yφ ) 2 (1 -2φ 2 ) + ((2y 3 -n)φ 2 -2yφ φ)(1 + φ 2 )]φ φ 2(1 + φ 2 ) 2 + (-φ + yφ ) 2 2y(1 + φ 2 ) (n|φ | 2 - |φ | 4 1 + φ 2 ) - n|φ | 4 2y(1 + φ 2 ) + (-φ + yφ ) 2 |φ | 4 y(1 + φ 2 ) 2 + 2(y -1)|φ | 3 (-φ + yφ ) 1 + φ 2 = 0. (2.5)
By the definition of derivative, for all y ∈ B 1 , there exists a small positive constant δ < m 1 such that my ∈ B 1 , and

|φ (y)| < | φ ((m + 1)y) -φ (y) my | + δ ≤ |φ ((m + 1)y)| + |φ (y)| my + δ. (2.6)
We observe that it follows from the inequality ab ≤ a p p + b q q with a, b > 0 and

1 p + 1 q = 1 that |φ 2 (y)φ (y)| ≤ 2 3 |φ| 3 + 1 3 |φ | 3 , (2.7
+ |(φ (y)) 2 (-φ(y) + yφ (y)) 2 | my +δ|(-φ(y) + yφ (y)) 2 φ (y)| ≤ c m -1 ,δ (|φ| 3 + |φ | 3 + |φ| 4 + |φ | 4 ), (2.9)
where c m -1 ,δ is a positive constant depending on m -1 and δ.

Similarly,

| (φ -yφ ) 2 φ φ 2 2(1 + φ 2 ) 2 | ≤ c m -1 ,δ (|φ| 4 + |φ| 6 + |φ | 2 + |φ | 4 + |φ | 6 ), (2.10)
| ((2y 3 -n)φ 2 -2yφ φ)φ φ 2(1 + φ 2 ) | ≤ c m -1 ,δ (|φ| 4 + |φ| 6 + |φ | 2 + |φ | 4 + |φ | 6 ), (2.11)
Thus by (2.9)-(2.11), we have

| [(-φ + yφ ) 2 (1 -2φ 2 ) + ((2y 3 -n)φ 2 -2yφ φ)(1 + φ 2 )]φ φ 2(1 + φ 2 ) 2 | ≤ c m -1 ,δ (|φ| 2 + |φ| 4 + |φ | 2 + |φ | 4 + |φ | 6 ).
(2.12)

Again by Young's inequality and the inequality ab ≤ a p p + b q q with a, b > 0 and 1 p + 1 q = 1, we can derive

| (-φ + yφ ) 2 2y(1 + φ 2 ) (n|φ | 2 - |φ | 4 1 + φ 2 )| |φ| 2 + |φ | 2 + |φ| 4 + |φ | 4 , (2.13)
| n|φ | 4 2y(1 + φ 2 ) | |φ | 4 , (2.14)
| (-φ + yφ ) 2 |φ | 4 y(1 + φ 2 ) 2 | |φ| 2 + |φ| 4 + |φ | 6 , (2.15)
| 2(y -1)|φ | 3 (-φ + yφ ) 1 + φ 2 | |φ| 4 + |φ | 4 . (2.16)
Hence, by (2.5) and (2.12)-(2.16), there exists a positive constant c m -1 ,δ depending on m -1 and δ such that

d dy (( n 2 -y 2 )|φ | 2 ) ≤ c m -1 ,δ |φ| 2 + |φ | 2 + |φ| 4 + |φ | 4 + |φ | 6 .
(2.17)

Since n ≥ 3, n 2 -y 2 > 0. Integrating (2.17) over (ỹ -δ, y) with y ∈ (ỹ -δ, ỹ + δ), ( n 2 -y 2 )|φ (y)| 2 -( n 2 -y 2 )|φ (ỹ -δ)| 2 ≤ c m -1 ,δ y ỹ-δ |φ| 2 + |φ | 2 + |φ| 4 + |φ | 4 + |φ | 6 dy ≤ c m -1 ,δ ỹ+δ ỹ-δ |φ| 2 + |φ | 2 + |φ| 4 + |φ | 4 + |φ | 6 dy.
(2.18)

Using Poincaré inequality, it follows from (2.18) that

( n 2 -y 2 )|φ | 2 ≤ ( n 2 -y 2 )||φ (ỹ -δ)| 2 + c m -1 ,δ ỹ+δ ỹ-δ |φ | 2 + |φ | 4 + |φ | 6 dy + c m -1 ,ỹ,δ , (2.19)
where c m -1 ,ỹ,δ is a positive constant depending on m -1 , ỹ and δ. Since it needs φ(ỹ

-δ) = 0 in the proof of Poincaré inequality, c m -1 ,ỹ,δ appears in (2.19). Integrating (2.19) over (ỹ -δ, ỹ + δ), we derive φ 2 0 ≤ 2δ|φ (ỹ -δ)| 2 + 2δc m -1 ,ỹ,δ ( φ 2 0 + φ 4 0 + φ 6 0 ) + 2δc m -1 ,ỹ,δ , (2.20)
where c m -1 ,ỹ,δ is a positive constant depending on m -1 , ỹ and δ. Let z = φ 2 0 > 0. Inequality (2.20) gives us

P 0 (z) := -2δc m -1 ,ỹ,δ (z + z 2 + z 3 ) + z -c m -1 ,ρ,δ,|φ(ỹ-δ)| ≤ 0, (2.21)
where c m -1 ,ỹ,δ,|φ(ỹ-δ)| is a positive constant depending on m -1 , ỹ, δ and |φ(ỹ -δ)|.

Since we have chosen 0 < δ < m 1, there exists a suitable δ such that 2δc m -1 ,ỹ,δ < 1. Furthermore, there is a greatest root z0 of the function of P 0 (z) such that z ≤ z0 , i.e. φ 2 0 ≤ z0 . So for any y ∈ (ỹ -δ, ỹ + δ), it holds

|φ| = | y ỹ-δ φ dy + φ(ỹ -δ)| ≤ | ỹ+δ ỹ-δ φ dy| + |φ(ỹ -δ)| ≤ φ 0 + |φ(ỹ -δ)| ≤ z 1 2 0 + |φ(ỹ -δ)|, which means there exists a positive constant M 0 such that |φ| ≤ M 0 . (2.22)
On the other hand, combining with (2.17) and (2.22), we have

d dy |φ | 2 (φ ) 6 + (φ ) 4 + (φ ) 2 + M 4 0 + M 2 0 , (2.23) Thus by (2.23), 2|φ | 2 d dy |φ | 2 ((φ ) 6 + (φ ) 4 + (φ ) 2 + M 4 0 + M 2 0 ) |φ | 2 , (2.24)
and an integration of (2.24) yields

|φ | 2 ỹ-δ ds (s 3 + s 2 + s + M 4 0 + M 2 0 ) y ỹ-δ |φ | 2 dy z0 . (2.25) Note that +∞ ỹ-δ ds (s 3 + s 2 + s + M 4 0 + M 2 0 ) = +∞.
Hence (2.25) holds only for

|φ | ≤ M 0 , y ∈ (ỹ -δ, ỹ + δ). ( 2
|φ | ≤ M i , y ∈ B i ,
Thus we conclude that there exists a positive constant

M = max{M 0 , M 1 , . . . , M N } such that |φ | ≤ M i , y ∈ [y 0 , y 0 ].
Next we prove φ is bounded in (0, y 0 ) and (y 0 , 1). For any y ∈ (0, y 0 ) or y ∈ (y 0 , 1), we claim that

|φ (y)| ≤ max{z 0 , |φ (y 0 )|}, (2.27)
where z 0 is a positive constant such that the polynomial with order 6

P 1 (z) := a 1 z 6 + a 2 z 5 + a 3 z 4 + a 4 z 3 + a 5 z 2 + a 6 z + a 7 ≤ 0 holds for |z| ≤ z 0 . a i (i = 1, 2, . . . , 7
) denote the coefficients, and a 1 , a 7 > 0. Assume there exists y 1 such that |φ (y 1 )| > |φ (y 0 )| with y 1 > y 0 . Then there must be |φ (y)| ≤ z 0 for all y ∈ [y 0 , y 1 ]. To prove this, assume that there exists a point y 2 ∈ (y 0 , y 1 ] such that |φ (y 2 )| = sup y∈[y 0 ,y 1 ] |φ (y)|. For a sufficient small positive constant δ and for all y ∈ (y 2 -δ, y 2 ], it follows from (2.5) that

0 ≤ d dρ (( n 2 -y 2 )|φ | 2 ) = -(y + n 2 2y )|φ | 2 - [(-φ + yφ ) 2 (1 -2φ 2 ) + ((2y 3 -n)φ 2 -2yφ φ)(1 + φ 2 )]φ φ 2(1 + φ 2 ) 2 - (-φ + yφ ) 2 2y(1 + φ 2 ) (n|φ | 2 - |φ | 4 1 + φ 2 ) + n|φ | 4 2y(1 + φ 2 ) - (-φ + yφ ) 2 |φ | 4 y(1 + φ 2 ) 2 - 2(y -1)|φ | 3 (-φ + yφ ) 1 + φ 2 , which gives (y + n 2 2y )|φ | 2 + [(-φ + yφ ) 2 (1 -2φ 2 ) + ((2y 3 -n)φ 2 -2yφ φ)(1 + φ 2 )] (1 + φ 2 ) 2 d dy |φ | 2 + (-φ + yφ ) 2 |φ | 2 2y(1 + φ 2 ) (n - |φ | 2 1 + φ 2 ) - n|φ | 4 2y(1 + φ 2 ) + (-φ + yφ ) 2 |φ | 4 y(1 + φ 2 ) 2 + 2(y -1)|φ | 3 (-φ + yφ ) 1 + φ 2 ≤ 0, y ∈ (y 2 -δ, y 2 ],
which means that

(y + n 2 2y )|φ | 2 + [(-φ + yφ ) 2 (1 -2φ 2 ) + ((2y 3 -n)φ 2 -2yφ φ)(1 + φ 2 )] (1 + φ 2 ) 2 d dy |φ | 2 - n|φ | 4 2y(1 + φ 2 ) + 2(y -1)|φ | 3 (-φ + yφ ) 1 + φ 2 ≤ 0, y ∈ (y 2 -δ, y 2 ]. (2.28)
By setting z := |φ |, the inequality (2.28) tells us that there exists a polynomial with order 6 such that

P 1 (z) = a 1 z 6 + a 2 z 5 + a 3 z 4 + a 4 z 3 + a 5 z 2 + a 6 z + a 7 ≤ 0, (2.29)
where

a 1 = y + n(n -1) 2y + 2sgn(φ )y(y -1), a 2 = 2(1 -y)φ, a 3 = 2y + n 2 y -(n + y 2 ) dz 2 dy + 2sgn(φ )(y -1)y, a 4 = [2sgn(φ )y dz 2 dy + 2(1 -y)]φ - n 2y , a 5 = y + n 2 2y + (y 2 -n 2 -φ 2 + 2y 3 ) dz 2 dy , a 6 = -2yφ(1 + y) dz 2 dy , a 7 = φ 2 dz 2 dy . Note that y ∈ [0, 1], n ≥ 3 and dz 2 dy > 0 in (y 2 -δ, y 2 ]
. The coefficients a 1 > 0 and a 7 > 0. Thus if z is unbounded, then inequality (2.29) does not hold. Hence there exists a positive constant z 0 such that (2.29) holds for |z| ≤ z 0 , i.e. |φ | ≤ z 0 . So (2.27) holds, and |φ | is bounded in (0, 1).

To see the boundedness of |φ | in [δ.1), by (2.4) and the boundedness of |φ | in [δ, 1), we have

y|( n 2 -y 2 ) + [(-φ + yφ ) 2 (1 -2φ 2 ) + ((2y 3 -n)φ 2 -2yφ φ)(1 + φ 2 )] 2(1 + φ 2 ) 2 ||φ | ≤ c, (2.30)
where c denotes a postive constant.

So |φ | is bounded in [δ, 1) for 0 < δ 1. Otherwise, if |φ | is unbounded for y ∈ (0, δ), (2.30) means that |φ | is at most |φ | ∼ y -1+ν for ν > 0. Because if |φ | ∼ y -1 or |φ | ∼ y -ν with ν > 1, then |φ | is unbounded in (0, δ).
This contradicts with |φ | being bounded in (0, 1). Hence, yφ is bounded in (0, 1).

