Appendix A. Some Useful Inequalities

This section starts by presenting the well-known Gronwall’s lemma:

Lemma A.1

Let \(f: } \longrightarrow }\) be a positive continuous function and \(g: } \longrightarrow }\) be a positive integrable function such that

$$\begin f(t)\leqslant A + \int _^ f(s)g(s)ds, \end$$

for some \(A\geqslant 0\) for every \(t\in [0,T]\). Then

$$\begin f(t)\leqslant A\exp \left( \int _^g(s)ds\right) , \end$$

(A.1)

for every \(t\in [0,T]\).

The second result to be presented is related to another pointwise bounds that were presented for Luli et al. in [31] for the study of the global problem in Sect. 3:

Lemma A.2

Under the assumption (3.11) and (3.12), there exists a universal constant \(C_3\),

$$\begin \begin&~ \left\Vert \frac}})^} }}(t,x) \right\Vert _(\Sigma _t)} \\&~ \qquad \leqslant C_3 \left( \left\Vert \frac}})^} }}(t,x) \right\Vert _(\Sigma _t)} + \left\Vert \frac})^}}\partial _x }(t,x)\right\Vert _(\Sigma _t)} \right) , \end \end$$

$$\begin \begin&~ \left\Vert \frac})^}} L}(t,x) \right\Vert _(\Sigma _t)} \\&~ \qquad \leqslant C_3 \left( \left\Vert \frac)^}} L}(t,x) \right\Vert _(\Sigma _t)} + \left\Vert \frac)^}}L\partial _x }(t,x)\right\Vert _(\Sigma _t)} \right) , \end \end$$

and

$$\begin \begin&~ \left\Vert \frac}})^} }\phi (t,x) \right\Vert _(\Sigma _t)} \\&~ \qquad \leqslant C_3 \left( \left\Vert \frac}})^} }\phi (t,x) \right\Vert _(\Sigma _t)} + \left\Vert \frac})^}}\partial _x \phi (t,x)\right\Vert _(\Sigma _t)} \right) , \end \end$$

$$\begin \begin&~ \left\Vert \frac})^}} L\phi (t,x) \right\Vert _(\Sigma _t)} \\&~ \qquad \leqslant C_3 \left( \left\Vert \frac)^}} L\phi (t,x) \right\Vert _(\Sigma _t)} + \left\Vert \frac)^}}L\partial _x \phi (t,x)\right\Vert _(\Sigma _t)} \right) . \end \end$$

Appendix B. Ending of Proof of Theorem 1.1

In this section, we describe the details of the estimates for the second equation in (3.1) that complete the proof of Theorem 1.1.

For simplicity, in Sect. 3 we worked with the first equation of system (3.1). Now we prove the estimates for the second equation.

Proof

The first step is the following: Using (3.3) and (3.4) in the second equation of (3.1), we obtain:

(B.1)

As in Sect. 3, fix \(\delta \in (0,1),\) under the assumptions (3.11)–(3.13) for all \(t\in [0,T^]\), we assume that the solution remains regular, to later show that these bounds are maintained, with a better constant.

Consider \(k=0,1\). Using (3.8) on (3.17), with \(\psi =\partial ^k_x \phi \), and taking the sum over \(k=0,1\), we obtain

$$\begin & \overline}}(t)+\overline}}(t) \leqslant 2C_0 \overline}}(0)\nonumber \\ & \quad +2C_0 \iint _ \left( \left( 1+\left| u \right| ^2\right) ^ | }\phi |+ \left( 1+\left| } \right| ^2\right) ^|L \phi |\right) \nonumber \\ & \qquad \, 2 |\coth \left( \lambda +}\right) | |Q_0\left( \phi ,\tilde\right) | \nonumber \\ & \quad +4C_0\iint _ \left( \left( 1+\left| u \right| ^2\right) ^ | }\partial _x\phi |+ \left( 1+\left| } \right| ^2\right) ^| L\partial _x \phi |\right) \nonumber \\ & \qquad \,|\coth \left( \lambda +}\right) ||(Q_0\left( \partial _x\phi , }}\right) + Q_0(\phi ,\partial _x}}) )| \nonumber \\ & \quad +4C_0 \iint _ \left( \left( 1+\left| u \right| ^2\right) ^ | }\partial _x\phi |+ \left( 1+\left| } \right| ^2\right) ^| L\partial _x \phi |\right) \nonumber \\ & \qquad \, |\partial _x} }\,}}^2\left( \lambda +}\right) ||Q_0\left( \phi ,}}\right) | \nonumber \\ & =: 2C_0\overline}(0)}+ 2C_0\sum _^8 I_j, \nonumber \\ \end$$

