Degeneracy in finite time of 1D quasilinear wave equations II

We consider the large time behavior of solutions to the following nonlinear wave equation: $\partial_{t}^2 u = c(u)^{2}\partial^2_x u + \lambda c(u)c'(u)(\partial_x u)^2$ with the parameter $\lambda \in [0,2]$. If $c(u(0,x))$ is bounded away from a positive constant, we can construct a local solution for smooth initial data. However, if $c(\cdot )$ has a zero point, then $c(u(t,x))$ can be going to zero in finite time. When $c(u(t,x))$ is going to 0, the equation degenerates. We give a sufficient condition that the equation with $0\leq \lambda<2$ degenerates in finite time.

Throughout this paper, we assume that c ∈ C ∞ ((−1, ∞)) ∩ C([−1, ∞)) satisfies that c(θ) > 0 for all θ > −1, for all x ∈ R. A typical example of c(θ) satisfying (1.2)-(1.4) is c(θ) = (1 + θ) a with a > 0. The assumptions (1.2) and (1.5) enable us to regard the equation in (1.1) as a strictly hyperbolic equation near t = 0. By the standard local existence theorem for strictly hyperbolic equations, the local solution of (1.1) with smooth initial data uniquely exists until the one of the following two phenomena occurs. The first one is the blow-up: The second is the degeneracy of the equation: c(u(s, x)) = 0.
When the equation degenerates, the standard local well-posedness theorem does not work since the equation loses the strict hyperbolicity. In general, for non-strictly hyperbolic equations, the persistence of the regularity of solutions does not hold (see Remark 1.5). The aim of this paper is to give a sufficient condition for the occurrence of the degeneracy of the equation with 0 ≤ λ < 2. The main theorem of this paper is the following. Theorem 1.1. Let 0 ≤ λ < 2, (u 0 , u 1 ) ∈ H 2 (R) × H 1 (R) and u 1 ≡ 0. Suppose that the initial data (u 0 , u 1 ) and c satisfy that (1.2)-(1.5) and (1.6) Then there exists T * > 0 such that a solution of (1.1) exists uniquely and satisfies that Furthermore, if 0 < λ < 2, then lim tրT * c(u(t, x 0 )) = 0 (1.8) for some x 0 ∈ R.
If λ = 2, then the equation in (1.1) is formally equivalent to the following conservation system: This conservation system is called p-system and describes several phenomena of the wave propagation in nonlinear media including the electromagnetic wave in a transmission line, shearing-motion in elastic-plastic rods and 1 dimensional gas dynamics (see Ames and Lohner [1] and Zabusky [24]). In addition to the assumptions of Theorem 1.1, if R u 1 (x)dx > −2 0 −1 c(θ)dθ is assumed, then (1.1) with λ = 2 has a global smooth solution such that the equation does not degenerate (e.g. Johnson [7] and Yamaguchi and Nishida [23]). On the other hand, in [18,19] (see Remark 1.5 in [18] and Theorem 4.1 in [19]), the author has shown that the degeneracy (1.8) occurs in finite time, if R u 1 (x)dx < −2 0 −1 c(θ)dθ. Namely, these results say that −2 0 −1 c(θ)dθ is a threshold of R u 1 (x)dx separating the global existence of solutions (such that the equation does not degenerate) and the degeneracy of the equation under the assumption (1.6). If (1.6) is not satisfied, then solutions can blow up in finite time (e.g. Klainerman and Majda [9], Manfrin [13] and Zabusky [24]). The main theorem of this paper implies that the degeneracy in finite time of the equation in (1.1) can occurs regardless of R u 1 (x)dx, when 0 ≤ λ < 2.
When λ = 1, the equation in (1.1) is called variational wave equation. As its name suggests, the equation with λ = 1 has a variational structure. The variational wave equation has some physical backgrounds including nematic liquid crystal and long waves on a dipole chain in the continuum limit (see [5]). In [4,5], Glassey, Hunter and Zheng have shown that solutions can blow up in finite time, if (1.6) is not satisfied (see also Remark 1.4). There are a lot of papers devoted to the global existence of weak solutions to variational wave equations (e.g. Bressan and Zheng [2] and Zhang and Zheng [25,26,27]).
