# Existence and uniqueness of periodic pseudospherical surfaces emanating from
Cauchy problems

Nilay Duruk Mutlubas  Faculty of Engineering and Natural Sciences, Sabanci
University, Turkey<EMAIL_ADDRESS>Igor Leite Freire  Department
of Mathematical Sciences, Loughborough University
LE11 3TU Epinal Way, Loughborough, United Kingdom
<EMAIL_ADDRESS>Departamento de Matemática, Universidade Federal de
São Carlos
Rodovia Washington Luís, Km 235, 13565-905
São Carlos-SP, Brasil
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Abstract: We study implications and consequences of well-posed solutions of
Cauchy problems of a Novikov equation describing pseudospherical surfaces. We
show that if the co-frame of dual one-forms satisfies certain conditions for a
given periodic initial datum, then there exists exactly two families of
periodic one-forms satisfying the structural equations for a surface. Each
pair then defines a metric of constant Gaussian curvature and a corresponding
Levi-Civita connection form. We prove the existence of universal connection
forms giving rise to second fundamental forms compatible with the metric. The
main tool to prove our geometrical results is the Kato’s semi-group approach,
which is used to establish well-posedness of solutions of the Cauchy problem
involved and ensure $C^{1}$ regularity for the first fundamental form and the
Levi-Civita connection form.

2020 AMS Mathematics Classification numbers: 35B10, 53A05, 58J60, 35A30.

Keywords: Equations describing pseudospherical surfaces; First fundamental
form; Second fundamental form; Cauchy problems; Kato’s approach

###### Contents

  1. 1 Introduction
    1. 1.1 Novelty and challenges of the manuscript
    2. 1.2 Outline of the manuscript
  2. 2 Notation, notions and preliminaries
    1. 2.1 Notation
    2. 2.2 Structure equations and pseudospherical surfaces
    3. 2.3 Equations describing pseudospherical surfaces
    4. 2.4 Sobolev spaces and a few of functional analysis
    5. 2.5 Semigroup Approach
  3. 3 Proof of theorem 1.2
  4. 4 Proof of theorem 1.3
  5. 5 Proof of theorem 1.1
  6. 6 Concluding remarks

##  1 Introduction

In [34] Sasaki made a remarkable observation, showing that solutions of
integrable equations solved by the AKNS $2\times 2$ method [1] give rise to
metrics of pseudospherical surfaces with Gaussian curvature ${\cal K}=-1$, see
[34, section 2] and [6, section 1] for further details.

Later on, Chern and Tenenblat, in their seminal paper [6], introduced the
notion of equations describing pseudospherical surfaces (PSS equation) and
gave a systematic way for finding them.

The works by Sasaki [34] and Chern and Tenenblat [6] showed a deep connection
between integrability and differential geometry of pseudospherical surfaces,
unsurprisingly, leading to a new notion of integrability, see [27, Definition
2] and [29, page 245].

Notwithstanding its relevance in terms of integrability, the work by Chern and
Tenenblat made an in-depth investigation on certain very peculiar equations
having the following property: with some exceptions (that we will discuss
later), their solutions give rise to metrics with constant Gaussian curvature.
This fact per se has been known for a long time for the Sine-Gordon equation,
see [32, Section 1], but its systematic study, implications and applications
to other equations, potential links with integrable systems and construction
of conserved quantities made [6] a paramount work.

The metric and the Gaussian curvature are intrinsic properties of a surface,
but alas insufficient to completely describe it. To this end, we need further
information provided by its second fundamental form. While the first
fundamental form (metric) can be though as the way a two-dimensional being
walks on the surface (an intrinsic aspect), the second fundamental form tells
us how the surface behaves from, or looks like to, an observer located outside
it.

Given the importance of the second fundamental form and the relevance of the
observations and ideas introduced in [34] and [6], respectively, it is
somewhat surprising that it was taken nearly three decades from [6] until the
first works [16, 17, 18] concerning second fundamental forms of the surfaces
defined by the solutions of PSS equations.

Although the work by Chern and Tenenblat was born in the context of
integrability of differential equations, and most of the follow-up works, not
to say all, were concerned with these connections, see [5, 27, 28, 29, 30,
31], along time the integrability aspects were put aside and the research
carried out has been more focused on geometric aspects and classification of
equations describing PSS, see [4, 9, 16, 17, 18, 3, 38] and references
therein.

A considerable number of relevant PSS equations can be seen as dynamical
systems in certain Banach spaces, and from the point of view of analysis of
PDEs, qualitative aspects of their solutions are obtained from Cauchy
problems, meaning that not only the equation is relevant, but also a condition
satisfied by a given solution of the equation at a given time, very often
$t=0$ (initial condition or datum). Usually, the regularity of the initial
datum determines that of the corresponding solution of the equation.

From a geometric perspective, solutions emanating from Cauchy problems
involving an equation for an unknown $u=u(x,t)$ can be seen as follows: given
a certain curve $x\mapsto(x,0,u_{0}(x))$, can we find a solution $u$ for the
equation such that the given curve belongs to the graph of $u$? Moreover, what
does the regularity of the curve say about the graph of $u$? Is $u$ the only
solution of the equation whose graph contains the given curve?

Despite being a topic mostly concerned with analysis of PDEs, the paragraph
above shows that the problem of existence and uniqueness of solutions (well-
posedness) of PDEs makes sense in the context of PSS equations. Surprisingly,
it seems this topic has been out of the agenda of the literature of PSS
equations. The purpose of our paper is to shed light on it.

The main motivation for us to undertake the research reported in the present
work are recent results reported in [33] concerned with the equation

$u_{t}-u_{txx}=\partial_{x}(2-\partial_{x})(1+\partial_{x})u^{2},$ (1.1)

which was discovered in [24] and recently has been proved to be geometrically
integrable [10], meaning that its solutions describe a non-trivial family of
pseudospherical surfaces.

Equation (1.1) was studied in [12, 21, 22, 23] from the point of view of
qualitative analysis, such as existence and uniqueness of solutions. More
recently, in [33, Theorem 5.1] results from [21, 23] were combined with [10]
to prove the existence of $C^{\omega}$ (metrics for) pseudospherical surfaces
arising from the solutions of (1.1).

The aforementioned result proved in [33], despite being established for
$C^{\omega}$ solutions, strongly indicates the possibility of relating Cauchy
problems and pseudospherical surfaces. Actually, it made such a connection,
but by considering solutions emanating from an initial datum with $C^{\omega}$
regularity. The question is: Could we consider the same problem replacing a
$C^{\omega}$ initial datum by one with lower regularity? Can we consider a
periodic initial datum?

In line with the comments above, the vast majority of works in the field of
PSS equations considers explicit or implicitly $C^{\infty}$ solutions of the
PSS equations, which technically avoid problems regarding regularity (that is,
how much smooth the object is) and lead to $C^{\infty}$ metrics. A simple
question then arises: What may happen if we consider solutions with regularity
other than $C^{\infty}$?

The answer to the questions above is given in our first result.

######  Theorem 1.1.

Let $u_{0}\in H^{4}(\mathbb{S})$ be a non-trivial and non-constant initial
datum, with $u-u_{0}^{\prime\prime}>0$, and consider the Cauchy problem

$\left\\{\begin{array}[]{l}u_{t}-u_{txx}=\partial_{x}(2-\partial_{x})(1+\partial_{x})u^{2},~{}~{}~{}~{}~{}~{}x\in\mathbb{R},~{}~{}~{}~{}~{}t>0,\\\
\\\ u(x,0)=u_{0}(x),~{}~{}~{}~{}~{}~{}x\in\mathbb{R},\\\ \\\
u(x,t)=u(x+1,t),~{}~{}~{}~{}~{}~{}x\in\mathbb{R},~{}~{}~{}~{}~{}t>0.\end{array}\right.$
(1.2)

Then there exists triads of $C^{1}$ one-forms
$\omega_{1},\,\omega_{2},\,\omega_{3}$, with

$\omega_{i}=f_{i1}dx+f_{i2}dt,\quad 1\leq i\leq 3,$ (1.3)
$f_{p1}=\mu_{p}f_{11}+\eta_{p},\quad 1\leq p\leq 2,$ (1.4)

where $\mu_{p},\,\eta_{p}\in\mathbb{R}$, such that the forms (1.3) are defined
on $U=\mathbb{R}\times(0,\infty)$, periodic with respect to $x$, and define a
PSS whenever $\nabla u\neq(0,0)$, with $\omega_{3}$ being the Levi-Civita
connection of the metric determined by $\omega_{1}$ and $\omega_{2}$.

Moreover, fixed a pair $\\{\omega_{1},\,\omega_{2}\\}$ and $p\in U$, there
exists connection forms $\omega_{13}=a\omega_{1}+b\omega_{2}$,
$\omega_{23}=b\omega_{1}+c\omega_{2}$, where $a,\,b,\,c$ are $C^{\infty}$
functions defined on an open neighborhood $V\subseteq U$ of $p$, such that
$\\{\omega_{1},\omega_{2},\omega_{13},\,\omega_{23}\\}$ defines a PSS of
Gaussian curvature ${\cal K}=-1$.

In section 2 we shall present all pertinent definitions and notions, but for
now it suffices to say that a function belonging to $H^{4}(\mathbb{S})$ is a
real valued periodic function, with period $1$, of class $C^{3}$.

Our theorem 1.1 can be seen as an existence and uniqueness result for PSS
surfaces. In fact, it says that from solutions of equation (1.1) whose graphs
contain the regular curve $x\mapsto(x,0,u_{0}(x))$, with $u_{0}\in
H^{4}(\mathbb{S})$, we can obtain an open set $V\subseteq\mathbb{R}^{2}$ in
which we have two possible choices to define a first fundamental form for a
PSS surface with Gaussian curvature ${\cal K}=-1$. Moreover, we can locally
define connection forms on each point of $V$. This fact, jointly with Bonnet
theorem, tells us that we can locally define a PSS surface embedded in
$\mathbb{R}^{3}$.

A key point to understand and prove theorem 1.1 is determining whether the
problem (1.2) is well-posed. To this end, recognising the presence of the
Helmholtz operator $\Lambda^{2}=1-\partial_{x}^{2}$ in (1.1), we can rewrite
the problem (1.2) in an alternative form, given by

$\left\\{\begin{array}[]{l}u_{t}-2uu_{x}=\partial_{x}\Lambda^{-2}(u^{2}+(u^{2})_{x}),~{}~{}~{}~{}~{}~{}x\in\mathbb{R},~{}~{}~{}~{}~{}t>0,\\\
\\\ u(x,0)=u_{0}(x),~{}~{}~{}~{}~{}~{}x\in\mathbb{R},\\\ \\\
u(x,t)=u(x+1,t),~{}~{}~{}~{}~{}~{}x\in\mathbb{R},~{}~{}~{}~{}~{}t>0.\end{array}\right.$
(1.5)

######  Theorem 1.2.

Let $u_{0}\in H^{s}(\mathbb{S})$, $s>3/2$ be a given initial datum. Then there
exists a maximal time of existence $T>0$, depending on $u_{0}$, such that
there is a unique solution $u$ to (1.5) satisfying $u\in
C^{0}(H^{s}(\mathbb{S}),[0,T))\cap C^{1}(H^{s-1}(\mathbb{S}),[0,T))$.
Moreover, the map $u_{0}\in H^{s}(\mathbb{S})\rightarrow u$, is continuous
from $H^{s}(\mathbb{S})$ to $C^{0}(H^{s}(\mathbb{S}),[0,T))\cap
C^{1}(H^{s-1}(\mathbb{S}),[0,T))$ and $T$ is independent of $s$.

The inverse of the Helmholtz operator, denoted by $\Lambda^{-2}$ and acting on
a function $f$, is defined by the convolution $g\ast f$, where

$g(x)=\frac{\cosh{(x-\lfloor x\rfloor-1/2)}}{2\sinh{(1/2)}}$ (1.6)

and $\lfloor\cdot\rfloor$ denotes the greatest integer function.

Some previous results in the literature had already proved well-posedness
results concerning periodic solutions of the problem (1.5), see [23], but it
was either shown that $u\in C^{0}([0,T),H^{s}(\mathbb{S}))$ or $u$ is
$C^{\omega}$ in both variables on a certain domain, see [23, Theorem 1.1] and
[23, Theorem 1.4], respectively. Although these results show the existence and
uniqueness of solutions for a very large class of functions, they are
unsuitable for our purposes because we need solutions with $C^{1}$ regularity
with respect to $t$.

The importance of our theorem 1.2 comes from just the fact that it ensures we
have $C^{1}$ solutions in both variables. Actually, this is a consequence of
the Sobolev Lemma. In particular, it tells us that the solutions granted by
theorem 1.2 are strong solutions for the (non-local) first order PDE in (1.5).
A natural question then arises: is a strong solution of the equation in (1.5)
also a strong solution of (1.1)? In general the answer is no! However,
requiring enough regularity of the initial datum we can find solutions for the
Cauchy problem (1.5) that are also strong solutions for (1.1), and therefore,
simultaneously strong solutions for both formulations of the equation.

