# Global Well-posedness of a Class of
Singular Hyperbolic Cauchy Problems††thanks: Dedicated to Bhagawan Sri Sathya
Sai Baba on the Occassion of His 96th Birthday.

Rahul Raju Pattar , N. Uday Kiran
Department of Mathematics and Computer Science
Sri Sathya Sai Institute of Higher Learning, Puttaparthi, India
<EMAIL_ADDRESS>(Corresponding<EMAIL_ADDRESS>
###### Abstract

The goal of this paper is to establish a global well-posedness, cone condition
and loss of regularity for singular hyperbolic equations with coefficients in
$L^{1}((0,T];C^{\infty}(\mathbb{R}^{n}))\cap
C^{1}((0,T];C^{\infty}(\mathbb{R}^{n}))$ and Cauchy data in an appropriate
Sobolev space tailored to a metric on the phase space. The coefficients are
unbounded near the singular hyperplane $t=0$ and polynomially growing as
$|x|\to\infty.$ The singular behavior is characterized by the blow-up rate of
the coefficients and their first $t$-derivatives near $t=0.$ In order to study
the interplay of the singularity in $t$ and unboundedness in $x$, we consider
a class of metrics on the phase space. Our methodology relies on the use of
the Planck function associated to the metric to subdivide the extended phase
and to define an infinite order pseudodifferential operator for the
conjugation. We also give some counterexamples.
Keywords: Singular Hyperbolic Cauchy Problem $\cdot$ Loss of Regularity
$\cdot$ Global Well-posedness $\cdot$ Metric on the Phase Space $\cdot$
Pseudodifferential Operators
MSC(2010): 35L81 $\cdot$ 35L15 $\cdot$ 35S05 $\cdot$ 35B65 $\cdot$ 35B30

##  1 Introduction

We consider the singular hyperbolic Cauchy problems of the form

$u_{tt}+A(t,x,D_{x})u_{t}+B(t,x,D_{x})u=f,\quad(t,x)\in(0,T]\times{\mathbb{R}}^{n},\;T<\infty,$

where $u$ is a function of $t,$ valued in a Sobolev space while $A$ and $B$
are linear partial differential operators some of whose coefficients tend to
infinity in some sense as $t\to 0.$ See [3] and the references therein for a
discussion on singular hyperbolic Cauchy problems. A study of such problems is
motivated by applications in physics [6], in the study of blow-up solutions of
quasilinear problems [7, Section 4] and also in the study certain degenerate
Cauchy problems [3]. One of the prominent features of singular hyperbolic
Cauchy problems is the loss of regularity index of the solutions in relation
to the initial data defined in an appropriate Sobolev space.

In order to describe the loss of regularity and well-posedness results from
the literature, consider the following model strictly hyperbolic equation

$\begin{cases}\begin{aligned}
&\partial_{t}^{2}u-a(t,x)\partial_{x}^{2}u+\sum_{\begin{subarray}{c}j,l=0\\\
j+l\leq
1\end{subarray}}^{1}b_{j,l}(t,x)\partial_{x}^{j}\partial_{t}^{l}u=f(t,x),\quad(t,x)\in(0,T]\times{\mathbb{R}},\\\
&u(0,x)=f_{1}(x),\;\partial_{t}(0,x)=f_{2}(x).\end{aligned}\end{cases}$ (1.1)

The operator coefficients are in
$L^{1}\left((0,T];C^{\infty}({\mathbb{R}})\right)\cap
C^{1}\left((0,T];C^{\infty}({\mathbb{R}})\right)$ and the singular behavior of
the above Cauchy problem is described by the following estimates

$\begin{rcases}\begin{aligned} |\partial_{x}^{\beta}\partial_{t}a(t,x)|&\leq
C_{\beta}^{(1)}\;\omega(x)^{2}\Phi(x)^{-|\beta|}\;\frac{1}{t^{q}}|\ln
t|^{(\tilde{\gamma}-1){\bf I}_{q}},\\\ |\partial_{x}^{\beta}a(t,x)|&\leq
C_{\beta}^{(2)}\;\omega(x)^{2}\Phi(x)^{-|\beta|}\;\frac{1}{t^{p}}|\ln
t|^{\tilde{\gamma}{\bf I}_{q}},\\\ |\partial_{x}^{\beta}b_{j,l}(t,x)|&\leq
C_{\beta}^{(3)}\;\omega(x)^{j}\Phi(x)^{-|\beta|}\;\frac{1}{t^{r}},\end{aligned}\end{rcases}$
(1.2)

with $\beta\in\mathbb{N}_{0},C_{\beta}^{(i)}>0,i=1,2,3,$
$q\in[1,\infty),p\in[0,1),p\leq q-1$ and $r\in[0,1).$ The function ${\bf
I}_{q}$ is such that ${\bf I}_{q}\equiv 1$ if $q=1$ else ${\bf I}_{q}\equiv
0.$ The functions $\omega(x)$ and $\Phi(x)$ are positive monotone increasing
in $|x|$ such that $1\leq\omega(x)\lesssim\Phi(x)\lesssim\langle
x\rangle=(1+|x|^{2})^{1/2}.$ They specify the structure of the differential
equation in the space variable. These functions will be discussed in detail in
Section 2.

In this work, our main interest is on the optimality of loss when the
coefficients are singular in time and unbounded in space. In particular, our
interest is either blow-up or infinitely many oscillations near $t=0$ and
polynomial growth in $x$. Such equations have a well-known behavior of a loss
of derivatives when the coefficients are bounded in space together with all
their derivatives (see for example [5, 4]). Along with the extension of the
results to a global setting ($x\in{\mathbb{R}}^{n}$ and the coefficients are
allowed to grow polynomially in $x$), we also investigate the behavior at
infinity of the solution in relation to the coefficients. By a loss of
regularity index of the solution in relation to the initial datum we mean a
change of indices in an appropriate Sobolev space. See Table 1 for the well-
posedness results from the literature in the context of singular hyperbolic
Cauchy problems.

| Order of Singularity
---
at $t=0$
Regularity in $t$ of coefficients |  | Growth in $x$
---
of coefficients
Loss of Regularity index for Solution | Ref.
$p$ | $q$ | $\tilde{\gamma}$ | $r$ | $\omega$ | $\Phi$
$0$ | $1$ | $(0,1)$ | 0 | $C^{1}((0,1])$ | $\omega(x)$ | $\Phi(x)$ |  | arbitrarily
---
small
[15]
$0$ | $1$ | $1$ | 0 | $C^{1}((0,1])$ | 1 | 1 | Finite | [4]
$0$ | $1$ | $1$ | 0 | $C^{1}((0,1])$ | $\omega(x)$ | $\Phi(x)$ | Finite |  [13]
$0$ | $1$ | $[1,\infty)$ | 0 | $C^{2}((0,T])$ | $\omega(x)$ | $\Phi(x)$ |  | Ranging from
---
Finite to Infinite
[14]
$[0,1)$ | $(1,\infty)$ | - | 0 | $C^{1}((0,1])$ | 1 | 1 | Infinite | [5]
$\Big{[}0,\frac{1}{2}\Big{)}$ | $\left(1,\frac{3}{2}\right)$ | - | $[0,1)$ | $C^{1}((0,1])$ | $\omega(x)$ | $\Phi(x)$ | Infinite | **
Table 1: Well-posedness results for singular hypebolic Cauchy problems. (**)
refers to Theorem 3.1 of this paper.

In [4], Cicognani discussed the well-posedness of (1.1) for the case
$\omega(x)=\Phi(x)=1$ and $p=0,q=1,\tilde{\gamma}=1,r=0$ in (1.2), where the
author reports well-posedness in $C^{\infty}({\mathbb{R}})$ for the Cauchy
problem (1.1) with a finite loss of derivatives. Colombini et al. [5]
considered the Cauchy problem (1.1) with operator coefficients independent of
$x$ and singular behavior prescribed by the parameters
$p\in[0,1),q\in(1,\infty)$ and $r=0$. They report well-posedness in Gevrey
space $G^{s},1\leq s<\frac{q-p}{q-1},$ with infinite loss of derivatives. We
study in [11] the case of $p=0,q\in\left(1,\frac{3}{2}\right)$ and $r=0$ with
generic structure functions $\omega$ and $\Phi$ in (1.2) and report infinite
loss of both derivatives and decay.

In this paper our interest is in the operator coefficients that are $L^{1}$
integrable in $t$ but singular near $t=0$ and are of polynomial growth in $x$.
In particular, our interest is
$p\in\big{[}0,\frac{1}{2}),q\in\left(1,\frac{3}{2}\right),p\leq q-1,r\in[0,1)$
and polynomial growth in $x$ prescribed by $\omega(x),\Phi(x)$ in (1.2). An
example of a coefficient $a(t,x)$ satisfying (1.2) is given below.

######  Example 1.1.

Let $T=1$, $\kappa_{1}\in[0,1]$ and $\kappa_{2}\in(0,1]$ such that
$\kappa_{1}\leq\kappa_{2}$. Then,

$a(t,x)=\langle x\rangle^{2\kappa_{1}}\left(2+\cos\langle
x\rangle^{1-\kappa_{2}}\right)\left(\frac{1}{t^{1/4}}\left(2+\sin\left(\frac{1}{t^{1/8}}\right)\right)\right)$

satisfies the estimates (1.2) for $\omega(x)=\langle x\rangle^{\kappa_{1}}$,
$\Phi(x)=\langle x\rangle^{\kappa_{2}},$ $p=\frac{1}{4},$ $q=\frac{11}{8}$.
The example shows that singular coefficients can also have infinitely many
oscillations near $t=0.$

In order to study the interplay between the singularity in time and
unboundedness in space one needs to consider an appropriate metric on the
phase space [9, 10]. In our case, we study the Cauchy problem (1.1) by
considering the metrics

$g_{\Phi,k}=\Phi(x)^{-2}|dx|^{2}+\langle\xi\rangle_{k}^{-2}|d\xi|^{2},$ (1.3)

where $\langle\xi\rangle_{k}=(k^{2}+|\xi|^{2})^{1/2},$ for sufficiently large
$k$ chosen appropriately. These metrics are discussed in Section 2.1. From the
estimates (1.2), note that $\omega$ and $\Phi$ are associated with the weight
and metric respectively, and they specify the structure of the differential
equation in the space variable. These functions will be discussed in detail in
Section 2.2. We report that the solution not only experiences a loss of
derivatives but also a decay in relation to the initial datum defined in a
Sobolev space modelled by the infinite order pseudodifferential operator

$e^{\Lambda(t)\Theta(x,D_{x})}.$ (1.4)

Here $\Lambda\in C([0,T])$ and the symbol of the operator $\Theta(x,D_{x})$ is
given by $h(x,\xi)^{-1/\sigma}=$ $(\Phi(x)\langle\xi\rangle_{k})^{1/\sigma}$
where $h(x,\xi)$ is the Planck function related to the metric $g_{\Phi,k}$ in
(1.3) and $3\leq\sigma<(q-p)/(q-1)$. The operator $\Theta(x,D_{x})$ explains
the quantity of the loss by linking it to the metric on the phase space and
the singular behavior while $\Lambda(t)$ gives a scale for the loss. Hence, we
call the conjugating operator as loss operator.