(B.2)

In this case, the integrals \(I_j, i\in \\) are defined as follows:

$$\begin & I_1:= 2C_0 \iint _ \left( \left( 1+\left| u \right| ^2\right) ^ | }\phi |\right) |\coth \left( \lambda +}\right) | |Q_0\left( \phi ,}}\right) | \nonumber \\ & I_2:= 2C_0 \iint _ \left( \left( 1+\left| } \right| ^2\right) ^|L \phi |\right) |\coth \left( \lambda +}\right) | |Q_0\left( \phi ,}}\right) | \nonumber \\ & I_3: = 2C_0\iint _ \left( \left( 1+\left| u \right| ^2\right) ^ | }\partial _x\phi |\right) |\coth \left( \lambda +}\right) ||(Q_0\left( \partial _x\phi , \tilde\right) |\nonumber \\ & I_4:= 2C_0\iint _ \left( \left( 1+\left| u \right| ^2\right) ^ | }\partial _x\phi |\right) |\coth \left( \lambda +}\right) ||Q_0\left( \phi ,\partial _x}}\right) )| \nonumber \\ & I_5: = 2C_0\iint _ \left( \left( 1+\left| } \right| ^2\right) ^ L\partial _x\phi |\right) |\coth \left( \lambda +}\right) ||(Q_0\left( \partial _x\phi , \tilde\right) |\nonumber \\ & I_6:= 2C_0\iint _ \left( \left( 1+\left| } \right| ^2\right) ^ | L\partial _x\phi |\right) |\coth \left( \lambda +}\right) ||Q_0\left( \phi ,\partial _x}}\right) )| \nonumber \\ & I_7:= 2C_0 \iint _ \left( \left( 1+\left| u \right| ^2\right) ^ | }\partial _x\phi |\right) |\partial _x} }\,}}^2\left( \lambda +}\right) ||Q_0\left( \phi ,}}\right) | \nonumber \\ & I_8:= 2C_0 \iint _ \left( \left( 1+\left| } \right| ^2\right) ^| L\partial _x \phi |\right) |\partial _x} }\,}}^2\left( \lambda +}\right) ||Q_0\left( \phi ,}}\right) |. \nonumber \\ \end$$

(B.3)

The goal is to control the right-hand side of the above estimate. Essentially, we have eight terms to control, but several are equivalent and we only need to consider essentially two cases. Indeed, it will be sufficient to bound the terms corresponding to \(} \partial _x \phi \) and \(} \phi ,\) since by symmetry, the procedure for the other terms will be analogous. First, we start to bound the term \(I_7\) that represents the most attention, given that it has different sub-terms to estimate, recalling that we define \(\varphi (x)= \left( 1+|x|^2\right) ^,\) with \(0<\delta \ll 1.\)

Taking into account (3.5), (3.13) and (3.15)–(3.16), and writing \(\partial _x \tilde= \frac(L-} )}} \), we get

$$\begin \begin I_7&\lesssim C_0\iint _ \varphi (u) |} \partial _x \phi | |L }}||L \phi | |} }}| + C_0\iint _ \varphi (u) |} \partial _x \phi | |L \tilde|^2|} \phi | \\&\quad +C_0\iint _ \varphi (u) |} \partial _x \phi | |L }}|^2|L \phi | + C_0\iint _ \varphi (u) |} \partial _x \phi | |} }}||} \phi | | L }}| \\&:= I_+ I_+I_+I_. \end \nonumber \\ \end$$

(B.4)