In fact, whenever we consider an initial datum in $H^{4}(\mathbb{S})$, the
corresponding solution provided by theorem 1.2 not only is a $C^{1}$ solution
(in both variables), but the Sobolev Lemma also implies that $x\mapsto
u(x,t)$, $t$ fixed, is $C^{3}$, meaning that the solution emanating from (1.5)
is a strong, or classical, solution for (1.1), which makes sense to be
considered in the study of PSS and differential equations.

Although the regularity of the initial datum is enough to make the
corresponding solution of (1.5) a strong solution of (1.1), it is insufficient
to guarantee that the solution is global, in the sense that it is defined for
every $t>0$. We can have global solutions requiring little more from the
initial datum.

######  Theorem 1.3.

If $u_{0}\in H^{4}(\mathbb{S})$ is a non-trivial initial datum, and
$u_{0}(x)-u_{0}^{\prime\prime}(x)>0$, $x\in\mathbb{R}$, then the solution of
the problem (1.5) exists for any $t>0$. Moreover, $u\in
C^{1}(\mathbb{R}\times(0,\infty))$ and $x\mapsto u(x,t)$ is a $C^{3}$ periodic
function, for each fixed $t>0$.

It is important to note that in view of the Sobolev Lemma, $H^{4}(\mathbb{S})$
is continuously and densely embedded in $H^{s}(\mathbb{S})$, for
$s\in(3/2,4)$. Therefore, both theorems 1.2 and 1.3 tell us that the problem
(1.2) has only one solution $u$.

###  1.1 Novelty and challenges of the manuscript

We study the problem of PSS determined by the solutions of a given equation
from the perspective of well-posedness of Cauchy problems, which as far as we
know, has not been considered yet.

Let us highlight the relevance of our results by discussing the following
problem: suppose we know a curve from the graph of a (unknown) solution of a
PSS equation. Can we precisely describe the corresponding PSS?

Let us exemplify by considering the function $u_{c}(x,t):=e^{x-ct}$. Any
member of the family ${\cal U}=\\{u_{c},\,c\in\mathbb{R}\\}$ is a solution of
(1.1) defined for $(x,t)\in\mathbb{R}\times(0,\infty)$, see [33, page 5]. For
any $u\in{\cal U}$, let
$\text{Gr}(u)=\\{(x,t,u(x,t)),\,\,x\in\mathbb{R},\,\,t>0\\}.$

Consider the curve $\Gamma$, given by $x\mapsto(x,0,e^{x})$ and let
$\partial\text{Gr}(u)$ denote the boundary of a set $\text{Gr}(u)$. Then it is
easy to see that $\Gamma\subseteq\partial\text{Gr}(u)$, for any $u\in{\cal
U}$. In particular, we have

$\Gamma\subseteq\bigcap_{u\in{\cal U}}\partial\text{Gr}(u).$

If we allow $t=0$ in the domain of $u_{c}$, then the curve $\Gamma$ belongs to
the corresponding graph, but for our purposes it is enough to consider it
lying in the boundary, the latter being disjoint from the graph.

On the other hand, (1.1) is a PSS equation (see [10, Theorem 1]), and for any
member $u_{c}$ of ${\cal U}$ we can construct a PSS $U_{c}$ in an intrinsic
way (note that these solutions satisfy all required conditions for the
existence of a PSS, see the comments after Theorem 1 in [10]).

Due to the fact that the curve $\Gamma$ belongs to the boundary of any graph
of the solutions $u_{c}$, we cannot determine any specific PSS surface only
knowing $\Gamma$.

Our theorem 1.1 gives a rather different answer to the same question. In
geometric terms, it says that given a curve $\Gamma$ of the form
$x\mapsto(x,0,u_{0}(x))$, as long as $u_{0}\in H^{4}(\mathbb{S})$, we can
precisely and intrinsically describe a PSS among all infinite surfaces
emanating from all solutions of the PDE (1.1).

To address this problem we make use of techniques of existence and uniqueness
of solutions for PDEs that can be seen as dynamical systems in certain Banach
spaces. In view of this approach, we deal with solutions of the equation that
are less regular than those usually considered in the literature of the PSS
equations. One of the difficulties to be overcome is concerned with the
regularity of the one-forms $\omega_{1}$ and $\omega_{2}$ defining the metric
of the corresponding PSS. Most of the books in differential geometry require
$C^{\infty}$ forms, although some of them require at least $C^{2}$ regularity.
As we will better discuss in the next section, for the classical theory of
curves and surfaces, we can have PSS surfaces from $C^{1}$ forms satisfying
the structure equations for a surface.

Last but not least, one of the challenges of this paper is that its main
result is geometric, but the tools for proving it comes from modern approaches
to prove qualitative aspects of solutions of Cauchy problems. For this reason,
we tried our best to make clear and explain the technical aspects of each
area, so that the readers can have a better reading and appreciation of our
work.

###  1.2 Outline of the manuscript

Since this is a paper focusing on Analysis and Geometry, in the next section
we provide an overview about PSS and functional analysis. We also fix the
notation, present essential concepts and revisit Kato’s semi-group approach,
which is the main tool for proving theorems 1.2 and 1.3, whose demonstrations
are given in sections 3 and 4, respectively. Theorem 1.1 is proved in section
5, whereas our conclusions are given in section 6.

##  2 Notation, notions and preliminaries

In this section we introduce and fix the notation used throughout the
manuscript. Given its plural and diverse aspects, we also present basic facts
and concepts from differential geometry of surfaces and functional analysis,
which are the main pillars of the work. Most of the geometric content can be
better explored in [2, Chapter 5], [7, Chapter 4] and [37, Chapter 2], whereas
our main references for functional analysis are [13, Chapter 3] and [36,
Chapter 4].

###  2.1 Notation

Given a function $u=u(x,t)$, by $u(x,\cdot)$ we mean the function $t\mapsto
u(x,t)$, for fixed $x$, whereas $u(\cdot,t)$ denotes the function $x\mapsto
u(x,t)$, for fixed $t$.

Let $I,J$ two open, non-empty subsets of $\mathbb{R}$. We say that $u\in
C^{0}(I\times J)$ if $u=u(x,t)$ is continuous with respect to both variables
$(x,t)\in I\times J$. Partial derivative of $u$ with respect to its first
argument will be denoted by $u_{x}$ or $\partial_{x}u$, whereas $u_{t}$ or
$\partial_{t}u$ will denote partial derivative with respect to the second
argument. Higher order derivatives can be considered using the standard
conventions.

For a positive integer $k$, we say that $u\in C^{k}(I\times J)$ if all partial
derivatives of $u$ up to order $k$ (including the mixed ones) are continuous.
Given a positive integer $n$, we denote by $u_{(n)}$ the set of ordered $n-th$
derivatives of $u$. Also, we say that $u$ is $C^{k}$ whenever all of its
partial derivatives up to order $k$ are continuous on the domain of $u$.

By $C^{3,1}(I\times J)$ we mean the set of function $u:I\times
J\rightarrow\mathbb{R}$ such that $u$, $u_{x}$, $u_{t}$, $u_{xx}$, $u_{xt}$,
$u_{xxx}$ and $u_{xxt}$ belong to $C^{0}(I\times J)$.

Let $X$ be a Banach space of real valued functions and $I\subseteq\mathbb{R}$.
The set $C^{0}(I,X)$ denotes collection of continuous functions such that
$u(t,\cdot)\in X$. More generally, given a positive integer $n$, we say that
$u\in C^{n}(I,X)$ if $\partial_{t}^{k}u(t,\cdot)\in C^{0}(I,X)$, $0\leq k\leq
n$.

###  2.2 Structure equations and pseudospherical surfaces

Let $\langle\cdot,\cdot\rangle$ be the usual inner product in $\mathbb{R}^{3}$
and denote the pair $(\mathbb{R}^{3},\langle\cdot,\cdot\rangle)$ by
$\mathbb{E}^{3}$, that is, the Euclidean space.

We recall that a surface is a two dimensional manifold in $\mathbb{E}^{3}$,
which we generally denote by $\cal M$. Given a point $p\in\cal M$, the tangent
and the co-tangent spaces to $\cal M$ at $p$ are denoted by $T_{p}\cal M$ and
$T^{\ast}_{p}\cal M$, respectively.

Let $\\{e_{1},e_{2}\\}$ be vector (sufficiently differentiable) valued
functions on $\cal M$, such that at each point $p\in\cal M$, we have:
$\\{e_{1},e_{2}\\}$ is orthonormal with respect to inner product
$\langle\cdot,\cdot\rangle$; $\text{Span}\\{e_{1},e_{2}\\}=T_{p}\cal M$;
$\\{\omega_{1},\omega_{2}\\}$ is the dual bases of $\\{e_{1},e_{2}\\}$. In
particular, $\text{Span}\\{\omega_{1},\omega_{2}\\}=T_{p}^{\ast}\cal M$.

Let $\\{\omega_{1},\omega_{2}\\}$ be the corresponding dual of the basis
$\\{e_{1},e_{2}\\}$. Since $\langle e_{i},e_{j}\rangle$ is either $0$ or $1$,
depending on whether $i=j$ or not, we have

$\langle de_{i},e_{j}\rangle+\langle e_{i},de_{j}\rangle=0,$ (2.1)

where $d(\cdot)$ denotes the usual differential, and we can then define one-
forms

$\omega_{ij}=\langle de_{i},e_{j}\rangle,$ (2.2)

called connection forms, and from (2.1) we see that
$\omega_{ij}=-\omega_{ji}$.

A one-form $\omega$ can be written as $\omega=f(x,t)dx+g(x,t)dt$, where $f$
and $g$ are certain functions, called coefficients of the form $\omega$. We
say that $\omega$ is of class $C^{k}$ if and only if both $f$ and $g$ are
$C^{k}$ functions.

Let $\otimes$ and $\wedge$ be the tensor and wedge products (for further
details, see [7, page 39], respectively. The dual forms $\omega_{1}$ and
$\omega_{2}$, jointly with the connection forms, satisfy the following
relations:

$d\omega_{1}=\omega_{2}\wedge\omega_{21},\quad
d\omega_{2}=\omega_{1}\wedge\omega_{12},$ (2.3)
$\omega_{1}\wedge\omega_{13}+\omega_{2}\wedge\omega_{23}=0,$ (2.4)

and

$d\omega_{12}=\omega_{13}\wedge\omega_{32},\quad
d\omega_{13}=\omega_{12}\wedge\omega_{23},\quad
d\omega_{23}=\omega_{21}\wedge\omega_{13}.$ (2.5)

It is important to note that the connection form $\omega_{12}$ is completely
determined by the forms $\omega_{1}$ and $\omega_{2}$, and is known as the
Levi-Civita (connection form). For this reason, it is common write
$\omega_{3}:=\omega_{12}$. Moreover, we can define the Gaussian curvature as
being the function ${\cal K}$ satisfying the relation

$d\omega_{3}=-{\cal K}\,\omega_{1}\wedge\omega_{2}.$ (2.6)

Equation (2.6) is called the Gauss equation, and it reflects the fact that the
Gaussian curvature is intrinsically determined by the surface, whereas we can
rewrite equations (2.3) in terms of form $\omega_{3}$, which reads

$d\omega_{1}=\omega_{3}\wedge\omega_{2},\quad
d\omega_{2}=\omega_{1}\wedge\omega_{3}.$ (2.7)

Equations (2.6)-(2.7) are called structure equations of the surface $\cal M$.

######  Definition 2.1.

Let $\omega_{1}$, $\omega_{2}$, $\omega_{13}$, and $\omega_{23}$ be given one-
forms on a surface $\cal M$ in $\mathbb{E}^{3}$, such that
$\\{\omega_{1},\omega_{2}\\}$ is LI, and $p\in\cal M$. The first and second
fundamental forms of $\cal M$ are defined, on each $T_{p}\cal M$, by
$I(v)=\omega_{1}(v)^{2}+\omega_{2}(v)^{2}$ and
$II(v)=\omega_{13}(v)\omega_{1}(v)+\omega_{23}(v)\omega_{2}(v)$, for each
$v\in T_{p}\cal M$.

Commonly one writes the first and the second fundamental forms as
$I=\omega_{1}^{2}+\omega_{2}^{2}$ and
$II=\omega_{13}\omega_{1}+\omega_{23}\omega_{2}$, with the convection
$\alpha\beta=\alpha\otimes\beta$ and $\alpha^{2}=\alpha\alpha$, for any
(one-)forms $\alpha$ and $\beta$.