When λ = 0, the equation in (1.1) describes the wave of entropy in superfluids (e.g. Landau and Lifshitz [11]). This equation is the one dimensional version of which has been studied in Lindbald [15]. In [15], Lindblad has shown that solutions exists globally in time with small initial data.
In [8,17], Kato and the author have shown that the equation in (1.1) with c(θ) = 1 + θ and λ = 0, 1 degenerates in finite time, if initial data are smooth, compactly supported and satisfy (1.5) and (1.6). The main theorem of this paper removes the compactness condition on initial data and extends the result in [8,17] to (1.1) with more general c(θ) and 0 ≤ λ < 2. In [17,18], the generalization on λ has already been pointed out without a proof. In fact, applying the method in [17,18] to the equation in (1.1), we can generalize the result in [17,18] to (1.1) with 0 ≤ λ < 2 and c(θ) = 1 + θ. However the compactness condition plays a crucial role in [8,17], since we use the following estimates for bounded solutions under the assumption that initial data are compactly supported: Furthermore, taking c(u) = 1 + u, we use the following estimate in [17,18]: 11) which is shown by the fundamental theorem of calculus and the finiteness of the propagation speed. We can not obtain the above estimates directly, if initial data are not compactly supported. The first idea of the proof of Theorem 1.1 is the use of Riemann invariant, which is a major tool for the study of 2×2 conservation systems. When we use the Riemann invariant in the reduction from (1.1) to a first order system, x)) 2 dx appears as a force term in the first order system (see (3.1) and (3.2)). The second idea for the proof is to divide the situation into the two cases that 2 dxds is bounded, then (1.10) holds and (1.9) can be shown by the use of the Riemann invariant. Hence we can use a variation of the method in [8,17]. The key for the generalization on c(·) is the use of 2 dxds is not bounded, we can use the method in [18]. In the case that R u 1 (x)dx ∈ L 1 (R), the Riemann invariant can not be defined in general. Theorem 1.1 with u 1 ∈ L 1 (R) can be shown by applying the same argument as in [18].
Addition to the assumptions of Theorem 1.1, if initial data are compactly suppurated, then (1.8) holds for (1.1) with 0 ≤ λ < 2, which can be shown by the finiteness of the propagation speed (see [8,17]). Our method does not work for the case that λ = 0 (see Remark 3.2). Remark 1.3. Under the assumptions (1.2)-(1.5), there is still no global existence result of (1.1) for 0 ≤ λ < 2. In stead of the assumptions, we assume that c ∈ C ∞ (R) satisfies that for some positive constants c 1 and c 2 . Under these assumptions and (1.6), Zhang and Zheng [25] have shown that (1.1) has global smooth solutions with λ = 1. This global existence result has been extended to 0 ≤ λ ≤ 2 in the author's paper [17]. , we can expect that blow-up solutions exist for 0 < λ ≤ 2, if (1.6) is not satisfied, since the right hand sides of the first and the second equations in (2.4) contain λR 2 and λS 2 respectively, which seems to derive the singularity formation. However, the existence of the blow-up solution is still open, since the proofs of the blow-up theorems for λ = 1 and 2 rely on structures of the equation. When λ = 0, if c(·) is uniformly positive, then it seems possible that (1.1) has global smooth solution for any smooth initial data, although a complete proof or a counterexample for this problem is also open. Remark 1.5. It is known that a loss of the regularity appears for solutions to the following non-strictly hyperbolic equation: where h is a constant and l ∈ N. Namely, in general, (u, ∂ t u) does not belong to Taniguchi and Tozaki [20], Yagdjian [22] and Qi [16]). From this fact, we can expect that solutions of (1.1) have a singularity when the equation degenerates.
