Refined regularity for the blow-up set at non characteristic points for the complex semilinear wave equation

In this paper, we consider a blow-up solution for the complex-valued semilinear wave equation with power non-linearity in one space dimension. We show that the set of non characteristic points $I_0$ is open and that the blow-up curve is of class $C^{1,\mu_0}$ and the phase $\theta$ is $C^{\mu_0}$ on this set. In order to prove this result, we introduce a Liouville Theorem for that equation.


The problem and known results
We consider the following complex-valued one-dimensional semilinear wave equation x u + |u| p−1 u, u(0) = u 0 and u t (0) = u 1 , where u(t) : x ∈ R → u(x, t) ∈ C, u 0 ∈ H 1 loc,u and u 1 ∈ L 2 loc,u ,with The Cauchy problem for equation (1) in the space H 1 loc,u × L 2 loc,u follows from the finite speed of propagation and the wellposedness in H 1 × L 2 . See for instance Ginibre, Soffer and Velo [8], Ginibre and Velo [9], Lindblad and Sogge [13] (for the local in time wellposedness in H 1 × L 2 ). Existence of blow-up solutions follows from ODE techniques or the energy-based blow-up criterion of [12]. More blow-up results can be found in Caffarelli and Friedman [6], Alinhac [1] and [2], Kichenassamy and Littman [11], [10] Shatah and Struwe [24]).
If u is a blow-up solution of (1), we define (see for example Alinhac [1]) a continuous curve Γ as the graph of a function x → T (x) such that the domain of definition of u (or the maximal influence domain of u) is From the finite speed of propagation, T is a 1-Lipschitz function. The timeT = inf x∈R T (x) and the graph Γ are called (respectively) the blow-up time and the blow-up graph of u.
We denote by I 0 the set of non characteristic points.
Given some (x 0 , T 0 ) such that 0 < T 0 ≤ T (x 0 ), we introduce the following self-similar change of variables: If T 0 = T (x 0 ), then we write w x 0 instead of w x 0 ,T 0 . This change of variables transforms the backward light cone with vertex (x 0 , T (x 0 )) into the infinite cylinder (y, s) ∈ (−1, 1) × [− log T (x 0 ), +∞). The function w x 0 (we write w for simplicity) satisfies the following equation for all |y| < 1 and s ≥ − log T 0 : This equation will be studied in the space which is the energy space for w.
Let us define By the argument of Antonini and Merle [3], which works straightforwardly in the complex case, we see that E is a Lyapunov functional for equation (5). Similarly, some arguments of the real case, can be adapted with no problems to the complex case, others don't. Let us first briefly state our main result in [4] , then we give the main results of this paper.
In [4], we proved the existence of the blow-up profile at non-characteristic points. More precisely, this is our statement (see Theorem 4 page 5895 in [4]).
There exist positive µ 0 and C 0 such that if u a solution of (1) with blow-up curve Γ : {x → T (x)} and x 0 ∈ R is non-characteristic (in the sense (3)), then there exists d ∞ ∈ (−1, 1) and κ(d, y) is given by the following: Moreover, we have E(w(s), ∂ s w(s)) ≥ E(κ 0 , 0) as s → +∞ and In the real case, relying on the existence of a blow-up profile, together with Liouville type Theorem, Merle and Zaag could prove the openness of the set of non-characteristic points I 0 and the C 1 regularity of the blow-up curve restricted to I 0 . Later, in [23] Nouaili improved this by showing the C 1,α regularity of T . In this paper, we aim at showing the same result. In fact, the situation is more delicate since we have to deal with the regularity of the phase, a new further with respect to the real case. More precisely, this is our main result: Note that the holder parameter µ 0 is the same as the parameter displayed in the exponential convergence to the profile given in (9).
This paper is organized as follows: -In section 2, we give the proof of Theorem 1 assuming the Liouville Theorem. -In section 3, we state briefly some previous results for the complex-valued solution of (1), then give the outline of the proof of the Liouville Theorem.