Since φ is integrable on (0, δ) and (1 -δ, 1) with 0 < δ 1, lim ρ→0 + φ(ρ) and lim ρ→1 -φ(ρ) exist and we denote them by φ 0 + and φ 1 -, respectively. Equation (2.4) can be rewritten as

y( n 2 -y 2 )φ = Φ(y, φ, φ ), (2.31)
where

Φ(y, φ, φ ) = - n 2 2 φ - y[(-φ + yφ ) 2 (1 -2φ 2 ) + ((2y 3 -n)φ 2 -2yφ φ)(1 + φ 2 )]φ 2(1 + φ 2 ) 2 - (-φ + yφ ) 2 2(1 + φ 2 ) (nφ - φ 3 1 + φ 2 ) + nφ 3 2(1 + φ 2 ) - (-φ + yφ ) 2 φ 3 (1 + φ 2 ) 2 - 2y(y -1)φ 2 (-φ + yφ ) 1 + φ 2 .
(2.32)

To see lim y→0 + φ (y) existence, let φ 0 + = lim inf y→0 + φ (y) and φ

0 + = lim sup y→0 + φ (y), if φ 0 + < φ 0 + , then we can take y = 0, φ = φ 0 + and φ = (φ * ) 0 + ∈ ( φ 0 + , φ 0 + ) in (2.32) such that Φ(0, φ 0 + , (φ * ) 0 + ) = - n 2 2 (φ * ) 0 + - φ 2 0 + 2(1 + ((φ * ) 0 + ) 2 ) (n(φ * ) 0 + + 2((φ * ) 0 + ) 3 1 + ((φ * ) 0 + ) 2 ) + n((φ * ) 0 + ) 3 2(1 + ((φ * ) 0 + ) 2 ) = 0.
Without loss of generality, assume Φ(0, φ 0 + , (φ * ) 0 + ) > 0. By the continuity of Φ and 1+φ 2 > 0, we get φ > 0 with φ (y) = (φ * ) 0 + in y ∈ (0, δ). This means φ is an increasing function in y ∈ (0, δ). Thus there must be φ (y) > (φ * ) 0 + or φ (y) ≤ (φ * ) 0 + for all y ∈ (0, y * ), where y * is a point in (0, δ). Both cases contradict with φ 0 + < (φ * ) 0 + < φ 0 + . So φ 0 + = φ 0 + , i.e. lim y→0 + φ (y) existence. Using the same process, we can obtain the existence of lim y→1 -φ (y) by observing (2.31). Consequently, let y → 0 + in (2.31),

- n 2 2 φ (0 + ) - φ 2 (0 + ) 2(1 + (φ (0 + )) 2 ) (nφ (0 + ) + 2(φ (0 + )) 3 1 + (φ (0 + )) 2 ) + n(φ (0 + )) 3 2(1 + (φ (0 + )) 2 ) = 0.

Existence of solutions to a nonlinear ODE

Assume that φ is a solution of (2.4) with initial data (2.3). Multiplying both sides of (2.4) by y α with α > 1, then integrating it,

φ (y) = y -α-1 ( n 2 -y 2 ) -1 y 0 s α [F (s, φ, φ , φ ) -( n 2 2 + 2s 2 )φ + (α + 1)( n 2 -s 2 )]ds, (2.33)
where

F (s, φ, φ , φ ) = - s[(-φ + sφ ) 2 (1 -2φ 2 ) + ((2s 3 -n)φ 2 -2sφ φ)(1 + φ 2 )]ψ 2(1 + φ 2 ) 2 - (-φ + sφ ) 2 2(1 + φ 2 ) (nφ - φ 3 1 + φ 2 ) + nφ 3 2(1 + φ 2 ) - (-φ + sφ ) 2 φ 3 (1 + φ 2 ) 2 - 2s(s -1)φ 2 (-φ + sφ ) 1 + φ 2 . (2.34) Let ψ(y) = φ (y) -φ 0 . Then ψ(0) = 0 and φ(y) = φ 0 + φ 0 y + y 0 ψ(s)ds. Define L(ψ) := y -α-1 ( n 2 -y 2 ) -1 y 0 s α [F (s, φ 0 + φ 0 s + s 0 ψ(s )ds , ψ, ψ ) -( n 2 2 + 2s 2 -(α + 1)( n 2 -s 2 ))ψ]ds. (2.35)
So finding the solution of (2.33) is equivalent to finding the fixed point ψ(y) of

L(ψ(y)) = ψ(y). (2.36) Lemma 2.2. Equation (2.36) with smooth initial data (2.3) has a unique solution φ ∈ C 2 ((0, 1])∩ C 1 ([0, 1]). Moreover, for a fixed 0 < µ 1, the solution φ ∈ C ∞ ([µ, 1]).
Proof. Recall that φ = φ 0 + φ 0 s + s 0 ψ(s )ds . By Lemma 2.1, there is a postive constant M 0 depending on n, α, φ 0 , φ 0 and the bound of |φ|, |φ | and yφ (y) in [0, 1] such that

|F (s, φ, ψ, ψ ) -( n 2 2 + 2s 2 -(α + 1)( n 2 -s 2 ))ψ| = | - [(-φ + sψ) 2 (1 -2ψ 2 ) + ((2s 3 -n)ψ 2 -2sψφ)(1 + ψ 2 )]ψ 2(1 + ψ 2 ) 2 - (-φ + sψ) 2 2(1 + ψ 2 ) (nψ - ψ 3 1 + ψ 2 ) + nψ 3 2(1 + ψ 2 ) - (-φ + sψ) 2 ψ 3 (1 + ψ 2 ) 2 - 2(s -1)ψ 2 (-φ + sψ) 1 + ψ 2 -( n 2 2 + 2s 2 -(α + 1)( n 2 -s 2 ))ψ| ≤ M 0 , which gives (for n ≥ 3) |L(ψ)| ≤ M 0 |y -α-1 ( n 2 -y 2 ) -1 y 0 s α ds| ≤ M 0 ( n 2 -1) -1 (1 + α) -1 .
So there exists a

M 1 = M 0 ( n 2 -1) -1 (1 + α) -1 such that L(ψ) ∈ C M 1 ([0, 1]) := {ψ ∈ C 2 ([0, 1]) : |ψ| ≤ M 1 }.
If there are ψ 1 and ψ 2 in C M 1 ([0, 1]), then by (2.35) and Lemma 2.1, direct computation shows that

|F (s, φ 1 , ψ 1 , ψ 1 ) -F (s, φ 2 , ψ 2 , ψ 2 )| ≤ | - [(-φ 1 + sψ 1 ) 2 (1 -2ψ 2 1 ) + ((2s 3 -n)ψ 2 1 -2sψ 1 φ 1 )(1 + ψ 2 1 )]ψ 1 2(1 + ψ 2 1 ) 2 + [(-φ 2 + sψ 2 ) 2 (1 -2ψ 2 2 ) + ((2s 3 -n)ψ 2 2 -2sψ 2 φ 2 )(1 + ψ 2 2 )]ψ 2 2(1 + ψ 2 2 ) 2 | +| - (-φ 1 + sψ 1 ) 2 2(1 + ψ 2 1 ) (nψ 1 - ψ 3 1 1 + ψ 2 1 ) + (-φ 2 + sψ 2 ) 2 2(1 + ψ 2 2 ) (nψ 2 - ψ 3 2 1 + ψ 2 2 )| +| nψ 3 1 2(1 + ψ 2 1 ) - nψ 3 2 2(1 + ψ 2 2 ) | + | - (-φ 1 + sψ 1 ) 2 ψ 3 1 (1 + ψ 2 1 ) 2 + (-φ 2 + sψ 2 ) 2 ψ 3 2 (1 + ψ 2 2 ) 2 | +| - 2(s -1)ψ 2 1 (-φ 1 + sψ 1 ) 1 + ψ 2 1 + 2(s -1)ψ 2 2 (-φ 2 + sψ 2 ) 1 + ψ 2 2 | |φ 1 -φ 2 | + |ψ 1 -ψ 2 |, which gives |L(ψ 1 ) -L(ψ 2 )| ≤ y -α-1 ( n 2 -y 2 ) -1 y 0 s α [|F (s, φ 0 + φ 0 s + s 0 ψ 1 (s )ds , ψ 1 , ψ 1 ) -F (s, φ 0 + φ 0 s + s 0 ψ 2 (s )ds , ψ 2 , ψ 2 )| +| n 2 2 + 2s 2 -(α + 1)( n 2 -s 2 )||ψ 1 -ψ 2 |]ds y -α-1 ( n 2 -y 2 ) -1 y 0 s α ( s 0 |ψ 1 -ψ 2 |ds + |ψ 1 -ψ 2 |)ds 1 α + 1 ψ 1 -ψ 2 .
So we can choose a suitable positive constant α such that

|L(ψ 1 ) -L(ψ 2 )| ≤ β ψ 1 -ψ 2 ,
where β ∈ (0, 1) is a positive constant depending on α -1 .

Hence by Banach fixed point theorem, there is a unique solution ψ ∈ C 2 ([0, 1]) such that (2.36) holds.

To see the smoothness of solution in [µ, 1] with a fixed 0 < µ 1, by (2.4),

φ = -( n 2 -y 2 ) -1 [ n 2 2y φ + [(-φ + yφ ) 2 (1 -2φ 2 ) + ((2y 3 -n)φ 2 -2yφ φ)(1 + φ 2 )]φ 2(1 + φ 2 ) 2 + (-φ + yφ ) 2 2y(1 + φ 2 ) (nφ - φ 3 1 + φ 2 ) - nφ 3 2y(1 + φ 2 ) + (-φ + yφ ) 2 φ 3 y(1 + φ 2 ) 2 + 2(y -1)φ 2 (-φ + yφ ) 1 + φ 2 ],
it is obviously that there is no singular point in the right hand side of above equation, and which is smooth in [µ, 1]. So we conculde that φ is smooth in [µ, 1].

Lemma 2.3. The self-similar profile φ(y) of (2.4) has at least 1th derivative at the origin diverges as t → T -.

Proof. Since quasilinear ODE (2.4) has intial data (2.3), the solution constructed in Lemma 2.2 is a non-trivial solution. Moveover, the solution can not be of the form φ(y) = c 0 + c 1 y, where c 0 and c 1 = 0 are two constants. By contradition, one can substitute it into (2.4), then

1 2y [n 2 c 1 + c 2 0 1 + c 2 1 (nc 1 - c 3 1 1 + c 2 1 ) - nc 3 1 1 + c 2 1 + 2c 2 0 c 3 1 (1 + c 2 1 ) 2 ] + 2(y -1)c 2 0 c 2 1 1 + c 2 1 = 0,
which is not possible unless c 1 = 0. This contradicts with c 1 = 0.

Remark 2.4. Lemma 2.3 tells us that

∂ ∂r v( r T -t )| r=0 = (T -t) -1 dφ dy (0) = c 1 (T -t) -1 -→ ∞, as t → T -.

Regularity property near y = 0

Since there are singular coefficients in the nonlinear term of equation (2.2), we have to analyze the smooth property of solution to equation (2.2) near y = 0.

Lemma 2.5. Let β > 1. Assume that φ is a solution of equation (2.2) with y ∈ (0, µ), for a fixed 0 < µ 1. Then φ(y) is smooth at least as y β+1 in (0, µ).

Proof. We make a rescaling to check the smoothness of φ(y) near 0. Let φ = y -β φ , then φ = -βy -β-1 φ + y -β φ and φ = y 0 s β φdy by φ 0 = 0, for β > 1. Substituting φ into equation (2.2), we get a first order ODE

y β ( n 2 -y 2 ) φ + y β-1 [β( n 2 -y 2 ) + n 2 2 β] φ + (- y 0 s β φds + y β+1 φ) 2 (1 -2y β φ2 )(βy β-1 φ + y β φ) 2(1 + y 2β φ2 ) 2 + [(2y 3 -n)y 2β φ2 -2y β+1 φ y 0 s β φds](βy β-1 φ + y β φ) 2(1 + y 2β φ2 ) + (- y 0 s β φds + y β+1 φ) 2 2(1 + y 2β φ2 ) (ny β-1 φ - y 3β-1 φ3 1 + y 2β φ2 ) - ny 3β-1 φ 2(1 + y 2β φ2 ) + y 3β-1 (- y 0 s β φds + y β+1 φ) 2 φ3 1 + y 2β φ2 + 2(y -1)y 2β φ2 (- y 0 s β φds + y β+1 φ) 1 + y 2β φ2 = 0,
which combining with φ bounded gives y β+1 φ y β φ, f or y ∈ (0, µ).

