BLOWUP TIME AND BLOWUP MECHANISM OF SMALL DATA SOLUTIONS TO GENERAL 2-D QUASILINEAR WAVE EQUATIONS

. For the 2-D quasilinear wave equation (cid:80) 2 i,j =0 g ij ( ∇ u ) ∂ 2 ij u = 0, whose coeﬃcients are independent of the solution u , the blowup result of small data solution has been established in [1, 2] when the null condition does not hold as well as a generic nondegenerate condition of initial data is assumed. In this paper, we are concerned with the more general 2-D quasilinear wave equation (cid:80) 2 i,j =0 g ij ( u, ∇ u ) ∂ 2 ij u = 0, whose coeﬃcients depend on u and ∇ u simultaneously. When the ﬁrst weak null condition is not fulﬁlled and a suitable nondegenerate condition of initial data is assumed, we shall show that the small data smooth solution u blows up in ﬁnite time, moreover, an explicit expression of lifespan and blowup mechanism are also established.

(ii) u ∈ C 2 (([0, T ε ] × R 2 ) \ {M ε }) and satisfies for t < T ε Remark 1. With respect to the blowup problem of small data solution to [9] we have shown that the blowup mechanism of smooth solution u is of ODE-type, which means ∇u start to develop singularities from the lifespan time T ε while u is still continuous up to T ε . Here Theorem 1.1 illustrates that the blowup mechanism of smooth solution to (1) is of geometric type, which means only ∇ 2 u develop singularities from T ε while u and ∇u are still continuous up to T ε . Our Theorem 1.1 is similar to the "lifespan theorems" in [1,2], where such 2-D nonlinear wave equations ∂ 2 t v − ∆v + 0≤i,j,k≤2 g k ij ∂ k v ∂ 2 ij v = 0 with g k ij some constants are studied when the null conditions are not fulfilled. [4,25,26] it was shown that the smooth solution exists globally. On the other hand, for the n-dimensional nonlinear wave equation (n = 2, 3) with coefficients depending only on the first order derivatives of the solution, ∂ 2 t u − c 2 (∂ t u)∆u = 0 and, more generally, , and the linear part n i,j=0 c ij ∂ 2 ij u is strictly hyperbolic with respect to time t, it is known that small data smooth solutions exist globally or almost globally if related null conditions hold (see [3,5,20,27,29,30] and the references therein), otherwise small data smooth solutions blow up in finite time (see [1,2,6,8,11,14,15,16,17,18,19,24,28] and so on).
Remark 3. In terms of the results in [23], one can only derive that the lifespan T ε of smooth solution to (1) satisfies T ε ≥ C ε for small ε > 0. On the other hand, similar to the proof of Theorem 2.3 in [24], where the 3-D quasilinear wave equations being quadratic forms are treated, one can further obtain T ε ≥ C ε 2 . Here we especially point out that a precise bound of ε 2 T ε has been given in our Theorem 1.1.

Remark 4.
If problem (1) admits a symmetric solution u(t, r) when ϕ 0 (x) and ϕ 1 (x) are symmetric, then by the similar arguments as in [10] and [17], we can prove (6) still holds even if assumption (ND) is removed.
Remark 6. In [6] and [28], for the 3-D quasilinear wave equations 3 i,j=0 g ij (∂u)∂ 2 ij u = 0, the authors utilize Christodoulou's geometric approach to study the blowup mechanism of small value smooth solutions under some other classes of restrictions on the initial data (u(0, x), ∂ t u(0, x)). In addition, for different blowup types of classical solutions to some other nonlinear equations, one can see [12], [13], [22] and the references therein.
Remark 7. Consider the general 3-D quasilinear wave equation When the corresponding weak null condition is not fulfilled, and an analogous (N D) condition for the 2-D quasilnear wave equation is posed, we have shown in [8] that the lifespan T ε of small data smooth solution u satisfies where the constant τ 0 is completely determined by the coefficients g ij and the data (u 0 (x), u 1 (x)). Otherwise, if the weak null condition holds, we have proved in [7] that the small data solution u exists globally.
Remark 8. For problem (1), so far the systematic results as in [7] and [8] for the general 3-D wave equations 3 i,j=0 g ij (u, ∇u)∂ 2 ij u = 0 have not been established. For examples, even for the cases of g ij (u, ∇u) = c ij +d ij u or g ij (u, ∇u) = c ij +d ij u 2 , the global existence or blowup results on the small data smooth solution u of (1) are not known yet.