Regularity of the blow-up curve
In this section, we give the outline of the proof of Theorem 1. In order to do so, we proceed in 4 steps: -In Step 1, we assume Theorem 2, and we study the differentiability of the blow-up curve at a given non characteristic point. -In Step 2, we give two geometrical results for a non characteristic point. -In Step 3, we use the results of the two previous parts to show that I 0 is open and that T is C 1 on this set. -In Step 4, adapting the strategy of Nouaili [23], we refine the result of Step 3 and prove that T is of class C 1,µ 0 and the phase θ is of class C µ 0 on I 0 .
Step 1: Differentiability of the blow-up curve at a given non characteristic point. In this step, we give the recall the result of the real case page 60 in [17] which remains valid in the complex case with no change. For the reader convenience we introduce the result and we give the outline of the proof.
Proof of Proposition 2.1. From translation invariance, we can assume that x 0 = T (x 0 ) = 0, we assume also that θ(0) = 1. In order to prove that T (x) is differentiable when x = 0 and that T ′ (0) = d(0), we proceed by contradiction. From the fact that T (x) is 1-Lipschitz, we assume that there is a sequence x n such that Up to extracting a subsequence and to considering u(−x, t) (also solution to (1)), we can assume that x n > 0. We recall the following where ± = −sgnλ, is a solution to (5).
Proof. The proof is the same as in the real (see page 63 in [17]), one can adapte it without difficulty.
We discuss within the sign of λ: Case λ < 0: Here, we will reach a contradiction using Corollary 2.2 and the fact that u(x, t) cannot be extended beyond its maximal influence domain D u defined by (2). Case λ > 0: Here, a contradiction follows from the fact that w xn (y, s) exists for all (y, s) ∈ (−1, 1) × [− log T (x n ), +∞) and satisfies a blow-up criterion (given in Theorem 2 page 1147 in the paper of Antonini and Merle [3], which is available also for a complex-valued solution) at the same time. Thus, (15) does not hold and T (x) is differentiable at x = 0 with T ′ (0) = d(0). This concludes the proof of Proposition 2.1.
Step 2: Openness of the set of x such that (9) holds We have from the dynamical study in self-similar variables (4) we have the following Proof. We proceed as in the real case in page 66 in [17], considering our two pamareters d and θ instead of one parameter d in the real case.
We claim: Lemma 2.4. (The slope of T (x) around 0 is less than (1 + |d(0)|)/2) It holds that Proof. The proof in the real case stay valid without any change in the complex case. In fact, we never use the profile of w,we use only a geometrical constuction. For more details see page 67 in [17].
Step 3: C 1 regularity of the blow-up set Let x 0 be a non characteristic point. One can assume that x 0 = T (x 0 ) = 0 from translation invariance. From [4] and Proposition 2.1, we know (up to replacing u(x, t) by −u(x, t)) that (9) holds with some d(0) ∈ (−1, 1) and θ(0) = 1, and that T (x) is differentiable at 0 with Using Lemma 2.4, we see that for all x ∈ [− η 0 20 , η 0 20 ], x is non characteristic in the sense (3). Using Proposition 2.1, we see that T is differentiable at x and T ′ (x) = d(x) where d(x) is such that (9) holds for w x . Using Lemma 2.3, we see from (17) Step 4: C 1,µ 0 regularity of the Blow-up curve and C µ 0 regularity the phase θ. In this step, we conclude the proof of Theorem 1. In order to do so, we use in addition to the techniques used in the real case in [23], which remains valid in our case, a decomposition into real and imaginary parts in some inequalities, which gives a new information concerning the regularity of the phase.
We introduce the following: Lemma 2.5. (Locally uniform convergence to the blow-up profile) There exist positive µ 0 = µ 0 (p) and C 0 = C 0 (p) such that for all x 0 ∈ R, there exist δ > 0, s * ∈ R, such that for all X ∈ (x 0 − δ, x 0 + δ) and s ≥ s * , Proof. The same idea used in the real case can be adapted to the complex case without any difficulty. It is to use the result of Lemma 2.3 to prove that the convergence in (9) is locally uniform with respect to x 0 . For more details see page 1544 [23].
We separate the real and imaginary part in (23), and, Thus, θ is C 1,µ 0 near x 0 . In addition, for x close enough to x 0 , Using (26) with (25), At this level, we reduce to the real case to conclude. We introduce a change of variables and prove that |f ′ (ξ)| ≤ C|ξ| µ 0 , which is equivalent to the fact that T is C 1,µ 0 .