Our methodology relies upon two important techniques: the subdivision of the
extended phase space into two regions and conjugation of a first order system
corresponding to the operator $P$ in (3.1) by the loss operator. Both these
techniques result in the change of the metric governing the operator where the
new metric is conformally equivalent to the one in (1.3). As seen in (5.3),
the characteristic roots corresponding to the operator $P$ showcase a stronger
singular behavior compared to the principal symbol. Due to the subdivision of
the phase space, this results in the change of the metric as demonstrated in
Lemma 5.1. This metric is of the form

$\tilde{g}_{\Phi,k}^{(1)}=(\Phi(x)\langle\xi\rangle_{k})^{2\delta^{\prime}}{g}_{\Phi,k},$
(1.5)

where $\delta^{\prime}=\frac{p}{q-p}.$ On the other hand, our work in [11,
Theorem 4.0.1] suggests that the conjugation by the loss operator changes the
metric to

$\tilde{g}_{\Phi,k}^{(2)}=(\Phi(x)\langle\xi\rangle_{k})^{2/\sigma}{g}_{\Phi,k}.$
(1.6)

As our approach to establish well-posedness is based on the energy estimates,
we consider the metric
$\tilde{g}_{\Phi,k}^{(2)}=\tilde{g}_{\Phi,k}^{(1)}\vee\tilde{g}_{\Phi,k}^{(2)}$
for the application of the sharp Gårding inequality [8, Theorem 18.6.14].

From the energy estimate used in proving the well-posedness we derive an
optimal cone condition for the solution of the Cauchy problem (3.1) in Section
6. Though the characteristics of the operator $P$ in (3.1) are singular in
nature, the $L^{1}$ integrability of the singularity guarantees that the
propagation speed is finite. The weight function governing the coefficients
influences the geometry of the slope of the cone due to which the cone
condition in our case is anisotropic in nature.

In Section 7, we give a set counterexamples showing how the lower order terms
influence the well-posedness and the loss of regularity when $L^{1}$
integrability condition is violated. We cover various cases such as no loss,
finite loss and nonuniqueness.

The paper is organized as follows. In Section 2, we describe the tools
necessary for our analysis. In Section 3, we define a Cauchy problem of our
interest and state the well-posedness result whose proof will be presented in
Section 5. In Section 4, we define appropriate generalized parameter dependent
symbol classes. In Section 6 we derive a cone condition, while in Section 7 we
provide the set of examples.

##  2 Tools

In this section we introduce the main tools of this paper. The first part is
devoted to a class of metrics on the phase space that govern the geometry of
the symbols in our consideration while the second for properties of structure
functions. In the third part, we define Sobolev spaces associated to the
metric. Lastly, we device a localization technique based on the Planck
function associated to the metric.

###  2.1 Our Choice of Metric on the Phase Space

In this section, we review some notation and terminology used in the study of
metrics on the phase space, see [9, Chapter 2] and [10] for details. Let us
denote by $\sigma(X,Y)$ the standard symplectic form on
$T^{*}{\mathbb{R}}^{n}\cong{\mathbb{R}}^{2n}$: if $X=(x,\xi)$ and
$Y=(y,\eta)$, then $\sigma$ is given by

$\sigma(X,Y)=\xi\cdot y-\eta\cdot x.$

We can identify $\sigma$ with the isomorphism of ${\mathbb{R}}^{2n}$ to
${\mathbb{R}}^{2n}$ such that $\sigma^{*}=-\sigma$, with the formula
$\sigma(X,Y)=\langle\sigma X,Y\rangle$. Consider a Riemannian metric $g_{X}$
on ${\mathbb{R}}^{2n}$ (which is a measurable function of $X$) to which we
associate the dual metric $g_{X}^{\sigma}$ by

$g_{X}^{\sigma}(Y)=\sup_{0\neq
Y^{\prime}\in{\mathbb{R}}^{2n}}\frac{\langle\sigma
Y,Y^{\prime}\rangle^{2}}{g_{X}(Y^{\prime})},\quad\text{ for all
}Y\in{\mathbb{R}}^{2n}.$

Considering $g_{X}$ as a matrix associated to positive definite quadratic form
on ${\mathbb{R}}^{2n}$, $g_{X}^{\sigma}=\sigma^{*}g_{X}^{-1}\sigma$. We define
the Planck function [10] which plays a crucial role in the development of
pseudodifferential calculus as

$h_{g}(x,\xi):=\sup_{0\neq
Y\in{\mathbb{R}}^{2n}}\Bigg{(}\frac{g_{X}(Y)}{g_{X}^{\sigma}(Y)}\Bigg{)}^{1/2}.$

The uncertainty principle is quantified as the upper bound $h_{g}(x,\xi)\leq
1$. In the following, we often make use of the strong uncertainty principle,
that is, for some $\kappa>0$, we have

$h_{g}(x,\xi)\leq(1+|x|+|\xi|)^{-\kappa},\quad(x,\xi)\in{\mathbb{R}}^{2n}.$

In general, we use the metrics of the form

$g_{\Phi,k}^{\rho,r}=\left(\frac{\langle\xi\rangle_{k}^{\rho_{2}}}{\Phi(x)^{\tilde{\rho}_{1}}}\right)^{2}|dx|^{2}+\left(\frac{\Phi(x)^{\tilde{\rho}_{2}}}{\langle\xi\rangle_{k}^{\rho_{1}}}\right)^{2}|d\xi|^{2}.$
(2.1)

Here $\rho=(\rho_{1},\rho_{2})$ ,
$\tilde{\rho}=(\tilde{\rho}_{1},\tilde{\rho}_{2})$ for
$\rho_{j},\tilde{\rho}_{j}\in[0,1]$, $j=1,2$ are such that
$0\leq\rho_{2}<\rho_{1}\leq 1$ and $0\leq\tilde{\rho}_{2}<\tilde{\rho}_{1}\leq
1$. The Planck function associated to the metric in (2.1) is
$\Phi(x)^{\tilde{\rho}_{2}-\tilde{\rho}_{1}}\langle\xi\rangle_{k}^{\rho_{2}-\rho_{1}}.$

###  2.2 Properties of the Structure Functions $\omega(x)$ and $\Phi(x)$

The functions $\omega(x)$ and $\Phi(x)$ are associated with weight and metric
respectively. They specify the structure of the differential equation. As
pseudodifferential calculus is the datum of the metric satisfying some local
and global conditions. In our case, it amounts to the conditions on $\Phi$.
The symplectic structure and the uncertainty principle also play a natural
role in the constraints imposed on $\Phi$. So we consider $\Phi$ to be a
monotone increasing function of $|x|$ satisfying the following conditions:

$\displaystyle 1\;\leq$ $\displaystyle\quad\Phi(x)$ $\displaystyle\lesssim
1+|x|$ (sub-linear) $\displaystyle|x-y|\;\leq$ $\displaystyle\quad r\Phi(y)$
$\displaystyle\implies C^{-1}\Phi(y)\leq\Phi(x)\leq C\Phi(y)$ (slowly varying)
$\displaystyle\Phi(x+y)$ $\displaystyle\lesssim\Phi(x)(1+|y|)^{s}$ (temperate)

for all $x,y\in{\mathbb{R}}^{n}$ and for some $r,s,C>0$.

For the sake of calculations arising in the development of symbol calculus
related to the metrics $g_{\Phi,k}$, we need to impose following additional
conditions:

$\displaystyle|\Phi(x)-\Phi(y)|\leq$ $\displaystyle\Phi(x+y)$
$\displaystyle\leq\Phi(x)+\Phi(y),$ $\displaystyle\quad(\text{Subadditive})$
$\displaystyle|\partial_{x}^{\beta}\Phi(x)|$
$\displaystyle\lesssim\Phi(x)\langle x\rangle^{-|\beta|},$
$\displaystyle\Phi(ax)$ $\displaystyle\leq a\Phi(x),\text{ if }a>1,$
$\displaystyle a\Phi(x)$ $\displaystyle\leq\Phi(ax),\text{ if }a\in[0,1],$

where $\beta\in\mathbb{Z}_{+}^{n}$. It can be observed that the above
conditions are quite natural in the context of symbol classes. In our work, we
need even the weight function $\omega$ to satisfy the above stated properties
of $\Phi$. In order to arrive at an energy estimate using the Sharp Gårding
inequality (see Section 5.3 for details), we impose the following condition

$\omega(x)\lesssim\Phi(x),\quad x\in{\mathbb{R}}^{n}.$

###  2.3 Sobolev Spaces

We now introduce the Sobolev space related to the metric $g_{\Phi,k}$ that is
suitable for our analysis.