From above inequality, we know that φ ∼ y. Thus y -β φ = φ ∼ y. Hence φ near y = 0 is smooth at least as y β+1 .

Remark 2.6. Since there is a singular coefficient y -1 in the right hand side of equation (2.2), the solution φ (y) may be not smooth near y = 0. If the smooth initial data φ(0, r) = φ 0 = 0, using the same proof process in Lemma 2.5, we can show φ behaves as y -1 near y = 0.

Obviously, this means that φ is not smooth at y = 0. To get the smoothness of βth derivative φ (β) , by setting φ(0, r) = φ 0 = 0, then Lemma 2.5 tells us that φ (β) (y) is smooth at least as y for β ∈ N + . Now we conclude our main result in this section.

Theorem 2.7. The quasilinear ODE (2.2) with smooth initial data (2.3) has a unique smooth solution φ ∈ C ∞ ((0, 1]).

Furthermore, we take self-similar solutions of the form

v(t, r) = κ(T -t)φ( r T -t ),
where T is a positive constant, and κ ∈ (0, 1) is a positive constant which can be seen as a parameter. Inserting this Ansatz into (1.6), the profile function φ satisfies a quasilinear ODE

( n 2 -y 2 )φ + n 2 2y φ + κ 2 2 (-φ + yφ ) 2 1 + φ 2 (φ + ny -1 φ - κ 2 (φ φ 2 + y -1 φ 3 ) 1 + κ 2 φ 2 ) - nκ 2 (φ φ 2 + y -1 φ 3 ) 2(1 + κ 2 φ 2 ) - κ 4 (-φ + yφ ) 2 (φ φ 2 + y -1 φ 3 ) (1 + κ 2 φ 2 ) 2 + 2κ 2 φ (-φ + yφ )((y -1)φ + y 2 φ ) 1 + κ 2 φ 2 = 0, (2.37)
with smooth initial data (2.3). Then using the similar proof of Theorem 2.7, we obtain the following result.

Theorem 2.8. Let κ ∈ (0, 1). The quasilinear ODE (2.37) with smooth initial data (2.3) has a unique smooth solution φ ∈ C ∞ ((0, 1]).

3 Nonlinear stability of self-similar solutions

Preliminaries

In this section, we study the nonlinear stability of self-similar solution of radial type graph equation (1.6) to hyperbolic mean curvature flow (1.1). We solve this problem inside backward light cone

L T := {(t, r), t ∈ (0, T ), r ∈ (0, T -t]},
and do not transform it into asymptotic stability of φ(y) under self-similar coordinates. In self-similar coordinates, the backward light cone L T is transformed into an infinite cylinder and the blow up point is shifted towards ∞. Then the stability of self-similar solution is equivalent to the asymptotic stability of steady solution. One can see R.Donninger [START_REF] Donninger | On stable self-similar blow up for equivariant wave maps[END_REF] for this idea on the stability of self-similar solutions to wave map. Otherwise, if we consider the stability of self-similar solution under self-similar coordinates, there are some troubles in solving global existence of one dimensional quasilinear wave equation. In order to overcome "loss of derivative", we have to study the stability of self-similar solution in backward light cone L T by using Nash-Moser iteration technique in weighted Sobolev space H 2,1 . Due to the involved structure of the equation (1.6), it is hard to carry out analysis of the linearized equation. So we have to get rid of the fraction term (1 + v 2 r ) 2 by multiplying both sides of equation (1.6) by (

1 + v 2 r ) 2 , equation (1.6) is transformed into -v tt + n 2 v rr + n 2 2r v r -v tt v 2 r (2 + v 2 r ) + 1 2 v 2 t (v rr + n r v r -2v rr v 2 r + n -3 r v 3 r ) + n 2 v 2 r (v rr + 2n -1 r v r + n -1 r v 3 r ) + 2v t v tr v r (1 + v 2 r ) = 0. ( 3
where the nonlinear term f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r) depends on the the coefficients φ, φ ,φ and singular coefficients 1 T -t and 1 r , which means that it is degenerate at the blow up point (t, r) = (T, 0). It contains the quadratic terms w 2 r , w 2 t , w tt w r , w t w rr , w t w r , w r w rr and w t w tr the cubic terms w 3 r , w tt w 2 r , w 2 t w rr , w r w 2 t , w 2 r w rr , w t w 2 r , w t w r w rr and w t w r w tr , the four terms w 4 r , w tt w 3 r , w 2 t w 2 r , w t w rr w 2 r , w 2 t w r w rr , w t w 3 r , w tr w 3 r , w t w 3 r and w t w tr w 2 r , the five terms w 5 r , w tt w 4 r , w rr w 2 t w 2 r , w 2 t w 3 r and w t w tr w 3 r . Moreover, it satisfies |f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r)|

( 1 T -t + 1 r ) (|w r | 2 + |w r | 3 + |w r | 4 + |w r | 5 + |w r | 6 + |w r | 8 + w 2 t + w 4 t + w 2 tt + w 2 tr + w 4 tr ) , (3.3
) which is proven in Appendix, and the coefficients

a(t) = 1 + κφ (2 + κ 2 φ 2 ), (3.4) b(t, r) = 1 2 [n + κ 2 (φ - r T -t φ ) 2 (1 -2κ 2 φ 2 ) + nκ 2 φ 2 ], (3.5)
c(t, r) = n 2 2r -4κ 2 r 2 (T -t) 3 φ φ (1 + κ 2 φ 2 ) + nκ 2 2r (φ -r T -t φ ) 2 -2κ 4 T -t φ φ (φ -r T -t φ ) 2 + 3κ 4 (n-3) 2r (φ -r T -t φ ) 2 φ 2 + nκ 2 T -t φ φ + 3κ 2 n(2n-1) 2r φ 2 + 5κ 4 n(n-1) 2r φ 4 + 2κ 2 r (T -t) 2 (-φ + r T -t φ )φ (1 + 3κ 2 φ 2 ), (3.6)
d(t, r) = κ 2 T -t φ (-φ + r T -t φ )(1 + 2κ 2 φ 2 ) + nκ 2 r φ (-φ + r T -t φ ) + κ 4 (n-3) r φ 3 (-φ + r T -t φ ) + 2κ 2 r (T -t) 2 φ φ (1 + κ 2 φ 2 ), (3.7) e(t, r) = 2κ 2 φ (-φ + r T -t φ )(1 + κ 2 φ 2 ). (3.8)
Remark 3.1. The coefficients 1 T -t and 1 r in the nonlinear terms of the equation (3.2) imply unboundedness of L T . Thus we have to construct an approximation solution to overcome the "loss of derivative" and the singular coefficients.

Energy estimate for the linearized graph equation

We denote the Fréchet derivative by D. Linearizing the nonlinear equation (3.2) around w, the linearized operator of (3.2) is given by

L[w](h) := -a(t)h tt + b(t, r)h rr + c(t, r)h r + d(t, r)h t + e(t,
T 0 T 0 D w f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r)hdrdt T 0 T -t 0 ( 1 T -t + 1 r )(h t + h tt + h r + h rr + h tr )drdt. (3.10)
The proof of (3.10) is postponed to the Appendix. Consider the linearized equation

L[w](h)(t, r) = f (t, r), (3.11)
with the initial data h(0, r) = 0, h t (0, r) = 0, (3.12)

and the boundary condition

h(t, r)| ∂Ω ≡ 0, (3.13)
where

f (t, r) ∈ H γ,k (Ω) for γ ≥ 1 and k ≥ 2. Obviously, ∂Ω = {r = T -t} ∪ {t = 0} ∪ {r = 0}, r = T -t is the hypotenuse of right-angled triangle domain Ω = (0, T ) × (0, T -t].
The boundary condition (3.13) means that the higher derivatives of h(t, r) is also zero on ∂Ω, i.e.

∂ k h(t, r)| ∂Ω ≡ 0, k = 0, 1, 2, . . . , (3.14)
where ∂ denotes ∂ t or ∂ r .

We need to carry out some priori estimates of linearized equation (3.11). Let 0 < R < 1 and k ≥ 3 be two fixed positive constant. Define

B R := {w ∈ C k 2 : w C k 2 ≤ R < 1}
, then we carry out some propri estimates of linearized equation (3.11) with initial data (3.12). Lemma 3.2. Assume that f (t, r) ∈ C k 2 and w ∈ B R . Let h be a regular solution to equation (3.11) with initial data (3.12) and boundary condition (3.13), then there holds

T 0 T -t 0 1 (T -t) 2 (|h t | 2 + |h r | 2 )drdt T 0 T -t 0 |f | 2 drdt.
(3.15)

Proof. Taking the inner product of the linearized equation (3.11) with e -1 T -t h t by observing (3.9), we get

- 1 2 d dt [e -1 T -t a(t)|h t | 2 + e -1 T -t b(t, r)|h r | 2 ] + d dr (e -1 T -t b(t, r)h t h r + 1 2 e -1 T -t e(t, r)|h t | 2 ) +e -1 T -t (d(t, r) + a (t) 2 - a(t) 2(T -t) 2 - 1 2 ∂ r e(t, r))|h t | 2 +e -1 T -t 1 2 (∂ t b(t, r) - b(t, r) (T -t) 2 )|h r | 2 = -e -1 T -t (c(t, r) -∂ r b(t, r))h r h t -e -1 T -t h t D w f (φ, φ , φ , w t , w tt , w r , w rr , w tr )h + f (t, r)e -1 T -t h t , (3.16)
where D w f (φ, φ , φ , w t , w tt , w r , w rr , w tr )h is given in (3.10). We should observe that the terms 1 T -t and 1 r caused some troubles when carrying out propri estimates. There are singular at point (r, t) = (0, T ). Since we consider nonlinear stability of self-similar solutions in function space C k 2 , the coefficient 1 T -t can be seen as the weighted. For the coefficient 1 r , if there is a term like 1 r w, then by using the initial data condition (3.12) and w ∈ B R , we have 1 r w ≈ w r with r near 0. Next we estimate each term in right hand sides of (3.16). By (3.5), direct computation shows that

∂ r b(t, r) = κ 2 φ T -t [r(φ - r T -t φ )(1 -2κ 2 φ 2 ) -2κφ (φ - r T -t φ ) 2 + nφ ],
which combining with (3.6), using Young's inequality and Hardy's inequality, r T -t ∈ [0, 1] and φ ∈ C 2 ([0, 1]), for any r ∈ (0, δ) with δ 1, we derive

T 0 T -t 0 e -1 T -t |(c(t, r) -∂ r b(t, r))h r h t |dr T -t 0 e -1 T -t | 1 r h r h t + 1 T -t h r h t |drdt T 0 T -t 0 e -1 T -t [|h rr h t | + 1 T -t (|h r | 2 + |h t | 2 )]drdt T 0 T -t 0 e -1 T -t [ d dt |h r | 2 + d dr |h r h t | + 1 T -t (|h r | 2 + |h t | 2 )]drdt T 0 T -t 0 e -1 T -t 1 T -t (|h r | 2 + |h t | 2 )drdt, (3.17) for any r ∈ [δ, T -t], note that T is a small constant, there is 1 T -t > 1 and T 0 T -t 0 e -1 T -t |(c(t, r) -∂ r b(t, r))h r h t | T 0 T -t 0 e -1 T -t | 1 r h r h t + 1 T -t h r h t |drdt T 0 T -t 0 e -1 T -t |h r h t + 1 T -t (|h r | 2 + |h t | 2 )|drdt T 0 T -t 0 1 T -t e -1 T -t (|h r | 2 + |h t | 2 )drdt. (3.18)
Moreover, for any r ∈ (0, δ), we have