This paper is organized as follows: In Section 2, we recall the local well-posedness and some properties of solutions of (1.1). In Sections 3 and 4, we show Theorem 1.1 in the cases that u 1 ∈ L 1 (R) and u 1 ∈ L 1 (R) respectively. Notation We denote Lebesgue space for 1 ≤ p ≤ ∞ and L 2 Sobolev space with the order m ∈ N on R by L p (R) and H m (R). For a Banach space X, C j ([0, T ]; X) denotes the set of functions f : [0, T ] → X such that f (t) and its k times derivatives for k = 1, 2, . . . , j are continuous. Various positive constants are simply denoted by C.

Preliminary
We recall the local well-posedness of (1.1) and some properties of solutions of (1.1). By applying the well-known local well-posedness Theorem (e.g Hughes, Kato and Marsden [6], Majda [12] or Taylor [21]), we can obtain the following theorem. (1.2) and (1.5) hold. Then there exist T > 0 and a unique solution u of (1.1) with c(u(s, x)) = 0 for some T * > 0.
We denote the maximal existence time of the solution u of (1.1) constructed in Theorem 2.1 by T * , that is, We set R(t, x) and S(t, x) as follows The functions R and S have been used in Glassey, Hunter and Zheng [4,5] and Zhang and Zheng [25]. We recall some properties of R and S proved in [17]. By (1.1), R and S are solutions to the system of the following first order equations: where R and S are the functions in (2.3) for the solution u of (1.1) constructed by Theorem 2.1.
Suppose that the assumptions of Theorem (1.1) are satisfied. Then we have for 0 < λ ≤ 2 where R and S are the functions in (2.3) for the solution u of (1.1) constructed by Theorem 2.1. Furthermore R(t) L ∞ and S(t) L ∞ are uniformly bounded with t ∈ [0.T * ) for 0 ≤ λ ≤ 2.
Lemmas 2.2 and 2.3 have been shown in the author's paper [17]. The proofs are essentially the same as in the case that λ = 1, which are proved in Zhang and Zheng [25]. In [17,18], it is assumed only p ≥ 2/λ for the inequality (2.6). But the proof in [17] is not collect for p < 2. In fact, in [17], the proof of (2.6) is based on the following inequality: If p < 2, the right hand side of this inequality is not negative except for λ = 2. However, in [17], we only use (2.6) for p ≥ 2. If λ = 2, (2.6) holds for all p ≥ 1. We note that (2.5) implies that ∂ t u(t, x) ≤ 0 for all (t, x) ∈ [0, T * ) × R.
3 Proof of Theorem 1.1 with u 1 ∈ L 1 (R) We show Theorem 1.1 in the case that u 1 ∈ L 1 (R). First, we show (1.7) by the contradiction argument. From Theorem 2.1 and Lemma 2.3, it is enough to show that T * < ∞. We set G(u) = u −1 c(θ)dθ for u ≥ −1 and µ = 2−λ and define the Riemann invariants (w 1 (t, x), w 2 (t, x)) and (v 1 (t, x), v 2 (t, x)) as follows: From (1.1), (w 1 (t, x), w 2 (t, x)) and (v 1 (t, x), v 2 (t, x)) satisfy that the following systems: y). Let x ± (t) be characteristic curves on the first and third equations of (3.1) respectively. That is, x ± (t) are solutions to the following differential equations: d dt x ± (t) = ±c(u(t, x ± (t))). and Case that t 0 Rẽ (s, y)dyds is bounded. By the contradiction argument, we show that T * is finite in the case that t 0 Rẽ (s, y)dyds is bounded on [0, ∞). We suppose that T * = ∞. (3.4) and (3.5) imply that and We fix an arbitrary number ε > 0. Since lim |x|→∞ u 0 (x) = 0, u 1 ∈ L 1 (R) and ∂ x w j (0, x) ≤ 0 for j = 1, 2, there exists a constant M 0 > 0 such that for any x ≤ −M 0 . Noting x ± (t) goes to −∞ as x ± (0) → −∞ for all t ≥ 0, since We note the positive constant M 1 can be chosen independently of t. Hence the equality 2G(u(t, x)) = w 1 (t, x) − w 2 (t, x) yields that Since G is invertible and G −1 is continuous, this inequality implies that From the above estimates of w 1 and G(u), we have By using (3.2), we have in the same way as in the derivation of (3.9) From the integration by parts and (1.1), it follows that F ′′ (t) = µ Rẽ (t, x)dx ≥ 0. Integrating this equality twice on [0, t] and dividing by t, we have By (3.23) and (3.10), we have Now we estimate the second term of the right hand side of (3.11). We setG(u) = u −1 c(θ)c ′ (θ)dθ. The Schwarz inequality implies that HenceG(u) can be defined for u ≥ −1. From (1.2) and (1.3), we have thatG ′ (u) = c(u)c ′ (u) > 0, from whichG(·) is invertible on [0, ∞) andG −1 (·) is continuous. The fundamental theorem of calculus yields that (3.12) Applying (3.8) to the first term of the right hand side of (3.12) and the Schwarz inequality to the second term, we havẽ

From (3.3) and (3.8), we have |x
where C M > 0 depends on M j for j = 1, 2, 3 and C * > 0 can be chosen independently of the three constants. Integrating the both sides of this inequality on [x − (t), x + (t)], we have While the Schwarz inequality implies that From this inequality, (3.11) and (3.13), we have
Putting t = t j in (3.14) and taking j → ∞, sinceG is continuous, we obtain which contradicts to the assumption that u 1 ≡ 0, if ε is sufficiently small. Therefore we obtain T * < ∞ in the case that t 0 Rẽ (s, y)dyds is bounded.
Remark 3.1. The strictly positivity of c ′ is only used in the estimate ofG. It is enough to assume that c ′ (θ) ≥ 0 for θ > 0 in the case that t 0 Rẽ (s, y)dyds is unbounded. In this case, we use the method in [18,19]. In [18,19], instead of (1.4), it is assumed that c ′ (θ) ≥ 0 for the occurrence of the degeneracy of the equation in (1.1) with λ = 2.
Case that t 0 Rẽ (s, y)dyds is unbounded. We suppose that T * = ∞. In this case, from the identity there exists a positive number T such that We setF (t) = − R u(t, x) − u(T, x)dx. We derive a estimate ofF (t) which contradicts to (3.17). Suppose that the plus and minus characteristic curves x ± (t) defined in (3.3) pass through (T, ∓M ) respectively. The characteristics x ± (t) are drawn on the (x, t) plane as follows: Figure 1: the two characteristic curves on the (x, t) plane From (3.16) and (3.17), these characteristic curves x + (t) and x − (t) do not cross for all t ≥ T . Hence it follows that lim t→∞ c(u(t, x ± (t))) = 0. (3.18) AndF can be divided as follows: Now we estimate x + (t)) − u(T, x + (t)))c(u(t, x + (t))). (3.20) From (3.5) and the definition of w 2 , the first term of the right hand side of (3.20) can be estimated as follows: From the boundedness of u, the second term of the right hand side of (3.20) can be estimated as (u(t, x + (t)) − u(T, x + (t)))c(u(t, x + (t))) ≥ −Cc(u(t, x + (t))).
Hence we have from (3.20) Using v 1 instead of w 2 , in the same way as in the derivation of (3.21), we get The boundedness of u(t, x) and |x While, in the same way as in the computation of F , we havẽ From the definitions of w 2 and v 1 , the second term of the right hand side of (3.25) can be written as follows: c(u(s, x + (s))) + c(u(s, x − (s)))ds.
From (3.18), the second term of the right hand side of the above inequality tends to 0 as t → ∞. Hence, taking t → ∞ in the above inequality, we have which contradicts to (3.17). Hence we have that T * < ∞ in the case t 0 Rẽ (s, y)dyds is unbounded.
Remark 3.2. In the case that λ = 0, since the boundedness of c(ũ)∂ xũ (t) L p is unknown for p = ∞, the above argument does not work. Hence the case that λ = 0 is excluded in (1.8).
Putting ε = 1/(1 + t), we have Since u 1 ∈ L 1 (R), the right hand side is going to infinity as t → ∞, which is a contradiction. Therefore we have T * < ∞.