We observe that everything done so far refers to a given surface $\cal M$ in
the euclidean space $\mathbb{E}^{3}$. A quite useful result for our purposes
is

######  Lemma 2.1.

Let $\omega_{1}$, $\omega_{2}$, $\omega_{12}$, $\omega_{13}$, and
$\omega_{23}$ be $C^{1}$ one-forms. Then they determine a local surface up to
a euclidean motion if and only if $\omega_{1}\wedge\omega_{2}\neq 0$ and
equations (2.3)–(2.5) are satisfied.

Lemma 2.1 (see [11, Theorem 10-19, page 232] and also [11, Theorem 10-18, page
232] for its proof) is a sine qua non result from classical differential
geometry of surfaces that enabled us to consider solutions $u\in
C^{3,1}(\mathbb{R}\times(0,\infty))$ and use them to prove our theorem 1.1.

######  Remark 2.1.

The relevance of lemma 2.1 for us is the following: it states that if a set of
given one-forms in $\mathbb{R}^{3}$ satisfies its conditions, then they
define, at least locally, a surface $\cal M$ in the euclidean space. Such a
result is sometimes fundamental theorem of surface theory, see [2, theorem 11,
page 143], or also called Bonnet theorem, see [7, theorem 4.39, page 127] or
[19, theorem 4.24, page 153].

Surfaces for which their Gaussian curvatures are constant and negative are
called pseudospherical surfaces [7, page 9].

###  2.3 Equations describing pseudospherical surfaces

If we take ${\cal K}=-1$ in the structure equations (2.6)–(2.7), we then have

$d\omega_{1}=\omega_{3}\wedge\omega_{2},\quad
d\omega_{2}=\omega_{1}\wedge\omega_{3},\quad
d\omega_{3}=\omega_{1}\wedge\omega_{2}.$ (2.8)

Sasaki’s observation [34] can be summed up as follows: if we denote

$\omega_{i}=f_{i1}dx+f_{i2}dt,\quad 1\leq i\leq 3,$ (2.9)

from the AKNS method [1] we can determine functions $f_{ij}$ for which the
corresponding triad of one-forms satisfies (2.7) and, as a consequence, they
determine a PSS in an intrinsic way.

Let $(x,t)$ be independent variables. A differential equation for a real
valued function $u=u(x,t)$ of order $n$ is generically denoted by

${\cal E}(x,t,u,u_{(1)},\cdots,u_{({n)}})=0.$ (2.10)

######  Definition 2.2.

A differential equation (2.10) is said to describe a pseudospherical surface,
or it is said to be of pseudospherical type, if it is a necessary and
sufficient condition for the existence of differentiable functions $f_{ij}$,
$1\leq i,j\leq 3$, such that the forms (2.9) satisfy the structure equations
of a pseudospherical surface (2.8).

In practical terms, given a triad of one-forms $\omega_{1}$, $\omega_{2}$ and
$\omega_{3}$, and an equation (2.10), we can check if they describe a PSS
surface in the following way: let us define a matrix of one-forms $\Omega$ by

$\Omega=\frac{1}{2}\begin{pmatrix}\omega_{2}&\omega_{1}-\omega_{3}\\\
\omega_{1}-\omega_{3}&-\omega_{2}\end{pmatrix}=:(\Omega_{ij}),\quad(\Omega\wedge\Omega)_{ij}:=(\sum_{k=1}^{2}\Omega_{ik}\wedge\Omega_{kj}),$
(2.11) $\Sigma:=d\Omega-\Omega\wedge\Omega,\quad d\Omega:=(d\Omega_{ij}).$
(2.12)

If, when restricted to the manifold determined by the solutions of (2.10), the
matrix $\Sigma$ vanishes, we say that (2.10) is a PSS equation, and the triad
$\\{\omega_{1},\,\omega_{2},\,\omega_{3}\\}$ satisfies the structure equations
of a PSS equation with Gaussian curvature ${\cal K}=-1$.

######  Example 2.1.

Let $m_{1}\in\\{-2,1\\}$, $\mu\in\mathbb{R}$ and consider the triad of one-
forms

$\begin{array}[]{lcl}\omega_{1}&=&\Big{(}u-u_{xx}\Big{)}dx+\Big{(}2u(u-u_{xx})+\psi\Big{)}dt,\\\
\\\ \omega_{2}&=&\Big{(}\mu(u-u_{xx})\pm
m_{1}\sqrt{1+\mu^{2}}\Big{)}dx+\displaystyle{\mu\big{(}2u(u-u_{xx})+\psi\big{)}}dt,\\\
\\\ \omega_{3}&=&\Big{(}\pm\sqrt{1+\mu^{2}}(u-u_{xx})+m_{1}\mu\Big{)}dx\\\ \\\
&&\pm\Big{(}\displaystyle{\sqrt{1+\mu^{2}}\big{(}2u(u-u_{xx})+\psi\big{)}}\Big{)}dt,\end{array}$
(2.13)

where

$\psi:=\frac{4}{m_{1}}uu_{x}-2u_{x}^{2}-2u^{2},$ (2.14)

and

${\cal E}=u_{t}-u_{txx}-4uu_{x}-2u_{x}^{2}-2uu_{xx}+6u_{x}u_{xx}+2uu_{xxx}.$
(2.15)

Note that ${\cal E}=0$ is nothing but (1.1). It is straightforward, but
lengthy, to confirm that (see [38, Theorem 4.5])

$\begin{array}[]{lcl}d\omega_{1}-\omega_{3}\wedge\omega_{2}&=&{\cal E}dx\wedge
dt,\quad d\omega_{2}-\omega_{1}\wedge\omega_{3}={\cal E}dx\wedge dt,\\\ \\\
d\omega_{3}-\omega_{1}\wedge\omega_{2}&=&\pm\sqrt{1+\mu^{2}}{\cal E}dx\wedge
dt.\end{array}$ (2.16)

Substituting (2.13)–(2.14) into (2.11), after reckoning we conclude that
(2.12) is given by

$\Sigma=\frac{{\cal E}}{2}\begin{pmatrix}1&1\pm\sqrt{1+\mu^{2}}\\\
1\pm\sqrt{1+\mu^{2}}&-1\end{pmatrix}.$ (2.17)

Therefore, $\Sigma=0$ if and only if ${\cal E}=0$, but ${\cal E}=0$ if and
only if the forms (2.13) satisfies the structure equations (2.8) for a PSS
with ${\cal K}=-1$ in view of (2.16). In particular, $u$ must be a solution of
(1.1).

It is important to highlight that a sine qua non condition for the existence
of a PSS defined on an open set $\Omega$ contained in the domain of a solution
$u$ is that $\omega_{1}\wedge\omega_{2}\neq 0$ whenever $(x,t)\in\Omega$,
otherwise the Gaussian curvature cannot be inferred from the Gauss equation
(2.6).

######  Definition 2.3.

Suppose that (2.10) is a PSS equation with corresponding one forms satisfying
(2.8). A solution $u$ of (2.10) for which $\omega_{1}\wedge\omega_{2}\neq 0$
is called generic, whereas those satisfying $\omega_{1}\wedge\omega_{2}=0$ are
said to be non-generic.

######  Example 2.2.

From the one-forms $\omega_{1}$ and $\omega_{2}$ in (2.13), we obtain

$\omega_{1}\wedge\omega_{2}=\pm\sqrt{1+\mu^{2}}\Big{(}2m_{1}uu_{xx}-4uu_{x}+2m_{1}u_{x}^{2}\Big{)}dx\wedge
dt.$ (2.18)

The condition $\omega_{1}\wedge\omega_{2}=0$ on an open set $\Omega$ contained
in the domain of $u$ is satisfied in the following circumstances:

  * •

For $m_{1}=-2$, then $\phi(x,t)=\pm\sqrt{ae^{-x}+b}$;

  * •

For $m_{1}=1$, then $\phi(x,t)=\pm\sqrt{ae^{2x}+b}$ or $\phi(x,t)=f(t)e^{x}$.

Above $a$ and $b$ are real constants, whereas $f\in C^{1}(\mathbb{R})$.

Let $u$ be a solution of (1.1), satisfying the condition $u(x+1,t)=u(x,t)$.
Then it is non-generic on an open set $\Omega$ if and only if it is constant.

###  2.4 Sobolev spaces and a few of functional analysis

Let ${\cal P}[0,1]$ be the collection of all periodic functions
$f:\mathbb{R}\rightarrow\mathbb{C}$ with period $1$. Given a positive integer
$k$, we denote by $f^{(n)}$ its $n-$th order derivative, while the set of
functions $f$ for which $f^{(n)}\in C^{0}(\mathbb{R})$, $0\leq n\leq k$, is
denoted by $C^{k}(\mathbb{R})$. If $k$ is a non-negative integer, we define
$C_{\text{per}}^{k}[0,1]=C^{k}(\mathbb{R})\cap{\cal P}[0,1]$. For the very
particular case $k=\infty$, we write ${\cal P}$ instead of
$C_{\text{per}}^{\infty}[0,1]$, with topological dual denoted by ${\cal
P}^{\prime}$. Recall that a member of ${\cal P}^{\prime}$ is a continuous
linear functional $f:{\cal P}\rightarrow\mathbb{C}$.

The Fourier transform of $f\in{\cal P}^{\prime}$ is defined by

$\hat{f}(k)=\frac{1}{2\pi}\int_{0}^{1}f(x)e^{-ikx}dx.$

Let $\ell^{2}(\mathbb{Z})$ be the collection of sequences
$\alpha=(\alpha_{n})_{n\in\mathbb{Z}}$ such that

$\sum_{n\in\mathbb{Z}}|\alpha_{n}|^{2}<\infty.$

Given $s\in\mathbb{R}$, we denote by $\ell^{2}_{s}(\mathbb{Z})$ as the
collection of $\alpha\in\ell^{2}(\mathbb{Z})$ such that

$\sum_{k=-\infty}^{\infty}(1+|k|^{2})^{s}|\alpha_{k}|^{2}<\infty,$

which has a structure of a Banach space when endowed with norm

$\|\alpha\|_{\ell^{2}_{s}}=\sqrt{\sum_{k=-\infty}^{\infty}(1+|k|^{2})|\alpha_{k}|^{2}}.$

The periodic Sobolev space of order $s$ is $H^{s}_{\text{per}}=\\{f\in{\cal
P}^{\prime};\,\,(\hat{f}(k))_{k\in\mathbb{Z}}\in\ell_{s}^{2}(\mathbb{Z})\\}$,
and the sesquilinear form

$\big{(}f\big{|}g\big{)}_{s}=\sum_{k=-\infty}^{\infty}(1+|k|^{2})^{2}\hat{f}(k)\overline{\hat{g}(k)}$

turns it into a Hilbert space. In particular, note that
$H^{0}_{\text{per}}=L^{2}[0,1]$.

Let us now define the following equivalence relation: given
$a,b\in\mathbb{R}$, we say that $a\sim b$ if $b=a+k$, for some integer $k$.
The quotient space $\mathbb{R}/\sim$ can be identified with the set $[0,1)$,
which we shall denote by $\mathbb{S}$. For this reason, henceforth we define
$H^{s}(\mathbb{S}):=H_{\text{per}}[0,1]$. The norm of $H^{s}(\mathbb{S})$ will
be denoted by $\|\cdot\|_{s}$, whereas $\|\cdot\|_{\infty}$ is reserved for
the norm in $L^{\infty}$.

Given two Banach spaces $X$ and $Y$, we write $X\hookrightarrow Y$ to mean
that $X$ is continuously and densely embedded in $Y$.

######  Lemma 2.2.

([13, Theorem 3.193, page 201]) Let $s,r\in\mathbb{R}$, with $s\geq r$. Then
$H^{s}(\mathbb{S})\hookrightarrow H^{r}(\mathbb{S})$ and
$\|f\|_{r}\leq\|f\|_{s},$ for any $f\in H^{s}(\mathbb{S})$. In particular,
$H^{s}(\mathbb{S})\hookrightarrow L^{2}[0,1]$ for any $s\geq 0$.

The next result is known as Sobolev Lemma, or also as Sobolev Embedding
Theorem.

######  Lemma 2.3.

([13, Theorem 3.195, page 204], [36, Proposition 3.3, page 329]) If $s>1/2$,
then $H^{s}(\mathbb{S})\hookrightarrow C_{\text{per}}^{0}[0,1]$ and
$\|f\|_{\infty}\leq c\|f\|_{s}$, for some constant $c$ depending only on $s$,
where $f\in H^{s}(\mathbb{S})$. More generally, if $s>1/2+m$, where $m$ is a
positive integer, then $H^{s}(\mathbb{S})\hookrightarrow
C_{\text{per}}^{m}[0,1]$.

We conclude our revision on Sobolev spaces by recalling the algebra property.