######  Definition 2.1.

The Sobolev space
$H_{\Phi,k}^{s,\varepsilon,\sigma}\left({\mathbb{R}}^{n}\right)$ for
$\sigma>2,\varepsilon\geq 0$ and
$s=\left(s_{1},s_{2}\right)\in{\mathbb{R}}^{2}$ is defined as

$H_{\Phi,k}^{s,\varepsilon,\sigma}\left({\mathbb{R}}^{n}\right)=\left\\{v\in
L^{2}\left(\mathbb{R}^{n}\right):\Phi(x)^{s_{2}}\langle
D\rangle_{k}^{s_{1}}\exp\\{\varepsilon\left(\Phi(x)\langle
D_{x}\rangle_{k}\right)^{1/\sigma}\\}v\in
L^{2}\left(\mathbb{R}^{n}\right)\right\\}$

equipped with the norm
$\|v\|_{\Phi,k;s,\varepsilon,\sigma}=\|\Phi(\cdot)^{s_{2}}\langle
D\rangle_{k}^{s_{1}}\exp\\{\varepsilon\left(\Phi(\cdot)\langle
D\rangle_{k}\right)^{1/\sigma}\\}v\|_{L^{2}}$. The operator
$\exp\\{\varepsilon\left(\Phi(x)\langle D_{x}\rangle_{k}\right)^{1/\sigma}\\}$
is an infinite order pseudodifferential operator with the Fourier multiplier
$\exp\\{\varepsilon\left(\Phi(x)\langle\xi\rangle_{k}\right)^{1/\sigma}\\}$.

The subscript $k$ in the notation
$H_{\Phi,k}^{s,\varepsilon,\sigma}\left({\mathbb{R}}^{n}\right)$ is related to
the parameter in the operator $\langle D\rangle_{k}=(k^{2}-\Delta_{x})^{1/2}.$

###  2.4 Subdivision of the Phase Space

One of the main tools in our analysis is the division of the extended phase
space $J=[0,T]\times{\mathbb{R}}^{2n},$ where $T>0,$ into two regions using
the Planck function $h(x,\xi)=(\Phi(x)\langle\xi\rangle_{k})^{-1}$ of the
metric $g_{\Phi,k}$ in (1.3). We use these regions in the proof of Theorem 3.1
(see Section 5.1) to handle the low regularity in $t$. To this end we define
the time splitting point $t_{x,\xi}$, for a fixed $(x,\xi)$, as the solution
to the equation

$t^{q-p}=Nh(x,\xi),$

where $N$ is the positive constant chosen appropriately later. Since
$3\leq\sigma<\frac{q-p}{q-1},$ we consider $\delta\in(0,1)$ such that

$\frac{1}{\sigma}=\frac{q-1+\delta}{q-p}.$ (2.2)

Implying

$\gamma:=1-\frac{1}{\sigma}=\frac{1-\delta-p}{q-p}.$ (2.3)

Using $t_{x,\xi}$ and (2.3) we define the interior region

$\displaystyle Z_{int}(N)$ $\displaystyle=\\{(t,x,\xi)\in J:0\leq t\leq
t_{x,\xi},\;|x|+|\xi|>N\\}$ (2.4) $\displaystyle=\\{(t,x,\xi)\in
J:t^{1-\delta-p}\leq N^{\gamma}h(x,\xi)^{\gamma},\;|x|+|\xi|>N\\}$

and the exterior region

$\displaystyle Z_{ext}(N)$ $\displaystyle=\\{(t,x,\xi)\in J:t_{x,\xi}<t\leq
T,\;|x|+|\xi|>N\\}$ (2.5) $\displaystyle=\\{(t,x,\xi)\in
J:t^{1-\delta-p}>N^{\gamma}h(x,\xi)^{\gamma},\;|x|+|\xi|>N\\}.$

We use these regions to define the parameter dependent global symbol classes
in Section 4.

##  3 Statement of the Main Result

We consider the Cauchy problem

$\begin{cases}P(t,x,\partial_{t},D_{x})u(t,x)=f(t,x),\qquad
D_{x}=-i\nabla_{x},\;(t,x)\in(0,T]\times{\mathbb{R}}^{n},\\\
u(0,x)=f_{1}(x),\quad\partial_{t}u(0,x)=f_{2}(x),\end{cases}$ (3.1)

with the strictly hyperbolic operator
$P(t,x,\partial_{t},D_{x})=\partial_{t}^{2}+b_{0}(t,x)\partial_{t}+a(t,x,D_{x})+b(t,x,D_{x})$
where

$a(t,x,\xi)=\sum_{i,j=1}^{n}a_{i,j}(t,x)\xi_{i}\xi_{j}\quad\text{ and }\quad
b(t,x,\xi)=i\sum_{j=1}^{n}b_{j}(t,x)\xi_{j}+b_{n+1}(t,x).$

Here, the matrix $(a_{i,j}(t,x))$ is real symmetric for all
$(t,x)\in(0,T]\times{\mathbb{R}}^{n}$, $a_{i,j}\in
L^{1}((0,T];C^{\infty}({\mathbb{R}}^{n}))\cap
C^{1}((0,T];C^{\infty}({\mathbb{R}}^{n}))$ and $b_{j}\in
L^{1}((0,T];C^{\infty}({\mathbb{R}}^{n}))$. We have the following assumptions
on $a(t,x,\xi)$, $b(t,x,\xi)$ and $b_{j}(t,x),j=0,n+1$ :

$\begin{rcases}\begin{aligned} a(t,x,\xi)&\geq
C_{0}\omega(x)^{2}\langle\xi\rangle_{k}^{2},\quad C_{0}>0,\\\
|\partial_{\xi}^{\alpha}\partial_{x}^{\beta}a(t,x,\xi)|&\leq
C^{|\alpha|+|\beta|}\alpha!(\beta!)^{\sigma}\frac{1}{t^{p}}\omega(x)^{2}\Phi(x)^{-|\beta|}\langle\xi\rangle_{k}^{2-|\alpha|},\\\
|\partial_{\xi}^{\alpha}\partial_{x}^{\beta}\partial_{t}a(t,x,\xi)|&\leq
C^{|\alpha|+|\beta|}\alpha!(\beta!)^{\sigma}\frac{1}{t^{q}}\omega(x)^{2}\Phi(x)^{-|\beta|}\langle\xi\rangle_{k}^{2-|\alpha|},\qquad\\\
|\partial_{\xi}^{\alpha}\partial_{x}^{\beta}b(t,x,\xi)|&\leq
C^{|\alpha|+|\beta|}\alpha!(\beta!)^{\sigma}\frac{1}{t^{r}}\omega(x)\Phi(x)^{-|\beta|}\langle\xi\rangle_{k}^{1-|\alpha|},\\\
|\partial_{x}^{\beta}b_{j}(t,x)|&\leq
C^{|\beta|}(\beta!)^{\sigma}\frac{1}{t^{r}}\Phi(x)^{-|\beta|},\quad
j=0,n+1,\end{aligned}\end{rcases}$ (3.2)

$q\in\left(1,\frac{3}{2}\right),p\leq
q-1,p\in\big{[}0,\frac{1}{2}\big{)},r\in\big{[}0,1\big{)},3\leq\sigma<(q-p)/(q-1)$
and $(t,x,\xi)\in[0,T]\times{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$. Note that
$C>0$ is a generic constant.

We now state the main result of this paper. Let $e=(1,1).$

######  Theorem 3.1.

Consider the strictly hyperbolic Cauchy problem (3.1) satisfying the
conditions in (3.2). Let the initial data $f_{j}$ belong to
$H^{s+(2-j)e,\Lambda_{1},\sigma}_{\Phi,k}$ and the right hand side $f\in
C([0,T];H^{s,\Lambda_{2},\sigma}_{\Phi,k}),$ $\Lambda_{j}>0,j=1,2$. Then,
there exist a continuous function $\Lambda(t)$ and positive constants
$\Lambda_{0}$ and $\delta^{*},$ such that there is a unique solution

$u\in C\left([0,T];H^{s+e,\Lambda(t),\sigma}_{\Phi,k}\right)\bigcap
C^{1}\left([0,T];H^{s,\Lambda(t),\sigma}_{\Phi,k}\right),$

for $\Lambda(t)<\Lambda^{*}=\min\\{\Lambda_{0},\Lambda_{1},\Lambda_{2}\\}.$
More specifically, the solution satisfies an a priori estimate

$\displaystyle\sum_{j=0}^{1}\|\partial_{t}^{j}u(t,\cdot)$
$\displaystyle\|_{\Phi,k;s+(1-j)e,\Lambda(t),\sigma}$ (3.3) $\displaystyle\leq
C\Bigg{(}\sum_{j=1}^{2}\|f_{j}\|_{\Phi,k;s+(2-j)e,\Lambda(0),\sigma}+\int_{0}^{t}\|f(\tau,\cdot)\|_{\Phi,k;s,\Lambda(\tau),\sigma}\;d\tau\Bigg{)}$

where $0\leq t\leq
T\leq(\delta^{*}\Lambda^{*}/\lambda)^{1/\delta^{*}},\;C=C_{s}>0$ and
$\Lambda(t)=\frac{\lambda}{\delta^{*}}\left(T^{\delta^{*}}-t^{\delta^{*}}\right)$
for a sufficiently large $\lambda$.

######  Remark 3.1.

Observe that we have $3\leq\sigma<(q-p)/(q-1)$ where as in [5, Theorem 2], it
is $1\leq\sigma<(q-p)/(q-1)$. The increase in the lower bound for $\sigma$ is
due to the application of sharp Gårding inequality in our context that
dictates $\sigma\geq 3$. This is discussed in Section 5.3. Due to this
increment in $\sigma$, we have $q\in\Big{(}1,\frac{3}{2}\Big{)}.$

##  4 Parameter Dependent Global Symbol Classes

We now define certain parameter dependent global symbols that are associated
with the study of the Cauchy problem (3.1). Let
$m=(m_{1},m_{2})\in\mathbb{R}^{2}$. Consider the metrics $g_{\Phi,k}$,
$\tilde{g}_{\Phi,k}^{(1)}$ and $\tilde{g}_{\Phi,k}^{(2)}$ as in (1.3) (1.5)
and (1.6). These metrics can be related to the metrics of the form (2.1) where
$(\rho,\tilde{\rho})$ are $((1,0),(1,0)),$
$((1-p/(q-p),p/(q-p)),(1-p/(q-p),p/(q-p)))$ and
$((1-1/\sigma,1/\sigma),(1-1/\sigma,1/\sigma)).$

######  Definition 4.1.

$G^{m_{1},m_{2}}(\omega,g_{\Phi,k}^{\rho,\tilde{\rho}})$ is the space of all
functions $a\in C^{\infty}(\mathbb{R}^{2n})$ satisfying

$|\partial_{\xi}^{\alpha}D_{x}^{\beta}a(x,\xi)|\leq
C_{\alpha\beta}\langle\xi\rangle_{k}^{m_{1}-\rho_{1}|\alpha|+\rho_{2}|\beta|}\omega(x)^{m_{2}}\Phi(x)^{-\tilde{\rho}_{1}|\beta|+\tilde{\rho}_{2}|\alpha|}$
(4.1)

Since $g_{\Phi,k}\leq\tilde{g}_{\Phi,k}^{(1)}\leq\tilde{g}_{\Phi,k}^{(2)},$ we
have

$G^{m_{1},m_{2}}(\omega,g_{\Phi,k})\subset
G^{m_{1},m_{2}}(\omega,\tilde{g}_{\Phi,k}^{(1)})\subset
G^{m_{1},m_{2}}(\omega,\tilde{g}_{\Phi,k}^{(2)}).$

Let $\mu\geq 1$ and $\nu\geq 1$.