T 0 T -t 0 |2[- 5r 2 (T -t) 3 φ φ - r 2 (T -t) 3 φ (2 + φ 2 ) + n -3 2r φ (-φ + r T -t φ ) 2 + n 2(T -t) φ + 1 T -t φ (-φ + r T -t φ ) 2 + 3n(2n -1) 2r φ + 5n(n -1) r φ 3 + 6r (T -t) 2 φ φ (-φ + r T -t φ )]e -1 T -t w r h t h r |drdt w rr L ∞ (0,T ) T 0 T -t 0 (1 + 1 T -t )e -1 T -t h t h r drdt T 0 T -t 0 1 T -t e -1 T -t (|h t | 2 + |h r | 2 )drdt, (3.19)
and for any r

∈ [δ, T -t), T 0
T -t 0 |2[- 5r 2 (T -t) 3 φ φ - r 2 (T -t) 3 φ (2 + φ 2 ) + n -3 2r φ (-φ + r T -t φ ) 2 + n 2(T -t) φ + 1 T -t φ (-φ + r T -t φ ) 2 + 3n(2n -1) 2r φ + 5n(n -1) r φ 3 + 6r (T -t) 2 φ φ (-φ + r T -t φ )]e -1 T -t w r h t h r |drdt w r L ∞ (0,T ) T 0 T -t 0 (1 + 1 T -t )e -1 T -t h t h r drdt T 0 T -t 0 1 T -t e -1 T -t (|h t | 2 + |h r | 2 )drdt. ( 3
T -t 0 e -1 T -t h t D w f (φ, φ , φ , w t , w tt , w r , w rr , w tr )hdrdt T 0 T -t 0 e -1 T -t 1 T -t (|h t | 2 + |h r | 2 )drdt, ( 0
T -t 0 1 2 d dt [e -1 T -t a(t)|h t | 2 + e -1 T -t b(t, r)|h r | 2 ]drdt - T 0 T -t 0 d dr (e -1 T -t b(t, r)h t h r + 1 2 e -1 T -t e(t, r)|h t | 2 )drdt + T 0 T -t 0 e -1 T -t (-d(t, r) - a (t) 2 + a(t) 2(T -t) 2 + 1 2 ∂ r e(t, r))|h t | 2 drdt + T 0 T -t 0 e -1 T -t 1 2 (-∂ t b(t, r) + b(t, r) (T -t) 2 )|h r | 2 drdt T 0 T -t 0 e -1 T -t ( 1 T -t + 1)(|h t | 2 + |h r | 2 ) + e -1 T -t |f | 2 drdt. (3.23) 0
On one hand, by (3.8), direct computation shows that

∂ r e(t, r) = 2κ 2 T -t φ (-φ + r T -t φ )(1 + κ 2 φ 2 ) + 2κ 2 T -t φ (-(1 + r T -t )φ + r T -t φ )(1 + κ 2 φ 2 ) + 4κ 4 T -t φ 2 φ (-φ + r T -t φ ). (3.24)
So by (3.4), (3.7) and (3.24), we have 2 is the leading term when we choose a suitable small positive contant T < 1. For a sufficient small postive constant κ, the leading term in (3.25)

-d(t, r) - a (t) 2 + a(t) 2(T -t) 2 + 1 2 ∂ r e(t, r) = - κ 2 T -t φ (-φ + r T -t φ )(1 + 2κ 2 φ 2 ) - κ 2 n r φ (-φ + r T -t φ ) - κ 4 (n -3) r φ 3 (-φ + r T -t φ ) - 2κ 2 r (T -t) 2 φ φ (1 + κ 2 φ 2 ) - κr 2(T -t) 2 φ (2 + κ 2 φ 2 ) -κφ (1 + κ 2 r (T -t) 2 φ φ ) + 2 (T -t) 2 (1 + κ 2 φ (2 + φ 2 )) + κ 2 T -t φ (-φ + r T -t φ )(1 + κ 2 φ 2 ) + κ 2 T -t φ (-(1 + r T -t )φ + r T -t φ )(1 + κ 2 φ 2 ) + 2κ 2 T -t φ 2 φ (-φ + r T -t φ ). (3.25) Since φ ∈ C ∞ ((0, 1]) and w ∈ B R , it is obviously that 1 (T -t)
is 2 (T -t) 2 (1 + κφ (2 + κ 2 φ 2 )) > 0.
Thus there are a small positive constant T > 0 and a sufficient small κ > 0 such that

-d(t, r) - a (t) 2 + a(t) 2(T -t) 2 + 1 2 ∂ r e(t, r) -c > 0, (3.26)
where c denotes a fixed positive constant.

On the other hand,

∂ t b(t, r) = - κ 2 r 2 (T -t) 3 φ (φ - r T -t φ )(1 -2κφ ) - κ 4 r (T -t) 2 φ φ (φ - r T -t φ ) 2 + κ 2 nr (T -t) 2 φ φ . which gives b(t, r) (T -t) 2 -∂ t b(t, r) = 1 2(T -t) 2 [n + κ 2 (φ - r T -t φ ) 2 (1 -2κ 2 φ 2 ) + nκ 2 φ 2 ] + κ 2 r 2 (T -t) 3 φ (φ - r T -t φ )(1 -2κφ ) + κ 4 r (T -t) 2 φ φ (φ - r T -t φ ) 2 - κ 2 nr (T -t) 2 φ φ . (3.27)
Obviously, the leading term in (3.27) is n (T -t) 2 for a sufficient small postive constant κ, φ ∈ C ∞ ((0, 1]) and w ∈ B R . So there is a small positive constant T such that

b(t, r) (T -t) 2 -∂ t b(t, r) -c > 0, (3.28)
where c denotes a fixed positive constant. Moreover, for a sufficient small postive constant κ, it follows from Lemma 2.1 that

a(t) > 0, b(t, r) > 0. (3.29)
So using (3.12)-(3.13), (3.26) and (3.28), there are a small positive contant T and a sufficient small postive constant κ such that

T 0 [e -1 r a(T -r)|h t (T -r, r)| 2 + e -1 r b(T -r, r)|h r (T -r, r)| 2 ]dr - T 0 T -t 0 ∂ r (e -1 T -t b(t, r)h t (t, r)h r (t, r) + 1 2 e -1 T -t e(t, r)|h t (t, r)| 2 )drdt + T 0 T -t 0 e -1 T -t [(-d(t, r) - a (t) 2 + a(t) 2(T -t) 2 + 1 2 ∂ r e(t, r) -c)|h t | 2 + 1 2 (-∂ t b(t, r) + b(t, r) (T -t) 2 -c)|h r | 2 ]drdt < C T 0 T -t 0 e -1 T -t |f | 2 drdt, (3.30)
where we use

T 0 T -t 0 h(t, r)drdt = T 0 T -r 0 h(t,
T -t 0 e -1 T -t [(-d(t, r) - a (t) 2 + a(t) 2(T -t) 2 + 1 2 ∂ r e(t, r) -c)|h t | 2 + 1 2 (-∂ t b(t, r) + b(t, r) (T -t) 2 -c)|h r | 2 ]drdt ≤ C T 0 T -t 0 e -1 T -t |f | 2 drdt, 0
and, combining with (3.26) and (3.28),

T 0 T -t 0 e -1 T -t 1 (T -t) 2 (|h t | 2 + |h r | 2 ) -C|f | 2 drdt < 0.
Hence, for a small T > 0, we can obtain (3.15) in the triangular domain (0, T ) × (0, T -t].

Next we plan to carry out higher order derivative estimates to the solution h of equation (3.31). For a fixed constant k ≥ 1, we denote mix derivatives of the time and spatial variable by ∂ t ∂ k r by D k+1 . Applying D k+1 to both sides of (3.11) to get

-a(t)∂ tt D k+1 h -a (t)h tt + b(t, r)∂ rr D k+1 h + c(t, r)∂ r D k+1 h + d(t, r)∂ t D k+1 h +e(t, r)∂ tr D k+1 h + D w f (φ, φ , φ , w t , w tt , w r , w rr , w tr )D k+1 h = f , (3.31
) where D w f (φ, φ , φ , w t , w tt , w r , w rr , w tr )h is given in (3.10), and

f = D k+1 f - k 1 +k 2 =k+1, k 2 ≤k (D k 1 d(t, r))∂ t D k 2 h - k 1 +k 2 =k+1, k 2 ≤k (D k 1 e(t, r))∂ tr D k 2 h - k 1 +k 2 =k+1, k 2 ≤k (D k 1 b(t, r))∂ rr D k 2 h - k 1 +k 2 =k+1, k 2 ≤k (D k 1 c(t, r))∂ r D k 2 h - k 1 +k 2 =k+1, k 2 ≤k (D k 1 D w f (φ, φ , φ , w t , w tt , w r , w rr , w tr ))D k 2 h.
From the structure of equation (3.31), it is easy to see that the left hand side of (3.31) has the same structure with (3.11). This means that we can employ the method of getting Lemma 3.2 to derive higher energy estimates of h. The following is our result. Lemma 3.3. Assume that f (t, r) ∈ C k 2 and w ∈ B R . Let h be a regular solution to equation (3.31) with initial data (3.12) and boundary condition (3.13), then there holds

T 0 T -t 0 1 (T -t) 2 (|D k+1 h t | 2 + |D k+1 h r | 2 )drdt T 0 T -t 0 |D k+1 f (t, r)| 2 drdt.
(3.32)

Proof. Taking the inner product of the linearized equation (3.31) with e -1 T -t D k+1 h t , we get

- 1 2 d dt e -1 T -t a(t)|∂ t D k+1 h| 2 + e -1 T -t b(t, r)|∂ r D k+1 h| 2 + d dr e -1 T -t b(t, r)(∂ t D k+1 h)(∂ r D k+1 h) + 1 2 e -1 T -t e(t, r)|∂ t D k+1 h| 2 +e -1 T -t d(t, r) + a (t) 2 - a(t) 2(T -t) 2 - 1 2 ∂ r e(t, r) |∂ t D k+1 h| 2 +e -1 T -t 1 2 ∂ t b(t, r) - b(t, r) (T -t) 2 |∂ r D k+1 h| 2 = -e -1 T -t c(t, r) -∂ r b(t, r) (∂ r D k+1 h)(∂ t D k+1 h) + a (t)e -1 T -t h tt ∂ t D k+1 h -e -1 T -t (∂ t D k+1 h)D w f (φ, φ , φ , w t , w tt , w r , w rr , w tr )D k+1 h + f e -1 T -t ∂ t D k+1 h. (3.33)
We observe that the highest order in any the component of f is k + 1 but k 2 ≤ k. So similar to get (3.23), observing w ∈ B R and φ ∈ C ∞ , applying Young's inequality and sup b>0

1 (T -t) b e -1
T -t = (be -1 ) b to each term in the right hand side of (3.33), there exists a positive constant c such that

- 1 2 d dt e -1 T -t a(t)|∂ t D k+1 h| 2 + e -1 T -t b(t, r)|∂ r D k+1 h| 2 + d dr e -1 T -t b(t, r)(∂ t D k+1 h)(∂ r D k+1 h) + 1 2 e -1 T -t e(t, r)|∂ t D k+1 h| 2 +e -1 T -t d(t, r) + a (t) 2 - a(t) 2(T -t) 2 - 1 2 ∂ r e(t, r) -c |∂ t D k+1 h| 2 +e -1 T -t 1 2 ∂ t b(t, r) - b(t, r) (T -t) 2 -c |∂ r D k+1 h| 2 e -1 T -t |D k+1 f | 2 , (3.34)
where we choose e -1 T -t as a weighted function to absorb the effect of 1 (T -t) b for any fixed constant b > 0.

Recall that (3.26) and (3.28), there exists a small T > 0 such that

-d(t, r) - a (t) 2 + a(t) 2(T -t) 2 + 1 2 ∂ r e(t, r) -c > 0, b(t, r) (T -t) 2 -∂ t b(t, r) -c > 0,
which combing with (3.12) and (3.14), then integrating (3.34) over the right-angled triangle domain (0, T ) × (0, T -t], we get

T 0 T -t 0 [e -1 T -t (- a (t) 2 -d(t, r) + a(t) 2(T -t) 2 + 1 2 ∂ r e(t, r) -c)|D k+1 h t | 2 +( 1 2 ∂ t (-b(t, r)e -1 T -t ) -ce -1 T -t )|D k+1 h r | 2 ]drdt < C T 0 T -t 0 e -1 T -t |D k+1 f (t, r)| 2 drdt,
which gives (3.32).