######  Lemma 2.4.

([13, Theorem 3.200, page 207]) If $s>1/2$, for any $f,g\in
H^{s}(\mathbb{S})$, we have $fg\in H^{s}(\mathbb{S})$ and their norm satisfies
the estimate $\|fg\|_{s}\leq c\|f\|_{s}\|g\|_{s},$ for some constant $c>0$
depending only on $s$.

###  2.5 Semigroup Approach

Let us revisit basic aspects of Kato’s theory [14, 15, 25], also known as
semigroup approach, which is our main tool for proving well-posedness of
solutions with the regularity we need to make them consistent with the
geometric nature of our problem.

Let $X$ be a Hilbert space, and let $u(\cdot,t)\in X$ such that

$u_{t}+A(u)u=f(u),~{}~{}~{}~{}t\geq 0,~{}~{}~{}~{}~{}u(0)=u_{0}.$ (2.19)

Let $Y\hookrightarrow X$ and $S:Y\rightarrow X$ be a topological isomorphism.
Assume that

  * (A1)

For any given $r>0$ it holds that for all $u\in\mathrm{B}_{r}(0)\subseteq Y$
(the ball around the origin in $Y$ with radius $r$), the linear operator
$A(u)\colon X\to X$ generates a strongly continuous semigroup $T_{u}(t)$ in
$X$ which satisfies
$\|T_{u}(t)\|_{\mathcal{L}(X)}\leq\mathrm{e}^{\omega_{r}t}$, for all
$t\in[0,\infty)$, for a uniform constant $\omega_{r}>0$;

  * (A2)

$A$ maps $Y$ into $\mathcal{L}(Y,X)$, more precisely the domain $D(A(u))$
contains $Y$ and the restriction $A(u)|_{Y}$ belongs to $\mathcal{L}(Y,X)$ for
any $u\in Y$. Furthermore $A$ is Lipschitz continuous in the sense that for
all $r>0$ there exists a constant $C_{1}$ which only depends on $r$ such that
$\|A(u)-A(v)\|_{\mathcal{L}(Y,X)}\leq C_{1}\,\|u-v\|_{X}$ for all
$u,~{}v\in\mathrm{B}_{r}(0)\subseteq Y$.

  * (A3)

For any $u\in Y$ there exists a bounded linear operator
$B(u)\in\mathcal{L}(X)$ satisfying $B(u)=SA(u)S^{-1}-A(u)$ and $B\colon
Y\to\mathcal{L}(X)$ is uniformly bounded on bounded sets in $Y$. Furthermore
for all $r>0$ there exists a constant $C_{2}$ which depends only on $r$ such
that $\|B(u)-B(v)\|_{\mathcal{L}(X)}\leq C_{2}\,\|u-v\|_{Y},$ for all
$u,~{}v\in\mathrm{B}_{r}(0)\subseteq Y$;

  * (A4)

For all $t\in[0,\infty)$, $f$ is uniformly bounded on bounded sets in $Y$.
Moreover, the map $f\colon Y\to Y$ is locally $X$-Lipschitz continuous in the
sense that for every $r>0$ there exists a constant $C_{3}>0$, depending only
on $r$, such that $\|f(u)-f(v)\|_{X}\leq C_{3}\,\|u-v\|_{X}$, for all
$u,~{}v\in\mathrm{B}_{r}(0)\subseteq Y$, and locally $Y$-Lipschitz continuous
in the sense that for every $r>0$ there exists a constant $C_{4}>0$, depending
only on $r$, such that $\|f(u)-f(v)\|_{Y}\leq C_{4}$ $\|u-v\|_{Y}$, for all
$u,~{}v\in\mathrm{B}_{r}(0)\subseteq Y$.

######  Lemma 2.5.

[14] Assume that (A1)-(A4) hold. Then for given $u_{0}\in Y$, there is a
maximal time of existence $T>0$, depending on $u_{0}$, and a unique solution
$u$ to (2.19) in $X$ such that $u=u(u_{0},.)\in C^{0}(Y,[0,T))\cap
C^{1}(X,[0,T)).$ Moreover, the solution depends continuously on the initial
data, i.e. the map $u_{0}\rightarrow u(u_{0},.)$ is continuous from $Y$ to
$C^{0}(Y,[0,T))\cap C^{1}(X,[0,T))$.

##  3 Proof of theorem 1.2

We begin by noticing that the equation in (1.5) is in the quasi-linear
equation form (2.19), where

$A(u)=-2u\partial_{x}$ (3.1)

and

$f(u)=\Lambda^{-2}\partial_{x}\big{(}u^{2}+(u^{2})_{x}\big{)}.$ (3.2)

Let

$(\Lambda^{s}f)(k):=\sum_{k\in\mathbb{Z}}(1+n^{2})^{s/2}\hat{f}(n)e^{ink}.$

For any $s,s^{\prime}\in\mathbb{R}$,
$\Lambda^{s}:H^{s^{\prime}}(\mathbb{S})\rightarrow
H^{s^{\prime}-s}(\mathbb{S})$ is an isomorphism [36, page 330]. Then, it is
natural to choose as Hilbert spaces
$X\coloneqq(H^{s-1}(\mathbb{S}),\|\cdot\|_{s-1})$ and
$Y\coloneqq(H^{s}(\mathbb{S}),\|\cdot\|_{s})$ with $s>\frac{3}{2}$, and
$S=\Lambda$ as well, and work with them using Kato’s approach. We aim at
proving lemmas ensuring the validity of the assumptions (A1)-(A4). For
convenience, in the remaining part of this section we simply write $H^{s}$ in
place of $H^{s}(\mathbb{S})$. Moreover, for a given function $g\in H^{r}$ with
$r>1/2$ let us denote by $M_{g}$ the corresponding multiplication operator on
$H^{r}$, i.e. $M_{g}\colon H^{r}\to H^{r},w\mapsto gw$. Since $H^{r}$,
$r>1/2$, is closed under multiplication, $M_{g}$ is continuous.

Now, we verify the assumptions needed for Theorem 1.2. We start with
assumption (A1):

######  Lemma 3.1.

Let $s>3/2$. For any given $r>0$, it holds that for all
$u\in\mathrm{B}_{r}(0)\subseteq H^{s}$, the linear operator $A(u)\colon
H^{s-1}\to H^{s-1}$, with domain $D(A(u))\coloneqq\\{w\in H^{s-1}\colon
A(u)w\in H^{s-1}\\}$, generates a strongly continuous semigroup $T_{u}(t)$ in
$X$ which satisfies
$\|T_{u}(t)\|_{\mathcal{L}(X)}\leq\mathrm{e}^{\omega_{r}t}$ for all
$t\in[0,\infty)$, for a uniform constant $\omega_{r}>0$. In particular, the
operator $A(u)$ given in (3.1), with domain $\mathcal{D}(A)=\\{\omega\in
H^{s-1}:A(u)\omega\in H^{s-1}\\}\subset H^{s-1}$ is quasi-m-accreative in
$H^{s-1}$if $u\in H^{s}$, $s>\frac{3}{2}$.

For convenience, it would be good to mention that the coefficient in (3.1)
does not affect the analysis. It just plays a role in constant estimation
which is not of our interest. Therefore, we will neglect it and keep the
operator form as $u\partial_{x}$. We prove this lemma in two steps. First step
is given as follows:

######  Lemma 3.2.

The operator $A(u)=u\partial_{x}$ in $L^{2}$, with $u\in H^{s}$,
$s>\frac{3}{2}$, is
quasi-m-accreative.

###### Proof.

A linear operator $A=A(u)$ in $X$ is quasi-m-accretive if and only if [15]:

  1. (a)

There is a real number $\beta$ such that
$(A\omega,\omega)_{X}\geq-\beta\|\omega\|_{X}^{2}$ for all $\omega\in D(A)$;

  2. (b)

The range of $A(u)+\lambda I$ is all of $X$ for some (or equivalently, all)
$\lambda>\beta$.

Note that if the above property (a) holds, then $A+\lambda I$ is dissipative
for all $\lambda>\beta$. Moreover, if $A$ is a closed operator, then
$A+\lambda I$ has closed range in $X$ for all $\lambda>\beta$. Hence, in order
to prove (b) in such a case, it is enough to show that $A+\lambda I$ has dense
range in $X$ for all $\lambda>\beta$.

First we show that $A$ is a closed operator in $L^{2}$. Let
$(v_{n})_{n\in\mathbb{N}}$ be a sequence in $D(A)$ with $v_{n}\to v$ in
$L^{2}$ and $Av_{n}\to w$ in $L^{2}$. Then $uv_{n}\in H^{1}$ for all
$n\in\mathbb{N}$ by definition of $D(A)$ since an alternative way of writing
the domain is $D(A)=\\{\omega\in L^{2}:u\omega\in H^{1}\\}$ and $v_{n}\in
D(A)$. Moreover, both $uv_{n}\to uv$ and $u_{x}v_{n}\to u_{x}v$ in $L^{2}$ by
the continuity of the multiplication $H^{r}\times L^{2}\to L^{2}$ for $r>1/2$.
Therefore,$(uv_{n})_{x}\to w+u_{x}v$ in $L^{2}$. Having sequences
$(uv_{n})_{n\in\mathbb{N}}$ and $((uv_{n})_{x})_{n\in\mathbb{N}}$ convergent
in $L^{2}$ implies that $(uv_{n})_{n\in\mathbb{N}}$ converges in $H^{1}$ with
the limit $uv$, thus $v\in D(A)$. Moreover the continuity of
$\partial_{x}\colon H^{1}\to L^{2}$ implies that
$\lim_{n\to\infty}(uv_{n})_{x}=(uv)_{x}$, therefore $w=(uv)_{x}-u_{x}v=Av$.

Now, we take the following $L^{2}$ inner product

$\displaystyle(A(u)\omega,\omega)_{0}$
$\displaystyle=(u\partial_{x}\omega,\omega)_{0}$

We refer to Lemma 3.3 to be stated below and use integration by parts to get:

$\displaystyle|(u\partial_{x}\omega,\omega)_{0}|=|-\frac{1}{2}(u_{x},\omega^{2})_{0}|\leq
C\|u_{x}\|_{L^{\infty}}\|w\|_{0}^{2}\leq\tilde{C}\|\omega\|_{0}^{2}.$

Having $\|u\|_{s}$ bounded allows us to choose
$\beta=\tilde{C}(\|u\|_{H^{s}})$ and to show that the operator satisfies the
inequality in (a). Thus, $A(u)+\lambda I$ is dissipative for all
$\lambda>\beta$. Moreover, recall that $A(u)$ is a closed operator. Therefore,
we now show that $A(u)+\lambda I$ has dense range in $L^{2}$ for all
$\lambda>\beta$.

It is known that if the adjoint of an operator has trivial kernel, then the
operator has dense range [26]. For $A(u)=u\partial_{x}$, the adjoint operator
can be expressed $A^{*}(u)=-u_{x}-u\partial_{x}$.

Observe that

$A^{*}(u)\omega=-u_{x}\omega-u\omega_{x}=-(u\omega)_{x}.$

Since $u_{x}\in L^{\infty}$ and $\omega\in L^{2}$, we have $u_{x}\omega\in
L^{2}$. Having also $A(u)\omega=u\omega_{x}\in L^{2}$ for $\omega\in D(A)$
reveals that $\mathcal{D}(A^{*})=\\{\omega\in L^{2}:A^{*}(u)\omega\in
L^{2}\\}$.

Assume that $A(u)+\lambda I$ does not have a dense range in $L^{2}$. Then,
there exists $0\neq z\in L^{2}$ such that $((A(u)+\lambda I)\omega,z)_{0}=0$
for all $\omega\in\mathcal{D}(A)$. Since $H^{1}\subset\mathcal{D}(A)$,
$\mathcal{D}(A)=\mathcal{D}(A^{*})$ is dense in $L^{2}$. It means that there
exists a sequence $z_{k}\in\mathcal{D}(A^{*})$ such that it converges to an
element $z\in L^{2}$. Recall that $D(A^{*})$ is closed. So,
$z\in\mathcal{D}(A^{*})$. Moreover,

$((A(u)+\lambda I)\omega,z)_{0}=(\omega,(A(u)+\lambda I)^{*}z)_{0}=0$

reveals that $(A^{*}(u)+\lambda I)z=0$ in $L^{2}$. Multiplying by $z$ and
integrating by parts, we get

$0=((A^{*}(u)+\lambda I)z,z)_{0}=(\lambda
z,z)_{0}+(z,A(u)z)_{0}\geq(\lambda-\beta)\|z\|_{0}^{2}~{}~{}~{}\forall\lambda>\beta$

and thus, $z=0$, which contradicts our assumption. It completes the proof of
(b). Therefore, the operator $A(u)$ is quasi-m-accreative. ∎

In the proof of Lemma 3.1 we use the fact that $C^{\infty}(\mathbb{S})$ is a
_core_ for $A$ in $H^{s-1}$, i.e. $A(u)v$ can be approximated by smooth
functions in $H^{s-1}$ ([8]):

######  Lemma 3.3.

Given $v\in D(A)$ there exists a sequence $(v_{n})_{n\in\mathbb{N}}$ in
$\mathcal{C}^{\infty}$ such that both $v_{n}\to v$ and $Av_{n}\to Av$ in
$H^{s-1}$.