######  Definition 4.2.

$AG_{\mu,\nu}^{m_{1},m_{2}}(\omega,g_{\Phi,k}^{\rho,\tilde{\rho}})$ is the
space of all functions $a\in C^{\infty}(\mathbb{R}^{2n})$ satisfying (4.1)
with $C_{\alpha\beta}=C^{|\alpha|+|\beta|}(\alpha!)^{\mu}(\beta!)^{\nu}$ for
some $C>0.$

######  Definition 4.3.

$AG_{\sigma}^{m_{1},m_{2}}(\omega,g_{\Phi,k}^{\rho,\tilde{\rho}})$ is the
space of all functions $a\in C^{\infty}(\mathbb{R}^{2n})$ satisfying (4.1)
when $h(x,\xi)\leq C_{1}|\alpha|^{-\sigma}$ with
$C_{\alpha\beta}=C_{2}^{|\alpha|+|\beta|}(\alpha!)(\beta!)^{\sigma}$ for some
positive constants $C_{1},C_{2}>0$

We denote the set of operators with symbols in
$G^{m_{1},m_{2}}(\omega,g_{\Phi,k}^{\rho,\tilde{\rho}})$ and
$AG_{\sigma}^{m_{1},m_{2}}(\omega,g_{\Phi,k}^{\rho,\tilde{\rho}})$ by
$OPG^{m_{1},m_{2}}(\omega,g_{\Phi,k}^{\rho,\tilde{\rho}})$ and
$OPAG_{\sigma}^{m_{1},m_{2}}(\omega,g_{\Phi,k}^{\rho,\tilde{\rho}}),$
respectively. As far as the calculi of these pseudodifferential operators are
concerned we refer to [12, Appendix II & III], [10, Section 6.3] and [1,
Appendix].

In our analysis, we require the following conjugation result.

######  Proposition 4.1.

Let $e^{\Lambda(t)\Theta(x,D_{x})}$ be as in (1.4) and $a(x,\xi)\in
AG_{\sigma,\sigma}^{m_{1},m_{2}}(\omega,\tilde{g}_{\Phi,k}^{(1)})$. Then,
there exists $\Lambda_{0}>0$ such that for $\Lambda(t)>0$ with
$\Lambda(t)<\Lambda_{0}$,

$e^{\Lambda(t)\Theta(x,D_{x})}a(x,D_{x})e^{-\Lambda(t)\Theta(x,D_{x})}=a(x,D_{x})+\sum_{j=1}^{3}r_{\Lambda}^{(j)}\left(t,x,D_{x}\right)$

where the symbols of $r_{\Lambda}^{(j)}\left(t,x,D_{x}\right)$ for $j=1,2,3$
are in
$C([0,T];AG_{\sigma,\sigma}^{-\infty,m_{2}-\gamma}(\omega,\tilde{g}_{\Phi,k}^{(2)})),$
$C([0,T];AG_{\sigma,\sigma}^{m_{1}-\gamma,-\infty}(\omega,\tilde{g}_{\Phi,k}^{(2)})),$
and
$C([0,T];AG_{\sigma,\sigma}^{-\infty,-\infty}(\omega,\tilde{g}_{\Phi,k}^{(2)})),$
respectively.

###### Proof.

Noting the fact that $\tilde{g}_{\Phi,k}^{(1)}\leq\tilde{g}_{\Phi,k}^{(2)}$,
the proof follows in similar lines to [11, Theorem 4.0.1]. ∎

Observe that the derivatives of $\sqrt{a(t,x,\xi)},$ characteristic roots of
operator $P$ in (3.1) show stronger singular behaviour compared to
$a(t,x,\xi)$ due to the singularity. Thus, to handle the singular behavior of
the characteristics, we have the following symbol classes.

######  Definition 4.4.

$AG_{\sigma}^{m_{1},m_{2}}\\{l;\delta_{1}\\}^{(1)}_{N}(\omega,g_{\Phi,k}^{\rho,\tilde{\rho}})$
for $l\in{\mathbb{R}}$ and $\delta_{1}\in[0,1)$is the space of all functions
$a\in C^{1}((0,T];G^{m_{1},m_{2}}(\omega,g_{\Phi,k}^{\rho,\tilde{\rho}}))$
satisfying

$|\partial_{\xi}^{\alpha}D_{x}^{\beta}a(t,x,\xi)|\leq
C^{|\alpha|+|\beta|}\alpha!(\beta!)^{\sigma}\langle\xi\rangle_{k}^{m_{1}-\rho_{1}|\alpha|+\rho_{2}|\beta|}\omega(x)^{m_{2}}\Phi(x)^{-\tilde{\rho}_{1}|\beta|+\tilde{\rho}_{2}|\alpha|}\left(\frac{1}{t}\right)^{\delta_{1}l},$
(4.2)

for all $(t,x,\xi)\in Z_{int}(N)$ and for some $C>0$ where
$\alpha,\beta\in\mathbb{N}^{n}_{0}.$

######  Definition 4.5.

$AG_{\sigma}^{m_{1},m_{2}}\\{l_{1},l_{2},l_{3};\delta_{1},\delta_{2}\\}^{(2)}_{N}(\omega,g_{\Phi,k}^{\rho,\tilde{\rho}})$
for $l_{1}.l_{3}\in{\mathbb{R}},l_{2}\in\\{0,1\\}$ and
$\delta_{1}\in[0,1),\delta_{2}\in(1,3/2)$ is the space of all functions $a\in
C^{1}((0,T];G^{m_{1},m_{2}}(\omega,g_{\Phi,k}^{\rho,\tilde{\rho}}))$
satisfying

$\displaystyle\small|\partial_{\xi}^{\alpha}D_{x}^{\beta}a(t,x,\xi)|$
$\displaystyle\leq
C^{|\alpha|+|\beta|}\alpha!(\beta!)^{\sigma}\langle\xi\rangle_{k}^{m_{1}-\rho_{1}|\alpha|+\rho_{2}|\beta|}\omega(x)^{m_{2}}$
(4.3)
$\displaystyle\qquad\Phi(x)^{-\tilde{\rho}_{1}|\beta|+\tilde{\rho}_{2}|\alpha|}\bigg{(}\frac{1}{t}\bigg{)}^{\delta_{1}(l_{1}+l_{2}(|\alpha|+|\beta|))+\delta_{2}l_{3}}$

for all $(t,x,\xi)\in Z_{ext}(N)$ and for some $C_{\alpha\beta}>0$ where
$\alpha,\beta\in\mathbb{N}^{n}_{0}.$

Given a $t$-dependent global symbol $a(t,x,\xi)$, we can associate a
pseudodifferential operator $Op(a)=a(t,x,D_{x})$ to $a(t,x,\xi)$ by the
following oscillatory integral

$\displaystyle a(t,x,D_{x})u(t,x)$
$\displaystyle=\iint\limits_{\mathbb{R}^{2n}}e^{i(x-y)\cdot\xi}a(t,x,\xi){u}(t,y)dy\textit{\dj}\xi$
$\displaystyle=(2\pi)^{-n}\int\limits_{\mathbb{R}^{n}}e^{ix\cdot\xi}a(t,x,\xi)\hat{u}(t,\xi)\textit{\dj}\xi,$

where $\textit{\dj}\xi=(2\pi)^{-n}d\xi$ and $\hat{u}$ is the Fourier transform
of $u$ in the space variable. The calculus for the operators with symbols of
form $a(t,x,\xi)=a_{1}(t,x,\xi)+a_{2}(t,x,\xi)$ such that

$\displaystyle a_{1}$ $\displaystyle\in
AG_{\sigma}^{\tilde{m}_{1},\tilde{m}_{2}}\\{\tilde{l};\delta_{1}\\}_{N_{1}}^{(1)}(\omega,g_{\Phi,k}^{\rho,\tilde{\rho}}),$
$\displaystyle a_{2}$ $\displaystyle\in
AG_{\sigma}^{m_{1},m_{2}}\\{l_{1},l_{2},l_{3};\delta_{1},\delta_{2}\\}_{N_{2}}^{(1)}(\omega,g_{\Phi,k}^{\rho,\tilde{\rho}}),$

for $N_{1}\geq N_{2},$ can be readily built by following the similar standard
arguments given in [13, Appendix], [12, Appendix II & III] and [1, Appendix].

##  5 Proof of Theorem 3.1

In this section, we give a proof of the main result, Theorem 3.1. There are
three key steps in the proof. First, we factorize the operator
$P(t,x,\partial_{t},D_{x})$. To this end, we begin with modifying the
coefficients of the principal part by performing an excision so that the
resulting coefficients are regular at $t=0$. Second, we reduce the original
Cauchy problem to a Cauchy problem for a first order system (with respect to
$\partial_{t}$). Lastly, using sharp Gårding’s inequality we arrive at the
$L^{2}$ well-posedness of a related auxiliary Cauchy problem, which gives
well-posedness of the original problem in the weighted Sobolev spaces
$H^{s,\varepsilon,\sigma}_{\Phi,k}$.