Near the blowup point M ε we can give a more accurate description on the behavior of solution u as in [1,2] and [8].
Remark 9. Theorem 1.2 provides a more accurate description of the solution u near the blowup point M ε for t ≤ T ε than Theorem 1.1 as in [1,2] and [8]. For instances, G, ∇G ∈ C(Φ(D 0 )) can be directly derived from (w(s, θ, τ ), v(s, θ, τ )) ∈ C 3 (D 0 ) and (H); in addition, by ∂ 2 σ G = ∂sv ∂sφ and u(t, it follows from a direct computation and condition (H) that there exists a positive constant C independent of ε such that 1 To prove Theorem 1.1, at first, as in [15,Chapter 6] and [8], by constructing a suitable approximate solution u a to (1) and then considering the difference of the exact solution u and u a , applying the Klainerman-Sobolev inequality in [21], and further establishing a delicate energy estimate, we finally obtain this lower bound of T ε . Next we derive the required upper bound of T ε . Motivated by the "geometric blowup" method of [1,2], as in [8], we introduce the blowup system of (1) to study the lifespan T ε and blowup mechanism. That is, by introducing a singular change Φ of coordinates in the domain D = {(σ, θ, τ ) : where φ(s, θ, τ 1 ) = s and ∂ s φ = 0 holds at some point, here σ = r − t, τ = ε √ t, and C 0 > 0 a fixed constant, and setting G(Φ) = w(s, θ, τ ) and (∂ σ G)(Φ) = v(s, θ, τ ), we obtain a nonlinear partial differential system on (φ, w, v) from the ansatz u = ε √ r G(r − t, θ, ε √ t) and the equation in (1). This blowup system for (1) can be shown to admit a unique smooth solution (φ, w, v) for τ ≤ τ ε , where the couple (φ, v) satisfies properties (H) and (7) of Theorem 1.2. This enables us to determine the blowup point at time t = T ε and give a complete asymptotic expansion of T ε as well as a precise description of the behavior of u close to the blowup point. In the process to treat the resulting blowup system, as in [1,2], we will use the Nash-Moser-Hörmander iteration method to overcome the difficulties introduced by the free boundary t = T ε and the complicated nonlinear blowup system. To this end, the linearized blowup system should be solved. However, due to the simultaneous appearances of u and ∇u in the coefficients g ij (u, ∇u), the resulting blowup system of (1) has some different features from that in [1,2] and [8] (one can see (52)-(53) in §3 below). For examples, compared with the linearized blowup system of ∂ 2 t u − ∆u + 0≤i,j,k≤2 g k ij ∂ k u∂ 2 ij u = 0 in [2], some coefficients α 1 and α 2 in (52) do not admit the smallness property, moreover, there are more terms to be treated in (53) than those in the corresponding (3.1.1 b ) of [2]. Thanks to the multipliers chosen in [1,2], through integration by parts we can derive the energy estimates of solutions to the linearized blowup system directly, and subsequently its solvability is shown. Based on this and the standard Nash-Moser-Hörmander iteration, the proof of Theorem 1.2 can be completed. In addition, compared with the treatments on the 3-D problem in [8], here the treatments on the 2-D problem (1) have some distinct features due to the different large time behaviors of solutions to 2-D and 3-D linear wave equations.
The paper is organized as follows: In §2, as in [15], we construct a suitable approximate solution u a (t, x) to (1) and establish related estimates, which allows us to obtain the required lower bound on the lifespan T ε . In §3, the blowup system of (1) is derived and solved, which enables us to prove Theorem 1.2. The proof of Theorem 1.1 is completed in §4.