###### Proof.

Let $v\in D(A)$ and fix $\rho\in C_{c}^{\infty}$, where $C_{c}^{\infty}$
denotes the set of $C^{\infty}$ functions with compact support, with $\rho\geq
0$ and $\int_{\mathbb{R}}\rho=1$. Given $n\geq 1$, let $\rho_{n}=n\rho(nx)$.
If we set $v_{n}\coloneqq\rho_{n}*v$, then $v_{n}\in C_{c}^{\infty}$ for
$n\geq 1$ and $v_{n}\to v$ in $H_{p}^{s-1}$. We have to prove that
$(uv_{n})_{x}\to(uv)_{x}$ in $H^{s-1}$. Since $v\in D(A)$, we have that
$uv_{x}\in H^{s-1}$ and hence $\rho_{n}*(uv_{x})\rightarrow uv_{x}$. Moreover,
since $uv_{n}\in H^{s}$ it follows that
$(uv_{n})_{x}=u_{x}v_{n}+u(v_{n})_{x}\in H^{s-1}$, hence we have that
$u_{x}v_{n}\rightarrow u_{x}v$ in $H^{s-1}$. Therefore

$\displaystyle(uv_{n})_{x}-(uv)_{x}$
$\displaystyle=u_{x}v_{n}-u_{x}v+\rho_{n}*(uv_{x})-uv_{x}+u(v_{n})_{x}-\rho_{n}*(uv_{x})$

holds true and it suffices to show that $u(v_{n})_{x}-\rho_{n}*(uv_{x})\to 0$
in $H^{s-1}$. To this end, denote

$P_{n}v\coloneqq u(v_{n})_{x}-\rho_{n}*(uv_{x}),\quad n\geq 1.$

We will show that there exists $K>0$ independent of $v$ such that

$\|P_{n}v\|_{s-1}\leq K\|v\|_{s-1},\quad n\geq 1.$ (3.3)

That will enable us to conclude that $P_{n}$ is uniformly bounded in $H^{s-1}$
by the uniform boundedness principle. When we approximate $v$ in $H^{s-1}$ by
smooth functions, and use this conclusion, we will be able to prove the
assertion $P_{n}\to 0$ for $v\in C_{c}^{\infty}$. Since the set of smooth
functions is dense in $H^{s-1}$ and $P_{n}$ are uniformly bounded, the proof
will be completed.

We first notice that

$\displaystyle P_{n}v(x)$
$\displaystyle=\int_{\mathbb{R}}(\rho_{n})_{y}(y)(u(x)-u(x-y))v(x-y)dy+(\rho_{n}*(u_{x}v))(x)$
$\displaystyle=n^{2}\int_{\mathbb{R}}\rho_{y}(ny)(u(x)-u(x-y))v(x-y)dy+(\rho_{n}*(u_{x}v))(x)$
$\displaystyle=n\int_{-1}^{1}\rho_{y}(y)(u(x)-u(x-\frac{y}{n}))v(x-\frac{y}{n})dy+(\rho_{n}*(u_{x}v))(x),$

where $\rm supp(\rho)\subset[-1,1]$. Moreover, using the mean value theorem,
we obtain the estimate

$\displaystyle\Big{|}n^{2}\int_{\mathbb{R}}\rho_{y}(ny)(u(x)-u(x-y))v(x-y)dy\Big{|}=\Big{|}n^{2}\int_{\mathbb{R}}\rho_{y}(ny)u_{x}(x_{0})yv(x-y)dy\Big{|}$
$\displaystyle=\Big{|}\int_{-1}^{1}\rho_{y}(y)u_{x}(x_{0})yv(x-y)dy\Big{|}\leq\|u_{x}\|_{L^{\infty}}\int_{-1}^{1}|\rho_{y}(y)|\,|y|\,|v(x-\frac{y}{n})|dy,$

for some $x_{0}\in(x,x-y)$. Let now
$C\coloneqq\sup_{x\in\mathbb{R}}\|u_{x}\|_{L^{\infty}}^{2}\int_{-1}^{1}|\rho_{y}(y)y|^{2}dy$.
Then the Cauchy-Schwarz inequality, Fubini’s theorem and the fact that the
operator $\Lambda^{s-1}$ commutes with integration yield that

$\displaystyle\|n^{2}\int_{\mathbb{R}}\rho_{y}(ny)(u(x)-u(x-y))v(x-y)dy\|^{2}_{s-1}$
$\displaystyle=\|\Lambda^{s-1}\int_{-1}^{1}\rho_{y}(y)u_{x}(x_{0})y(v(x-\frac{y}{n}))dy\|^{2}_{2}$
$\displaystyle=\int_{\mathbb{R}}\left|\int_{-1}^{1}\rho_{y}(y)u_{x}(x_{0})y\Lambda^{s-1}v(x-\frac{y}{n})dy\right|^{2}dx$
$\displaystyle\leq
C\int_{-1}^{1}\int|\Lambda^{s-1}v(x-\frac{y}{n})|^{2}dxdy\leq 2C\|v\|_{s-1}.$

Moreover, we obtain by Plancherel’s theorem that

$\displaystyle\|\rho_{n}*(u_{x}v)\|_{s-1}$
$\displaystyle=\|\Lambda^{s-1}(\rho_{n}*(u_{x}v))\|_{2}=\|\rho_{n}*\Lambda^{s-1}(u_{x}v))\|_{2}$
$\displaystyle\leq\|\Lambda^{s-1}(u_{x}v)\|_{2}$
$\displaystyle\leq\|u_{x}\|_{L^{\infty}}\|v\|_{s-1}.$

Therefore we conclude that

$\|P_{n}v\|_{s-1}\leq(\sqrt{2C}+\|u_{x}\|_{L^{\infty}})\,\|v\|_{s-1},~{}~{}n\geq
1.$ (3.4)

For $K=\sqrt{2C}+\|u_{x}\|_{L^{\infty}}$ in (3.3), proof is completed by the
estimate (3.4). ∎

Second step to prove Lemma 3.1 is making use of the following lemma proved in
[25]:

######  Lemma 3.4.

Let $X$ and $Y$ be two Banach spaces such that $Y$ is continuously and densely
embedded in $X$. Let $-A$ be the infinitesimal generator of the
$C_{0}$-semigroup $T(t)$ on $X$ and let $Q$ be an isomorphism from $Y$ onto
$X$. Then $Y$ is $-A$-admissible (i.e. $T(t)Y\subset Y$ for all $t\geq 0$, and
the restriction of $T(t)$ to $Y$ is a $C_{0}$-semigroup on $Y$) if and only if
$-A_{1}=-QAQ^{-1}$ is the infinitesimal generator of the $C_{0}$-semigroup
$T_{1}(t)=QT(t)Q^{-1}$ on $X$. Moreover, if $Y$ is $-A$-admissible, then the
part of $-A$ in $Y$ is the infinitesimal generator of the restriction $T(t)$
to $Y$.

Before we proceed with the proof of Lemma 3.1, we give the commutator estimate
which will be used:

######  Lemma 3.5 ([35]).

Let $m>0$, $s\geq 0$ and $3/2<s+m\leq\sigma$. Then for all $f\in H^{\sigma}$
and $g\in H^{s+m-1}$ one has $\|[\Lambda^{m},f]g\|_{s}\leq
C\|f\|_{\sigma}\,\|g\|_{s+m-1},$ where $C$ is a constant which is independent
of $f$ and $g$.

Proof of Lemma 3.1: Following the arguments in the proof of Lemma 3.2, we
first take the following $H^{s-1}$ inner product

$\displaystyle(A(u)\omega,\omega)_{s-1}$
$\displaystyle=(u\partial_{x}\omega,\omega)_{s-1}=(\Lambda^{s-1}u\partial_{x}\omega,\Lambda^{s-1}\omega)_{0}$
$\displaystyle=([\Lambda^{s-1},u]\partial_{x}\omega,\Lambda^{s-1}\omega)_{0}+(u\partial_{x}\Lambda^{s-1}\omega,\Lambda^{s-1}\omega)_{0}.$
(3.5)

Using Cauchy-Schwartz’s inequality and Lemma 3.5 with $m=s-1$, $\sigma=s$, we
get the following estimate for the first term of (3.5):

$\displaystyle|([\Lambda^{s-1},u]\partial_{x}\omega,\Lambda^{s-1}\omega)_{0}|$
$\displaystyle\leq
C\|u\|_{s}\|\partial_{x}\omega\|_{s-2}\|\omega\|_{s-1}\leq\tilde{C}\|\omega\|_{s-1}^{2},$

for some constant $\tilde{C}$ depending on $\|u\|_{s}$.

For the second term of (3.5), we again refer to Lemma 3.3 and use integration
by parts to get:

$\displaystyle|(u\partial_{x}\Lambda^{s-1}\omega,\Lambda^{s-1}\omega)_{0}|=|-\frac{1}{2}(u_{x},(\Lambda^{s-1}\omega)^{2})_{0}|\leq
C\|u_{x}\|_{L^{\infty}}\|w\|_{s-1}^{2}\leq\tilde{C}\|\omega\|_{s-1}^{2}.$

Choosing $\beta=\tilde{C}(\|u\|_{H^{s}})$, the operator satisfies the required
inequality.

Moreover, let $Q:=\Lambda^{s-1}$ and notice that $Q$ is an isomorphism of
$H^{s-1}$ to $L^{2}$ and $H^{s-1}$ is continuously and densely imbedded into
$L^{2}$ as $s>\frac{3}{2}$.
Define

$\displaystyle
A_{1}(u)=QA(u)Q^{-1}=\Lambda^{s-1}A(u)\Lambda^{1-s}=\Lambda^{s-1}u\partial_{x}\Lambda^{1-s}=\Lambda^{s-1}u\Lambda^{1-s}\partial_{x},$

and let $\omega\in L^{2}$ and $u\in H^{s}$, $s>\frac{5}{2}$. Then write
$B_{1}(u)=A_{1}(u)-A(u)$ and consider the following estimate:

$\displaystyle\|B_{1}(u)\omega\|_{0}$
$\displaystyle=\|[\Lambda^{s-1},A(u)]\Lambda^{1-s}\omega\|_{0}=\|[\Lambda^{s-1},u]\Lambda^{1-s}\partial_{x}\omega\|_{0}$
$\displaystyle\leq C\|u\|_{s}\|\Lambda^{1-s}\partial_{x}\omega\|_{s-2}\leq
C\|u\|_{s}\|\omega\|_{0},$

where we applied Lemma 3.5 with $m=s-1$ and $\sigma=s$. Hence, we obtain
$B_{1}(u)\in\mathcal{L}(L^{2}).$

Recall from Lemma 3.2 that $A(u)$ is quasi-m-accretive in $L^{2}$, i.e.
$-A(u)$ is the infinitesimal generator of a $C_{0}$-semigroup on $L^{2}$.
Thus, $A_{1}(u)=A(u)+B_{1}(u)$ is also the infinitesimal generator of a
$C_{0}$-semigroup in $L^{2}$ by means of a perturbation theorem for semigroups
(see [25]). Lemma 3.4 reveals that for $Y=H^{s-1}$, $X=L^{2}$ and
$Q=\Lambda^{s-1}$, $H^{s-1}$ is $A$-admissible. Hence, $-A(u)$ is the
infinitesimal generator of a $C_{0}$-semigroup on $H^{s-1}$. ∎

We continue with the proof of assumption (A2):

######  Lemma 3.6.

$A$ maps $H^{s}$ into $\mathcal{L}(H^{s},H^{s-1})$, more precisely the domain
$D(A(u))$ contains $H^{s}$ and the restriction $A(u)|_{H^{s}}$ belongs to
$\mathcal{L}(H^{s},H^{s-1})$ for any $u\in H^{s}$. Furthermore $A$ is
Lipschitz continuous in the sense that for all $r>0$ there exists a constant
$C_{1}$ which only depends on $r$ such that

$\|A(u)-A(v)\|_{\mathcal{L}(H^{s},H^{s-1})}\leq C_{1}\,\|u-v\|_{s-1}$ (3.6)

for all $u,~{}v$ inside $\mathrm{B}_{r}(0)\subseteq H^{s}$.