Recall that by Fubini’s Theorem the spacetime \(D_t\) in (3.7) is foliated by \(}_}}\) for \(u\in },\) and also by \(\\times \Sigma _t, t\in }.\) Using again Lemma 3.1, we obtain

$$\begin I_&\lesssim \iint _ \varepsilon \underbrace)^\varphi (u)^|}\partial _x \phi |\right) }_\underbrace(})|L\tilde|\right) }_L_x^2}\underbrace)^\varphi (u)^|}}}|\right) }_}\\&\lesssim \varepsilon \underbrace \dfrac}\partial _x \phi |^2}})^} \right) ^}_ \underbrace\left( \int _\varphi (})|L \tilde|^2 \right) ^}_ \\&\qquad \underbrace^ \left\Vert \dfrac}})^} |}}}|\right\Vert _(\Sigma _)}^ d\tau \right) ^.}_ \end$$

Let us study each of the factors \(T_j\). For \(T_1\), one has:

$$\begin T_1^2&\leqslant \int _}}\left[ \int _}_}} \dfrac}\partial _x\phi |^2}})^} ds\right] d} = \int _}} \dfrac)^}\underbrace}_}}} \varphi (u)|}\partial _x \phi |^2ds \right] }_}}_1(t)} d}\\ &\lesssim \int _}} \dfrac)^}d}, \end$$

since the integral is finite, we have \(T_1 \lesssim \varepsilon .\) The integral \(T_2\) is part of the energy norm \(}_0(t)\) in (3.9) then \(T_2 \lesssim \varepsilon .\) For the term \(T_3\) one can use the same argument as in [31]: using Lemma A.2 one gets

$$\begin T_3&\lesssim \left( \int _0^t \left\Vert \frac}})^} }}} (t,x) \right\Vert ^2_(\Sigma _)} +\int _0^t \left\Vert \frac})^} }\partial _x}}(t,x) \right\Vert _(\Sigma _)}^2 \right) ^ \\&\lesssim \left( \iint _ \frac)^}|}}}|^2 + \iint _ \frac)^}|}\partial _x }}|^2 \right) ^; \end$$

both terms above are of the same form as \(T_1\) and then we have that \(T_3 \lesssim \varepsilon .\) We conclude that \( I_ \lesssim \varepsilon ^4.\)

Now we control the integral \(I_\) in (B.4), using again Lemma 3.1, the assumption (3.12) and Cauchy–Schwarz inequality. We have:

$$\begin I_&= C_0\iint _ \varphi (u) |} \partial _x \phi | |L }}|^2|} \phi | \leqslant \iint _C_2 \varepsilon ^2 \frac})^}|} \partial _x \phi | \frac}})^}|} \phi |\\&\leqslant C_2\varepsilon ^2 \left( \iint _ \frac})}|} \partial _x \phi |^2 \right) ^ \left( \iint _ \frac})}|} \phi |^2 \right) ^ \lesssim \varepsilon ^4. \end$$

To finish with the term \(I_7\), we need to estimate the terms \(I_\) and \(I_\) in (B.4), which are similar in structure; for this case, we get:

$$\begin I_&= ~ I_+I_\lesssim \iint _ \varepsilon ^2 |}\partial _x \phi ||L\phi |+ \iint _ \varepsilon ^2 |}\partial _x \phi ||L}}|\\&= \iint _ \varepsilon ^2 \dfrac}})^}|}\partial _x\phi |\left( \dfrac})^}}|L\phi | + \dfrac})^}}|L}}|\right) \\&\lesssim \iint _ \varepsilon ^2 \left( \dfrac})}|}\partial _x\phi |^2 \right) + \iint _ \varepsilon ^2\left( \dfrac})}|L\phi |^2 \right) \\&\quad + \iint _ \varepsilon ^2\left( \dfrac})}|L}}|^2 \right) \\&\lesssim \int _}} \dfrac})}\underbrace}_}}} \varphi (u) |}\partial _x\phi |^2ds \right] }_}}_1(t)} d} + \int _}} \dfrac\underbrace} \varphi (}) |L\phi |^2ds \right] }_}}_0(t)} du\\&\quad + \int _}} \dfrac\underbrace} \varphi (}) |L}}|^2ds \right] }_}_0(t)} du \lesssim \varepsilon ^4. \end$$