###  5.1 Factorization

Consider the operator $a(t,x,D_{x})$ defined in (3.1). We modify its symbol
$a(t,x,\xi)$ in $Z_{int}(2)$, by defining

$\tilde{a}(t,x,\xi)=\varphi(t\Phi(x)\langle\xi\rangle_{k})\omega(x)^{2}\langle\xi\rangle_{k}^{2}+(1-\varphi(t\Phi(x)\langle\xi\rangle_{k}))a(t,x,\xi)$
(5.1)

for $\varphi\in C^{\infty}(\mathbb{R})\text{ , }0\leq\varphi\leq 1\text{ ,
}\varphi=1\text{ in }[0,1]\text{ , }\varphi=0\text{ in }[2,+\infty).$ Note
that $(a-\tilde{a})\in AG_{\sigma}^{2,2}\\{1;p\\}_{int,2}(\omega,g_{\Phi,k})$
and $(a-\tilde{a})\sim 0$ in $Z_{ext}(2).$ This implies that
$t^{p}(a-\tilde{a})$ for $t\in[0,T]$ is a bounded and continuous family in
$AG_{\sigma}^{2,2}(\omega,g_{\Phi,k})$. Observe that $a-\tilde{a}$ is $L^{1}$
integrable in $t$, i.e.,

$\displaystyle\int^{T}_{0}|(a-\tilde{a})(t,x,\xi)|dt$
$\displaystyle\leq\kappa_{0}^{\prime}\omega(x)^{2}\langle\xi\rangle_{k}^{2}\int_{0}^{(2/\Phi(x)\langle\xi\rangle_{k})^{1/(q-p)}}\frac{1}{t^{p}}dt$
(5.2) $\displaystyle\leq(\Phi(x)\langle\xi\rangle_{k})^{2-(1-p)/(q-p)}.$

as $\omega(x)\lesssim\Phi(x).$

Let $\tau(t,x,\xi)=\sqrt{\tilde{a}(t,x,\xi)}$. Denote the indicator functions
for the regions $Z_{int}(N_{1})$ and $Z_{ext}(N_{2})$ by $\chi_{1}(N_{1})$ and
$\chi_{2}(N_{2})$, respectively. It is easy to note that

  1. i)

$\tau(t,x,\xi)$ is $G_{\omega}$-elliptic symbol of order $(1,1)$ i.e. there is
$C>0$ such that for all
$(t,x,\xi)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}$ we have

$|\tau(t,x,\xi)|\geq C\omega(x)\langle\xi\rangle_{k}.$

  2. ii)

$\tau\in
AG_{\sigma}^{1,1}\\{0;0\\}_{1}^{(1)}(\omega,g_{\Phi,k})+AG_{\sigma}^{1,1}\\{1/2,1,0;p,0\\}_{1}^{(2)}(\omega,g_{\Phi,k})$.
More precisely, for $|\alpha|+|\beta|>0,$

$\begin{rcases}\begin{aligned} |\tau(t,x,\xi)|&\leq
C_{0}\omega(x)\langle\xi\rangle_{k}\left(\chi_{1}(1)+\chi_{2}(1)t^{-p/2}\right),\\\
{|\partial_{\xi}^{\alpha}D_{x}^{\beta}\tau(t,x,\xi)|}&\leq
C_{\alpha\beta}\omega(x)\Phi(x)^{-|\beta|}\langle\xi\rangle_{k}^{1-|\alpha|}\left(\chi_{1}(1)+\chi_{2}(1)t^{-p(|\alpha|+|\beta|)}\right).\end{aligned}\end{rcases}$
(5.3)

  3. iii)

$\partial_{t}\tau$ is such that for $|\alpha|+|\beta|>0$ we have

$\displaystyle{|\partial_{t}\tau(t,x,\xi)|}$ $\displaystyle\sim 0\text{ in
}Z_{int}(1),$ $\displaystyle{|\partial_{t}\tau(t,x,\xi)|}$ $\displaystyle\leq
C_{0}\;\omega(x)\langle\xi\rangle_{k}\left(\chi_{1}(2)\omega(x)\langle\xi\rangle_{k}t^{-p}+\chi_{2}(1)t^{-q}\right),$
$\displaystyle{|\partial_{\xi}^{\alpha}D_{x}^{\beta}\partial_{t}\tau(t,x,\xi)|}$
$\displaystyle\leq
C_{\alpha\beta}\omega(x)\Phi(x)^{-|\beta|}\langle\xi\rangle_{k}^{1-|\alpha|}\left(\chi_{1}(2)\omega(x)\langle\xi\rangle_{k}+\chi_{2}(1)t^{-q}\right)t^{-p(|\alpha|+|\beta|)}.$

By the definition of the time splitting point and the subdivision of the phase
space, we see that

$\begin{rcases}\begin{aligned} {|\partial_{t}\tau(t,x,\xi)|}&\sim 0\text{ in
}Z_{int}(1),\\\
{|\partial_{\xi}^{\alpha}D_{x}^{\beta}\partial_{t}\tau(t,x,\xi)|}&\leq
C_{\alpha\beta}\chi_{1}(2)\omega(x)\Phi(x)^{-|\beta|}\langle\xi\rangle_{k}^{1-|\alpha|}t^{-q}t^{-p(|\alpha|+|\beta|)},\end{aligned}\end{rcases}$
(5.4)

for $|\alpha|+|\beta|\geq 0.$ Hence, $\partial_{t}\tau\sim 0$ in $Z_{int}(1)$
and $\partial_{t}\tau\in
AG_{\sigma}^{1,1}\\{0,1,1;p,q\\}_{1}^{(2)}(\omega,g_{\Phi,k}).$

From the above properties of $\tau$ and by the definition of $\tilde{a}$ in
(5.1), we have the following two lemmas.

######  Lemma 5.1.

Let $\tilde{g}_{\Phi,k}^{(1)})$ be as in (1.5) and $\delta$ as in (2.3). Then,

  1. i)

$\tau\in
L^{\infty}\left([0,T];AG_{\sigma}^{1+\delta^{\prime}/2,1}(\omega\Phi^{\delta^{\prime}/2},\tilde{g}_{\Phi,k}^{(1)})\right)$,

  2. ii)

$t^{1-\frac{p}{2}}\tau\in
C\left([0,T];AG_{\sigma}^{1,1}(\omega,\tilde{g}_{\Phi,k}^{(1)})\right)$,

  3. iii)

$\tau^{-1}\in
C\big{(}[0,T];AG_{\sigma}^{-1,-1}(\omega,\tilde{g}_{\Phi,k}^{(1)})\big{)}$,

  4. iv)

$t^{1-\delta}\partial_{t}\tau\in
C\big{(}[0,T];AG_{\sigma}^{1+1/\sigma,1}(\omega\Phi^{1/\sigma},\tilde{g}_{\Phi,k}^{(1)})\big{)}$.

###### Proof.

The first claim follows from (5.3) and the observation that in $Z_{ext}(1)$

$\left(\frac{1}{t}\right)^{p}\leq\left(\frac{\Phi(x)\langle\xi\rangle_{k}}{N}\right)^{p/(q-p)},$
(5.5)

while the second and third claims are straight forward consequences of (5.3)
and (5.5). The fourth claim follows from (5.5) and the following estimate in
$Z_{ext}(1)$

$\displaystyle\frac{1}{t^{q}}$
$\displaystyle=\frac{1}{t^{1-\delta}}\;\frac{1}{t^{(q-p)/\sigma}}\leq\frac{1}{t^{1-\delta}}\;\left(\frac{\Phi(x)\langle\xi\rangle_{k}}{N}\right)^{1/\sigma}.$

∎

######  Lemma 5.2.

Let $\delta$ as in (2.3). Then,

  1. i)

$t^{1-\delta}(a(t,x,D_{x})-\tilde{a}(t,x,D_{x}))\in
C\Big{(}[0,T];OPAG_{\sigma}^{1+1/\sigma,1+1/\sigma}(\omega,g_{\phi,k})\Big{)},$

  2. ii)

$\tilde{a}(t,x,D_{x})-\tau(t,x,D_{x})^{2}\in
L^{\infty}\Big{(}[0,T];OPAG_{\sigma}^{1,1}(\omega,\tilde{g}_{\Phi,k}^{(1)})\Big{)},$

  3. iii)

$t^{r}b(t,x,D_{x})\in
C\Big{(}[0,T];OPAG_{\sigma}^{1,1}(\omega,g_{\phi,k})\Big{)}$

###### Proof.

The proof is a consequence of the fact that in $Z_{int}(2)$

$\displaystyle|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}(a-\tilde{a})(t,x,\xi)|$
$\displaystyle\leq
C_{\alpha\beta}\chi_{1}(2)\omega(x)^{2}\Phi(x)^{-|\beta|}\langle\xi\rangle_{k}^{2-|\alpha|}\frac{1}{t^{p}}$
$\displaystyle\leq
C_{\alpha\beta}\chi_{1}(2)\omega(x)^{1+1/\sigma}\Phi(x)^{-|\beta|}\langle\xi\rangle_{k}^{{1+1/\sigma}-|\alpha|}(\Phi(x)\langle\xi\rangle_{k})^{{}^{1-1/\sigma}}\frac{1}{t^{p}}$
$\displaystyle\leq
C_{\alpha\beta}\chi_{1}(2)\omega(x)^{1+1/\sigma}\Phi(x)^{-|\beta|}\langle\xi\rangle_{k}^{{1+1/\sigma}-|\alpha|}\frac{1}{t^{1-\delta-p}}\;\frac{1}{t^{p}}$
$\displaystyle\leq
C_{\alpha\beta}\chi_{1}(2)\omega(x)^{1+1/\sigma}\Phi(x)^{-|\beta|}\langle\xi\rangle_{k}^{{1+1/\sigma}-|\alpha|}\frac{1}{t^{1-\delta}}.$

The second and third claims follow directly from the definitions of
$\tilde{a}(t,x,D_{x})$ and $b(t,x,D_{x}).$ ∎

Let us define $\delta^{*}>0$ as

$\delta^{*}=\min\\{\delta,1-r,1-p\\}.$ (5.6)

We are interested in the factorization of the operator
$P(t,x,\partial_{t},D_{x})$. This leads to

