WELL-POSEDNESS OF LOW REGULARITY SOLUTIONS TO THE SECOND ORDER STRICTLY HYPERBOLIC EQUATIONS WITH NON-LIPSCHITZIAN COEFFICIENTS

. In this paper, we establish the local well-posedness of low regularity solutions to the general second order strictly hyperbolic equation of divergence form domain R n , where the coeﬃcients a 0 ,a j ,a jk ∈ L ∞ (Ω) ∩ LL (¯Ω) (1 ≤ j,k ≤ n ), b 0 ,c 0 ,b j ,c j ∈ L ∞ (Ω) ∩ C α (¯Ω) (1 ≤ j ≤ n ) for α ∈ ( 12 , 1), d ∈ L ∞ (Ω), ( u (0 ,x ) ,Xu (0 ,x )) ∈ ( H 1 − θ + β log ,H − θ + β log ) with θ ∈ (1 − α,α ), β ∈ R , and Xu = a 0 ∂ t u + n (cid:88) j =1 a j ∂ j u . Compared with previous references, except a little more general initial data in the space ( H 1 − θ + β log ,H − θ + β log ) (only β = 0 is considered as before), we improve both the lifespan of u up to the precise number T ∗ and the range of θ to the left endpoint 1 − α under some suitable conditions.