Preliminaries
In the following, we recall some results from [4], which we have used in this work. In the following Proposition we recall some dispersion estimates.  (5)) Consider w(y, s) a solution to (5) defined for all (y, s) ∈ (−1, 1) × [s 0 , +∞) for some s 0 ∈ R. Then: where E is defined in (8).
(ii) For all s ≥ s 0 + 1, Proof. The proof is the same as in the real case. See [3] for (i). For (ii), see Proposition 2.2 in [15] for a statement and the proof of Proposition 3.1 page 1156 in [14] for the proof.
Proof. The proof of this Proposition present more difficulties than the real case. In fact, in addition to the techniques used in the real case, we have used an ODE techniques for complexvalued equation, in particular, a decomposition w(y) = ρ(y)e iθ(y) with a delicate phase behavior θ(y). For more details see Section 2 in [4].

Proofs of Theorem 3 and Theorem 2
Proof of Theorem 3 assuming Theorem 2 . The proof is the same as in the real case. For the reader's convenience we recall it. Consider w(y, s) a solution to equation (5) defined for all (y, s) ∈ (− 1 δ * , 1 δ * ) × R for some δ * ∈ (0, 1) such that for all s ∈ R, (14) holds. If we introduce the function u(x, t) defined by then we see that u(x, t) satisfies the hypotheses of Theorem 2 with T * = x * = 0, in particular (11) holds. Therefore, the conclusion of Theorem 2 holds for u. Using back (28), we directly get the conclusion of Theorem 3. Now, we introduce the proof of the Theorem 2.
Proof of Theorem 2 . Consider a solution u(x, t) to equation (1) defined in the backward cone C x * ,T * ,δ * (see (3)) such that (11) holds, for some (x * , T * ) ∈ R 2 and δ * ∈ (0, 1). From the bound (11) and the resolution of the Cauchy problem of equation (1), we can extend the solution by a function still denoted by u(x, t) and defined in some influence domain D u of the form for some 1-Lipschitz function T (x) where one of the following cases occurs: -Case 1: For all x ∈ R, T (x) ≡ ∞.
-Case 2: For all x ∈ R, T (x) < +∞. In this case, since u(x, t) is known to be defined on C x * ,T * ,δ * (3), we have C x * ,T * ,δ * ⊂ D u , hence from (3) and (29) In this case, we will denote the set of non characteristic points by I 0 .
We will treat separately these two cases: Case 1: T (x) ≡ ∞. In the following, we give the behavior of wx ,T (s) as s → −∞.
Proofs of Proposition 3.3 and Corollary 3.4 . The proof is similar to the real case in [17], we have only to adapt it with respect to our set of stationary solutions S ≡ {0, e iθ κ(d, .), |d| < 1, θ ∈ R}.
In the following, we conclude the proof of Theorem 2, when case 1 holds.
Proof. In this case, u(x, t) is defined for all (x, t) ∈ R 2 . The conclusion is a consequence of the uniform bounds stated in the hypothesis of Theorem 2 and the bound for solutions of equation (5) in terms of the Lyapunov functional stated in (ii) of Lemma 3.1. Indeed, consider for arbitrary t ∈ R and T > t the function w 0,T defined from u(x, t) by means of the transformation (4). Note that w 0,T is defined for all (y, s) ∈ R 2 . If s = − log(T − t), then we see from (ii) in Lemma 3.1 and (32) that Using (4), this gives in the original variables Fix t and let T go to infinity to get u(x, t) = 0 for all x ∈ R, and then u ≡ 0, which concludes the proof of Corollary 3.5 and thus the proof of Theorem 2 in the case where T (x) ≡ +∞.

Case 2: T (x) < +∞
In this case also, we conclude by the same way as in the real case in [17]. For the reader's convenience we give the three important ideas used in order to conclude the proof: -In Step 1, we localize a non characteristic point for some slop δ 1 . -In Step 2, we give an explicit expression of w at non characteristic points. -In Step 3, we see that the set of non characteristic points is given by the hole space R.
-In Step 4, we use this three previous steps to conclude the proof when T (x) < +∞.
Step 1: Localization of a non characteristic point in a given cone with slope δ 1 > 1: We claim the following: For all x 1 ∈ R and δ 1 ∈ (δ * , 1), there exists x 0 = x 0 (x 1 , δ 1 ) such that In particular, x 0 is non characteristic.
Proof. In the proof we use a geometrical construction (see page 73 in [17]).
Remark: From this Proposition, we see that we have at least a non characteristic point: In fact, Taking x 1 = x * and δ 1 = 1+δ * 2 , x 0 ∈ R is non characteristic point (in the sense (3)).