$P=(\partial_{t}-i\tau(t,x,D_{x}))(\partial_{t}+i\tau(t,x,D_{x}))+b_{0}(t,x)\partial_{t}+(a-\tilde{a}+a_{1})(t,x,D_{x})$

where the operator $a_{1}(t,x,D_{x})$ is such that, for $t\in[0,T]$,

$a_{1}=-i[\partial_{t},\tau]+\tilde{a}-\tau^{2}+b\;\text{ and
}\;t^{1-\delta^{*}}a_{1}(t,x,D_{x})\in
OPAG_{\sigma}^{1+1/\sigma,1}(\omega\Phi^{1/\sigma},\tilde{g}_{\Phi,k}^{(1)}).$

###  5.2 First Order Pseudodifferential System

We will now reduce the operator $P$ to an equivalent first order $2\times 2$
pseudodifferential system. The procedure is similar to the one used in [13,
15, 4]. To achieve this, we introduce the change of variables
$U=U(t,x)=(u_{1}(t,x),u_{2}(t,x))^{T}$, where

$\begin{cases}u_{1}(t,x)=(\partial_{t}+i\tau(t,x,D_{x}))u(t,x),\\\
u_{2}(t,x)=\omega(x)\langle D_{x}\rangle_{k}u(t,x)-H(t,x,D_{x})u_{1},\\\
\end{cases}$ (5.7)

and the operator $H$ with the symbol $\sigma(H)(t,x,\xi)$ is such that

$\sigma(H)(t,x,\xi)=-\frac{i}{2}\omega(x)\langle\xi\rangle_{k}\frac{\Big{(}1-\varphi\Big{(}t\Phi(x)\langle\xi\rangle_{k}/3\Big{)}\Big{)}}{\tau(t,x,\xi)}.$

Note that by the definition of $H$, $\text{supp }\sigma(H)\cap\text{supp
}\sigma(a-\tilde{a})=\emptyset$ and we have

$\displaystyle\sigma(2iH(t,x,D_{x})\circ\tau(t,x,D_{x}))$ $\displaystyle\sim
0,\quad\text{ in }Z_{int}(3),$
$\displaystyle\sigma(2iH(t,x,D_{x})\circ\tau(t,x,D_{x}))$
$\displaystyle=\omega(x)\langle\xi\rangle_{k}(1+\sigma(K_{1})),\quad\text{ in
}Z_{ext}(3),$

where $\sigma(K_{1})\in
AG_{\sigma}^{-1,-1}\\{0;p\\}_{6}^{(1)}(\omega,g_{\Phi,k})+AG_{\sigma}^{-1,-1}\\{2,1,0;p,q\\}_{3}^{(2)}(\omega,g_{\Phi,k})$.
Then, the equation $Pu=f$ is equivalent to the first order $2\times 2$ system
:

$\displaystyle LU$ $\displaystyle=(\partial_{t}-\mathcal{D}+A_{0}+A_{1})U=F,$
(5.8) $\displaystyle U(0,x)$
$\displaystyle=(f_{2}+i\tau(0,x,D_{x})f_{1},\Phi(x)\langle D_{x}\rangle
f_{1})^{T},$

where

$\displaystyle F$ $\displaystyle=(f(t,x),-H(t,x,D_{x})f(t,x))^{T},$
$\displaystyle\mathcal{D}$
$\displaystyle=\text{diag}(i\tau(t,x,D_{x}),-i\tau(t,x,D_{x})),$
$\displaystyle A_{0}$ $\displaystyle=\begin{pmatrix}B_{0}H&B_{0}\\\
-HB_{0}H&\quad HB_{0}\end{pmatrix}=\begin{pmatrix}\mathcal{R}_{1}&B_{0}\\\
-\mathcal{R}_{3}&\mathcal{R}_{2}\end{pmatrix},$ $\displaystyle A_{1}$
$\displaystyle=\begin{pmatrix}B_{1}H+B_{3}&B_{1}+B_{4}\\\ B_{2}-HB_{3}&\qquad
i[M,\tau]M^{-1}-H(B_{1}+B_{4})\end{pmatrix}.$

The operators $M,M^{-1},B_{0},B_{1}$ and $B_{2}$ are as follows

$\displaystyle M$ $\displaystyle=\omega(x)\langle D_{x}\rangle_{k},\quad
M^{-1}=\langle D_{x}\rangle_{k}^{-1}\omega(x)^{-1},$ $\displaystyle B_{0}$
$\displaystyle=(a(t,x,D_{x})-\tilde{a}(t,x,D_{x}))\langle
D_{x}\rangle_{k}^{-1}\omega(x)^{-1},$ $\displaystyle B_{1}$
$\displaystyle=(-i\partial_{t}\tau(t,x,D_{x})+\tilde{a}(t,x,D_{x})-\tau(t,x,D_{x})^{2}+b(t,x,D_{x}))\langle
D_{x}\rangle_{k}^{-1}\omega(x)^{-1},$ $\displaystyle B_{2}$
$\displaystyle=2iH\tau-M+i[M,\tau]M^{-1}H+i[\tau,H]-HB_{1}H+\partial_{t}H$
$\displaystyle B_{3}$ $\displaystyle=b_{0}(1-i\lambda M^{-1}H),\qquad
B_{4}=ib_{0}\lambda M^{-1}.$

Here $b_{0}=b_{0}(t,x)$ is as in (3.1). By the definition of operator $H$, we
have $B_{0}H=\mathcal{R}_{1},HB_{0}=\mathcal{R}_{2}$,
$HB_{0}H=\mathcal{R}_{3}$ for $\mathcal{R}_{j}\in
G^{-\infty,-\infty}(\omega,g_{\Phi,k}),j=1,2,3,$ and the operator $2iH\tau-M$
is such that

$\sigma(2iH\tau-M)=\begin{cases}-\omega(x)\langle\xi\rangle_{k},&\text{ in
}Z_{int}(3),\\\ \omega(x)\langle\xi\rangle_{k}\sigma(K_{1}),&\text{ in
}Z_{ext}(3).\end{cases}$

Since $2p\leq q,$ we have

$AG_{\sigma}^{0,0}\\{2,1,0;p,q\\}_{N}^{(2)}(\omega,{g}_{\Phi,k})\subset
AG_{\sigma}^{0,0}\\{0,1,1;p,q\\}_{N}^{(2)}(\omega,{g}_{\Phi,k}).$

The symbols of operators $\mathcal{D},A_{0}$ and $A_{1}$ are in the following
symbol classes

$\begin{rcases}\begin{aligned} \sigma(\mathcal{D})&\in
AG_{\sigma}^{1,1}\\{0;0\\}_{2}^{(1)}(\omega,g_{\Phi,k})+AG_{\sigma}^{1,1}\\{1,1,0;p,0\\}_{1}^{(2)}(\omega,g_{\Phi,k})\\\
\sigma(A_{0})&\in
AG_{\sigma}^{1,1}\\{1;p\\}_{2}^{(1)}(\omega,{g}_{\Phi,k})+AG_{\sigma}^{-\infty,-\infty}\\{0,0,0;0,0\\}_{3}^{(2)}(\omega,{g}_{\Phi,k}),\\\
\sigma(A_{1})&\in
AG_{\sigma}^{1,1}\\{0;0\\}_{6}^{(1)}(\omega,{g}_{\Phi,k})+AG_{\sigma}^{0,0}\\{1;r\\}_{1}^{(1)}(\omega,{g}_{\Phi,k})\\\
&\qquad+AG_{\sigma}^{0,0}\\{0,1,1;p,q\\}_{1}^{(2)}(\omega,{g}_{\Phi,k})\end{aligned}\end{rcases}$
(5.9)

and thus, by Lemmas 5.1 \- 5.2 and the choice of $\delta^{*}$ as in (5.6),

$\begin{rcases}\begin{aligned} t^{1-\delta^{*}}\sigma(A_{0}(t))&\in
C\left([0,T];AG_{\sigma}^{1/\sigma,1/\sigma}(\omega,{g}_{\Phi,k})\right),\\\
t^{1-\delta^{*}}\sigma(A_{1}(t))&\in
C\left([0,T];AG_{\sigma}^{1/\sigma,1/\sigma}(\omega,\tilde{g}_{\Phi,k}^{(1)}\right).\end{aligned}\end{rcases}$
(5.10)

As ${g}_{\Phi,k})\leq\tilde{g}^{(1)}_{\Phi,k})\leq\tilde{g}^{(2)}_{\Phi,k}),$
from (5.10) we have

$t^{1-\delta^{*}}\sigma(A_{0}(t)),\;t^{1-\delta^{*}}\sigma(A_{1}(t))\in
C\left([0,T];AG_{\sigma}^{1/\sigma,1/\sigma}(\omega,\tilde{g}_{\Phi,k}^{(2)}\right).$
(5.11)

Let us choose $\lambda>0$ as large as possible so that

$|\sigma(A_{0}(t))|+|\sigma(A_{1}(t))|\leq\frac{\lambda}{t^{1-\delta^{*}}}(\Phi(x)\langle\xi\rangle_{k})^{1/\sigma}.$
(5.12)

###  5.3 Energy Estimate

In this section, we prove the estimate (3.3). Note that it is sufficient to
consider the case $s=(0,0)$ as the operator $\Phi(x)^{s_{2}}\langle
D\rangle^{s_{1}}L\langle D\rangle^{-s_{1}}\Phi(x)^{-s_{2}}$, where
$s=(s_{1},s_{2})$ is the index of the weighted Sobolev space, satisfies the
same hypotheses as $L$.