Here the Log-Lipschitz space in domainΩ (LL in brief) is defined as and the corresponding semi-norm |a| LL stands for the best constant C > 0 in (9). In addition, ||a|| LL =: ||a|| L ∞ + |a| LL .
For the requirements later on, we now introduce the following function spaces (see [7] or [16]).
So far there have been extensive results on the well-posedness of solutions for the second order strictly hyperbolic equations with non-regular coefficients (in the case of regular coefficients, one can see [11] or [15] for the well-posedness of solutions). For instances, with respect to the wave equation in divergence form: the authors in [12] prove that the Cauchy problem of (14) is well-posed in H 1 (Ω) × L 2 (Ω) if the coefficients a ij are Lipschitzian in the variable t and measurable in the variable x ∈ R n , and is well-posed in H s+1 (Ω) × H s (Ω) for any s ∈ R when a ij are Lipschitzian in the variable t and C ∞ b in x. Especially, for the latter case, the authors in [12] obtain the energy estimate of solution u without loss of derivatives.
If the coefficients a ij (t, x) = a ij (t) in (14) depend only on the time variable t and the Lipschitzian continuity of a ij (t) is not fulfilled, for example, in the case of a ij (t) ∈ C α with α < 1, it is proved in [4] that the Cauchy problem of (14) is ill-posed in C ∞ , furthermore, the well-posedness of (14) in Sobolev spaces may not be obtained. Precisely speaking, for x ∈ R 1 , it is shown in [6] that for any function φ with lim r→0 + φ(r) = +∞, there exists a function a(t) with such that the Cauchy problem of Here we point out that the functions in C α (α < 1) clearly satisfy condition (15) (at this time, the ill-posedness of solution u can be also referred to [5] and [10]).
On the other hand, another kind of non-Lipschitzian regularity, that is, Log-Lipschitzian regularity (9) which is smoother than C α with α < 1, has been concerned for the coefficients of (14). Indeed, in [4], when the coefficients a ij (t, x) = a ij (t) ∈ LL in (14), the authors obtain the C ∞ well-posedness of weak solution u to (14) with loss of derivatives. We remark that if the Lipschitzian continuity hypothesis of a ij (t) is not fulfilled, then the loss of regularity for the solution u of (14) cannot be avoided as indicated in [3].
Interesting results have also been obtained when the coefficients a ij (t, x) in (14) depend both on time variable t and space variable x. The authors in [13] and [14] establish the well-posedness of (14) in Gevery spaces when the coefficients a ij (t, x) are C α (0 < α < 1) in t and Gevery class in x. For a ij (t, x) ∈ LL in (t, x), the authors in [6] prove that there exists T > 0 such that for 0 ≤ t < T , the solution u of (14) satisfies the following estimate: This implies that the solution u to (14) for a ij (t, x) ∈ LL will have the loss of regularities with the development of time t. With respect to the wave equations in non-divergence form the authors in [6] also obtain C ∞ well-posedness with a loss of regularity increasing in time, where a ij are LL in t and smooth in x. In [8], for x ∈ R 1 and a 11 (t, x) is the authors have obtained the energy estimate similar to (16) for the weak solution u of (14). Recently, the 1-D result in [8] has been extended to the case of n > 1 by the authors of [9], where the related para-differential calculus with parameters is used to derive the positivity of some corresponding para-product operators.
With respect to the more general second order strictly hyperbolic equation in (1), the authors in [7] prove that for (u 0 , where λ > 0 is a constant depending on the LL-norms of the coefficients, the constants δ 0 and δ 1 of hyperbolicity and α in (5)- (8). Note that for b ∈ C α and v ∈ H s , bv ∈ H s generally holds only for |s| < α, and note H −λt−θ+ 1 2 log → H −λT * −θ for any t ∈ [0, T * ], then in order to let the first-order terms b j ∂ j u in (1.1) make sense in distribution, it is required to pose T * < α−θ λ . As derived in [7], the lifespan of solution u to (1) admits the form of θ1−θ λ , where the number θ 1 satisfies θ < θ 1 < α. Here we especially point out that although one can rewrite the lifespan T * in the form of α−θ λ for some other constant λ > 0, however, direct analysis easily shows that the constant λ has to depend on θ. Indeed, if taking In the present paper, we intend to improve the lifespan of solution u for (1) up to the precise number T * = α−θ λ for any θ ∈ (1 − α, α) and study the regularity of u up to the left endpoint θ = 1 − α under some suitable conditions. We next state the main results in this paper.
This paper is organized as follows. Some basic results including paradifferential calculus and positivity estimates are listed or established in Section 2. In Section 3, we rewrite the equation in (1) as a new system on (u, v) with v = Xu + c 0 u, and some a-priori estimates for smooth solution (u, v) is obtained. In Section 4, with the aid of a-priori estimates derived in Section 3, together with the mollifier argument, we establish the energy estimate for the weak solution of (1) and further complete the proofs of Theorem 1.2 and Theorem 1.3. In addition, Lemma A.1 in Appendix gives a validity illustration on the weak solution u to problem (1) under the suitable regularity assumptions in Theorem 1.2 and Theorem 1.3.
2. Paradifferential calculus. In this section, some useful results on paradifferential calculus will be listed or established. These results, especially Proposition 2, will play a key role in proving Theorem 1.2 and Theorem 1.3.
2.1. Dyadic partition of unity. As in [2] (page 212) or [1] (page 59), one can introduce the following dyadic partition of unity: ]. Then for all nonnegative integers k and l satisfying |k − l| > N 0 , one has C k ∩ C l = ∅.
From now on, we fix two functions χ and ϕ satisfying Lemma 2.1. The Fourier transform of an L 1 function f is defined bŷ f (ξ) = F(f )(ξ) =: R n e −ix·ξ f (x)dx and the inverse Fourier transform is given by In general, for f ∈ S , its Fourier transform is defined by <f , φ >=< f,φ > and its inverse Fourier transform is given by where φ ∈ S, and S stands for the Schwartz space.
The nonhomogeneous dyadic blocks ∆ k are given by The nonhomogeneous low-frequency cut-off operator are defined by For u ∈ S , one then has Next proposition shows some properties of ∆ k a and S k a when a ∈ LL (one can see Proposition 3.3 of [7]).
Paraproducts. For some positive integer N , the nonhomogeneous paraproduct of a and u is defined by The remainder R(a, u) is defined by R(a, u) =: and R a u =: T u a + R(a, u).
Proof. Since the proof is basically similar to that for u ∈ H s (see page 212 of [2]), we omit it here.
The next proposition illustrates some key properties of R a u and au for a ∈ LL and u ∈ H s+βlog .
where the constant C(β, s) > 0 depends on β and s, which is uniformly bounded when (β, s) lies in a compact subset of R × (0, 1).
The following commutator estimate can be found in [1] (see Lemma 2.97 of page 110). Lemma 2.3. Let θ be a C 1 (R n ) function such that (1 + |ξ|)θ ∈ L 1 . Then for a ∈ Lip with ∇a ∈ L p and b ∈ L q (p, q ≥ 1), we have that for any λ > 0, where 1 r = 1 p + 1 q , and k =: |z||(F −1 θ)(z)|. Proposition 3. If a ∈ L ∞ ∩ LL, and α 1 , α 2 ∈ {0, 1}, then for s, β ∈ R, we have where Proof. Since the operator ∂ α2 j Q −s β ∆ k commutes with ∆ l and the spectra of both S l−N a∆ l u and ∆ l u are contained in the annulus of size ≈ 2 l for l ≥ N , one immediately has where Φ l =: φ(2 −l D), and φ ∈ C ∞ 0 is supported in some fixed annulus. In addition, Note that the spectra of [∂ j Φ l , S l−N a]∆ k ∆ l u and [∂ α2 j Q −s β ∆ k Φ l , S l−N a]∆ l u are contained in some annulus. This, together with Lemma 2.3 and Proposition 1 , yields Then (35) and (36) are proved.
In particularly, we have the following corollary.