###### Proof.

The operator $A(u)|_{H^{s}}$ belongs to $\mathcal{L}(H^{s},H^{s-1})$ for any
$u\in H^{s}$, since $\partial_{x}\in\mathcal{L}(H^{s},H^{s-1})$ and
$M_{u}\in\mathcal{L}(H^{s-1})$. To see that the required estimate is
satisfied, let $u,v,w\in H^{s}$ be arbitrary. Then,

$\displaystyle\|(A(u)-A(v))w\|_{s-1}$
$\displaystyle=\|(u-v)\partial_{x}w\|_{s-1}\leq
C\|u-v\|_{s-1}\,\|\partial_{x}w\|_{s-1}$ $\displaystyle\leq
C\|u-v\|_{s-1}\,\|w\|_{s},$

where $C$ denotes a generic constant. This shows that for arbitrary $r>0$ one
can always find a constant $C_{1}$ such that (3.6) holds uniformly for all
$u,v\in\mathrm{B}_{r}(0)\subseteq H^{s}$. ∎

The last two assumptions (A3)-(A4) are proved by the help of the following
commutator and product estimates stated in [20, Proposition B.10.(2)] and [14,
Lemma A1], respectively:

######  Lemma 3.7.

Let $r>1/2$.

  1. (i)

If $-1/2<t\leq r+1$, there exists a constant $C_{r,t}>0$ such that

$\big{\|}[\Lambda^{t},M_{g}]h\big{\|}_{0}\leq
C_{r,t}\,\|g\|_{r+1}\,\|h\|_{t-1}$

for all $g\in H^{r+1}$ and $h\in H^{t-1}$.

  2. (ii)

If $-r<t\leq r$, there exists a constant $C_{r,t}>0$ such that $\|fg\|_{t}\leq
C_{r,t}\,\|f\|_{r}\,\|g\|_{t}$, for all $f\in H^{r}$ and $g\in H^{t}$.

Even though Lemma 3.7 is stated on the real line (in general
$\mathbb{R}^{m}$), it holds on periodic domain as well.

Now, we define a bounded linear operator and prove assumption (A3):

######  Lemma 3.8.

For any $u\in H^{s}$ there exists a bounded linear operator
$B(u)\in\mathcal{L}(H^{s-1})$ satisfying $B(u)=\Lambda A(u)\Lambda^{-1}-A(u)$
and $B\colon H^{s}\to\mathcal{L}(H^{s-1})$ is uniformly bounded on bounded
sets in $H^{s}$. Furthermore for all $r>0$ there exists a constant $C_{2}$
which depends only on $r$ such that

$\|B(u)-B(v)\|_{\mathcal{L}(H^{s-1})}\leq C_{2}\,\|u-v\|_{s}$ (3.7)

for all $u,v\in\mathrm{B}_{r}(0)\subseteq H^{s-1}$. Here, $A(u)$ is the
operator given by (3.1).

###### Proof.

Let $u\in H^{s}$. Since $\partial_{x}$ commutes with $\Lambda$ and
$\Lambda^{-1}$ we obtain that

$B(u)=\Lambda
u\partial_{x}\Lambda^{-1}-u\partial_{x}=[\Lambda,u\partial_{x}]\Lambda^{-1}=[\Lambda,u]\Lambda^{-1}\partial_{x}.$

Hence we can write $\Lambda^{s-1}B(u)$ as

$\displaystyle\Lambda^{s-1}[\Lambda,u]\Lambda^{-1}\partial_{x}$
$\displaystyle=\Lambda^{s}u\Lambda^{-1}\partial_{x}-\Lambda^{s-1}u\partial_{x}=[\Lambda^{s},u]\Lambda^{-1}\partial_{x}+u\Lambda^{s-1}\partial_{x}-\Lambda^{s-1}u\partial_{x}$
$\displaystyle=[\Lambda^{s},u]\Lambda^{-1}\partial_{x}+[u,\Lambda^{s-1}]\partial_{x}.$

Let now $\omega\in H^{s-1}$ and $u,v\in H^{s}$ be arbitrary. In view of the
above identity and Lemma 3.7, we obtain the following estimate

$\displaystyle\|(B(u)-B(v))\omega\|_{s-1}=\|\Lambda^{s-1}(B(u)-B(v))\omega\|_{0}$
$\displaystyle\leq\|[\Lambda^{s},u-v]\Lambda^{-1}\partial_{x}\omega\|_{0}+\|[u-v,\Lambda^{s-1}]\partial_{x}\omega\|_{0}$
$\displaystyle\leq
C\|u-v\|_{s}(\|\Lambda^{-1}\partial_{x}\omega\|_{s-1}+\|\partial_{x}\omega\|_{s-2})\leq
C\|u-v\|_{s}\|\omega\|_{s-1},$

where $C$ is a generic constant independent of $u,w$ and $w$. In particular,
this shows that $B(u)$ extends to a bounded linear operator on $H^{s-1}$ for
every $u\in H^{s}$ such that $B\colon H^{s}\to\mathcal{L}(H^{s-1})$ is
uniformly bounded on bounded sets in $H^{s}$. Furthermore, this estimation
proves that there exists a constant $C_{2}$ depending only on the radius of
the ball $\mathrm{B}_{r}(0)\subseteq H^{s}$ such that (3.7) is satisfied for
all $u,v\in B_{r}(0)$. ∎

The last assumption (A4) is proved in Lemma 3.9:

######  Lemma 3.9.

For all $t\in[0,\infty)$, $f$ is uniformly bounded on bounded sets in $H^{s}$.
Moreover, the map $f\colon H^{s}\to H^{s}$ is locally $H^{s-1}$-Lipschitz
continuous in the sense that for every $r>0$ there exists a constant
$C_{3}>0$, depending only on $r$, such that $\|f(u)-f(v)\|_{s-1}\leq
C_{3}\,\|u-v\|_{s-1}$ for all $u,v\in\mathrm{B}_{r}(0)\subseteq H^{s}$ and
locally $H^{s}$-Lipschitz continuous in the sense that for every $r>0$ there
exists a constant $C_{4}>0$, depending only on $r$, such that
$\|f(u)-f(v)\|_{s}\leq C_{4}\,\|u-v\|_{s}$ for all
$u,v\in\mathrm{B}_{r}(0)\subseteq H^{s}$.

###### Proof.

Recall that $f(u)=\Lambda^{-2}\partial_{x}\big{(}u^{2}+(u^{2})_{x}\big{)}.$
Therefore, with the help of Lemma 3.7

$\displaystyle\|f(u)-f(v)\|_{s-1}$ $\displaystyle\leq
C\|(u^{2}-v^{2})+(u^{2}-v^{2})_{x}\|_{s-2}$ $\displaystyle\leq
C\|(u+v)(u-v)+((u+v)(u-v))_{x}\|_{s-2}$ $\displaystyle\leq
C(\|(u+v)\|_{s-2}\|(u-v)\|_{s-1}+\|u+v\|_{s-1}\|u-v\|_{s-1})$
$\displaystyle\leq C_{3}\|u-v\|_{s-1}$

where $C_{3}$ is a constant depending on $\|u\|_{H^{s}}$ and $\|v\|_{H^{s}}$.
This proves $H^{s-1}$-Lipschitz continuity.

Similar arguments will show that we have the following estimates:

$\displaystyle\|f(u)-f(v)\|_{s}\leq$ $\displaystyle C_{4}\|u-v\|_{s}$ (3.8)

where $C_{4}$ is also a constant depending on $\|u\|_{H^{s}}$ and
$\|v\|_{H^{s}}$. Since we choose $u_{0}\in H^{s}$, this estimate actually
corresponds to the proof of continuous dependence on the initial data. Note
that boundedness of $f(u)$ on bounded subsets $\\{u\in H^{s}:\|u\|_{s}\leq
M\\}$ of $H^{s}$ (for all $M$) can be obtained from (3.8) by choosing $v=0$.
Hence, we get the estimates for (A4). ∎

##  4 Proof of theorem 1.3

In order to prove theorem 1.3 we need some estimates for $u,\,u_{x}$ and
$u_{xx}$. To this end, we need the next result.

######  Theorem 4.1.

Assume that $u_{0}\in H^{3}(\mathbb{S})$,
$m_{0}(x):=u_{0}(x)-u^{\prime\prime}_{0}(x)$, and let $u$ be the corresponding
solution of (1.5). If $m_{0}(x)\geq 0$, $x\in\mathbb{S}$, then
$m(x,t):=u(x,t)-u_{xx}(x,t)$ is non-negative for any $t$ as long as the
solution exists, and any $x\in\mathbb{S}$. Moreover, $u$ is also non-negative.
In particular, if $m_{0}>0$, then $u>0$.

An analogous result for non-periodic problems was proved in [21, Lemma 5.5],
and following the same steps we get the demonstration for Theorem 4.1. For
this reason it is omitted.

There is one more fact regarding the $L^{\infty}$ norm of $u_{x}$. We begin by
observing that

$\int_{\mathbb{S}}u_{xx}dx=0.$

This fact is enough to guarantee the existence of a point $\xi_{t}-1\in(0,1)$
such that $u_{x}(t,\xi_{t}-1)=0$, for each $t\in(0,T)$,.

######  Lemma 4.1.

If $u_{0}\in H^{3}(\mathbb{S})\cap L^{1}(\mathbb{S})$, is such that $m_{0}\geq
0$, then there exists a constant $K>0$ such that the solution of (1.5)
satisfies $\|u_{x}\|_{L^{\infty}(\mathbb{S})}\leq K$.

###### Proof.

Let us first assume that $\|m(\cdot,t)\|_{L^{1}}(\mathbb{S})$ is constant for
any $t$ as long as the solution exists. Assume $m_{0}$ does not change sign
and $m_{0}\geq 0$. Then,

$\displaystyle K_{1}$ $\displaystyle=$
$\displaystyle\|m_{0}\|_{L^{1}(\mathbb{S})}=\int_{\mathbb{S}}m_{0}(r)dr=\int_{\mathbb{S}}m(r,t)dr=\int_{\xi_{t}-1}^{\xi_{t}}m(r,t)dr$
$\displaystyle\geq$
$\displaystyle\int_{\xi_{t}-1}^{x}(u-u_{xx})(r,t)dr=\int_{\xi_{t}-1}^{x}u(r,t)dr-
u_{x}(x,t)\geq-u_{x}(x,t)$

holds for every $x\in[\xi_{t}-1,\xi_{t}]$. Here, we use Theorem 4.1, which
guarantee that $u$ does not change sign, under the assumption that $m_{0}$
does not change sign. Taking into account the final result, we observe that
$u_{x}$ is bounded from below.

Moreover,

$\displaystyle
K_{1}=\int_{\xi_{t}-1}^{\xi_{t}}m(r,t)dr\geq\int_{x}^{\xi_{t}}m(r,t)dr=\int_{x}^{\xi_{t}}u(r,t)dr+u_{x}(x,t)\geq
u_{x}(x,t).$

Hence, $u_{x}$ is bounded also from above. Therefore, we can conclude that
$\|u_{x}\|_{\infty}$ norm is bounded provided that $m$ does not change sign ,
i.e. $\|u_{x}\|_{\infty}\leq K$. The case $m_{0}\leq 0$ is proved in a similar
way and, therefore, is omitted.

We now complete the demonstration proving that
$\|m(\cdot,t)\|_{L^{1}(\mathbb{S})}$ is constant. We begin by noticing that
(1.1) is itself a conservation law, in the sense that

$\partial_{t}(u-u_{xx})=\partial_{x}\Big{(}(2-\partial_{x})(1+\partial_{x})u^{2}\Big{)}=\partial_{x}\Big{(}(1-\partial_{x}^{2})u^{2}+u^{2})\Big{)}.$

Integrating the relation above with respect to $x$ on $\mathbb{S}$, we obtain

$\frac{d}{dt}\int_{\mathbb{S}}(u-u_{xx})dx=\Big{(}(1-\partial_{x}^{2})u^{2}+u^{2})\Big{)}\big{|}_{\mathbb{S}}=0,$

meaning that the $\|m(\cdot,t)\|_{L^{1}(\mathbb{S})}=const.$. Since $m_{0}\in
L^{1}(\mathbb{S})$, we conclude that
$\|m(\cdot,t)\|_{L^{1}(\mathbb{S})}=\|m_{0}\|_{L^{1}(\mathbb{S})}.$ ∎

Now we start proving Theorem 1.3:

First, we rewrite the equation (1.5) in the following form

$u_{t}-2uu_{x}+u^{2}=\Lambda^{-2}(u^{2}+(u^{2})_{x})$

by using $\Lambda^{-2}(f(u))_{xx}=\Lambda^{-2}f(u)-f(u)$. Calling
$f(u)=u^{2}+(u^{2})_{x}$ and observing that $2uu_{x}=(u^{2})_{x}$, we get

$u_{t}+(1-\partial_{x})u^{2}=\Lambda^{-2}f(u).$ (4.1)

Now, we will differentiate (4.1) with respect to $x$, simplify and write

$u_{tx}+(\partial_{x}-\partial_{x}^{2})u^{2}=\Lambda^{-2}f(u)-u^{2}.$ (4.2)

We continue this process and get the following equations:

$u_{txx}+(\partial_{x}^{2}-\partial_{x}^{3})u^{2}=\Lambda^{-2}f(u)-f(u),$
(4.3)
$u_{txxx}+(\partial_{x}^{3}-\partial_{x}^{4})u^{2}=\partial_{x}\Lambda^{-2}f(u)-\partial_{x}f(u),$
(4.4)

$u_{txxxx}+(\partial_{x}^{4}-\partial_{x}^{5})u^{2}=\Lambda^{-2}f(u)-f(u)-\partial_{x}^{2}f(u).$
(4.5)

Moreover, we will multiply (4.1) by $u$, (4.2) by $u_{x}$, (4.3) by $u_{xx}$,
(4.4) by $u_{xxx}$, (4.5) by $u_{xxxx}$ and integrate all over $\mathbb{S}$.