In the following, we establish some lower bounds for the operator
$\mathcal{D}-A_{0}-A_{1}$. The symbol $d(t,x,\xi)$ of the operator
$\mathcal{D}(t)+\mathcal{D}^{*}(t)$ is such that

$d\in
AG_{\sigma}^{0,0}\\{0;0\\}_{2}^{(1)}(\omega,g_{\Phi,k})+AG_{\sigma}^{0,0}\\{1/2,1,0;p,q\\}_{1}^{(2)}(\omega,g_{\Phi,k}).$

It follows from the definition of $\delta^{*}$ and Lemma 5.1 that

$t^{1-\delta^{*}}d\in
C([0,T];AG_{\sigma}^{0,0}(\omega,\tilde{g}_{\Phi,k}^{(1)}).$

Thus

$2\operatorname{Re}\langle\mathcal{D}{U},{U}\rangle_{L^{2}}\geq-\frac{C_{1}}{t^{1-\delta^{*}}}\langle{U},{U}\rangle_{L^{2}},\quad
C_{1}>0.$ (5.13)

To control lower order terms, we make the following change of variable

$V(t,x)=e^{\Lambda(t)\Theta(x,D_{x})}{U}(t,x),$ (5.14)

where $\Lambda(t)=\frac{\lambda}{\delta^{*}}(T^{\delta^{*}}-t^{\delta^{*}})$
with $\lambda$ as in (5.12), and the operator $\Theta(x,D_{x})$ is as in
(1.4). From [11, Corollary 4.0.4],

$e^{\pm\Lambda(t)\Theta(x,D_{x})}e^{\mp\Lambda(t)\Theta(x,D_{x})}=I+R^{(\pm)}(t,x,D_{x}),$

where for sufficiently large $k$ the operators $I+R^{(\pm)}(t,x,D_{x})$ are
invertible. Let us denote the operators $I+R^{(+)}(t,x,D_{x}),$
$I+R^{(-)}(t,x,D_{x})$ and $e^{\pm\Lambda(t)\Theta(x,D_{x})}$ by
$\mathcal{R}(t,x,D_{x})$, $\tilde{\mathcal{R}}(t,x,D_{x})$ and
$E^{(\pm)}(t,x,D_{x}),$ respectively. Then, from (5.14),

$U(t,x)=\mathcal{R}^{(-)}(t,x,D_{x})e^{-\Lambda(t)\Theta(x,D_{x})}V(t,x)\text{
and }\;\|{U}(t)\|_{\Phi,k;s,\Lambda(t),\sigma}=\|V(t)\|_{L^{2}}.$

Then $Pu=f$ is equivalent to $L_{1}V=F_{1}$ where

$L_{1}=\partial_{t}-\mathcal{D}+\left(B+\frac{\lambda}{t^{1-\delta}}\Theta(x,D_{x})\right)$

$F_{1}(t,x)=\mathcal{R}^{-1}E^{(+)}\tilde{\mathcal{R}}F(t,x)$ and the operator
$B\left(t,x,D_{x}\right)$ is given by

$B=\mathcal{R}^{-1}E^{(+)}\left(\tilde{\mathcal{R}}\left(\partial_{t}\tilde{\mathcal{R}}^{-1}\right)+\tilde{\mathcal{R}}A\tilde{\mathcal{R}}^{-1}\right)E^{(-)}-\left(\mathcal{R}^{-1}E^{(+)}\tilde{\mathcal{R}}\mathcal{D}\tilde{\mathcal{R}}^{-1}E^{(-)}-\mathcal{D}\right)$

Observe that from Proposition 4.1 and from the Cauchy data given in conditions
of Theorem 3.1, we need
$\Lambda(t)<\Lambda^{*}=\min\left\\{\Lambda_{0},\Lambda_{1},\Lambda_{2}\right\\}$.
This implies
$T<\left(\frac{\delta^{*}}{\lambda}\Lambda^{*}\right)^{1/\delta^{*}}$. Then,
we have $t^{1-\delta^{*}}B\in
C\left([0,T];OPAG_{\sigma}^{\frac{1}{\sigma}e}\\{\omega,\tilde{g}^{(2)}_{\Phi,k}\\}\right).$
Choosing $\lambda$ sufficiently large, we obtain

$\operatorname{Re}\left\langle\left(\frac{\lambda}{t^{1-\delta^{*}}}\Theta(x,D_{x})+B\right)V,V\right\rangle_{L^{2}}\geq-
C_{2}\|V\|_{L^{2}},\quad C_{2}>0.$ (5.15)

The above estimate is the result of application of sharp Gårding inequality,
see [8, Theorem 18.6.14] for the metric $\tilde{g}_{\Phi,k}^{(2)}$ with the
Planck function $(\Phi(x)\langle\xi\rangle_{k})^{\frac{1}{\sigma}-\gamma}$. It
is important to note that the application of sharp Gårding inequality requires
$\sigma\geq 3$.

The estimate (3.3) on the solution $u$ can be established by proving that the
function $V(t,x)$ satisfies the a priori estimate

$\|V(t)\|_{L^{2}}^{2}\leq
C\left(\|V(0)\|_{L^{2}}^{2}+\int_{0}^{t}\left\|F_{1}(\tau,\cdot)\right\|_{L^{2}}d\tau\right),\quad
t\in[0,T],C>0.$ (5.16)

The Cauchy problem for the operator $L_{1}$ is given by

$\begin{rcases}\begin{aligned}
\partial_{t}V(t,x)&=\left(\mathcal{D}-\frac{\lambda}{t^{1-\delta}}\Theta(x,D_{x})-B\right)V(t,x)+F_{1}(t,x),\\\
V(0,x)&=e^{\Lambda(0)\Theta(x,D_{x})}U(0,x)\end{aligned}\end{rcases}$ (5.17)

Observe that

$\displaystyle\partial_{t}\|V(t)\|_{L^{2}}^{2}=$ $\displaystyle
2\operatorname{Re}\langle\partial_{t}V,V\rangle_{L^{2}}$ $\displaystyle=$
$\displaystyle
2\operatorname{Re}\langle\mathcal{D}V,V\rangle_{L^{2}}-2\operatorname{Re}\left\langle\left(\frac{\lambda}{t^{1-\delta}}\left(\Phi(x)\left\langle
D_{x}\right\rangle_{k}\right)^{1/\sigma}+B\right)V,V\right\rangle_{L^{2}}$
$\displaystyle+2\operatorname{Re}\langle F_{1},V\rangle.$

From (5.13) and (5.15) we have

$\frac{d}{dt}\|V(t)\|_{L^{2}}^{2}\leq\frac{C}{t^{1-\delta^{*}}}\|V(t)\|_{L^{2}}+C\left\|F_{1}(t,\cdot)\right\|_{L^{2}}$

Considering the above inequality as a differential inequality, we apply
Gronwall’s lemma and obtain that

$\|V(t)\|_{L^{2}}^{2}\leq
C^{\prime}e^{\frac{T^{\delta^{*}}}{\delta^{*}}}\left(\|V(0)\|_{L^{2}}^{2}+\int_{0}^{t}\left\|F_{1}(\tau,\cdot)\right\|_{L^{2}}^{2}d\tau\right)$

This proves the well-posedness of the Cauchy problem (5.17). Note that the
solution ${U}$ to (5.8) belongs to
$C\left([0,T];H_{\Phi,k}^{s,\Lambda(t),\sigma}\right)$. Returning to our
original solution $u=u(t,x)$ we obtain the estimate (3.3) with

$u\in C([0,T];H^{s+e,\Lambda(t),\sigma}_{\Phi})\cap
C^{1}([0,T];H^{s,\Lambda(t)\sigma}_{\Phi}).$

This concludes the proof.

##  6 Cone Condition

Existence and uniqueness follow from the a priori estimate established in the
previous section. It now remains to prove the existence of cone of dependence.

We note here that the $L^{1}$ integrability of the characteristics plays a
crucial role in arriving at the finite propagation speed. The implications of
the discussion in [16, Section 2.3 & 2.5] to the global setting suggest that
if the Cauchy data in (3.1) is such that $f\equiv 0$ and $f_{1},f_{2}$ are
supported in the ball $|x|\leq R$, then the solution to Cauchy problem (3.1)
is supported in the ball $|x|\leq R+c^{*}\omega(x)t^{1-\frac{p}{2}}$. The
constant $c^{*}$ is such that the quantity $c^{*}\omega(x)t^{-\frac{p}{2}}$
dominates the characteristic roots, i.e.,

$c^{*}=\sup\Big{\\{}\sqrt{a(t,x,\xi)}\omega(x)^{-1}t^{\frac{p}{2}}:(t,x,\xi)\in[0,T]\times{\mathbb{R}}^{n}_{x}\times{\mathbb{R}}^{n}_{\xi},\>|\xi|=1\Big{\\}}.$
(6.1)

Note that the support of the solution increases as $|x|$ increases since
$\omega(x)$ is monotone increasing function of $|x|$.

In the following we prove the cone condition for the Cauchy problem
$(\ref{eq1})$. Let $K(x^{0},t^{0})$ denote the cone with the vertex
$(x^{0},t^{0})$:

$K(x^{0},t^{0})=\\{(t,x)\in[0,T]\times{\mathbb{R}}^{n}:|x-x^{0}|\leq
c^{*}\omega(x)(t^{0}-t)^{1-\frac{p}{2}}\\}.$

Observe that the slope of the cone is anisotropic, that is, it varies with
both $x$ and $t$.