Corollary 1. Under the assumptions of Proposition 3 , one has
Proof. Since χ is a real radial function, we get that for l ≥ N , This means that F −1 χ(2 −l+N ξ) is a real value function. Note that a is also real, then where Φ l =: φ(2 −l D), φ ∈ C ∞ 0 is supported in some fixed annulus. Therefore, By Lemma 2.3 and Proposition 1 , it follows direct computation that Then (37) and (39) are shown. On the other hand, the proof of (38) is similar to that of (37), we omit it here.
Proof. Note that By Proposition 1 and Lemma 2.3 , one has that for large fixed N ∈ N, In addition,

Positivity estimates.
For ν ∈ N 0 , we define the modified paraproduct by T a u = S ν aS ν+N −1 u + k≥ν S k a∆ k+N u.

Direct computation yields
and au − T a u = j≥ν+1 ∆ j aS j+N −1 u.
If a is smooth, one then has However, this is not true for a ∈ LL since ∂ t a makes no sense as a function. This difficulty and the absence of positivity results of T a are overcome in [7] (see Definition 3.13 of page 193) by introducing the modified paraproduct T a u.
Proof. Note that By (1.24) of [6], one has, By the assumption on the number ν, we finish the proof of (44).
Proof. By (42) and Definition 2.4 , one has Since the spectra of ∆ i a∆ m u and S ν aS ν+N −1 u are contained in a ball, and the spectrum of S m a∆ m+N u lies in an annulus, one has By (1.24) of [6], ||a − a m || L ∞ ≤ C(1 + m)2 −m |a| LL , we arrive at Similiary, This, together with (46) and (47), yields (45).

A-priori estimate. It is easy to know that the equation in
wherẽ In this and the next section, we set where θ ∈ (θ, α) is a fixed number, and λ > 0 will be determined. We now establish some a-priori estimates for the smooth solution (u, v) of system (60), which play a key role in proving Theorem 1.2 and Theorem 1.3 .
To prove Theorem 3.1 , we require to define the following operators where s(t) = λt + θ.
Applying the operator ∆ k to L(u, v) in (60), and performing the L 2 inner product with Q 2 γ ∆ k v, we infer that Namely, Next we treat each term in both sides of (64). This process will be divided into three subsections.
3.1. Treatment on the left hand side of (64). We now deal with each term in the left hand side (written as LHS) of (64). I. Note that Note that Substituting (65), (67) and (68) into (64) yields where E k i = n j=1 E k j,i for i = −1, · · ·, −4.

3.2.
Treatment on the right hand side of (64). This process is divided into two parts: Part I is devoted to treating the second-order term, Part II is to deal with the left terms, which are essentially remainders.
Part I. Note thatã jl ∂ l u = Tã jl ∂ l u + Rã jl ∂ l u. Then 2Re n j,l=1 Set u γ =: Q γ u and then substituting (72) into (71) yields For the principal part II 1 , by utilizing the composition and adjoint operator of para-product, one has Similarly to the analysis of II 1 in the above, byã jk =ã kj , we have II 2 will be taken as a new error term. Indeed, direct computation yields To deal with III 1 , we introduce the notation b jl = a 0ãjl . Note that the positivity result as in Proposition 6 does not hold for T b jl due to the absence of some lowfrequent terms in paraproduct. To overcome this difficulty and the low-regularity of b jl in t, we shall substitute T b jl by T b jl . It follows from direct computation that where E k 10 = 2Re n j,l=1 Then we arrive at In terms of b jl = b lj , then and and Combining (71) and (73)-(79), we get 2Re n j,l=1 Part II. Note that the rest terms are just remainders in the right hand side of (64). In addition, we set −2Re Thus, the right hand side of (64) reads RHS of (64) = − d dt Re n j,l=1 This, together with (69), yields Summing up over k in (81), we arrive at

Estimates of remainders
Next we treat each E i respectively. For the simplicity of notations, we denote the constants which depend on β by D 1 , and D 2 stands for the constants depending both on α and β .
Finally, we show the uniqueness of solution u to problem (1). Set w = w 1 − w 2 , where w 1 and w 2 are two solutions to (1.1) and satisfy the same initial data. Note that the error solution w satisfies the energy inequality (20) whose right hand side vanishes. This derives w = 0 and then the proof of Theorem 1.2 is finished.