Let
$I(u)=\int_{\mathbb{S}}(u^{2}+u_{x}^{2}+u_{xx}^{2}+u_{xxx}^{2}+u_{xxxx}^{2})dx$.
Therefore, summing up equations (4.1)-(4.5) we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}I(u)+\int_{\mathbb{S}}(u(1-\partial_{x})u^{2}+u_{x}(\partial_{x}-\partial_{x}^{2})u^{2}+u_{xx}(\partial_{x}^{2}-\partial_{x}^{3})u^{2}$
$\displaystyle+u_{xxx}(\partial_{x}^{3}-\partial_{x}^{4})u^{2}+u_{xxxx}(\partial_{x}^{4}-\partial_{x}^{5})u^{2})dx$
$\displaystyle=\int_{\mathbb{S}}(u\Lambda^{-2}f(u)+u_{x}\Lambda^{-2}f(u)-u^{2}u_{x}+u_{xx}(\Lambda^{-2}f(u)-f(u))$
$\displaystyle+u_{xxx}(\partial_{x}\Lambda^{-2}f(u)-\partial_{x}f(u))+u_{xxxx}(\Lambda^{-2}f(u)-f(u)-\partial_{x}^{2}f(u))dx.$

Our main aim is to obtain $I(u)$, which is equivalent to $H^{4}$ norm of $u$,
within the equation so that Gronwall’s inequality is applicable and we get an
upper bound valid for all time. That bound will imply the global existence of
solution.

After integration by parts, we can rewrite the equality in the following form:

$\displaystyle\frac{1}{2}\frac{d}{dt}I(u)+\int_{\mathbb{S}}(u^{3}+2(u^{2})_{x}u_{xx}+(u^{2})_{xx}u_{xxx}+2(u^{2})_{xxx}u_{xxxx}$
$\displaystyle+(u^{2})_{xxxx}u_{xxxx}+(u^{2})_{xxxx}u_{xxxxx}+u^{2}u_{xxxx})dx=\int_{\mathbb{S}}u(\Lambda^{-2}f(u))dx.$

Since

$\displaystyle(u^{2})_{x}=2uu_{x},\quad(u^{2})_{xx}=2u_{x}^{2}+2uu_{xx},\quad(u^{2})_{xxx}=6u_{x}u_{xx}+2uu_{xxx},$
$\displaystyle(u^{2})_{xxxx}=6u_{xx}^{2}+8u_{x}u_{xxx}+2uu_{xxxx},$
$\displaystyle(u^{2})_{xxxxx}=20u_{xx}u_{xxx}+10u_{x}u_{xxxx}+2uu_{xxxxx},$

and integrating by parts once more, the integral becomes

$\displaystyle\frac{1}{2}\frac{d}{dt}I(u)+\int_{\mathbb{S}}(u^{3}-2(u^{2})_{xx}u_{x}-(u^{2})_{xxx}u_{xx}-2(u^{2})_{xxxx}u_{xxx}$
(4.6)
$\displaystyle-(u^{2})_{xxxxx}u_{xxx}-(u^{2})_{xxxxx}u_{xxxx}+(u^{2})_{xx}u_{xx})dx$
$\displaystyle=$
$\displaystyle\frac{1}{2}\frac{d}{dt}I(u)+\int_{\mathbb{S}}(u^{3}-4u_{x}^{3}-4uu_{x}u_{xx}-4uu_{xx}^{2}-2uu_{xx}u_{xxx}$
$\displaystyle-16u_{x}u_{xxx}^{2}-4uu_{xxx}u_{xxxx}-20u_{xx}u_{xxx}^{2}$
$\displaystyle-10u_{x}u_{xxx}u_{xxxx}-2uu_{xxx}u_{xxxxx}-20u_{xx}u_{xxx}u_{xxxx}-10u_{x}u_{xxxx}^{2}$
$\displaystyle-2uu_{xxxx}u_{xxxxx})dx=\int_{\mathbb{S}}u(\Lambda^{-2}f(u))dx.$

By Lemma 4.1, we have that $\|u_{x}\|_{\infty}$ is bounded. Moreover, we can
rewrite (1.1) as

$u_{t}-u_{txx}=4uu_{x}+2u_{x}^{2}+2uu_{xx}-6u_{x}u_{xx}-2uu_{xxx}.$

Multiplying equation above by $u$, noting that

$\displaystyle 2uu_{x}u_{xx}$ $\displaystyle=$
$\displaystyle\partial_{x}(uu_{x}^{2})-u_{x}^{3},\quad
u^{2}u_{xxx}=\partial_{x}(u^{2}u_{xx}-uu_{x}^{2})+u_{x}^{3},$ $\displaystyle
u^{2}u_{xx}$ $\displaystyle=$
$\displaystyle\partial_{x}(u^{2}u_{x})-2uu_{x}^{2},$

integrating over $\mathbb{S}$ and using the identities above, we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\mathbb{S}}(u^{2}+u_{x}^{2})dx=\int_{\mathbb{S}}(4u_{x}^{3}-2uu_{x}^{2})dx\leq
5\|u_{x}\|_{\infty}\int_{\mathbb{S}}(u^{2}+u_{x}^{2})dx,$

which implies

$\|u\|_{1}^{2}\leq\|u_{0}\|_{1}^{2}e^{10\int_{0}^{t}\|u_{x}\|_{\infty}d\tau}\leq\|u_{0}\|_{1}^{2}e^{At}=C_{0}^{2},$

for some optimal constant $A>0$ since $\|u_{x}\|_{\infty}$ norm is bounded.
This estimate is valid at any finite time, therefore is a global bound for
$H^{1}$ norm of $u$. The reason we provide this inequality is to verify that
$\|u\|_{\infty}$ is also bounded, since $\|u\|_{\infty}\leq\|u\|_{1}\leq
C_{0}$ by Sobolev embedding theorem.

Moreover, we can show that $H^{2}$ norm of $u$ will be bounded in finite time:
Let $J(u)=\int_{\mathbb{S}}(u^{2}+u_{x}^{2}+u_{xx}^{2})dx$. Following the
arguments done above for (4.1)-(4.3) and using integration by parts,

$\displaystyle\frac{1}{2}\frac{d}{dt}J(u)$ $\displaystyle+$
$\displaystyle\int_{\mathbb{S}}(2u^{3}+(u^{2})_{x}u_{x}+2(u^{2})_{x}u_{xx}+(u^{2})_{xx}u_{xx}+(u^{2})_{xx}u_{xxx})dx$
$\displaystyle\leq$ $\displaystyle\int_{\mathbb{S}}2u(\Lambda^{-2}f(u))dx.$

Since $\|u\|_{\infty}$ and $\|u_{x}\|_{\infty}$ are bounded,

$\displaystyle\frac{1}{2}\frac{d}{dt}J(u)$ $\displaystyle\leq$ $\displaystyle
2\max{(\|u\|_{\infty},\|u_{x}\|_{\infty})}J(u)+2\max{(\|u\|_{\infty})}\int_{\mathbb{S}}(\Lambda^{-2}f(u))dx.$

As it was given in (1.6), $\Lambda^{-2}f=g*f$. Now, we need to estimate the
$L^{\infty}-$norm of the integrand of the second term in order to obtain a
differential inequality and apply Gronwall’s inequality. For this purpose, we
first provide the following two estimates:

$\displaystyle\|g\|_{2}\leq\frac{1}{2}(\frac{e^{2}+2e-1}{e^{2}-2e+1})^{1/2}:=n_{2},\quad\|g\|_{\infty}\leq\frac{1}{2}(\frac{e+1}{e-1}):=n_{\infty}.$

Hence,

$\displaystyle\|g*u^{2}\|_{\infty}\leq\|g\|_{\infty}\|u^{2}\|_{1}\leq\|g\|_{\infty}\|u\|_{2}^{2}\leq
n_{\infty}C_{0},$
$\displaystyle\|g*(u^{2})_{x}\|_{\infty}\leq\|g\|_{2}\|(u^{2})_{x}\|_{2}\leq\|g\|_{2}\|u^{2}\|_{1}\leq
n_{2}C_{0}^{2}.$

These estimates, together with Gronwall’s inequality, provide the boundedness
of $J(u)$ which is equivalent to $H^{2}$ norm. Therefore, we will be able to
give an upper bound for $\|u_{x}\|_{\infty}$ norm as well since
$\|u_{x}\|_{\infty}\leq\|u_{x}\|_{1}<\infty.$

Proving that $H^{3}$ norm is bounded in finite time will be the last issue to
conclude the proof of Theorem 1.3:
Let $K(u)=\int_{\mathbb{S}}(u^{2}+u_{x}^{2}+u_{xx}^{2}++u_{xxx}^{2})dx$. Like
we did for $J(u)$, we can evaluate the following inequality:

$\displaystyle\frac{1}{2}\frac{d}{dt}K(u)$ $\displaystyle+$
$\displaystyle\int_{\mathbb{S}}(2u^{3}+(u^{2})_{x}u_{x}+2(u^{2})_{x}u_{xx}+(u^{2})_{xx}u_{xx}$
$\displaystyle+$
$\displaystyle(u^{2})_{xx}u_{xxx}+(u^{2})_{xxx}u_{xxx}+(u^{2})_{xxx}u_{xxxx})dx$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{S}}(2u(\Lambda^{-2}f(u))+u_{x}(\Lambda^{-2}f(u))+u_{xxx}\partial_{x}(\Lambda^{-2}f(u)))dx,$

and

$\displaystyle\frac{1}{2}\frac{d}{dt}K(u)$ $\displaystyle\leq$ $\displaystyle
2\max{(\|u\|_{\infty},\|u_{x}\|_{\infty}})K(u)+\int_{\mathbb{S}}(2u(\Lambda^{-2}(u^{2}))+2u_{x}(\Lambda^{-2}(u^{2})_{x}))dx$
$\displaystyle\leq$ $\displaystyle
2\max{(\|u\|_{\infty},\|u_{x}\|_{\infty}})K(u)+2\max{(\|u\|_{\infty},\|u_{x}\|_{\infty}})\int_{\mathbb{S}}(\Lambda^{-2}f(u))dx.$

Similar arguments reveal that $K(u)$ is bounded in finite time and hence,
$\|u_{xx}\|_{\infty}\leq\|u_{xx}\|_{1}<\infty.$

Recalling the equality (4.6), and Theorem 4.1 which guarantees that $u>0$, we
evaluate

$\displaystyle\frac{1}{2}\frac{d}{dt}I(u)$ $\displaystyle=$
$\displaystyle-(\int_{\mathbb{S}}(u^{3}-4u_{x}^{3}-4uu_{x}u_{xx}-4uu_{xx}^{2}-2uu_{xx}u_{xxx}$
$\displaystyle-16u_{x}u_{xxx}^{2}-4uu_{xxx}u_{xxxx}-20u_{xx}u_{xxx}^{2}$
$\displaystyle-10u_{x}u_{xxx}u_{xxxx}-2uu_{xxx}u_{xxxxx}-20u_{xx}u_{xxx}u_{xxxx}-10u_{x}u_{xxxx}^{2}$
$\displaystyle-2uu_{xxxx}u_{xxxxx})dx)+\int_{\mathbb{S}}u(\Lambda^{-2}f(u))dx$
$\displaystyle\leq$
$\displaystyle-(\int_{\mathbb{S}}(-u^{3}-4u_{x}^{3}-4uu_{x}u_{xx}-4uu_{xx}^{2}-2uu_{xx}u_{xxx}$
$\displaystyle-16u_{x}u_{xxx}^{2}-4uu_{xxx}u_{xxxx}-20u_{xx}u_{xxx}^{2}$
$\displaystyle-10u_{x}u_{xxx}u_{xxxx}-2uu_{xxx}u_{xxxxx}-20u_{xx}u_{xxx}u_{xxxx}-10u_{x}u_{xxxx}^{2}$
$\displaystyle-2uu_{xxxx}u_{xxxxx})dx)+\int_{\mathbb{S}}u(\Lambda^{-2}f(u))dx$
$\displaystyle\leq$
$\displaystyle\max{(\|u\|_{\infty},\|u_{x}\|_{\infty},\|u_{xx}\|_{\infty})}I(u)+\int_{\mathbb{S}}u(\Lambda^{-2}f(u))dx$
$\displaystyle\leq$
$\displaystyle\max{(\|u\|_{\infty},\|u_{x}\|_{\infty},\|u_{xx}\|_{\infty})}I(u)+\max{(\|u\|_{\infty})\int_{\mathbb{S}}}(\Lambda^{-2}f(u))dx.$

Therefore,

$\displaystyle\frac{1}{2}\frac{d}{dt}I(u)$ $\displaystyle\leq$ $\displaystyle
K_{2}I(u)+K_{3}$

for some optimal constants $K_{2}$, $K_{3}$. Gronwall’s inequality implies
$I(u)\leq[I(0)+K_{3}t]e^{K_{2}t}$, which is valid for any finite time $0<t\leq
T$. Since we find an upper bound for $\|u\|_{4}$, this completes the proof of
Theorem 1.3.

##  5 Proof of theorem 1.1

The proof of theorem 1.1 is divided in three major parts, namely,

  * P1

Existence of $C^{1}$ periodic one-forms $\omega_{1},\omega_{2}$ and
$\omega_{3}$ satisfying (1.4);

  * P2

Existence of a domain $V$, depending on the initial datum, containing open
sets endowed with a PSS structure;

  * P3

Existence of local connection forms $\omega_{13},\omega_{23}$.

$\bullet$ Existence of $C^{1}$ periodic one-forms $\omega_{1}$, $\omega_{2}$
and $\omega_{3}$.

Example 2.1 exhibits two triads of one forms (2.13) satisfying the condition
(1.4).

For solutions $u$ emanating from an initial datum $u_{0}\in
H^{4}(\mathbb{S})$, with $u_{0}-u_{0}^{\prime\prime}>0$, theorem 1.2 implies
that $u\in C(H^{4}(\mathbb{S}),[0,T))\cap C^{1}(H^{3}(\mathbb{S}),[0,T))$,
whereas Theorem 1.3 informs us that $u$ is defined on
$U=\mathbb{R}\times(0,\infty)$. Moreover, $u_{t}(\cdot,t)\in
H^{3}(\mathbb{S})\subseteq C^{2}_{\text{per}}(\mathbb{R})$ and $u(\cdot,t)\in
H^{4}(\mathbb{S})\subseteq C^{3}_{\text{per}}(\mathbb{R})$ in view of the
Sobolev Lemma (see lemma 2.3). Therefore, $u\in C^{3,1}(\mathbb{R})$ and then
$f_{ij}\in C^{1}(\mathbb{R})$ and is periodic in the variable $x$, where
$f_{ij}$ are the coefficients of the forms given in (2.13).

=$\bullet$ Existence of a domain $V$, depending on the initial datum,
containing open sets endowed with a PSS structure;

From example 2.2, a non-generic solution $u$ of (1.1) can only be periodic if
it is constant. Since the initial datum satisfies the condition
$u_{0}-u_{0}^{\prime\prime}>0$, by theorem 4.1 we know that $u>0$ and it
cannot be constant on $U$.

Let us then suppose the existence of an open set $\Omega\subseteq U$ for which
$u\big{|}_{\Omega}=k$, where $k>0$ is a real number. Without loss of
generality, we may assume that $\Omega\subseteq(0,1)\times(0,\infty)=:U_{1}$
in view of the periodicity of $u$ with respect to $x$. For some $p\in
U_{1}\setminus\Omega$ and $\epsilon>0$, we have $\nabla
u(p)=(u_{x}(p),u_{t}(p))\neq(0,0)$ and $u\big{|}_{B_{\epsilon}(p)}$ is non-
constant, where $B_{\epsilon}(p)$ denotes the disc of centre $p$ and radius
$\epsilon$. As a result, $U_{1}$ has at least one connected component $V$ (and
$U$ as well), with $B_{\epsilon}(p)\subseteq V$, endowed with a PSS structure
determined by the forms $\omega_{1}$ and $\omega_{2}$.

$\bullet$ Existence of connection forms defined everywhere
$\omega_{1}\wedge\omega_{2}\neq 0$.

Henceforth we assume that the open sets under consideration are those that
$\nabla u\neq(0,0)$ everywhere. Let us denote these sets generically by $V$.

Our proof is based on, and follows, that made by Castro Silva and Kamran [4,
Proposition 3.7].

We have two possible choices for the form $\omega_{2}$. For this reason, fix
one of them and consider the frame $\\{\omega_{1},\omega_{2}\\}$. Let $a$, $b$
and $c$ functions such that

$\omega_{13}=a\omega_{1}+b\omega_{2},\quad\omega_{23}=b\omega_{1}+c\omega_{2}.$
(5.1)

Our task is to find functions locally defined on any open set of $U$ for which
(5.1) and the Levi-Civita connection form $\omega_{3}$ given in (2.13) satisfy
(2.8).

In [4, Proposition 3.7] it was shown that the connection forms (5.1) for an
equation of the type

$u_{t}-u_{txx}=\lambda uu_{xxx}+G(u,u_{x},u_{xx}),$ (5.2)

satisfy a certain set of differential equations, see [4, Theorem 2.4] and also
[3, Theorem 3.4]. The function $G$ has somewhat a specific dependence on its
arguments, and also some parameters $\mu$, $m_{1}$ and $m_{2}$. In [10,
Theorem 1] it was shown that (1.1) is a PSS equation, and one of the steps for
that demonstration was just to show that it falls in the class considered in
[3, Theorem 3.4]. In particular, the mentioned parameters are
$\mu\in\mathbb{R}$, $m_{1}\in\\{-2,1\\}$ and $m_{2}=0$.

Therefore, in view of [10, Equation (7)] and [3, Theorem 3.4], we fall either
into [4, Proposition 3.7, case ii.)] or [4, Proposition 3.7, case iii.)],
depending on whether $\mu=0$ or $\mu\neq 0$.

According to [4, Equation (181)], the functions $a$, $b$ and $c$ in (5.1) take
the form (recall that $m_{2}=0$) $a=\phi_{1}(z)$, $b=\phi_{2}(z)$, and
$c=\phi_{3}(z)$, $z=m_{1}x$, for some real valued and smooth functions
$\phi_{1}$, $\phi_{2}$ and $\phi_{3}$ to be determined, satisfying the
condition

$\phi_{1}\phi_{3}\neq 0.$ (5.3)

The Codazzi-Mainardi equations give (see [4, Equations (182)–(183)])

$\phi_{1}^{\prime}+\mu\phi_{2}^{\prime}-\phi_{1}-2\mu\phi_{2}+\phi_{3}=0$
(5.4)

and

$\phi_{2}^{\prime}+\mu\phi_{3}^{\prime}+\mu\phi_{1}-2\phi_{2}-\mu\phi_{3}=0,$
(5.5)

whereas the Gauss equation reads

$\phi_{1}\phi_{3}-\phi_{2}^{2}=-1.$ (5.6)

Equation (5.6), jointly with condition (5.3), imply that $b\neq 0$ everywhere.

From (5.4) we obtain $\phi_{3}$ in terms of $\phi_{1}$, $\phi_{2}$ and their
first derivatives. Substituting the result into (5.5) and integrating once, we
obtain (see [4, Equation (184)])

$\mu\phi_{1}^{\prime}=(1+\mu^{2})\phi_{2}-\mu^{2}\phi_{2}^{\prime}-\beta
e^{2z},$ (5.7)

where $\beta\in\mathbb{R}$ is a constant.

We now divide our proof in two different cases.

Case $\mu=0$. From (5.7) we conclude that $\phi_{2}=\beta e^{2z},$ whereas
(5.4) gives

$\phi_{3}=\phi_{1}-\phi_{1}^{\prime}.$ (5.8)

Substituting $\phi_{2}$ and $\phi_{3}$ into the Gauss equation (5.6) we obtain
the following Abel differential equation of the second kind

$\phi_{1}\phi_{1}^{\prime}=\phi_{1}^{2}-\beta^{2}e^{4z}+1.$

Under the change $\phi_{1}=e^{z}w$, where $w$ is another function of $z$, we
obtain the following simpler ODE

$ww^{\prime}=e^{-2z}-\beta^{2}e^{2z},$

that, after solving, substituting back the result for $\phi_{1}$, and
proceeding some manipulation, gives

$\phi_{1}(z)=\pm e^{z}\sqrt{\gamma-\beta^{2}e^{2z}-e^{-2z}}=\pm\sqrt{\gamma
e^{2z}-1-\beta^{2}e^{4z}}.$ (5.9)

Substituting (5.9) into (5.8) and going back to the original functions $a$,
$b$ and $c$, we obtain

$\begin{array}[]{lcl}a(x,t)&=&\displaystyle{\pm\sqrt{\gamma
e^{2m_{1}x}-\beta^{2}e^{4m_{1}x}-1}},\quad b(x,t)=\displaystyle{\beta
e^{2m_{1}x}},\\\ \\\
c(x,t)&=&\pm\displaystyle{\frac{\beta^{2}e^{2m_{1}x}-1}{\sqrt{\gamma
e^{2m_{1}x}-\beta^{2}e^{4m_{1}x}-1}}}.\end{array}$ (5.10)

]Case $\mu\neq 0$. From (5.7) we can write $\phi_{1}^{\prime}$ in terms of
$\phi_{2}$ and $\phi_{2}^{\prime}$. Substituting it into (5.6) we obtain

$\phi_{3}=\phi_{1}+\phi,\quad\phi=\displaystyle{\frac{\mu^{2}-1}{\mu}\phi_{2}-\frac{\beta}{\mu}e^{2z}.}$
(5.11)

Substituting (5.11) into the Gauss equation (5.6) we obtain

$\phi_{1}^{2}+\phi\phi_{1}-\phi_{2}^{2}=-1,$

which, after solved for $\phi_{1}$, yields

$\phi_{1}=\frac{-\phi\pm\sqrt{\Delta}}{2},\quad\Delta=\phi^{2}-4(1-\phi_{2}^{2}),$
(5.12)

which are well defined as long as $\Delta\geq 0$.

Substituting (5.12) into (5.11) and the result into (5.7) we obtain the
following ODE for $\phi_{2}$

$[(1+\mu^{2})\sqrt{\Delta}\pm(\mu^{2}-1)\phi\pm
4\mu\phi_{2}]\phi_{2}^{\prime}-2(1+\mu^{2})\sqrt{\Delta}\phi_{2}\mp 2\beta
e^{2z}\phi=0.$ (5.13)

It was shown in [4, page 36] that the coefficient of $\phi_{2}^{\prime}$
cannot vanish, otherwise we would conclude that $\phi_{1}=\phi_{3}$ and
$\phi_{2}=0$, which contradicts (5.6). By continuity, such a coefficient does
not change its sign. Without loss of generality, we may assume it to be
positive and the ODE above takes the form $b^{\prime}=g(z,b)$.

Given a point $p=(x_{0},t_{0})\in V$, arguing exactly as [4, page 37] we
conclude that the ODE to $b$ subject to $b(x_{0}/m_{1})=t_{0}$ has a unique
(local) solution, that guarantee the local existence of $b$ in a neighborhood
of each point of $V$. $\square$

######  Remark 5.1.

The uniqueness of the forms (2.13) follows from [3, Theorem 3.4], see also
[10, page 760], and the fact that the solution $u$ of (1.1) is unique in view
of theorem 1.2.

##  6 Concluding remarks

In this paper we studied an equation whose solutions define metrics for a PSS
from the point of view of geometric analysis. More precisely, we used tools of
semi-group theory to establish well-posedness of solutions and then study the
corresponding surface qualitatively.

From the point of view of analysis, our Theorem 1.2 ensures well-posedness of
solutions, whereas our Theorem 1.3 ensures enough regularity of the solution
in order to ensure geometric relevance in the one-forms given in Theorem 1.1.
In particular, this last theorem can be seen as a sort of existence and
uniqueness theorem for periodic surfaces emanating from a given initial datum,
which can be associated to a certain curve in the three-dimensional space.

## Acknowledgements

N. D. Mutlubaş is supported by the Turkish Academy of Sciences within the
framework of the Outstanding Young Scientists Awards Program (TÜBA-
GEBIP-2022). I. L. Freire is thankful to Enrique Reyes, Marcio Fabiano da
Silva and Stefano Nardulli for stimulating discussions and support. He is also
grateful to CNPq (grant nº 310074/2021-5) and FAPESP (grants nº 2020/02055-0
and 2022/00163-6) for financial support. He is also grateful to the Department
of Mathematical Sciences for warm hospitality and support during the
development of this work.

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