######  Proposition 6.1.

The Cauchy problem (3.1) has a cone dependence, that is, if

$f\big{|}_{K(x^{0},t^{0})}=0,\quad
f_{i}\big{|}_{K(x^{0},t^{0})\cap\\{t=0\\}}=0,\;i=1,2,$ (6.2)

then

$u\big{|}_{K(x^{0},t^{0})}=0.$ (6.3)

###### Proof.

Consider $t^{0}>0$, $C^{*}>0$ and assume that (6.2) holds. We define a set of
operators
$P_{\varepsilon}(t,x,\partial_{t},D_{x}),0\leq\varepsilon\leq\varepsilon_{0}$
by means of the operator $P(t,x,\partial_{t},D_{x})$ in (3.1) as follows

$P_{\varepsilon}(t,x,\partial_{t},D_{x})=P(t+\varepsilon,x,\partial_{t},D_{x}),\>t\in[0,T-\varepsilon_{0}],x\in{\mathbb{R}}^{n},$

and $\varepsilon_{0}<T-t^{0}$, for a fixed and sufficiently small
$\varepsilon_{0}$. For these operators we consider Cauchy problems

$\displaystyle P_{\varepsilon}v_{\varepsilon}$ $\displaystyle=f,$
$\displaystyle t\in[0,T-\varepsilon_{0}],\;x\in{\mathbb{R}}^{n},$
$\displaystyle\partial_{t}^{k-1}v_{\varepsilon}(0,x)$
$\displaystyle=f_{k}(x),\qquad$ $\displaystyle k=1,2.$

Note that $v_{\varepsilon}(t,x)=0$ in $K(x^{0},t^{0})$ and $v_{\varepsilon}$
satisfies an a priori estimate (3.3) for all $t\in[0,T-\varepsilon_{0}]$.
Further, we have

$\displaystyle P_{\varepsilon_{1}}(v_{\varepsilon_{1}}-v_{\varepsilon_{2}})$
$\displaystyle=(P_{\varepsilon_{2}}-P_{\varepsilon_{1}})v_{\varepsilon_{2}},\qquad$
$\displaystyle t\in[0,T-\varepsilon_{0}],\;x\in{\mathbb{R}}^{n},$
$\displaystyle\partial_{t}^{k-1}(v_{\varepsilon_{1}}-v_{\varepsilon_{2}})(0,x)$
$\displaystyle=0,\qquad$ $\displaystyle k=1,2.$

Since our operator is of second order, for the sake of simplicity we denote
$b_{j}(t,x)$, the coefficients of lower order terms, as $a_{0,j}(t,x),1\leq
j\leq n,$ while $b_{0}(t,x)$ and $b_{n+1}(t,x)$ are denoted as $a_{1,0}(t,x)$
and $a_{0,0}(t,x),$ respectively. Let $a_{i,0}(t,x)=0,\;2\leq i\leq n.$
Substituting $s-e$ for $s$ in the a priori estimate, we obtain

$\displaystyle\sum_{j=0}^{1}\|\partial_{t}^{j}(v_{\varepsilon_{1}}-v_{\varepsilon_{2}})(t,\cdot)\|_{\Phi,k;s-je,\Lambda(t),\sigma}$
(6.4) $\displaystyle\leq
C\int_{0}^{t}\|(P_{\varepsilon_{2}}-P_{\varepsilon_{1}})v_{\varepsilon_{2}}(\tau,\cdot)\|_{\Phi,k;s-e,\Lambda(\tau),\sigma}\;d\tau$
$\displaystyle\leq
C\int_{0}^{t}\sum_{i,j=0}^{n}\|(a_{i,j}(\tau+\varepsilon_{1},x)-a_{i,j}(\tau+\varepsilon_{2},x))D_{ij}v_{\varepsilon_{2}}(\tau,\cdot)\|_{\Phi,k;s-e,\Lambda(\tau),\sigma}\;d\tau,$

where $D_{00}=I,D_{10}=\partial_{t},D_{i0}=0,i\geq
2,D_{0j}=\partial_{x_{j}},j\neq 0$ and
$D_{ij}=\partial_{x_{i}}\partial_{x_{j}},i,j\neq 0$. Using the Taylor series
approximation in $\tau$ variable, we have

$\displaystyle|a_{i,j}(\tau+\varepsilon_{1},x)-a_{i,j}(\tau+\varepsilon_{2},x)|$
$\displaystyle=\Big{|}\int_{\tau+\varepsilon_{2}}^{\tau+\varepsilon_{1}}(\partial_{t}a_{i,j})(r,x)dr\Big{|}$
$\displaystyle\leq\omega(x)^{2}\Big{|}\int_{\tau+\varepsilon_{2}}^{\tau+\varepsilon_{1}}\frac{dr}{r^{q}}\Big{|}$
$\displaystyle\leq\omega(x)^{2}|E(\tau,\varepsilon_{1},\varepsilon_{2})|,$

where

$E(\tau,\varepsilon_{1},\varepsilon_{2})=\frac{1}{q-1}\left((\tau+\varepsilon_{1})^{-q+1}-(\tau+\varepsilon_{2})^{-q+1}\right).$

Note that $\omega(x)\lesssim\Phi(x)$ and $E(\tau,\varepsilon,\varepsilon)=0$.
Then right-hand side of the inequality in (6.4) is dominated by

$C\int_{0}^{t}|E(\tau,\varepsilon_{1},\varepsilon_{2})|\|v_{\varepsilon_{2}}(\tau,\cdot)\|_{\Phi,k;s+e,\Lambda(\tau),\sigma}\;d\tau,$

where $C$ is independent of $\varepsilon$. By definition, $E$ is
$L_{1}$-integrable in $\tau$.

The sequence $v_{\varepsilon_{k}}$, $k=1,2,\dots$ corresponding to the
sequence $\varepsilon_{k}\to 0$ is in the space

$C\Big{(}[0,T^{*}];H^{s,\Lambda(t),\sigma}_{\Phi,k}\Big{)}\bigcap
C^{1}\Big{(}[0,T^{*}];H^{s-e,\Lambda(t),\sigma}_{\Phi,k}\Big{)},\quad
T^{*}>0,$

and $u=\lim\limits_{k\to\infty}v_{\varepsilon_{k}}$ in the above space and
hence, in $\mathcal{D}^{\prime}(K(x^{0},t^{0}))$. In particular,

$\langle u,\varphi\rangle=\lim\limits_{k\to\infty}\langle
v_{\varepsilon_{k}},\varphi\rangle=0,\;\forall\varphi\in\mathcal{D}(K(x^{0},t^{0}))$

gives (6.3) and completes the theorem. ∎

##  7 Counterexamples

In this section, we show by a set of counterexamples how the lower order terms
of operator $P$ in (3.1) influence the loss of regularity and well-posedness
when $L^{1}$ integrability condition is violated. We cover various cases such
as no loss, finite loss and nonuniqueness. All the examples in this section
correspond to the case $\omega(x)=\Phi(x)=1,\;x\in{\mathbb{R}}$ in (1.2).

Following example shows that one can encounter finite loss when the
coefficients of lower order terms are not $L^{1}$ integrable.

######  Example 7.1.

$\begin{cases}\begin{aligned}
&\left(\partial_{t}^{2}-\partial_{x}^{2}+\frac{1}{2t}\left(\partial_{t}-(4m+1)\partial_{x}\right)\right)u(t,x)=0,\\\
&\;u(0,x)=u_{0}(x),\;\;\partial_{t}u(0,x)=(4m+1)\partial_{x}u_{0}(x),\end{aligned}\end{cases}$
(7.1)

for some $m\in\mathbb{N}_{0}.$ Note that the above problem corresponds to the
case $p=q=0$ and $r=1$ in (3.2). The solution to the above Cauchy problem is
given by

$u(t,x)=\sum_{j=0}^{m}C_{j}^{(m)}t^{j}\partial_{x}^{j}u_{0}(x+t),$

for $C_{j}^{(m)}$ of the form

$C_{0}=1,\quad
C_{j}^{(m)}=\frac{(-2)^{j}}{j!}\;\frac{(m)_{j}}{\left(-\frac{1}{2}\right)_{j}},\quad
j\geq 1,$

where $(y)_{j},y\in{\mathbb{R}},$ is the $j^{th}$ falling factorial of $y$
[18] given by

$(y)_{j}=y(y-1)\cdots(y-j+1).$

For example, when $m=2,$ $u(t,x)$ is of the form

$u(t,x)=u_{0}(x+t)+8t\partial_{x}u_{0}(x+t)+\frac{16}{3}t^{2}\partial_{x}^{2}u_{0}(x+t).$

We observe loss of derivatives: if $u_{0}\in H^{s},$ then $u(t,\cdot)\in
H^{s-m}$ where $H^{s}$ is the usual Sobolev space with $s\in{\mathbb{R}}.$

From Theorem 3.1 we see that $L^{1}$ integrability condition on lower order
terms is sufficient to ensure that they do not influence the well-posedness
and loss of regularity of the solution. The following example shows that the
condition is not necessary.

######  Example 7.2.

$\begin{cases}\begin{aligned}
&\left(\partial_{t}^{2}-\partial_{x}^{2}-\frac{2}{t}\partial_{x}\right)u(t,x)=0\\\
&u(0,x)=0,\;\partial_{t}u(0,x)=u_{0}(x).\end{aligned}\end{cases}$ (7.2)

This corresponds to the case $p=q=0$ and $r=1$ in (1.2). The solution to the
above Cauchy problem is given by

$u(t,x)=tu_{0}(x+t).$

The following example demonstrates that when the coefficient of the top order
term is oscillatory but in $C^{1}((0,T])\cap W^{1,1}((0,T])$ and that of the
lower order term is in $C^{1}((0,T])\cap L^{1}((0,T])$, one may have no loss.

######  Example 7.3.

$\begin{cases}\begin{aligned}
&\left(\partial_{t}^{2}-\big{(}2+\sin\sqrt{t}\big{)}^{2}\partial_{x}^{2}-\frac{\cos\sqrt{t}}{2\sqrt{t}}\partial_{x}\right)u(t,x)=0\\\
&u(0,x)=u_{0}(x),\;\partial_{t}u(0,x)=2\partial_{x}u_{0}(x).\end{aligned}\end{cases}$
(7.3)

This corresponds to the case $p=0,q=\frac{1}{2}$ and $r=\frac{1}{2}$ in (1.2).
The solution to the above Cauchy problem is given by

$u(t,x)=u_{0}\left(x+\int_{0}^{t}(2+\sin\sqrt{s})ds\right).$

The following example demonstrates that one may encounter nonuniqueness when
the lower order terms are not $L^{1}$ integrable.

######  Example 7.4.

$\begin{cases}\begin{aligned}
&\left(\partial_{t}^{2}-\partial_{x}^{2}-\frac{1}{t}\left(\partial_{t}+3\partial_{x}\right)\right)u(t,x)=0\\\
&u(0,x)=0,\;\partial_{t}u(0,x)=0.\end{aligned}\end{cases}$ (7.4)

This corresponds to the case $p=q=0$ and $r=1$ in (1.2). The solution to the
above Cauchy problem is given by

$u(t,x)=t^{2}u_{0}(x+t),$

for any function $u_{0}(x).$

######  Remark 7.1.

Observe that the form of solutions to the Cauchy problems in all the above
examples suggests that the Cauchy data only propagates along one
characteristic i.e., they are all one-way waves [2].

## Acknowledgements

The first author is funded by the University Grants Commission, Government of
India, under its JRF and SRF schemes.

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