Throughout the paper, we will use the following notations: Z denotes one of the Klainerman vector fields in R + t × R 2 , i.e., 2. The lower bound of the lifespan T ε . In this section, we will establish the estimate on the lower bound of T ε for the smooth solution to problem (1). As in the proof of [15, Theorem 6.5.3], by constructing the approximate solution u a of (1), and then estimating the difference of u a and the solution u, we can derive the lower bound of T ε by continuity induction argument. Our new gradients in this procedure are how to construct the approximate solution and look for the precise blowup time for the nonlinear profile equation of (1) and subsequently treat the solution u itself and ∇u other than only treat the derivatives of solutions in [15]. Although some related procedures are analogous to those in [8], for reader's convenience, we shall give some detailed proofs. Set the slow time variable τ = ε √ 1 + t, and assume the solution u of (1) can be approximated by where (ω 0 ,ω 1 ,ω 2 ) = (−1, ω 1 , ω 2 ), and F 0 (q, ω) has been defined in (2). Before studying the blowup problem of (8), we require cite the following two lemmas, whose proofs are almost analogous to Lemma 2.1-2.2 in [8] and are then omitted here.
Remark 10. Note that as in the analysis of [8], one can show that assumption (N D) is generic.
According to the Chapter 6 of [15], we know that In addition, from the expression of V (q, ω, τ ) we can conclude that if and We now start to construct an approximate solution to (1) for 0 ≤ τ = ε √ 1 + t < τ 0 . Let w 0 be the solution of the following linear wave equation: we have Proof. We divide this proof into the following three cases. (14)- (15) and the explicit expression of V , then Moreover, by the Theorem 6.2.1 of [15], we know for any constant l > 0, if r ≥ lt, On the other hand, according to the fact of Since the support of J 4 on the variable q lies in (−∞, − 1 3ε ), and apply the fact for any φ(t, r) ∈ C 1 we can get the estimate |Z α J 4 | ≤ C α,b ε 3 (1 + t) − 1 2 . All the above analysis yield Collecting the estimates above, we arrive at
For the latter requirements, we list a conclusion which has been shown in [24].
Based on the preparations above, next we establish Proposition 1. For sufficiently small ε > 0 and 0 < τ = ε √ 1 + t ≤ b < τ 0 , then (1) has a C ∞ solution which admits for all |α| ≤ 2, Proof. Set v = u − u a . Then one has We make the induction hypothesis, for some T ≤ b 2 ε 2 − 1, which further implies for |α| ≤ 2 and t ≤ T , To prove the validity of (21), we will show for sufficiently small ε > 0, and then utilize the continuity method to obtain ε √ 1 + T = b. Applying Z α on two hand sides of (20) yields where Next we establish the estimate of ∂Z α v(t, ·) L 2 from the equation (24). Define the energy Multiplying ∂ t Z α v (|α| ≤ 4) on two hand sides of (24) and integrating by parts, and noting the fact of |∂ β u| = |∂ β u a + ∂ β v| ≤ C b ε(1 + t) − 1 2 (|β| = 1, 2) from the construction of u a and assumption (21), then we arrive at We now treat each term in the integration |α|≤4 R 2 |F ||∂ t Z α v|dx.
We now give some illustrations on the existence of local solution to (41). According to the analysis in §2, we know that P (G) = 0 can be solved with the corresponding initial data on t = τ 2 1 /ε 2 in the strip where C 0 > 0 is a large constant, τ 1 > 0 is a fixed constant and η > 0 is a small constant satisfying η < τ 0 − τ 1 .
By the expression of > 0 for small ε and smooth function φ, and then by implicit function theorem one obtains where E is a smooth function on its arguments. With the initial data φ(s, θ, τ 1 ) = s, (42) has a unique solutionφ for a sufficient small η > 0. Set Then (φ,w,v) is a local solution to the blowup system (41) since the local existence of G is known by (38). Moreover, from the uniqueness result on the solution u of (1) for t ∈ [0, (τ 1 + η) 2 /ε 2 ] we know thatv and φ − s are smooth and flat on {s = M }.
In order to solve the blowup system (41), as in [1,2], we will use the Nash-Moser-Hörmander iteration method under the restriction (H). For this end, we will divide this into the following five steps.
3.1. Structure of the linearized blowup system. Denote (φ,ẇ,v) by the corresponding unknown solution to the linearized blowup system of (41). Similar to Theorem 3 in [2], setż =ẇ − vφ and note that 2 i,j,k=0 e k ij ∂ k u∂ 2 ij u does not satisfy the null condition, then it follows from a direct computation that the linearized system of (41) can be changed into such forms where and a i (i = 0, 1, · · · , 9) are smooth functions.
On the other hand, in terms of Proposition II.2 in [2],v can be determined by the first equation I 1 (φ,ẇ,v) =ḟ 3 in the linearized blowup system of (41).