Putting all estimates together for \(I_7,\) we conclude that \(I_7 \lesssim \varepsilon ^4.\) A similar result is obtained for \(I_8.\)

Now we treat the term \(I_1+I_3+I_4\) from (B.2). We have from (3.5) and (3.15)–(3.16),

$$\begin&\iint _ \varphi (u ) | }\partial _x\phi | \left( |L\partial _x\phi ||}}}|+ |} \partial _x \phi | |L}}| + |L\phi ||} \partial _x }} |+ |} \phi ||L \partial _x }} |\right) \\&\quad \quad \quad +\iint _ \left( \varphi (u)| }\phi |\right) \left( |L\phi ||}\tilde|+ |L}}| |}\phi |\right) . \end$$

Using the condition (3.13), the situation matches Case 1 developed in [31]. All these integrals can be written as

$$\begin \sim \iint _ \left( \varphi (u) |}\partial _x }||L\phi ||}\partial _x \phi | + \varphi (u) |}\partial _x }||L\partial _x \phi | |}\phi |\right) . \end$$

Then, we can use the estimate (3.23) in Sect. 3 to conclude the bounds on these terms, which again are of order \(\varepsilon ^3.\) \(\square \)

Appendix C. Classical Solution: Local Theory

As we can see, Proposition 1.1 does not directly provide us with a classical solution for the initial value problem (1.17). In order to obtain such a classical solution, we need an initial data with sufficient regularity, which allows us to control the terms associated with the nonlinearity. The idea of the proof still has the same structure.

Recall that the initial value problem for (1.17) can be written in vector form as follows

$$\begin \partial _ \left( m^\partial _\Psi \right) =F(\Psi ,\partial \Psi )\\ (\Psi ,\partial _t \Psi )|_}=(\Psi _0, \Psi _1) \in \mathcal }}. \end\right. } \end$$

(C.1)

Where \(m^\) are the components of the Minkowski metric with \(\alpha , \beta \in \left\ \), and

$$\begin (\Psi ,\partial _t \Psi )\in \mathcal }}:=H^3(})\times H^(}) \times H^2(}) \times H^2(}). \end$$

(C.2)

We are also going to impose the following condition on the initial data

$$\begin \left\Vert \left( \Psi _0,\Psi _1\right) \right\Vert _}}} \leqslant \dfrac, \end$$

(C.3)

where the assumptions on the constant \(D\geqslant 1\) will be indicated below.

The following proposition shows that the equation (C.1), in terms of the function \(}\) introduced in (1.16), is locally well-posed in the space \(L^([0,T]; \mathcal }})\) with the norm in this space defined by

$$\begin \left\Vert (\Psi ,\partial _t \Psi ) \right\Vert _([0,T]; })} = \sup _ \left( \left\Vert \Psi \right\Vert _})\times H^(})}+ \left\Vert \partial _t \Psi \right\Vert _})\times H^2(})} \right) , \end$$

with \((\Psi ,\partial _t \Psi )\) introduced in (1.18). The result is the following.

Proposition C.1

If \((\Psi _0, \Psi _)\) satisfies the condition (C.3) with an appropriate constant \(D\geqslant 1\), then:

(1)

(Existence and uniqueness of local-in-time solutions). There exists

$$\begin T=T\left( \left\Vert \left( }_0, \phi _0 \right) \right\Vert _}) \times H^3(})}, \left\Vert \left( }_1, \phi _1 \right) \right\Vert _}) \times H^2(})},\lambda \right) > 0, \end$$

such that there exists a (classical) solution \(\Psi \) to (C.1) with

$$\begin (\Psi ,\partial _t \Psi )\in L^([0,T];\mathcal }}). \end$$

Moreover, the solution is unique in this function space.

(2)

(Continuous dependence on the initial data). Let \(\Psi _^, \Psi _^\) be sequence such that \(\Psi _^ \longrightarrow \Psi _\) in \(H^3(})\times H^(})\) and \(\Psi _^ \longrightarrow \Psi _\) in \(H^2(})\times H^2(})\) as \(i \longrightarrow \infty .\) Then taking \(T>0\) sufficiently small, we have

$$\begin & \left\Vert \left( \Psi ^-\Psi , \partial _t(\Psi ^-\Psi ) \right) \right\Vert _([0,T]; H^s(})\times H^(}) )\times L^([0,T]; H^(}) \times H^(}))}\\ & \quad \longrightarrow 0. \end$$

as \(i \longrightarrow \infty \) for every \(1\leqslant s < 3.\) Here \(\Psi \) is the solution arising from data \((\Psi _0,\Psi _1)\) and \(\Psi ^\) is the solution arising from data \(\left( \Psi _0^,\Psi _1^ \right) .\)

Proof of Proposition C.1

(1). This part of the Proposition is proved by Picard’s iteration. Using a density argument it is sufficient to assume the initial data \((\Psi _0,\Psi _1)\in }^4\) (\(}\) being the Schwartz class), along with condition (C.3). Define a sequence of smooth functions \(\Psi ^,\) with \(i \geqslant 1\) such that

$$\begin \Psi ^=(0,0), \end$$

and for \(i \geqslant 2,\) \(\Psi ^\) is iteratively defined as the unique solution to the system

$$\begin \partial _ \left( m^\partial _\Psi ^\right) =F\left( \Psi ^,\partial \Psi ^\right) \\ \left( \Psi ^,\partial _t \Psi ^\right) |_}=(\Psi _0, \Psi _1) \in }. \end\right. } \end$$

(C.4)

It is important to note that from (1.18) and (C.3) we can assure that for \(j=1,2,\)

$$\begin \sum _^ \sup _}|\partial ^_ F_j|(x,p) \leqslant C_\lambda }. \end$$

(C.5)

Indeed, this can be seen from the fact that for \((x,p)=(x_1,x_2,p_1,p_2,p_3,p_4)\) and \(|x|\leqslant \frac,\)

$$\begin F_1(x,p)= & 2\sinh (2\lambda +2x_1)\left( p_4^2-p_3^2\right) ,\quad \\ F_2(x,p)= & \dfrac \left( p_3 p_1 - p_2p_4 \right) . \end$$

Define bounded functions in the class \(C^1\).

It is important to note that condition (C.5) allows this iterative definition of the functions \(\Psi ^\) to be possible, since it maintains each component of F with the required regularity, see [12]. First, it will be shown that for a sufficiently small \(T>0,\) the sequence \((\Psi ,\partial _t \Psi )\) is uniformly (in i) bounded in \(L^([0,T]; \mathcal }})\), then it will be shown that it is also a Cauchy sequence. For the first part, the idea is to use the energy estimates (2.2), and we want to prove that there is a constant \( 0 < A \leqslant \frac\) such that

$$\begin \left\Vert \left( \Psi ^,\partial _t \Psi ^ \right) \right\Vert _([0,T];\mathcal }})} \leqslant A, \end$$

(C.6)

implies that

$$\begin \left\Vert \left( \Psi ^,\partial _t \Psi ^ \right) \right\Vert _([0,T];\mathcal }})} \leqslant A. \end$$

The energy estimation (2.2) allows us to write for (C.1) the following estimate:

$$\begin \begin \sup _ \left\Vert \left( \Psi ^,\partial _t \Psi ^\right) \right\Vert _}}}&\leqslant C(1+T)(\left\Vert \left( \Psi _0, \Psi _1\right) \right\Vert _}}}) \\&\quad +C(1+T) \int _^ \Bigg (\left\Vert F_1\left( \Psi ^,\partial \Psi ^\right) \right\Vert _})}\\&\quad +\left\Vert F_2\left( \Psi ^,\partial \Psi ^ \right) \right\Vert _})}\Bigg )(t)dt. \end\nonumber \\ \end$$

(C.7)

With this estimate, our goal is to bound the integral on the right hand side of the inequality above. That is, we want to prove that there exists \(B=B(A,F)>0\) such that for \(t\in [0,T],\) we have

$$\begin \sum _^ ||\partial _x^nF\left( \Psi ^,\partial \Psi ^\right) ||_(t) \leqslant B. \end$$

(C.8)

For this, we will use the conditions (C.3) for each \(F_j\) which is satisfied by the hypothesis in (C.6), if \(B_1=\max \}, C_} \},\) and using chain rule we get

$$\begin \begin \sum _^ ||\partial _x^nF\left( \Psi ^,\partial \Psi ^\right) ||_&\leqslant B_1 + B_1||\partial _x \Psi ^||_\\&\quad + B_1||\partial \partial _x \Psi ^||_+ B_1||\partial \Psi ^||^2_\\&\quad + B_1||\partial _x^2 \Psi ^||_\\&\quad + B_1||\partial \partial _x \Psi ^\cdot \partial \partial _x \Psi ^ ||_+ ||\partial \partial _x^2 \Psi ^||_ \\&\leqslant B, \end \end$$

where \(B=B(B_1,A,\lambda )\), which results in the following estimate

$$\begin \begin \sup _ \left\Vert \left( \Psi ^,\partial _t \Psi ^ \right) \right\Vert _}}} \leqslant C(1+T)\left( \left\Vert \left( \Psi _0, \Psi _1 \right) \right\Vert _}}}+2BT \right) , \end \end$$

(C.9)

we can choose \(T> 0\) sufficiently small such that

$$\begin 2BT \leqslant \left\Vert \left( \Psi _0,\Psi _1\right) \right\Vert _}}}, \end$$

so

$$\begin \left\Vert \left( \Psi ^,\partial _t \Psi ^\right) \right\Vert _([0,T]; \mathcal }})} \leqslant 2C \left\Vert (\Psi _0,\Psi _1) \right\Vert _}}}. \end$$

If we choose \(D > 4C\) in (C.3) and \(A:= 2C||(\Psi _0,\Psi _1)||_}}}\leqslant \frac \leqslant \frac.\) We have thus shown the desired implication.

In Sect. 2 we showed that the last sequence is of Cauchy type in the larger space \(L^([0,T]; })\). Therefore, the sequence is Cauchy on \(L^([0,T]; \mathcal H),\) and hence convergent. That is, there exists \((\Psi , \partial _t \Psi )\) in \(L^([0,T]; })\). The uniform bounds (on i) in \(L^([0,T], \mathcal }})\) guarantee that the limit in fact lies in the smaller space \(L^([0,T], \mathcal }}),\) that is, for almost \(t\in [0,T],\) \((\Psi ^, \partial _t \Psi ^)(t)\in \mathcal }}\), uniform in i, and therefore by Banach–Alaoglu’s Theorem, there is a weak limit in \( \mathcal }}\) (up to a subsequence). But the uniqueness of the limit ensures that this limit must agree with \((\Psi , \partial _t \Psi )(t).\) This concludes the proof of existence.

Finally, for the continuous dependence on initial data, we prove in Sect. 2 that taking \(i \longrightarrow \infty ,\) we get

$$\begin \sup _ \left\Vert \left( \Psi ^-\Psi , \partial _t \Psi ^-\partial _t \Psi \right) \right\Vert _}}} \longrightarrow 0. \end$$

To obtain the result in general for \(1 \leqslant s < 3,\) simply observe that

$$\begin \begin&\sup _ ||\left( \Psi ^-\Psi ,\partial _t \Psi ^-\partial _t \Psi \right) ||_\times H^\times H^}(t) \\&\quad \leqslant C\sup _\left( ||\left( \Psi ^-\Psi ,\partial _t \Psi ^-\partial _t \Psi \right) ||_\times L^\times L^}(t) \right) ^}\\&\quad \quad \times \left( ||\left( \Psi ^-\Psi ,\partial _t \Psi ^-\partial _t \Psi \right) ||_\times H^\times H^}(t) \right) ^} \longrightarrow 0. \end \end$$

This last property ends the proof of Proposition 1.1. \(\square \)