Existence of solutions to the linearized graph equation

Theorem 3.4. Let k ≥ 3 be a fixed constant. Assume that f (t, r) ∈ C k 2 , φ ∈ C ∞ and w ∈ B R . Then equation (3.11) with intial data (3.12) and boundary condition (3.13) admits a unique solution

h ∈ C k 2 = 2 i=0 C i ((0, T ); H 2,k-i (Ω)).
Moreover, it satisfies that

h(t, r) C k 2 ≤ f (t, r) C k 2 . (3.35)
Proof. Since b(t, > 0 in (3.5), the linearized equation (3.11) is a strictly hyperbolic linear equation. Thus we can take a standard fixed point iteration process. Let h = (h, h t ). Then linearized equation (3.11) can be rewritten as

∂ t h + A(t, r) h = F (t, r),
where F (t, r) = (0, f (t, r)) and the matrix A(t, r) is

A(t, r) := 0 1 A 1 (t, r) A 2 (t, r) ,
the coefficients A 1 (t, r) and A 2 (t, r) are determinded by (3.9). By the standard fixed point iteration and a priori estimate (3.32), one can prove the approximation problem

h(m) = h0 - t 0 (A(s, r) h(m-1) + F (s, r))dt has a Cauchy sequence { h(m) } m∈Z + in C k 2 ,
≤ f (t, r) C k 2 .
For the existence of solution for general linear wave equation, see [START_REF] Sogge | Lectures on Nonlinear Wave Equations[END_REF].

A nonlinear perturbation theory

In what follows, we are to deal with the nonlinear equation by constructing a Nash-Moser iteration scheme. This scheme has been used in [START_REF] Yan | The motion of closed hypersurfaces in the central force fields[END_REF] with initial data w(0, r) = w 0 (r), w r (0, r) = w 1 (r). For convenience, we assume that w(0, r) = 0, w r (0, r) = 0.

(3.38) Remark 3.5. If the initial data is not zero, i.e.

w(0, r) = w 0 (r), w r (0, r) = w 1 (r).

We introduce an auxiliary function w(t, r) = w(t, r) -w 0 (r) -tw 1 (r), then we have vanishing initial data w(0, r) = 0, wr (0, r) = 0, and the equation (3.36) is equivalent to -a(t) wtt + b(t, r) wrr + c(t, r) wr + d(t, r) wt + e(t, r) wtr +b(t, r)(w 0 + tw 1 ) + c(t, r)(w 0 + tw 1 ) + d(t, r)w 1 + e(t, r)w 1 + f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r) = 0, with w(t, r) = w(t, r) + (w 0 (r) + tw 1 (r)).

In lth iteration step, the error term e (l) (t, r) should be

e (l) (t, r) = -a(t) w(l) tt + b(t, r) w(l) rr + c(t, r) w(l) r + d(t, r) w(l) t + e(t, r) w(l)
tr +b(t, r)(w 0 + tw 1 ) + c(t, r)(w 0 + tw 1 ) + d(t, r)w 1 + e(t, r)w 1 + f (φ, φ , φ , w 0 , w 1 ),

where f (φ, φ , φ , w 0 , w 1 ) is the term containing all the terms w 0 and w 1 , which is from the nonlinear term f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r). In the rest steps, there are the same with the zero initial data case, and the convergence of approximation scheme holds.

Define J (w) := -a(t)w tt + b(t, r)w rr + c(t, r)w r + d(t, r)w t + e(t, r)w tr + f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r), (3.39) if we construct a w(t, r) ∈ B R such that J (w) = 0, then w(t, r) is a regular solution of quasilinear wave equation (3.36) with initial data (3.38). In fact, we treat the initial value problem (3.36)-(3.38) iteratively as a small perturbation of the initial value problem for a linear hyperbolic system. Linearizing (3.39), we obtain the linearized operator

D w J (w)h = L[w](h),
where

L[w](h) is defined in (3.9). Define R[w](h) := J (w + h) -J (w) -D w J (w)h.
Then the following tame estimate holds.

Lemma 3.6. Let k ≥ 1 be a fixed postive constant. Assume that φ ∈ C ∞ ((0, 1)) and w ∈ B R . There is

R[w](h) C k 2 h 2 C k+4 2 , h ∈ B R . (3.40)
Proof. For any w ∈ B R , by (3.3), we have

w p H 2,k ≤ w 2 H 2,k ,
and the order of leading term in R[w](h) is 2, and we only need to obtain the lowest exponent of h with the highest order of derivatives 2 for convenience. There is bad coefficient 1 r in R[w](h). Obviously, this coefficient is singular at r = 0. Due to the zero initial data and Hardy's inequality, we can get T -t 0 h rr drdt. Thus this causes the highest order of derivatives changed into 3. Hence applying Young's inequality, we can obtain our result by direction computation. Lemma 3.6 tells us that there is a loss of derivatives in the nonlinear terms. Now we introduce a family of smooth operators (see [START_REF] Alinhac | Existence d'ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels[END_REF][START_REF] Hörmander | The boundary problems of physical geodesy[END_REF]), which takes the following important properties. Proposition 3.7. There is a family {Π θ } θ≥1 of smoothing operators in the space H γ,k (Ω) acting on the class of functions vanishing in the past such that

Π θ w H γ,k 1 (Ω) ≤ c θ (s 1 -s 2 ) + w H γ,k 2 (Ω) , k 1 , k 2 ≥ 0, (3.41) Π θ w -w H γ,k 1 (Ω) ≤ c θ s 1 -s 2 w H γ,k 2 (Ω) , 0 ≤ k 1 ≤ k 2 , d dθ Π θ w H γ,k 1 (Ω) ≤ c θ s 1 -s 2 -1 w H γ,k 2 (Ω) , k 1 , k 2 ≥ 0,
where c is a positive constant and (k 1 -k 2 ) + := max(0, k 1 -k 2 ). Moreover, there is another family of smoothing operators acting on functions defined on the boundary ∂Ω, which satisfying above three properties with norm • H γ,k (∂Ω) .

In our iteration scheme, we set

θ = N l = 2 l , l = 0, 1, 2, . . . , (3.42)
then by (3.41), we have Assume that a suitable "0th step" approximation solution w (0) = 0 of (3.44) has been chosen.

Π N l w H γ,k 1 N k 1 -k 2 l w H γ.k 2 , k 1 ≥ k 2 > 0, γ ≥ 1. ( 3
The "lth step" approximation solution is denoted by

w (l) = w (0) + l i=1 h (i) .
Then we want to construct the "l + 1th step" approximation solution w (l+1) . By (3.44), it holds

J N l+1 (w (l) + h (l+1) ) = e (l) + ∂ w (l) J N l+1 (w (l) )h (l+1) + R l+1 = e (l+1) , (3.45)
where e (l) := -a(t)w

(l) tt + b(t, r)w (l) rr + c(t, r)w (l) r + d(t, r)w (l)
t + e(t, r)w

(l) tr + Π N l f (φ, φ , φ , w (l) t , w (l)
tt , w (l) r , w (l) rr , w

tr , t, r).

(3.46)

D w (l) J N l+1 (w (l) )h (l+1) = L[w (l) ](h), (3.47) R l+1 = R[w (l) ](h) := J (w (l) + h) -J (w (l) ) -D w (l) J (w (l) )h, (3.48)
The following result is a direct application of Theorem 3.4, which gives the existence of the "l + 1 th" step approximation solution.

Lemma 3.8. There exists a solution h (l+1) of the initial value problem

D w (l) J N l+1 (w (l) )h (l+1) + e (l) = 0, h (l+1) 0 = 0, h (l+1) 1
= 0, where e (l) defined in (3.46). Furthermore, it holds

||h (l+1) || C k 2 ||e (l) || C k 2 . (3.49) For 3 ≤ k < k 0 ≤ k, set k l := k + k - k 2 l , (3.50)
k l+1 := k l -k l+1 = k - k 2 l+1 . ( 3
w ∞ = w (0) + ∞ l=1 h (l) ∈ C k 0 2 .
Proof. The proof is based on the induction. For any l = 0, 1, 2, . . ., we claim that there exists a constant 0 < s < 1 such that

||h (l+1) || C k l+1 2 < s 2 l < 1, (3.53) ||e (l+1) || C k l+1 2 ≤ s 2 l+1 , (3.54)
w (l+1) ∈ B R . (3.55)
We choose a fixed sufficient small w (0) > 0 such that

||w (0) || C k 0 2 1, ||e (0) || C k 0 2 1. (3.56)
For the case l = 0, by (3.49), we have

||h (1) || C k 1 2 ||e (0) || C k 1 2 ||e 0 || C k 0 2 < 1, ( 3
||e (1) || C k 1 2 ≤ ||R 1 || C k 1 2 ||h (1) || 2 C k 1 2 ||e (0) || 2 C k 0 2 . ( 3
(l+1) || C k l+1 2 ||R l+1 || C k l+1 2 2 3(l+1) ||h (l+1) || 2 C k l+1 2 2 3(l+1) ||e (l) || 2 C k l+1 2 1+2 2 3(l+1)+2•3l ||e (l-1) || 2 2 C k l-1 2
. . .

(24 ||e (0) || C k 0 2 ) 2 l+1 ,
which implies that we can choose a fixed sufficient small > 0 such that 0 < 24 ||e (0) || C k 0 2 < 1, then (3.53)-(3.55) holds. Here we use the fact that 2 3(l+1)+2•3l+2 2 •3(l-1)+...+3•2 l < 24 2 l+1 . Therefore, we derive the error term lim l→+∞ ||e (l) || C k 0 2 = 0, which implies that system (3.36) with zero initial data has a solution

w ∞ = w 0 + ∞ l=1 h (l) ∈ C k 0 2 .
Next we prove the uniqueness of w ∞ . Assume that there exist two solutions of equation (3.36) with the zero initial data. We denote two solutions of (3.36)

A Appendix

This appendix gives the proof of (3.3) and (3.10). To see (3.3), direct computation shows that f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r) = f 1 + f 2 + f 3 + f 4 + f 5 , (A.1)

where Proof. Note that (T, 0) is the blow-up point and r T -t ∈ (0, 1] in backward light cone. Singular coefficients are 1 T -t and 1 r in (A.1). Since φ ∈ C ∞ ((0, 1]) and its smooth property near y = 0 in Lemma 2.5, φ, φ and φ are bounded in (0, 1]. Hence this result can be directly obtained by applying Young's inequality to estimate each terms in (A.1).

f 1 = (1 -
Again by direct computation, we have D w f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r)h = A 1 (t, r)h t + A 2 (t, r)h tt + A 3 (t, r)h r +A 4 (t, r)h

Remark 1 . 3 .

 13 Here κ is a positive parameter taking values in (0, 1]. The solution v(t, r) = κ(T -t)φ( r T -t ) is different with the scaling invariance of solution in (1.8). By the scaling invariance

  .20) Thus using the same idea of getting (3.17)-(3.20), and (3.10), for r ∈ (0, 1), by (3.12)-(3.13) and integration by parts, T

  r)dtdr in the right-angled triangle domain (0, T ) × (0, T -t], and C is a positive constant. Furthermore, by (3.14), (3.29) and (3.30), T

Lemma 3 . 9 .

 39 .51) By (3.50)-(3.51), it follows thatk 0 > k 1 > . . . > k l > k l+1 > . . . . (3.52) For φ ∈ C ∞ ((0, 1)), there exists a positive constant k 0 ≥ 3 such that the quasilinear wave equation (3.36) with zero initial data (3.38) admits a unique solution

  .57) using (3.40), (3.43), (3.48) and (3.49), it holds

2 < 1 , 2 ≤l+1 2 ||e (l) || C k l 2 <

 21222 .58) Thus (3.57)-(3.58) gives (3.53) l=0 -(3.55) l=0 by choosing suitable small > 0. Also we get w (1) ∈ B R . Assume that (3.53)-(3.55) holds for l, i.e. ||h (l) || C k l ||e (l) || C k l s 2 l , (3.59) w (l) ∈ B R . Next we prove (3.53)-(3.55) holds for l + 1. Using (3.49) and (3.59), we have ||h (l+1) || C k cs 2 l < 1. (3.60) On the other hand, by (3.43), (3.48) and (3.49), observing that N l = 2 l , there exists a fixed postive constant k 0 ≥ 3 such that ||e

2 1+2 2 3 ( 2 C k l 2 .k 0 2 ,|| R|| C k 0 2 -

 232222 one hand, by (3.61)-(3.63), it holds L[w 2 ](ψ) = -R, (3.64) and, using the same method as for deriving (3.49), hand, by (3.43), (3.48) and (3.49) and (3.64)-(3.65), we derive l+1)+2•3l || R|| 2 . . (24 ) 2 l+1 || R|| 2 l+1C so for a suitable small > 0, and observing ψ ∈ B R , we get lim l→+∞ → 0, which gives the uniqueness of w ∞ .

φ 2 )w r +κ( 1 T 3 r + 2 ( 1 -κ 2 φ 2 -φ + κ 3 T 2 + 4 rφ 2 + κ( 1 Tw 2 r w 2 t- 2 t+ 2 r + w 3 r w t +2κw 2 r 0 T -t 0 D

 21321232421222200 rr + A 5 (t, r)h tr , whereA 1 (t, r) := 2 κ( n 2r φ + 1 2(T -t) φ + κ 2 T -t φ 2 φ + κ 2 (n-3) 2r φ 3 ) + ( n 2r + 2κ 2 T -t φ φ + 3κ 2 (n-3) 2r -t φ + 3(n-3) 2r φ )w 2 r + (n-3) 2r w κφ w r -2w 2 r )w rr w t +κ (-φ + r T -t φ )( n r -2κφ -4κ 2 T -t φ φ + 3κ 2 (n-3) r φ 2 + κ( 2 T -t φ + 3(n-3) r φ -2 κ )w r + n-3 r w 2 r ) + 2r (T -t) 2 φ (3κ 2 φ 2 + φ w r + w 2 r ) w r + κ(1 -2κ 2 φ 2 )(-φ + r T -t φ )w rr +2 κφ + κ 3 φ 2 (-3φ + ( 3r T -t + 1)φ ) + (1 + 3κ 2 φ 2 )w r + 3κφ w 2 r + w 3 r w tr , A 2 (t, r) := -κφ (4κ 2 φ 2 + κφ + 4) -(2 + 5κ 2 φ 2 )w r -4κφ w 2 r -w 3 r w r , A 3 (t, r) := 2 -κ 2 r 2 (T -t) 3 φ (2 + φ + κ 2 φ 2 ) + ( κ 3 (n-3) 2r -t φ )(-φ + r T -t φ ) 2 + κn 2(T -t) φ + 3κn(2n-1) 2r φ (1 + 5κ 2 (n-1) 3(2n-1) φ 2 ) + 6κ 3 r (T -t) 2 φ φ (-φ + r T -t φ ) w r +3 -4κr 2 (T -t) 3 φ + κ 2 (n-3) 2r (-φ + r T -t φ ) 2 + n(2n-1)2r+ 5κ 2 n(n-1) r φ 2κ 2 r (T -t) 2 φ (-φ + r T -t φ ) w 2 r +4κ -r 2 (T -t) 3 φ + 5κ 3 n(n-1) +κ (-φ + r T -t φ ) n r -4κ 2 T -t φ φ + 3κ 2 (n-3) r φ 2 + 2κ( 2 T -t φ + 3(n-3) r φ ) +4w rr (κφ -w r ) + 3(n-3) r w 2 r + 12r (T -t) 2 φ φ w r + 6r (T -t) 2 φ w 2 r w t +2 n 2r + 2κ 2 T -t φ φ + 3κ 2 (n-3) 2r -t φ + 3(n-3) 2r φ )w r -(κφ + w r )w rr +3(n-3) 2r κφ (4κ2 φ 2 + κφ + 4) + 2(2 + 5κ 2 φ 2 -3κφ w r -w 2 r )w r w tt + κφ (n -2(-φ + κ 2 r T -t φ ) 2 ) + 2( n 2 -κ 2 (-φ + r T -t φ ) 2 )w r w rr +2 (1 + 3κ 2 φ 2 )w t + 3κ (-φ + r T -t φ )(2κφ + w r ) + w t (κ -1 w r + 2φ ) w r w tr , A 4 (t, r) := κ(-φ + r T -t φ ) 1 -2κ 2 φ 2 -4κφ w r -2w 2 r w t + 1 -κ 2 φ 2 -2κφ w r -w 2 r w κφ (n -2(-φ + κ 2 r T -t φ ) 2 ) + ( n 2 -κ 2 (-φ + r T -t φ ) 2 )w r w r , A 5 (t, r) := 2 κφ + κ 3 φ 2 (3(-φ + r T -t φ ) + φ ) + (1 + 3κ 2 φ 2 )w r + 3κφ w (-φ + r T -t φ )(3κφ + w r ).Lemma A.2. Let φ ∈ C ∞ ((0, 1]) and w ∈ B R . Then one has T w f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r)hdrdt t + h tt + h r + h rr + h tr )drdt.

  number and B i denotes an open interval. As getting (2.26), we can prove in each B i , there holds

.26) Furthermore, by Heine-Borel's theorem, [y 0 , y 0 ] ⊂ N i=1 B i with i = 1, 2, . . . , N , N is a finite

  r)h tr +D w f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r)h,(3.9) where the linearized term D w f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r) depends on the the coefficients φ, φ ,φ , singular coefficients 1 T -t and 1 r and the linear term coefficients w r , w t , w rr , w tt , w tr , the quadratic term coefficients w 2 r , w 2 t , w tt w r , w t w rr , w t w r , w r w rr and w t w tr the cubic term coefficients w 3 r , w tt w 2 r , w 2 t w rr , w r w 2 t , w 2 r w rr , w t w 2 r , w t w r w rr and w t w r w tr , the four term coefficients w 4 r , w tt w 3 r , w 2 t w 2 r , w t w rr w 2 r , w 2 t w r w rr , w t w 3 r , w tr w 3 r , w t w 3 r and w t w tr w 2 r , it satisfies that

  T -t h t D w f (φ, φ , φ , w t , w tt , w r , w rr , w tr )hdrdt

		T		T -t	e -1
	0		0
	0	T	0	T -t	e -1 T -t	1 T -t	(|h t | 2 + |h r | 2 )drdt.	(3.22)
	Integrating (3.16) over the triangular domain (0, T ) × (0, T -t], by (3.17)-(3.18) and (3.19)-
	(3.20), for any r ∈ (0, T -t),

.21) and for any r ∈ [δ, T -t) T

  whose limit is h which solves the linearized equation (3.11). The estimate (3.35) is directly from the estimate (3.32) by observing f (t, r) C k 0

  tt + b(t, r)w rr + c(t, r)w r + d(t, r)w t + e(t, r)w tr + f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r) = 0,(3.36) where f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r) is defined in (3.2). So by (3.10), we have |f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r)| | 2 + |w r | 3 + |w r | 4 + |w r | 5 + |w r | 6 + |w r | 8

	we get the following one dimensional quasilinear wave equation
	-a(t)w ( T -t 1	+	1 r	t + w 4 t + w 2 tt + w 2 tr + w 4 tr ) |w r + w 2	(3.37)
					. Rescaling in (3.2) amplitude as
				w(t, r) → w(t, r),	> 0,

  tt + b(t, r)w rr + c(t, r)w r + d(t, r)w t + e(t, r)w tr + Π N l f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r).

	.43)
	We approximate (3.39) by
	J N l (w) := -a(t)w (3.44)

  by w 1 and w 2 in B R , respectively. Letψ = w 1 -w 2 ∈ B R . Then ψ satisfies -a(t)ψ tt + b(t, r)ψ rr + c(t, r)ψ r + d(t, r)ψ t + e(t, r)ψ tr + Π N l [f (φ, φ , φ , (w 1 ) t , (w 1) tt , (w 1 ) r , (w 1 ) rr , (w 1 ) tr , t, r) -f (φ, φ , φ , (w 2 ) t , (w 2 ) tt , (w 2 ) r , (w 2 ) rr , (w 2 ) , (w 2 ) tt , (w 2 ) r , (w 2 ) rr , (w 2 ) tr , t, r) -D w 2 f (φ, φ , φ , (w 1 ) t , (w 2 ) tt , (w 2 ) r , (w 2 ) rr , (w 2 ) tr , t, r)ψ].

tr , t, r)] = 0.

(3.61) Define L[w 2 ](ψ) := -a(t)ψ tt + b(t, r)ψ rr + c(t, r)ψ r + d(t, r)ψ t + e(t, r)ψ tr + Π N l D w 2 f (φ, φ , φ , (w 1 ) t , (w 2 ) tt , (w 2 ) r , (w 2 ) rr , (w 2 ) tr , t, r)ψ,

(3.62)

and R := Π N l [f (φ, φ , φ , (w 1 ) t , (w 1 ) tt , (w 1 ) r , (w 1 ) rr , (w 1 ) tr , t, r)

-f (φ, φ , φ , (w 2 ) t

  2κ 2 φ 2 )(-κφ + κr T -t φ )w rr + (-κφ + κr T -t φ )( n r -4κ 2 T -t φ φ + 3κ 2 (n-3) -t) 2 φ φ 2 w r w t + ( κ 2(T -t) φ + nκ 2r φ + κ 3 T -t φ 2 φ + κ 3 (n-3) -κ 2 φ 2 )w rr +( n 2r + 2κ 2 T -t φ φ + 3κ 2 (n-3) 2r φ 2 -2κφ w rr -w rr w r + n-3 2r w 2 r )w r w 2 t f 2 = -κφ (4κ 2 φ 2 + κφ + 4)w r + (2 + 5κ 2 φ 2 )w 2 r + 4κφ w 3 r + w 4 r w tt , f 3 = -5κ 2 r 2 (T -t) 3 φ φ -κr 2 (T -t) 3 φ (2 + κ 2 φ 2 ) + κ 3 (n-3) 2κr (T -t) 2 φ w r w t w 2 r , f 4 = κφ (n -2κ 2 (-φ + r T -t φ ) 2 ) + ( n 2 -κ 2 (-φ + r T -t φ ) 2 )w r -4κ 2 φ (-φ + rT -t φ )w t w r w rr , andf 5 = 6φ 2 κ 3 (-φ + r T -t φ ) + (1 + 2κ)φ + 2(1 + 3κ 2 φ 2 )w r w t + 6κ 2 φ (-φ + r T -t φ + w t )w 2 For φ ∈ C ∞ ((0, 1]) one has |f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r)| | 2 + |w r | 3 + |w r | 4 + |w r | 5 + |w r | 6 + |w r | 8 + w 2 t + w 4 t + w 2 tt + w 2 tr + w 4 tr .

						r	φ 2 )
		+ 6κ 3 r (T 2r	φ 3 )
		+(-κr 2 (T -t) 3 φ + 5κ 4 n(n-1) 2r	φ 4 )w 4 r + n(n-1) 2r w 5 r + (1 2r φ (-φ + r T -t φ ) 2 + nκ 2(T -t) φ
		+ κ 3 T -t φ (-φ + r T -t φ ) 2 + 3κn(2n-1) 2r	φ + 5κ 3 n(n-1)
	(	1 T -t	+	1 r	) |w r

r φ 3 + 6κ 3 r (T -t) 2 φ φ (-φ + r T -t φ ) w 2 r + -4κr 2 (T -t) 3 φ + κ 2 (n-3) 2r (-φ + r T -t φ ) 2 + n(2n-1) 2r + 5κ 2 n(n-1) r φ 2 + 2κ 2 r (T -t) 2 φ (-φ + r T -t φ ) w 3 r +κ 2 (-φ + r T -t φ )( 2 T -t φ + 3(n-3) r φ -2κw rr ) + 6r (T -t) 2 φ φ + κ( 1 T -t φ + 3(n-3) 2r φ )w t + κ(n-3) r (-φ + r T -t φ ) + r +2κ(-φ + r T -t φ + κ -1 w t )w 3 r w tr .

Lemma A.1.

Acknowledgements

The first author (PLF) was partially supported by the Innovative Training Networks (ITN) grant 642768 (ModCompShock). The research of the second author (WY) was supported by the Natural Science Foundation of Fujian Province of China, No. 2017J05014, and NSFC No 11771359 and was realized during a visit at Sorbonne University.

Proof. Since φ ∈ C ∞ ((0, 1]) and w ∈ B R , by H 1 ((0, T )) ⊂ L ∞ ((0, T )), we derive

Similarly, we can estimate each terms in D w f (φ, φ , φ , w t , w tt , w r , w rr , w tr , t, r)h. Hence by observing 1 T -t and 1 r are singular coefficient coefficients, the result holds in backward light cone.


Camille Laporte

Emily Tubbs
email: tubbsemily@gmail.com

Maxime Pierron

Amanda Gallego

Anaïck Moisan

Frédéric Lamarche

Tamara Lozano

Andrea Hernandez

Cécile Cottet-Rousselle

Anne-Sophie Gauchez

Virginie Persoons

Frédéric Bottausci

Caroline Fontelaye

François Boizot

Sandrine Lablanche

Florence Rivera

Improved human islets' viability and functionality with mesenchymal stem cells and arg-gly-asp tripeptides supplementation of alginate micro-encapsulated islets in vitro

Keywords: diabetes mellitus type 1, islets of Langerhans transplantation, mesenchymal stem cells, capsules, bioprocessing

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Pancreatic islet transplantation is a promising cell therapy for the treatment of type 1 diabetes. To date, according to the Collaborative Islets Transplant Registry (CITR) more than 1000 patients have been transplanted within clinical trials around the world [START_REF] Investigators | CITR 0th Annual Report (Internet)[END_REF]. Currently, islet transplantation is restricted to the severest form of diabetes. The main challenge to overcome in order to generalize islet transplantation to the overall population of patients with type 1 diabetes is to avoid immunosuppressive treatments currently responsible for severe adverse events [START_REF] Hafiz | Immunosuppression and procedure-related complications in 26 patients with type 1 diabetes mellitus receiving allogeneic islet cell transplantation[END_REF].

To address this challenge, islets' immuno-isolation in alginate microcapsules has been suggested to be a promising approach [START_REF] Darquy | Immunoisolation of pancreatic B cells by microencapsulation. An in vitro study[END_REF] allowing to reduce or even to avoid the use of immunosuppressants [START_REF] Elliott | Live encapsulated porcine islets from a type 1 diabetic patient 9.5 yr after xenotransplantation[END_REF].

Islet encapsulation has to guarantee islet homeostasis while ensuring efficient immunoprotection.

However, in vivo studies highlight the limits of such strategy. Several parameters including lack of biomaterials biocompatibility [START_REF] De Vos | Polymers in cell encapsulation from an enveloped cell perspective[END_REF], quality of encapsulated cells [START_REF] Dufrane | Six-month survival of microencapsulated pig islets and alginate biocompatibility in primates: proof of concept[END_REF], properties of the final capsule [START_REF] Weaver | Synthetic poly(ethylene glycol)-based microfluidic islet encapsulation reduces graft volume for delivery to highly vascularized and retrievable transplant site[END_REF] and the site of transplantation [START_REF] Zhu | Selection of Implantation Sites for Transplantation of Encapsulated Pancreatic Islets[END_REF] need to be controlled to ensure islet survival within the capsule, limited immune reaction against the capsule and finally allow islet encapsulation success. The selection of new biomaterials, the composition of the capsule itself and the way of processing them are key elements that could enhance the immunoprotective proprieties of the capsules, contribute to improve the immune-response towards the grafted capsules, and enhance the viability and functionality of the encapsulated islet and ultimately improving transplantation outcomes.

Studies suggest that reconstitution of a favorable microenvironment within the microcapsule could improve the survival of the encapsulated islets within the capsules [START_REF] Wang | Maintenance of beta-cell function and survival following islet isolation requires re-establishment of the islet-matrix relationship[END_REF]. Among the strategies to build such favorable microenvironment, Mesenchymal Stem Cells (MSCs) co-encapsulation with islets or capsule enrichment with component of the native pancreatic extracellular matrix (ECM) are of interest. MSCs are multipotent adult stem cells largely described for their therapeutic application in cell and regenerative therapies [START_REF] Patel | Therapeutic Potential of Mesenchymal Stem Cells in Regenerative Medicine[END_REF]. MSCs evidenced cytoprotective effects that have been demonstrated to improve the viability [START_REF] Jung | Bone marrow-derived mesenchymal stromal cells support rat pancreatic islet survival and insulin secretory function in vitro[END_REF] and insulin secretory capabilities of pancreatic islets [START_REF] Wang | Maintenance of beta-cell function and survival following islet isolation requires re-establishment of the islet-matrix relationship[END_REF] in vitro [START_REF] Johansson | Pancreatic Islet Survival and Engraftment Is Promoted by Culture on Functionalized Spider Silk Matrices[END_REF] and to enhance islet transplantation outcomes in vivo [START_REF] Montanari | Multipotent mesenchymal stromal cells enhance insulin secretion from human islets via N-cadherin interaction and prolong function of transplanted encapsulated islets in mice[END_REF] [START_REF] Johansson | Pancreatic Islet Survival and Engraftment Is Promoted by Culture on Functionalized Spider Silk Matrices[END_REF]. MSCs are known to prevent the formation of pericapsular fibrotic tissue when co-encapsulated with islets thanks to immunomodulatory properties [START_REF] Vaithilingam | Co-encapsulation and cotransplantation of mesenchymal stem cells reduces pericapsular fibrosis and improves encapsulated islet survival and function when allografted[END_REF]. Moreover, the addition in the microcapsule of the tripeptide arginineglycine-aspartate RGD, a component of the ECM, has been shown to improve the viability and the functionality of C2C12 cells [START_REF] Santos | The synergistic effects of the RGD density and the microenvironment on the behavior of encapsulated cells: In vitro and in vivo direct comparative study[END_REF]. Encapsulation of MSCs in alginate hydrogels microcapsules coupled with RGD motifs improves the MSCs viability as well as their secretion capacities [START_REF] Ho | Increased Survival and Function of Mesenchymal Stem Cell Spheroids Entrapped in Instructive Alginate Hydrogels[END_REF].

To ensure in vivo immuno-protection, the capsule needs to resist to mechanical solicitation and to protect the encapsulated islets from the immune system, i.e. by controlling the capsule permeability.

Mechanical properties and permeability of alginate-based microcapsules are controlled by the concentration and composition of alginate, cross-linking ions and time of gelation [START_REF] Mørch | Binding and leakage of barium in alginate microbeads[END_REF][START_REF] Martinsen | Alginate as immobilization material: I. Correlation between chemical and physical properties of alginate gel beads[END_REF]. Current standard encapsulation technologies do not allow the use of highly viscous alginate polymers that could be necessary to process microcapsules with stronger mechanical resistance and more selective permeability than the current standard. Thus, low viscosity alginate are mainly used to process capsules and to ensure mechanical stability of such alginate-based microcapsules, barium crosslinking ion is often used despite the risk of chronic barium toxicity (EPA) for recipients [START_REF] Mørch | Binding and leakage of barium in alginate microbeads[END_REF][START_REF] Martinsen | Alginate as immobilization material: I. Correlation between chemical and physical properties of alginate gel beads[END_REF].

Here, we hypothesized that the utilization of M-rich alginate in combination RGD-G rich alginate and MSCs improve the islets encapsulation outcomes in vitro. To our knowledge, our study describes for the first time the beneficial effect of RGD-G rich and MSCs double supplementation in M-rich alginate microbeads on insulin secretion capacity of human islets (Patent EP19305415.2).

Material and methods

Human islets isolation and culture

The human pancreatic islets were provided by Cell Therapy and Engineering Unit of EFS Auvergne Rhone Alps or by the Integrated Islet Distribution Program (IIDP) for Human Islet Research. The islets were isolated from a pancreas from a brain-dead donor. For EFS donors, the procedure was approved by the National Agency for Medicines and Health Products Safety (ANSM) in accordance with the French bioethics law (Act No.2004-800 of August 6, 2004, as amended by Act No.2011-814 of July 7, 2011). For IIDP donors, documented consent was provided for the use of human islets for research, patients' information is de-identified and batches of isolated human islets together with analytical data were labeled with a SVI number or Lot number, respectively, to ensure the anonymity of the donor. These procedures were carried out with review and approval by appropriate regulatory authorities. A summary of the human pancreatic islets used in this study is shown in Table 1. Islets were cultured in CMRL 1066 medium, 1g/L Glucose (Gibco, Thermo Fisher Scientific, Waltham, Mass., USA) supplemented with 10% FBS, 10mM Hepes, 1mM Sodium Pyruvate, 2mM L-Glutamine, 100mg.UI.mL -1 Penicillin, 100 μg.mL -1 streptomycin, in a 5% CO2 humidified atmosphere at 37°C.

Islet

Human mesenchymal stem cells isolation and culture

MSCs were cultured as described previously [START_REF] Laporte | Human mesenchymal stem cells improve rat islet functionality under cytokine stress with combined upregulation of heme oxygenase-1 and ferritin[END_REF].

Human mesenchymal stem cells characterization

MSCs were immunophenotyped and differentiated using induction StemPro media according manufacturer instruction as described previously [START_REF] Laporte | Human mesenchymal stem cells improve rat islet functionality under cytokine stress with combined upregulation of heme oxygenase-1 and ferritin[END_REF].

Encapsulation method

Islets were encapsulated alone or with MSCs at a ratio of 150 MSCs/islet per capsule by microencapsulation based on microfluidics allowing the handling of highly viscous polymer. A microfluidic card in Cyclic Olefin Copolymer (COC) with channels from 200 to 550µm, was developed (Supp.Fig. 1). For microcapsules' detailed production, please refer to supplementary methods.

Alginate based solution are prepared as follow: a batch solution of SLM100 4% w/w and MVG GRGDSP 10 mg/mL (RGD, Novamatrix, Sandvika, Norway). An intermediate solution is made by mixing the two batch solutions. MSCs are harvested and washed in calcium-enriched physiological serum. Islets and MSCs at a ratio of 150MSCs/islet are washed in alginate buffer solution, centrifuged 3min at 150G and mix gently with a pipette tip. The reference microcapsule is composed of pancreatic islets entrapped in 2.5% SLM100 alginate. The capsule composite product composition is: SLM100 2.5%w/w, RGD-G rich alginate 0.5mg/mL, 150 MSCs/islet and pancreatic islets.

Rheological characterization of the fluids have been performed on 1.5ml sample with 1/60 conerheometer (Malvern Bohlin Gemini), 20°C.

Human pancreatic islets and microcapsules size distribution were performed by image analysis (ImageJ). For each batch, a minimum of 100 islets and 50 microcapsules were characterized.

Biocompatibility

Please refer to supplementary methods.

Measurement of insulin-secreting capacities of encapsulated islets

Twenty-four hours after encapsulation, and for each condition tested, 100 islets were incubated as described previously [START_REF] Laporte | Human mesenchymal stem cells improve rat islet functionality under cytokine stress with combined upregulation of heme oxygenase-1 and ferritin[END_REF]. The total insulin content was extracted after dissociation of the capsule with a solution of trisodium citrate 1% by incubation in a solution of acid ethanol 75%/vol; 0.18N HCL (Sigma-Aldrich, St. Louis, MO, USA).

Human insulin concentration in each collected media was measured using commercial radioimmunoassay kit BI-INS-IRMA (Cisbio Bioassay, Codolet, France).

Viability analysis by confocal microscopy

• Cell staining and acquisition

The viability of the encapsulated islets was determined 24 hours after encapsulation by SYTO 13 (λex/λem=488/509nm) (Thermo Fisher Scientific, Waltham, Massachusetts, United States) and Propidium iodide (PI) (λex/λem=545/630nm) (Interchim, Montluçon, France) double staining, as described previously [START_REF] Chabert | New Automatized Method of 3D Multiculture Viability Analysis Based on Confocal Imagery: Application to Islets and Mesenchymal Stem Cells Co-Encapsulation[END_REF].

• Islet viability quantification

The quantification of the nuclei of SYTO 13-labeled live cells and the nuclei of PI-labeled dead cells allowed to assess the islet viability as follows:

Cell viability(%)=(number of green nuclei)/(number of green + red nuclei)*100

In order to discriminate the viability of islets from the viability of MSCs within the capsule, an image analysis macro was developed on the ImageJ interface to discriminate the islets from the MSCs, on size criteria. The islets were selected on the transmission image as regions of interest that were applied to the images acquired in fluorescence (data not shown). The quantization of the red and green fluorescent nuclei within this region of interest was performed automatically on the selected elements.

Statistics

The islets viability and functionality data were analyzed with a 1-way or 2-way ANOVA, according to the number of conditions. In case of significance, an omnibus variance test was followed by a Holm-Sidak post-hoc. The data were expressed as mean ±SEM.

Results

Microcapsules fabrication and characterization

Rheological characterization of cell-free alginate based capsules solutions exhibiting non-Newtonian behaviour were performed before microcapsules manufacturing (Fig. 1). Fitting data with the Carreau-Yasuda model for shear-thinning solutions, viscosity at zero shear rate were respectively determined to be 8.3 and 12.7Pa.s for reference and composite capsules solutions (Fig. 1A). Such high-viscosity solutions were then processed inside the microfluidic flow-focusing device to produce microcapsules and a maximum production rate of 120 microcapsules per minute was achieved. To optimize the ratio of filled microcapsules, i.e. each microcapsule should contain at least one islet per capsule. Human pancreatic islets number and size distribution were characterized through image analysis, and consequently, volume of biomaterials adapted for each encapsulation experiment.

Microfluidic device capsule production working range (i.e. droplet regime, in-between the stable and jetting pressure) was characterized for the three following mix: reference solution (SLM100 2.5% w/w), MSC-free composite solution (SLM100 2.5% w/w, RGD-G rich alginate 0.5mg/mL), composite solution (SLM100 2.5% w/w, RGD-G rich alginate 0.5mg/mL, 150 MSCs/islet). As shown on Figure 1B, the adjunction of such quantity of mesenchymal stem cells does not change the rheological properties of the composite solutions. For each type of solution leading to the fabrication of alginate, RGD-alginate and MSC-alginate microcapsules, working ranges of capsules production were determined. The adjunction of MSCs to RGD-alginate to obtain 150 MSC/islet do not drastically modify the working range of the device. Such characterization allows determining manufacturing parameters to get a constant mean diameter of the microcapsule. In the present case, couple of applied pressures were set to obtain capsule with diameter ranging around 450 to 500µm with a coefficient of variation inferior to 5% (data not shown). Although the dispersion of islets size is similar between donors, human pancreatic islets size distribution is variable from one donor to the other [START_REF] Buchwald | Quantitative Assessment of Islet Cell Products: Estimating the Accuracy of the Existing Protocol and Accounting for Islet Size Distribution[END_REF] ranging from 30 to 270µm. Most of the population is inferior to 100µm with a peak around 40µm (Fig. 2A). As expected, the few large islets enhance the polydispersity of the microcapsule diameter (Fig. 2B).

Biocompatibility and immune response towards the microcapsules

Due to human islets scarcity, we chose to determine the biocompatibility of the process and final products on cell-free microcapsules. Indeed, the human islets and MSCs added to the microcapsules are at clinical grade and by definition biocompatible. We analyzed contamination, cytotoxicity, hemocompatibility and immunocompatibility assays (Supp.Fig. 2). Evaluation showed that the procedure of fabrication of the microcapsules avoids microbiological contamination validated by the absence of microorganism with a low content of endotoxin (<0.5EU/mL, data not shown), and no significant impairment of cell viability (Supp.Fig. 2A), nor apoptosis induction nor induction of ROS production was observed (Supp.Fig. 2B). The microcapsules did not impact complement system activation (Supp.Fig. 2C). The cytokine profile of human PBMCs incubated with the microcapsules did not show remarkable changes (Supp.Fig. 1D). Microcapsules did not stimulate the proliferation of peripheral blood mononuclear cells (PBMCs) after 7 days (Supp.Fig. 1E) and did not induce the activation of caspase-1. Taken together, cell-free microcapsules did not show evidence of toxicity suggesting their biocompatibility under the conditions tested.

Human pancreatic islets viability and functionality

Composite capsules improved encapsulated islets' viability compared to the reference capsule.

We developed a barium-free biocompatible composite capsule of RGD G-rich alginate in combination with MSCs. This capsule composition improves the islets encapsulation outcomes in vitro and appears as a new promising cell therapy product for type I diabetes.

To our knowledge, it is the first time that such high viscosity mix of biomaterials (viscosity at zero shear rate: 12.7Pa.s) is processed within microfluidic devices for cell encapsulation [START_REF] Prüße | Comparison of different technologies for alginate beads production[END_REF] without deleterious effect on cells. Mixing of MSCs and pancreatic islets within such highly viscous biomaterials is complex and requires soft handling not to dissociate pancreatic islets. Currently, this step is still manual. Protocols and tools have been optimized and implemented to produce reproducible microcapsules (CV<10%) despites spreading of pancreatic islet distribution. Such variation in capsule diameter affects the amount of MSCs co-encapsulated with pancreatic islets. This variation of MSCs numbers inside the capsule was characterized through images analysis and found to be inferior to 30%. Interestingly, when islet diameter is inferior to 30% of the capsule diameter, capsule size spreading is not affected (CV<5%, data not shown).

Processing of highly viscous mix of polymers allows the production of barium-free microcapsules that sustain handling, shipping and characterization. However, frequency rate of production still need to be increased in order to quickly encapsulate a whole batch of islets for clinical trials. Parallelization of such production is realistic and has been already done in the laboratory. Additionally, the frequency of production could be increased by reducing the viscosity of the cell loaded solutions. It can be done by maintaining the solutions to be processed at 37°C and using the shear thinning properties of solutions, i.e. increasing the shear stress when producing the microcapsules (Fig. 2A).

The biocompatibility study performed demonstrates that both microcapsules' production and composition are safe considering that neither contamination nor immune response have been detected.

This study demonstrates that compared to encapsulation using conventional alginate, supplementing the capsule composition with both MSCs and RGD-G-rich alginate is beneficial for human pancreatic islets viability and functionality. This beneficial effect could be related to the specific cytoprotective functions of MSCs through paracrine effectors such as growth factors and cytokines [START_REF] Vaithilingam | Co-encapsulation and cotransplantation of mesenchymal stem cells reduces pericapsular fibrosis and improves encapsulated islet survival and function when allografted[END_REF] but also to an improvement of islet adhesion with the presence of RGD motifs, preventing the viability loss associated with anoïkis [START_REF] Pinkse | Integrin Signaling via RGD Peptides and Anti-1 Antibodies Confers Resistance to Apoptosis in Islets of Langerhans[END_REF]. Another hypothesis is that the mechanical properties of the microenvironment conferred by the composite capsule, i.e. mix of M and RGD-G rich alginate, is beneficial to pancreatic cells as previously shown [START_REF] Naujok | Effect of substrate rigidity in tissue culture on the function of insulin-secreting INS-1E cells[END_REF].

The present study combined both components RGD and MCSs with human islets in monodispersed microcapsules. Even if it is difficult to conclude on the separate and isolated effect of these two components, our study shows that the combination MSCs and RGD-G rich motifs improved islets outcome. Human islets scarcity prevented us from further evaluating the isolated effect of the biomaterials used on the functionality and viability of the islets, thus limiting our conclusions on the synergistic effect of RGD-G and M-rich alginate and MSCs. However Ho et al. [START_REF] Ho | Increased Survival and Function of Mesenchymal Stem Cell Spheroids Entrapped in Instructive Alginate Hydrogels[END_REF] showed that the presence of RGD in microcapsules containing MSCs increases the viability of MSCs as well as their secreting capacity, especially VEGF. This study suggests that when entrapped in an adhesive biomaterial, MSCs improve survival and proangiogenic growth factor secretion. Johansson et al. [START_REF] Johansson | Pancreatic Islet Survival and Engraftment Is Promoted by Culture on Functionalized Spider Silk Matrices[END_REF] showed that the RGD motifs are described as enhancing viability and functionality of the encapsulated islets suggesting a synergistic effect of the two biomaterials via a direct islet stimulation by RGD-G rich alginate and an indirect islet stimulation by the functionalization of MSCs by the RGD-G rich motifs.

In vitro results from our study and data from the literature suggest that RGD-G rich alginate and MSCs could improve the outcome of islets encapsulation. The final validation of this combination of biomaterials will have to be further confirmed in vivo.

Authors' contributions

SL, CL, ET wrote the overall manuscript and designed viability and functionality experiments. CL and ET performed islet viability (confocal acquisition and image analysis) functionality assays, statistical analysis. MP performed islet encapsulations, took part in the writing. AG, TL and AH performed biocompatibility assays and participated to manuscript writing. AM-VP provided the MSCs and performed the surface antigen analysis. FL performed GSIS experiments and provided assistance on the cell culture platform. CCR provided assistance on the imaging platform. ASG was in charge of the insulin measurement by radioimmunoassay, helped with analysis. FBOI and CF fabricated microfluidic chips for cell encapsulation. FR-FB designed microfluidic chips, rheological and capsules characterization, encapsulation experiments, coordinated work between teams, and participate to writing the manuscript. All authors read and approved the final manuscript.

Figure 1 :

 1 Figure 1: Rheological characterization of biomaterials (A) and phase diagram of the microfluidic

Figure 2 :

 2 Figure 2: Example of islet diameter distribution for five different donors (A) and reference

Table 1 : Summary of pancreatic islets´ information used in this study.

 1

			Donor	Viability at	Purity of	Time between isolation
		isolation	Age	reception	preparation	and encapsulation
		center		(%)	(%)	(days)
	Donor 1	EFS	59	71	90	2-6
	Donor 2	IIDP	47	52	90	6
	Donor 3	EFS	27	86	30	2
	Donor 4	EFS	54	85	80	2
	Donor 5	IIDP	47	76	90	4
	Donor 6	IIDP	59	68	95	7
	Donor 7	EFS	58	90	50	2
	Donor 8	IIDP	63	66	90	7

Acknowledgements

CL was supported by a 3-year Ph.D. fellowship by the foundation "Agir Pour les Maladies Chroniques". Authors thank the Grenoble Biology and Pathology Institute Grenoble Alps who provided insulin secretion data, and M.Orhant-Prioux, in charge of mesenchymal stem cells amplification. This research was performed in the context of BIOCAPAN H2020 project, authors thank the consortium.

Data availability

The raw data required to reproduce these findings are available on request.

We first confirmed the multipotency of the MSCs lot used for this study (Supp.Fig. 3A) by differentiating them into cells of adipogenic lineage (lipid droplets stained with Oil Red O, Supp.Fig. 3B), chondrogenic (glycosaminoglycans that stained with Alcian blue, (Supp.Fig. 3C) and osteogenic (calcic deposits that stained with Alizarin Red S, Supp.Fig. 3D). In addition, immunophenotyping confirmed the MSCs identity (data not shown).

We then investigated the viability of islets encapsulated in reference and composite microcapsules.

Encapsulation in reference capsules induced a significant decrease of 24.8% ±4.7% (p=0.005) of islets viability compared to non-encapsulated islets (naked islets) (Fig. 3). However, the encapsulation of islets in the composite capsules induced a significant increase of 29.9% ±5.7% (p=0.02) in islets viability compared to islets encapsulated alone in reference capsule (Fig. 3).

Islet functionality was improved by the composite capsule composition.

A significant decrease in the insulin secretion capability was observed when the islets were encapsulated in the reference capsule (alginate 2.5%): the percentage of insulin secreted under low glucose exposition dropped from 1.6% ±0.59% for naked islets to 0.6% ±0.88% for encapsulated islets and under high glucose stimulation from 3.1% ±0.61% for naked islets to 1.8% ±0.38% of insulin secreted for the encapsulated islets (p=0.016) (Fig. 4). We observed a significant increase in the insulin secretion capacity when islets were encapsulated in the composite capsule compared to the reference capsule. Indeed, the percentage of insulin secreted under low glucose exposition significantly increased from 0.6% ±0.88% for islets in reference capsule to 1.1% ±0.27% for islets in composite capsules, and from 1.8% ±0.38% for the reference capsule to 4.7% ±1.29% for islets in composite capsules under high glucose stimulation (p=0.017) (Fig. 4).

Discussion

Declaration of interest

Authors declare no conflict of interest.