For the convenience to obtain the weighted energy estimate on (43)-(44) later on, as in [2] we choose a "nearly horizontal" surface Σ through {τ = τ 1 , s = M } instead of the initial plane {τ = τ 1 }, Σ being a characteristic surface of the operator Z 1 ∂ s − ε 2 ∂ sφ Q whose corresponding coefficients are computed on (φ,v,w). Note that if we define the characteristic surface Σ by τ = ψ(s, θ) + τ 1 , then ψ satisfies (45) For small ε > 0 it is easy to know that (45) has a smooth solution ψ(s, θ) in the domain D S . Choosing a truncation function χ ∈ C ∞ (R) with χ(t) = 1 for t ≤ 1 2 , and χ(t) = 0 for t ≥ 1, and making the following transformation then we will work later on in the following domain which is actually an unknown domain since we do not know what the precise τ ε is so far. By (46), the characteristic surface Σ becomes {T = 0}.
3.2. The construction of an approximate solution to (41). As the first step to use Nash-Moser-Hörmander iteration method, it is required to construct an approximate solution (φ a , w a , v a ) of (41) such that φ a satisfies (H) at some point.
Here we specially point out that although (52) and (53) are somewhat similar to the linearized equations (3.1.1 a ) and (3.1.1 b ) in [2], however, the coefficients α 1 and α 2 in (52) are only bounded quantities other than the ones of O(ε 2 ) in (3.1.1 a ) of [2], in addition, there are more terms in (53) than those in (3.1.1 b ) of [2] due to the simultaneous appearances of the solution u and its first order derivatives ∇u in the coefficients of (1). Due to the differences among (52)-(53) and (3.1.1 a ) − (3.1.1 b ) in [2], we will derive the energy estimates of solutions to (52)-(53) directly by choosing suitable multipliers and integrating by parts other than change the main part of (52) into a third order scalar equation to derive the related estimates by introducing a new unknown functionk withż = Zk as in [1,2].
In the process of solving (52)-(53), we require to choose a subdomain D 3 of D 2 which is an influence domain of the first order differential operator Z, contains the point (σ, θ 0 , 1) and is bounded by the planes {x = −C 0 }, {x = M }, {y = 0}, {ρ = 0}, {ρ = 1}, S + and S − . Here, S + and S − do not intersect in D 2 , and their normal directions are (−η, ν, 1) and (−η, −ν, 1), ν is some appropriate positive constant. In addition, it is assumed that we are given a smooth function φ in D 3 and a constant λ in (49) close to φ 0 and λ 0 respectively, where the function φ also satisfies (H) for some point (x 0 ,ȳ 0 , 1) (this can be achieved by the implicit function theorem established in [1] in terms of the properties of φ 0 satisfying (H) at the point (σ, θ 0 , 1)).
3.5. The tame estimate and solvability of (52)-(53). Note that ρ = 0 is a characteristic surface of the operator ZS − ε 2 (Sφ)N , then As in [1], set then we can obtain the following energy estimate.
Proof. Set P = ZS − ε 2 AN and choose the multiplier Mż = aSż + dZż as in [1], where the functions a and d will be determined later on. Then through integrations by parts we have with +ε 8 z 0 (∂ y s 0 )AaN 2 + ε 4 AaN 3 (∂ y s 0 ) + 2ε 6 AdN 2 (∂ y z 0 ) + ε 2 ∂ y (AdN 3 ) and Choosing a = A −1 δ 2 g and d = −δ 2 g, and using (H) condition of φ, (54) and the geometric property of D 3 as in the proof of Proposition 3.3 in [2], we can obtain that by (57) and a careful computation Note that for any ϑ > 0, there exists C > 0 such that for all h ≥ 1 and smooth ψ satisfying ψ| t=0 = 0, the following inequality holds In addition, Substituting (60) into the right hand side of (58) and using (59) for ϑ = 2 and the functionż instead of ψ, we have for sufficiently large h here we point out that the "largeness" of α 1 and α 2 does not play essential role since the parameter h > 0 can be chosen larger.
Based on Lemma 3.1-Lemma 3.2, by the standard Picard iteration and fixed point theorem we can obtain the following conclusion as in Proposition 3.4 of [2]: