# Hopf-Like Bifurcation in a Wave Equation at a Removable Singularity

Nemanja Kosovalić  Shopify (formerly University of South Alabama)
<EMAIL_ADDRESS>and  Brian Pigott  Wofford College
<EMAIL_ADDRESS>
###### Abstract.

It is shown that a one-dimensional damped wave equation with an odd time
derivative nonlinearity exhibits small amplitude bifurcating time periodic
solutions, when the bifurcation parameter is the linear damping coefficient is
positive and accumulates to zero. The upshot is that the singularity of the
linearized operator at criticality which stems from the well known small
divisor problem for the wave operator, is entirely removed without the need to
exclude parameters via Diophantine conditions, nor the use of accelerated
convergence schemes. Only the contraction mapping principle is used.

###### Key words and phrases:

damped wave equation, small amplitude time periodic solution, removable
singularity, bifurcation, viscoelastic damping

###### 1991 Mathematics Subject Classification:

(Primary) (Secondary)

##  1\. Introduction

Time periodic solutions of nonlinear wave equations, have been a large focus
of research spanning the last fifty years, with major interest in non-
dissipative systems, or dissipative systems subject to a periodic forcing. In
the last thirty years, the emphasis of many works addressing the bifurcation
of time periodic solutions of nonlinear wave equations is the resolution of
small divisor problems. For instance see [6, 5] and [4, 3], to name a few. In
this paper, we consider the problem of bifurcating small amplitude time
periodic solutions for nonlinear wave equations when the (positive) linear
damping coefficient is used as the bifurcation parameter, and is allowed to be
arbitrarily small. In order for nontrivial time periodic solutions to persist,
we consider a nonlinear negative damping as in, for instance, the
investigation of the Beam equation in [9], to balance out the positive linear
damping. The linearized problem at criticality (when the damping coefficient
is zero) naturally contains a singularity owing to the small divisor problem
stemming from the wave operator. The upshot of this work is that the
singularity can be entirely removed without the use of techniques such as
Nash-Moser iteration, nor the exclusion of parameters via Diophantine
conditions. In order to remove the singularity, we exploit three critical
features of our problem: (i) An asymptotic expansion for the lower bound of
the damping parameter in terms of amplitude, (ii) the fact a certain integral
vanishes (see Lemma 4.1), and (iii) the regularizing effect of Laplacian
damping. This allows us to solve the infinite dimensional part of the problem
using merely the contraction mapping principle. Our work is motivated by [2]
which proves Lyapunov Center Theorems for various nonlinear wave equations
using the contraction mapping principle but differs in two ways: We have an
explicit bifurcation parameter, so we are really proving a Hopf Bifurcation
Theorem and not a Lyapunov Center Theorem, and we do not require any
Diophantine conditions to restrict parameters.

We consider the following damped nonlinear wave equation with Dirichlet
boundary conditions:

$\left\\{\begin{array}[]{ll}\partial_{t}^{2}u-\partial_{x}^{2}u-\alpha\partial_{t}\partial_{x}^{2}u+mu=\Big{(}\partial_{t}u\Big{)}^{2p+1},&t\in\mathbf{R},x\in\mathbf{T}=\mathbf{R}/2\pi\mathbf{Z},\\\
u(t,0)=u(t,\pi)=0.\end{array}\right.$ (1)

In (1) $p\geq 1$ is taken to be an integer, $\alpha>0$ and $m>0$ is
irrational.

Before stating the main results we make some clarifying remarks concerning our
choices of the nonlinearity, damping, and the question of the stability of the
obtained periodic solutions:

  1. (1)

Nonlinearity \- Due to the presence of positive linear damping, in order for
non-trivial periodic solutions to persist, we need the nonlinearity to restore
energy to the system, and the odd time derivative nonlinearity is the simplest
autonomous mechanism which achieves this. See the energy argument in [9] for
details. Such velocity dependent nonlinearities arise in the phenomena of
self-excited vibrations; see the introduction of [9] and the references
therein. In fact, our results can be extended to include higher order
polynomial perturbations on top of the present nonlinearity, which do not
contain any derivatives. On a related note the Lemma 4.1 which concerns the
vanishing of a certain integral, is really just another way of stating that
monomial expansions of trigonometric functions have finitely many nonzero
Fourier modes, and has nothing special to do with our choice of nonlinearity.

  2. (2)

Damping \- The choice of Laplacian damping (sometimes called Kelvin-Voigt
damping) in this work is used to facilitate removing the singularity. It is
not clear if our results can be obtained for non-Laplacian linear damping
using the same techniques. On the other hand, our choice of damping naturally
arises in modelling the vibrations of viscoelastic materials, as an internal
friction of the vibrating material. See e.g. the introduction of [1] for a
discussion of damping mechanisms in wave equations.

  3. (3)

Stability \- We do not obtain any results concerning the stability of the
periodic solutions here, but can comment on the direction of bifurcation.
Although the solutions we obtain persist for arbitrarily small positive values
of the damping coefficient, a change of variable $t\to-t$ allows us to get
solutions for negative values of the damping coefficient for the nonlinearity
$-1\cdot(\partial_{t}u)^{2p+1}$. Provided the principle of exchange of
stability holds (see Section I.12 of the monograph [8]), then we expect the
latter branch of periodic solutions to be stable.

###  1.1. Statement of Results

Our main result is

######  Theorem 1 (Small Damping Bifurcation).

Let $k_{0}\geq 2$ be given. Then there is some $M=M(p,m,k_{0})>0$ so that for
each $\rho\in(0,M)$ there is some $\alpha\gtrsim_{m,p,k_{0}}\rho^{2p}$ and
$\omega\approx\sqrt{1+m}$ so that the nonlinear wave equation (1) has a
$C^{k_{0}}$ in $(t,x)$ $\frac{2\pi}{\omega}$-time periodic solution of the
form

$u(t,x)=\rho\cos(\omega t)\sin(x)+v(\omega t,x)$

where $(t,x)\mapsto v(t,x)$ is a $2\pi$-time periodic function, and
$v(t,x)=O(\rho^{2p+\frac{1}{2}})$ with respect to a suitable Sobolev norm.

###  1.2. Organization

The paper is organized as follows. In Section 2 we introduce the basic
notation that will be used throughout the paper as well as the Hilbert space
$X^{s}$ in which we will work. We discuss the periodic solution ansatz and
develop some key properties of the linearized operator in Section 3. In
Section 4 we introduce the Lyapunov-Schmidt decomposition, solve the range
equation and the bifurcation equations, and prove Theorem 1.

##  2\. Notation

We begin by introducing some notation that we use throughout the paper:

  * •

$A\lesssim B$ means that there is a universal constant $C$ so that $A\leq CB$.
In cases where we wish to indicate that the implicit constant $C$ depends on a
variable $q$ we will write $A\lesssim_{q}B$; that is, $A\lesssim_{q}B$ means
$A\leq C(q)B$.

  * •

$A\sim B$ means that $A\lesssim B$ and $B\lesssim A$.

  * •

$x\approx y$ means that there is an $\epsilon>0$ for which $|x-y|<\epsilon$.

  * •

$\displaystyle\mathcal{L}(X,Y)$ is the set of bounded linear maps from $X\to
Y$ (where $X$ and $Y$ are Banach spaces).

  * •

$\mathscr{B}_{X}(x,r)$ denotes the ball of radius $r>0$ centered at the point
$x$ in the metric space $X$. When the space $X$ is clear from the context, we
will write $\mathscr{B}(x,r)$.

  * •

We use the symbol $D$ to denote the Fréchet derivative for mappings between
Banach spaces.

###  2.1. The Hilbert Space $X^{s}$

We recall the definition of the space $X^{s}$ that was introduced in [10]. The
space $X^{s}$ is the subset of $H^{s}(\mathbf{T}^{2})$ consisting of all real-
valued functions that can be written in the form

$u(t,x)=\sum_{\begin{subarray}{c}n\in\mathbf{Z}\\\ k\geq
1\end{subarray}}c_{n,k}e^{int}\sin(kx).$

That is,

$X^{s}:=\left\\{\sum_{\begin{subarray}{c}n\in\mathbf{Z}\\\ k\geq
1\end{subarray}}c_{n,k}e^{int}\sin(kx)\ \Big{|}\ \overline{c_{n,k}}=c_{-n,k},\
\sum_{\begin{subarray}{c}n\in\mathbf{Z}\\\ k\geq
1\end{subarray}}(n^{2s}+k^{2s})|c_{n,k}|^{2}<\infty\right\\}.$ (2)

We recall that if $s\geq 2$, then $X^{s}$ is a closed subspace of
$H^{s}(\mathbf{T}^{2})$ whose members are real-valued continuous functions. In
fact, the space $X^{s}$ is a real Hilbert space whose inner product is given
by

$\langle
u,v\rangle_{X^{s}}=\int_{\mathbf{T}^{2}}D_{t}^{s}u(t,x)D_{t}^{s}v(t,x)dtdx+\int_{\mathbf{T}^{2}}D_{x}^{s}u(t,x)D_{x}^{s}v(t,x)dtdx$

with corresponding norm given by $\|u\|_{X^{s}}^{s}=\langle
u,u\rangle_{X^{s}}$. We point out that this norm is equivalent to

$\sum_{\begin{subarray}{c}n\in\mathbf{Z}\\\ k\geq
1\end{subarray}}(n^{2s}+k^{2s})|c_{n,k}|^{2}.$

As $k\geq 1$ in the index of summation in the definition of $X^{s}$ and its
norm, we find that this norm is also equivalent to the norm in
$H^{s}(\mathbf{T}^{2})$ for members of $X^{s}$.

We point out that if $u\in X^{s}$, then $u(t,x)$ can be written in the form

$u(t,x)=\sum_{k\geq 1}c_{k}\sin(kx)+\sum_{\begin{subarray}{c}n\geq 1\\\ k\geq
1\end{subarray}}\Big{(}d_{n,k}\cos(nt)\sin(kx)+e_{n,k}\sin(nt)\sin(kx)\Big{)},$

where the Fourier coefficients $c_{k},d_{n,k},e_{n,k}\in\mathbf{R}$.

####  2.1.1. Sobolev Embedding and Banach Algebra

With this notation, we have the following Sobolev embedding theorem: If $k\geq
0$ and $s>k+1$ then $H^{s}(\mathbf{T}^{2})\subset C^{k}(\mathbf{T}^{2})$ and
$\|u\|_{C^{k}}\lesssim\|u\|_{H^{s}}$ and hence the same holds for $X^{s}$ in
place of $H^{s}$. Further if $s>1$ then $H^{s}(\mathbf{T}^{2})$ is a Banach
algebra via multiplication of functions, which is inherited by space $X^{s}$.
That is, for $v,w\in X^{s}$

$\|vw\|_{X^{s}}\leq 2^{2s}\|v\|_{X^{s}}\|w\|_{X^{s}}.$ (3)

##  3\. Motivating Calculations and Periodic Solution Ansatz

We seek a solution of (1) which is periodic in $t$. As in [10, 11, 12] we
begin with the ansatz for the Lyapunov-Schmidt reduction in $X^{s}$. To find a
$\frac{2\pi}{\omega}$-periodic solution of (1) it suffices to find a
$2\pi$-periodic solution of the following problem:

$\left\\{\begin{array}[]{ll}\omega^{2}\partial_{t}^{2}u^{\omega}-\partial_{x}^{2}u^{\omega}-\omega\alpha\partial_{t}\partial_{x}^{2}u^{\omega}+mu^{\omega}=\omega^{2p+1}\big{(}\partial_{t}u^{\omega}\big{)}^{2p+1}\\\
u^{\omega}(t,0)=u^{\omega}(t,\pi)=0\end{array}\right.$ (4)

(We require, of course, that $\omega>0$.) In the sequel we will omit the
$\omega$ superscript and write $u$ for the solution of (4).

That having been said in order to prove Theorem 1, it suffices to prove the
following auxiliary result:

######  Theorem 2 (Small Damping Bifurcation).

Let $k_{0}\geq 2$ be given. Then there is some $M=M(p,m,k_{0})>0$ so that for
each $\rho\in(0,M)$ there is some $\alpha\gtrsim_{m,p,k_{0}}\rho^{2p}$ and
$\omega\approx\sqrt{1+m}$ so that (4) has a $C^{k_{0}}$ in $(t,x)$ $2\pi$-time
periodic solution of the form

$u(t,x)=\rho\cos(t)\sin(x)+O(\rho^{2p+\frac{1}{2}}).$

Upon linearizing (4) we look for solutions that have the form

$u(t,x)=\sum_{n\in\mathbf{Z}}\sum_{k\geq 1}c_{n,k}e^{int}\sin(kx).$ (5)

Because the solution is $\mathbf{R}$-valued the coefficients $c_{n,k}$ satisfy
$\overline{c_{n,k}}=c_{-n,k}$. Inserting the ansatz (5) into the linearized
equation (around the zero steady state $u=0$) yields

$-n^{2}\omega^{2}+k^{2}+i\alpha\omega nk^{2}+m=0.$

In taking real and imaginary parts in this equation we find the following
system of equations:

$\displaystyle-n^{2}\omega^{2}+k^{2}+m$ $\displaystyle=0$ (6)
$\displaystyle\alpha\omega nk^{2}$ $\displaystyle=0.$ (7)

Notice that (7) is automatically satisfied if $\alpha=0$. Turning to (6) we
see that if we take $n=\pm 1,k=1$, then $\omega=\sqrt{1+m}$. It follows that
if $m>0$ is taken to be irrational, then $\omega_{1}:=\sqrt{1+m}$ is
irrational. With this in mind we fix $m>0$ to be irrational .

Consider the linear operator

$L_{\omega,\alpha}u:=\omega^{2}\partial_{t}^{2}u-\partial_{x}^{2}u-\omega\alpha\partial_{t}\partial_{x}^{2}u+mu.$
(8)

With this notation we find that (4) can be written

$L_{\omega,\alpha}u=\omega^{2p+1}\big{(}\partial_{t}u\big{)}^{2p+1}.$

######  Lemma 3.1.

The kernel of $L_{\omega_{1},0}$ in the space $X^{s}$ is two-dimensional:

$\ker L_{\omega_{1},0}=\textnormal{Span}\\{e^{it}\sin(x),e^{-it}\sin(x)\\}.$

###### Proof.

A function $u(t,x)$ of the form (5) satisfies $L_{\omega_{1},0}u=0$ if and
only if $m-n^{2}\omega_{1}^{2}+k^{2}=0$. Upon solving for $\omega_{1}^{2}$ and
recalling that $\omega_{1}=\sqrt{1+m}$ we find that

$\frac{m+k^{2}}{n^{2}}=1+m.$

This can be rearranged to read $k^{2}-n^{2}=m(n^{2}-1)$. Since
$m\in(\mathbf{R}\setminus\mathbf{Q})^{+}$ we must have $n=\pm 1$ and, in that
case, $k=1$. ∎

We write $X^{s}=\ker L_{\omega_{1},0}\oplus V^{s}$, the space $V^{s}$ being
the orthogonal complement of $\ker L_{\omega_{1},0}$ (with respect to the
$L^{2}$ inner product). Taking $A\subset\mathbf{Z}\times\mathbf{N}$ to be the
set

$A:=\\{(n,k)\in\mathbf{Z}\times\mathbf{N}\ |\ (n,k)\neq(\pm 1,1)\\},$

we see that $u\in V^{s}$ if

$u(t,x)=\sum_{(n,k)\in A}c_{n,k}e^{int}\sin(kx).$

Consider the inverse operator $L_{\omega,\alpha}^{-1}$. Formally we have

$L_{\omega,\alpha}^{-1}f(t,x)=\sum_{(n,k)\in
A}\frac{f_{n,k}}{k^{2}+m-\omega^{2}n^{2}+i\omega\alpha
nk^{2}}e^{int}\sin(kx).$

It is natural that we take the domain of $L_{\omega,\alpha}^{-1}$ to be

$\mathscr{D}^{s}(L_{\omega,\alpha}^{-1})=\\{f\in X^{s}\ |\ \textnormal{supp
}\widehat{f}\subseteq A\\}=V^{s},$

where $\textnormal{supp }\widehat{f}$ denotes the support of the Fourier
transform of $f$.

We set

$\vartheta(n,k,m,\omega,\alpha):=k^{2}+m-\omega^{2}n^{2}+i\omega\alpha
nk^{2},$

so that

$L_{\omega,\alpha}^{-1}f(t,x)=\sum_{(n,k)\in
A}\frac{f_{n,k}}{\vartheta(n,k,m,\omega,\alpha)}e^{int}\sin(kx).$ (9)

Note that

$|\vartheta(n,k,m,\omega,\alpha)|^{2}=(k^{2}-\omega^{2}n^{2}+m)^{2}+\omega^{2}\alpha^{2}n^{2}k^{4}.$
(10)

In Section 4 below we will write our solution $u(t,x)$ of (4) as

$u(t,x)=\frac{\rho}{2}(e^{it}+e^{-it})\sin(x)+v(t,x),$

where $\rho\in\mathbf{R}$ and $v\in V^{s}$. The details of this decomposition
will be discussed further below.

It is at this stage that we make the following key assumptions regarding the
values of the parameters.

######  Assumption 1.

Recall that we assume $m>0$ is irrational. There exist $0<W_{0}<W_{1}$ such
that $W_{0}<\omega_{1}=\sqrt{1+m}<W_{1}$ and $1<W_{0}^{2}\leq\omega^{2}\leq
W_{1}^{2}<m+2$.

######  Assumption 2.

The parameter $\rho\in(0,1)$ satisfies
$0<W_{0}^{2p}\theta\rho^{2p}/4\leq\alpha$, where $\theta=\theta(p)>0$ is a
constant (depending only on $p$) that will be determined below.

######  Assumption 3.

Given $k_{0}\geq 2$ from Theorem 2, we assume that the Sobolev index

$s\in[k_{0}+10,k_{0}+20].$

We conclude this section with the following result concerning the inverse
operator $L_{\omega,\alpha}^{-1}$ which we will require in the next section.
In what follows we will often use the notation $\Pi_{k>K}$ to denote the
projection onto Fourier modes for which $k>K$.

######  Lemma 3.2.

Let $s\geq 2$.

$L_{\omega,\alpha}:V^{s}\to V^{s-3},$ (11)

while

$L_{\omega,\alpha}^{-1}:V^{s}\to V^{s+1}.$ (12)

Let $m>0$ be irrational. Then under Assumptions 1 and 2 above, if
$\rho\in(0,1)$ is sufficiently small (depending only on $m,p$) we have the
bound

$\|L_{\omega,\alpha}^{-1}\|_{\mathcal{L}(V^{s},V^{s+1})}\lesssim\frac{1}{\rho^{2p}}.$
(13)

where the implicit constant only depends on $m,p$.

Furthermore, let $K$ be any positive integer. Under the same assumptions we
also have the bound

$\|L_{\omega,\alpha}^{-1}\Pi_{k>K}\|_{\mathcal{L}(\Pi_{k>K}V^{s},V^{s+1})}\lesssim\frac{1}{K^{2}\rho^{2p}}+2.$
(14)

where the implicit constant still only depends on $m,p$.

###### Proof.

We will show the last bound since the second one will follow similarly. Note
that if $v\in V^{s}$ then we have by the expansion (9)

$\|L_{\omega,\alpha}^{-1}v\|^{2}_{V^{s+1}}=\sum_{(n,k>K)\in
A}\frac{(n^{2(s+1)}+k^{2(s+1)})|v_{n,k}|^{2}}{|\vartheta(n,k,m,\omega,\alpha)|^{2}}$

and hence it suffices to show that the quantities

$\frac{n^{2}}{|\vartheta(n,k,m,\omega,\alpha)|^{2}}\quad\text{and}\quad\frac{k^{2}}{|\vartheta(n,k,m,\omega,\alpha)|^{2}}.$

are bounded by an appropriate constant depending only on $\rho,p,$ and $m$.

Recall that

$|\vartheta(n,k,m,\omega,\alpha)|^{2}=(k^{2}-\omega^{2}n^{2}+m)^{2}+\omega^{2}\alpha^{2}k^{4}n^{2}.$

We first establish the desired bounds on

$\frac{n^{2}}{|\vartheta(n,k,m,\omega,\alpha)|^{2}}$

If $n=0$ there is nothing to prove. Otherwise $|n|>0$ and we first note that

$\frac{n^{2}}{|\vartheta(n,k,m,\omega,\alpha)|^{2}}\leq\frac{n^{2}}{\omega^{2}\alpha^{2}k^{4}n^{2}}=\frac{1}{\omega^{2}k^{4}\alpha^{2}}<\frac{1}{K^{4}\alpha^{2}}\lesssim_{m,p}\frac{1}{K^{4}\rho^{4p}}$
(15)

where we used Assumptions 1 and 2.

It remains to establish a bound on

$\frac{k^{2}}{|\vartheta(n,k,m,\omega,\alpha)|^{2}}.$

First suppose that

$\frac{k^{4}}{(k^{2}-\omega^{2}n^{2}+m)^{2}}\leq 4.$

As $k^{2}\leq k^{4}$ we are done, since

$\frac{k^{2}}{|\vartheta(n,k,m,\omega,\alpha)|^{2}}\leq\frac{k^{2}}{(k^{2}-\omega^{2}n^{2}+m)^{2}}\leq\frac{k^{4}}{(k^{2}-\omega^{2}n^{2}+m)^{2}}.$

On the other hand if

$\frac{k^{4}}{(k^{2}-\omega^{2}n^{2}+m)^{2}}>4$

then

$k^{2}>2|k^{2}+m-\omega^{2}n^{2}|\geq 2k^{2}+2m-2W_{1}^{2}n^{2}$

and hence

$k^{2}\leq 2W_{1}^{2}n^{2}-2m\leq 2(2+m)n^{2}-2m<2(2+m)n^{2},$

where we used that $W_{1}^{2}<m+2$ from Assumption 1.

It follows that in this case

$\frac{k^{2}}{|\vartheta(n,k,m,\omega,\alpha)|^{2}}<\frac{2(2+m)n^{2}}{|\vartheta(n,k,m,\omega,\alpha)|^{2}}\lesssim_{m,p}\frac{1}{K^{4}\rho^{4p}},$

where the last part follows from (15). Hence

$\displaystyle\frac{n^{2}}{|\vartheta(n,k,m,\omega,\alpha)|^{2}}\quad,\quad\frac{k^{2}}{|\vartheta(n,k,m,\omega,\alpha)|^{2}}\leq\max\left\\{4,C(m,p)\frac{1}{K^{4}\rho^{4p}}\right\\}$
(16)

for some constant $C(m,p)>0$ and (14) follows. We note that to obtain (13) we
can repeat the same argument but omit the addition of 2 by taking
$\rho\in(0,1)$ sufficiently small depending only on $m,p$ so that the maximum
of the two quantities in (16) is not 4. The reason we did not do this above
for (14) is to avoid the dependence of $\rho$ on $K$. ∎

##  4\. The Lyapunov-Schmidt Reduction

As in Section 3 we write $X^{s}=\ker L_{\omega_{1},0}\oplus V^{s}$, the
orthogonal sum being with respect to the $L^{2}(\mathbf{T}^{2})$ inner
product. Denote by

$\Pi_{\omega_{1},0}:X^{s}\to\ker
L_{\omega_{1},0}\qquad\text{and}\qquad\Pi_{V^{s}}:X^{s}\to V^{s},$

the relevant projection operators. If $u(t,x)\in X^{s}$ solves (4), then we
write

$u(t,x)=\frac{\rho}{2}(e^{it}+e^{-it})\sin(x)+v(t,x),$

where $v\in V^{s}$ and $\rho>0$ is a real amplitude parameter. As the
projection operators commute with $L_{\omega,\alpha}$ we obtain the system

$\displaystyle L_{\omega,\alpha}\rho\cos(t)\sin(x)$
$\displaystyle=\Pi_{\omega_{1},0}\omega^{2p+1}\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v(t,x)]\Big{)}^{2p+1}$
(17) $\displaystyle L_{\omega,\alpha}v(t,x)$
$\displaystyle=\Pi_{V^{s-1}}\omega^{2p+1}\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v(t,x)]\Big{)}^{2p+1}.$
(18)

Equation (18) is referred to as the range equation while equation (17) is
called the bifurcation equation.

###  4.1. Solution of the Range Equation

The Implicit Function Theorem is not applicable as in the usual Hopf
Bifurcation Theorem because when the damping $\alpha=0$ there is a genuine
small divisor problem since the numbers $k^{2}+m-\omega^{2}n^{2}$ accumulate
to zero. The novelty of this work is to show that this singularity can be
removed owing to a certain integral vanishing (see Lemma 4.1), in addition to
having a suitable lower bound for $\alpha$ in terms of $\rho$ (see Assumption
2), and also exploiting the regularizing effect of the Laplacian damping. The
removal of this singularity invokes neither Diophantine conditions on the
temporal frequency $\omega$, nor any Nash-Moser iteration. We can accomplish
this by solving the range equation using the contraction mapping principle.
This is a surprising fact since the solutions we will find have damping
parameter which accumulates to $\alpha=0$.

Note that (18) can be written as

$v=L_{\omega,\alpha}^{-1}\Pi_{V^{s-1}}\omega^{2p+1}\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v]\Big{)}^{2p+1}.$
(19)

We begin with some Lemmas we will need to solve the range equation (19).

######  Lemma 4.1.

There is an integer $K_{p}\geq 1$ such that for each $k>K_{p}$ we have

$\int_{\mathbf{T}}\sin^{2p+1}(x)\sin(kx)dx=0.$

###### Proof.

Using the formula

$\sin^{2p+1}(x)\sin(kx)=i2^{-2p-2}(e^{-ikx}-e^{ikx})i^{2p+1}(e^{-ix}-e^{ix})^{2p+1}$

as well as the Binomial Theorem to expand the term $(e^{-ix}-e^{ix})^{2p+1}$
as a sum of exponentials multiplied by constants, it is not difficult to see
that we obtain the desired result as long as $k>2p+1$ since the exponents in
the exponentials do not vanish, and using periodic boundary conditions. ∎

In what follows we will often use the notation $\Pi_{k>K_{p}}$ to denote the
projection onto Fourier modes for which $k>K_{p}$.

######  Lemma 4.2.

We have that

$\Pi_{k>K_{p}}\Pi_{V^{s}}\sin^{2p+1}(t)\sin^{2p+1}(x)=0.$

###### Proof.

For the first part, as a consequence of Lemma 4.1 we find that if $k>K_{p}$,
then for any $n\in\mathbf{Z}$

$\displaystyle\int_{\mathbf{T}^{2}}\sin^{2p+1}(t)\sin^{2p+1}(x)e^{int}\sin(kx)dtdx$
$\displaystyle=$
$\displaystyle\left(\int_{\mathbf{T}}e^{int}\sin^{2p+1}(t)dt\right)\left(\int_{\mathbf{T}}\sin^{2p+1}(x)\sin(kx)dx\right)=0,$

where we have used Fubini’s Theorem to factor the integrals. It follows that

$\Pi_{k>K_{p}}\Pi_{V^{s}}\sin^{2p+1}(t)\sin^{2p+1}(x)=0.$

∎

For $(n,k)\in A$, we define

$Q(n,k)=\frac{n^{2}}{(k^{2}+m-\omega^{2}n^{2})^{2}}\quad\text{and}\quad
R(n,k)=\frac{k^{2}}{(k^{2}+m-\omega^{2}n^{2})^{2}}.$ (20)

######  Lemma 4.3.

Let $K>K_{p}$ be a positive integer and let $(n,k)\in A$ with $k\leq K$. Then
provided $0<W_{1}^{2}-W_{0}^{2}$ is small enough (depending on $K,m,p$) there
is a constant $C(K,m,p)$ which is independent of $\rho$ such that

$Q(n,k),R(n,k)\leq C(K,m,p).$

###### Proof.

We begin by focusing on the bound for $Q(n,k)$. If $n=0$ there is nothing to
prove. Otherwise if $n\neq 0$ and

$n^{2}\geq\left(\frac{K^{2}+1+m}{W_{0}^{2}}\right)|n|$

then

$\frac{\omega^{2}n^{2}-(k^{2}+m)}{|n|}\geq\frac{W_{0}^{2}n^{2}-(K^{2}+m)}{|n|}\geq(K^{2}+1+m)-\frac{(K^{2}+m)}{|n|}\geq
1$

Hence

$\frac{n^{2}}{(\omega^{2}n^{2}-(k^{2}+m))^{2}}\leq 1.$

On the other hand if $n\neq 0$ and

$n^{2}<\left(\frac{K^{2}+1+m}{W_{0}^{2}}\right)|n|$

hence by Assumption 1 since $W_{0}>1$,

$|n|<\left(\frac{K^{2}+1+m}{W_{0}^{2}}\right)<K^{2}+1+m.$

It follows both $|n|,k$ are bounded above by constants which only depend on
$K,m,p$; that is $1\leq|n|,k\leq A(K,m,p)$ for some $A(K,m,p)>0$. In this case
we can choose $W^{2}_{1}-W^{2}_{0}>0$ small enough so that

$|\omega^{2}-(1+m)|<\min_{|n|,k\leq
A(K,m,p)}\frac{1}{2}\left|\frac{k^{2}+m}{n^{2}}-(1+m)\right|$

Then it is easily seen that

$|\omega^{2}n^{2}-(k^{2}+m)|\geq\min_{|n|,k\leq
A(K,m,p)}\frac{1}{2}\left|\frac{k^{2}+m}{n^{2}}-(1+m)\right|>0,$

where the last part is positive since $m$ is assumed irrational. Hence the
quantities $Q(n,k)$ are bounded, above by a constant depending only on $m,p$.
Lastly, the statement concerning the quantities $R(n,k)$ follows because
$k^{2}$ is bounded and we just showed the denominators are bounded away from
zero. ∎

######  Lemma 4.4.

Let $K>K_{p}$ be given. Then provided $0<W_{1}^{2}-W_{0}^{2}$ is small enough
(depending on $K,m,p$) there is some $C(K,m,p)>0$ independent of $\rho$ such
that

$\displaystyle\|L_{\omega,\alpha}^{-1}\Pi_{k\leq K}\|_{\mathcal{L}(\Pi_{k\leq
K}V^{s-1},V^{s})}$ $\displaystyle\leq C(K,m,p).$

That is, the inverse restricted to the said region is bounded independently of
$\rho$.

###### Proof.

This follows from Lemma 4.3 (recall also the beginning of the proof of Lemma
3.2). ∎

Our strategy will be to decompose the range equation according to the spatial
frequency $k$, thereby enabling us to obtain a contraction. Let

$\mathcal{A}(v)=L_{\omega,\alpha}^{-1}\Pi_{V^{s-1}}\omega^{2p+1}\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v(t,x)]\Big{)}^{2p+1}.$
(21)

We begin with a proposition which establishes a bound for the above nonlinear
map at zero.

######  Proposition 4.1.

If $0<W_{1}^{2}-W_{0}^{2}$ is small enough (depending on $m,p$) then there is
some constant $C(p,m,k_{0})>0$ depending only on $p,m,k_{0}$ so that

$\|\mathcal{A}(0)\|_{V^{s}}\leq C(p,m,k_{0})\rho^{2p+1}.$

###### Proof.

We let $K_{p}$ be the parameter from Lemma 4.2 and $K=K_{p}+1$. We decompose
$\mathcal{A}$ as follows:

$\mathcal{A}(v)=\mathcal{A}_{\leq K}(v)+\mathcal{A}_{>K}(v),$ (22)

where

$\displaystyle\mathcal{A}_{\leq K}(v)$
$\displaystyle=L_{\omega,\alpha}^{-1}\Pi_{k\leq
K}\Pi_{V^{s-1}}\omega^{2p+1}\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v(t,x)]\Big{)}^{2p+1},$
$\displaystyle\mathcal{A}_{>K}(v)$
$\displaystyle=L_{\omega,\alpha}^{-1}\Pi_{k>K}\Pi_{V^{s-1}}\omega^{2p+1}\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v(t,x)]\Big{)}^{2p+1}.$

In particular

$\displaystyle\|\mathcal{A}(0)\|_{V^{s}}$
$\displaystyle\leq\|\mathcal{A}_{\leq
K}(0)\|_{V^{s}}+\|\mathcal{A}_{>K}(0)\|_{V^{s}}$
$\displaystyle=\|L_{\omega,\alpha}^{-1}\Pi_{k\leq
K}\Pi_{V^{s-1}}\omega^{2p+1}\Big{(}-\rho\sin(t)\sin(x)\Big{)}^{2p+1}\|_{V^{s}}$
$\displaystyle+\|L_{\omega,\alpha}^{-1}\Pi_{k>K}\Pi_{V^{s-1}}\omega^{2p+1}\Big{(}-\rho\sin(t)\sin(x)\Big{)}^{2p+1}\|_{V^{s}}$
$\displaystyle=\|L_{\omega,\alpha}^{-1}\Pi_{k\leq
K}\Pi_{V^{s-1}}\omega^{2p+1}\Big{(}-\rho\sin(t)\sin(x)\Big{)}^{2p+1}\|_{V^{s}}$
$\displaystyle\leq C(p,m,k_{0})\rho^{2p+1}$

having used Lemmas 4.2, 4.4, the Banach algebra property 3, and Assumptions 1
and 3.

∎

######  Proposition 4.2.

Let $\rho>0$ and $0<W_{1}^{2}-W_{0}^{2}$ be sufficiently small (depending on
$p,m,k_{0}$). Then

$\mathcal{A}:V^{s}\to
V^{s}\qquad\text{and}\qquad\mathcal{A}:\mathscr{B}_{V^{s}}(0,\rho^{2p+\frac{1}{2}})\to\mathscr{B}_{V^{s}}(0,\rho^{2p+\frac{1}{2}}).$

Moreover, if $\rho$ is sufficiently small (depending on $p,m,k_{0},$) and
$v,w\in\mathscr{B}_{V^{s}}(0,\rho^{2p+\frac{1}{2}})$, then

$\|\mathcal{A}(v)-\mathcal{A}(w)\|_{V^{s}}\leq\frac{1}{2}\|v-w\|_{V^{s}}.$
(23)

###### Proof.

From Lemma 3.2 it follows that $\mathcal{A}:V^{s}\to V^{s}$. We first prove
(23). Note that from 21 we have

$D\mathcal{A}(v)\zeta=L_{\omega,\alpha}^{-1}\Pi_{V^{s-1}}\left[\omega^{2p+1}(2p+1)\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v]\Big{)}^{2p}\partial_{t}\zeta\right]$

It suffices to show that for
$v\in\mathscr{B}_{V^{s}}(0,\rho^{2p+\frac{1}{2}})$,
$\|D\mathcal{A}(v)\|_{\mathcal{L}(V^{s},V^{s})}\leq\frac{1}{2}$, whence the
contraction estimate holds thanks to the Mean Value Inequality. Let $K$ be a
positive integer which will be chosen later.

$\displaystyle D\mathcal{A}(v)\zeta$
$\displaystyle=L_{\omega,\alpha}^{-1}\Pi_{V^{s-1}}\left[\omega^{2p+1}(2p+1)\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v]\Big{)}^{2p}\partial_{t}\zeta\right]$
$\displaystyle=L_{\omega,\alpha}^{-1}\Pi_{k\leq
K}\Pi_{V^{s-1}}\left[\omega^{2p+1}(2p+1)\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v]\Big{)}^{2p}\partial_{t}\zeta\right]$
$\displaystyle+L_{\omega,\alpha}^{-1}\Pi_{k>K}\Pi_{V^{s-1}}\left[\omega^{2p+1}(2p+1)\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v]\Big{)}^{2p}\partial_{t}\zeta\right]$
$\displaystyle=(D\mathcal{A}(v)\zeta)_{k\leq K}+(D\mathcal{A}(v)\zeta)_{k>K},$

where the terms of the last line are defined in the corresponding way by the
terms from the line preceding it.

If $v\in\mathscr{B}_{V^{s}}(0,\rho^{2p+\frac{1}{2}})$ then for any $\zeta\in
V^{s}$ using (3), Lemma 3.2, Assumptions 1 and 3, as well as
$\|\partial_{t}\zeta\|_{V^{s-1}}\lesssim\|\zeta\|_{V^{s}}$, we have

$\displaystyle\|(D\mathcal{A}(v)\zeta)_{k>K}\|_{V^{s}}\leq$ $\displaystyle
C(p,m,k_{0})(\frac{1}{K^{2}\rho^{2p}}+2)\left\|\Pi_{V^{s-1}}\left[\Big{(}-\rho\sin(t)\sin(x)+\partial_{t}v\Big{)}^{2p}\partial_{t}\zeta\right]\right\|_{V^{s-1}}$
$\displaystyle\leq
C(p,m,k_{0})\frac{1}{K^{2}\rho^{2p}}\left\|(-\rho\sin(t)\sin(x)+\partial_{t}v\Big{)}^{2p}\right\|_{V^{s-1}}\left\|\partial_{t}\zeta\right\|_{V^{s-1}}$
$\displaystyle+2C(p,m,k_{0})\left\|(-\rho\sin(t)\sin(x)+\partial_{t}v\Big{)}^{2p}\right\|_{V^{s-1}}\left\|\partial_{t}\zeta\right\|_{V^{s-1}}$
$\displaystyle\leq
C(p,m,k_{0})\frac{\rho^{2p}}{K^{2}\rho^{2p}}\left\|\zeta\right\|_{V^{s}}+2C(p,m,k_{0})\rho^{2p}\left\|\zeta\right\|_{V^{s}}$
$\displaystyle=$ $\displaystyle
C(p,m,k_{0})\frac{1}{K^{2}}\left\|\zeta\right\|_{V^{s}}+2C(p,m,k_{0})\rho^{2p}\left\|\zeta\right\|_{V^{s}}.$

######  Remark 1.

The constants $C$ above may change from line to line but only depend on the
parameters in the parentheses.

Hence we can take $K$ to be large enough (depending only on $p,m,k_{0}$) so
that $C(p,m,k_{0})\frac{1}{K^{2}}\leq\frac{1}{8}$ as well as $\rho\in(0,1)$
small enough (depending on $p,m,k_{0}$ to ensure
$2C(p,m,k_{0})\rho^{2p}\leq\frac{1}{8}$.

Having selected $K=K(p,m,k_{0})$, next we have by Lemma 4.4, using
$v\in\mathscr{B}_{V^{s}}(0,\rho^{2p+\frac{1}{2}})$, (3), Assumptions 1 and 3,
and $\|\partial_{t}\zeta\|_{V^{s-1}}\lesssim\|\zeta\|_{V^{s}}$ ,

$\displaystyle\|(D\mathcal{A}(v)\zeta)_{k\leq K}\|_{V^{s}}\leq
C(p,m,k_{0})\left\|\Pi_{V^{s-1}}\left[\Big{(}-\rho\sin(t)\sin(x)+\partial_{t}v\Big{)}^{2p}\partial_{t}\zeta\right]\right\|_{V^{s-1}}$
$\displaystyle\leq C(p,m,k_{0})\rho^{2p}\left\|\zeta\right\|_{V^{s}}.$

We choose $\rho\in(0,1)$ (depending only on $p,m,k_{0}$) small enough so that
$C(p,m,k_{0})\rho^{2p}\leq\frac{1}{4}$. It follows that

$\sup_{v\in\mathscr{B}_{V^{s}}(0,\rho^{2p+\frac{1}{2}})}\|D\mathcal{A}(v)\|_{\mathcal{L}(V^{s},V^{s})}\leq\frac{1}{2}.$

Finally, to complete the first part of the proof suppose that
$v\in\mathscr{B}_{V^{s}}(0,\rho^{2p+\frac{1}{2}})$ . We first observe from
Proposition 4.1 that

$\|\mathcal{A}(0)\|_{V^{s}}\leq C(m,p,k_{0})\rho^{2p+1}$

and hence using the above we have

$\displaystyle\|\mathcal{A}(v)\|_{V^{s}}\leq\|\mathcal{A}(v)-\mathcal{A}(0)\|_{V^{s}}+\|\mathcal{A}(0)\|_{V^{s}}$
$\displaystyle\leq\frac{1}{2}\|v\|_{V^{s}}+\|\mathcal{A}(0)\|_{V^{s}}\leq\frac{1}{2}\rho^{2p+\frac{1}{2}}+C(m,p,k_{0})\rho^{2p+1}\leq\rho^{2p+\frac{1}{2}}$

provided

$C(m,p,k_{0})\rho^{\frac{1}{2}}\leq\frac{1}{2}$

which can be accomplished if $\rho\in(0,1)$ is chosen small enough, depending
only on $m,p,k_{0}$.

∎

Altogether we are lead to the following result.

######  Proposition 4.3.

Under Assumptions 1, 2, and 3 if $\rho>0$ and $0<W_{1}^{2}-W_{0}^{2}$ are
sufficiently small (depending on $p,m,k_{0}$) the range equation (18) (as well
as (19)) has a solution $v(\rho,\omega,\alpha)$ satisfying

$\|v\|_{V^{s}}\leq\rho^{2p+\frac{1}{2}}.$ (24)

Using Proposition 4.2 we see that the map $\mathcal{A}$ is a contraction on
$\mathscr{B}_{V^{s}}(0,\rho^{2p+\frac{1}{2}})$, and hence has a fixed point
inside this ball which is a solution of the range equations (18), (19) which
clearly depends on $(\rho,\omega,\alpha)$.

Next we derive estimates on the derivatives of the nonlinear map defined in
(21) as well as the solution of the range equation granted above, which will
be handy for solving the bifurcation equations downstream.

######  Lemma 4.5.

Let $v$ be the solution of the range equation granted in Proposition 4.3.
Omitting the dependence on $\rho,\omega$ for the sake of brevity we have that
under Assumptions 1, 2, and 3 if $\rho>0$ and $0<W_{1}^{2}-W_{0}^{2}$ are
sufficiently small (depending on $p,m,k_{0}$) then

$\Big{\|}\partial_{\alpha}\mathcal{A}(\alpha,v)\Big{|}_{v=v(\alpha)}\Big{\|}_{V^{s}}\leq
C(p,m,k_{0})\rho^{\frac{1}{2}}$

for some $C(p,m,k_{0})>0$ where $\mathcal{A}$ is given in (21), but we call
out the dependence on $\alpha$ explicitly here.

###### Proof.

We see that for general $v\in V^{s}$

$\displaystyle\partial_{\alpha}\mathcal{A}(\alpha,v)=(\partial_{\alpha}L_{\omega,\alpha}^{-1})\Pi_{V^{s-1}}\omega^{2p+1}\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v(t,x)]\Big{)}^{2p+1}$
(25)

and it is not difficult to see that for any $v\in V^{s}$

$\displaystyle\partial_{\alpha}L_{\omega,\alpha}^{-1}v=\sum_{n,k}\frac{-i\omega
nk^{2}}{(k^{2}+m-\omega^{2}n^{2}+i\omega n\alpha k^{2})^{2}}v_{nk}.$

See the appendix of [12] for details. We establish the following properties:

$\mbox{if}\ v\in V^{s}\ \mbox{then}\
\partial_{\alpha}L_{\omega,\alpha}^{-1}v\in
V^{s+1},\;\mbox{and}\;\|\partial_{\alpha}L_{\omega,\alpha}^{-1}\|_{\mathcal{L}(V^{s},V^{s+1})}\leq
C(p,m,k_{0})\frac{1}{W_{0}\rho^{4p}}$

To see this it suffices to show that the quantities

$\displaystyle
n^{2}\cdot\frac{\omega^{2}n^{2}k^{4}}{((k^{2}+m-\omega^{2}n^{2})^{2}+(\omega
n\alpha
k^{2})^{2})^{2}},k^{2}\cdot\frac{\omega^{2}n^{2}k^{4}}{((k^{2}+m-\omega^{2}n^{2})^{2}+(\omega
n\alpha k^{2})^{2})^{2}}$ (26)

are bounded by an appropriate constant. It is easily checked that for both we
have the upper bound $\frac{1}{\omega^{2}\alpha^{4}}$ from which the second
property above follows from Assumption 2 (and taking the square root).

Finally, by (25)

$\displaystyle\partial_{\alpha}\mathcal{A}(\alpha,v)$ $\displaystyle=$
$\displaystyle(\partial_{\alpha}L_{\omega,\alpha}^{-1})\Pi_{V^{s-1}}\omega^{2p+1}\Big{(}(-\rho\sin(t)\sin(x))^{2p+1}$
$\displaystyle\qquad\qquad\qquad\quad\qquad+\sum_{j=1}^{2p+1}\binom{2p+1}{j}(-\rho\sin(t)\sin(x))^{2p+1-j}(\partial_{t}v)^{j}\Big{)}$
$\displaystyle=(\partial_{\alpha}L_{\omega,\alpha}^{-1})\Pi_{V^{s-1}}\omega^{2p+1}(-\rho\sin(t)\sin(x))^{2p+1}$
$\displaystyle+(\partial_{\alpha}L_{\omega,\alpha}^{-1})\Pi_{V^{s-1}}\omega^{2p+1}\Big{(}\sum_{j=1}^{2p+1}\binom{2p+1}{j}(-\rho\sin(t)\sin(x))^{2p+1-j}(\partial_{t}v)^{j}\Big{)}$
$\displaystyle=:(I)+(II)$

For the first term $(I)$ we see that by Lemma 4.2 we see that we need to only
consider at most $K_{p}+1$ modes in $k$, that is,

$(I)=(\partial_{\alpha}L_{\omega,\alpha}^{-1})\Pi_{k\leq
K_{p}+1}\Pi_{V^{s-1}}\omega^{2p+1}(-\rho\sin(t)\sin(x))^{2p+1}.$

To obtain a bound for this we observe that the quantities in 26 are in fact
bounded by $D(p,m)Q(n,k)^{2},D(p,m)R(n,k)^{2}$ respectively for some constant
$D(p,m)>0$, where $Q(n,k),R(n,k)$ are the rational expressions defined in (20)
and the latter are, by Lemma 4.3, bounded by a constant depending only on
$p,m$. This means that $\|(I)\|_{V^{s}}\leq D(p,m)\rho^{2p+1}$ for some
possibly different $D(p,m)>0$.

Turning to $(II)$ and recalling we are interested in the case $v=v(\alpha)$ we
have that since $\|v(\alpha)\|_{V^{s}}\leq\rho^{2p+\frac{1}{2}}$

$\displaystyle\Big{\|}(\partial_{\alpha}L_{\omega,\alpha}^{-1})\Pi_{V^{s-1}}\omega^{2p+1}\Big{(}\sum_{j=1}^{2p+1}\binom{2p+1}{j}(-\rho\sin(t)\sin(x))^{2p+1-j}(\partial_{t}v)^{j}\Big{)}\Big{\|}_{V^{s}}$
$\displaystyle\leq
E(p,m,k_{0})\frac{1}{\rho^{4p}}\sum_{j=1}^{2p+1}\rho^{2p+1-j}(\rho^{2p+\frac{1}{2}})^{j}\leq
E(p,m,k_{0})\rho^{\frac{1}{2}}$

for some $E(p,m,k_{0})>0$. Hence

$\Big{\|}\partial_{\alpha}\mathcal{A}(\alpha,v)\Big{|}_{v=v(\alpha)}\Big{\|}_{V^{s}}\leq
D(p,m)\rho^{2p+1}+E(p,m,k_{0})\rho^{\frac{1}{2}}\leq
2E(p,m,k_{0})\rho^{\frac{1}{2}}$

giving the desired result if $\rho\in(0,1)$ is small enough depending only on
$p,m,k_{0}$. ∎

######  Proposition 4.4.

Omitting the dependence on $\rho,\omega$ for the sake of brevity we have that
under Assumptions 1, 2, and 3 if $\rho>0$ and $0<W_{1}^{2}-W_{0}^{2}$ are
sufficiently small (depending on $p,m,k_{0}$) then
$\|\partial_{\alpha}v(\alpha)\|_{V^{s}}\leq C(p,m,k_{0})\rho^{\frac{1}{2}}$

###### Proof.

We have

$v(\alpha)-\mathcal{A}(\alpha,v(\alpha))=0,$

where $\mathcal{A}$ is given in (21) but we call out the dependence on
$\alpha$ explicitly here. We remark that the invertibility of

$\partial_{v}\Big{(}v-\mathcal{A}(\alpha,v)\Big{)}\Big{|}_{v=v(\alpha)}=\Big{(}I_{V^{s}}-\partial_{v}\mathcal{A}(v)\Big{)}\Big{|}_{v=v(\alpha)}$

and the fact that

$\Big{\|}\Big{(}I_{V^{s}}-\partial_{v}\mathcal{A}(v)\Big{)}^{-1}\Big{|}_{v=v(\alpha)}\Big{\|}_{\mathcal{L}(V^{s},V^{s})}\leq
2$

follows from a Neumann series argument using the estimate (24) on $v$, the
same argument as in the first part of the proof of Proposition 4.2 (where we
showed there that
$\left\|\partial_{v}\mathcal{A}(v)\Big{|}_{v=v(\alpha)}\right\|_{\mathcal{L}(V^{s},V^{s})}\leq\frac{1}{2})$,
and taking $\rho>0$ to be sufficiently small.

Hence it follows that

$\displaystyle\partial_{\alpha}v(\alpha)=-\Big{(}I_{V^{s}}-\partial_{v}\mathcal{A}(v)\Big{)}^{-1}\Big{|}_{v=v(\alpha)}\partial_{\alpha}\mathcal{A}(\alpha,v)\Big{|}_{v=v(\alpha)}\implies$
$\displaystyle\|\partial_{\alpha}v\|_{V^{s}}\leq
2\Big{\|}\partial_{\alpha}\mathcal{A}(\alpha,v)\Big{|}_{v=v(\alpha)}\Big{\|}_{V^{s}}\leq
C(p,m,k_{0})\rho^{\frac{1}{2}}$

for some $C(p,m,k_{0})>0$ by Lemma 4.5. ∎

###  4.2. Solution of the Bifurcation Equation

To solve the bifurcation equation (17) we first reformulate the problem as a
pair of equations. The left side of (17) can be calculated explicitly:

$L_{\omega,\alpha}\rho\cos(t)\sin(x)=(1-\omega^{2}+m)\rho\cos(t)\sin(x)-\omega\alpha\rho\sin(t)\sin(x).$
(27)

One can also rewrite the right side of (17) using Lemma 3.1:

$\displaystyle\omega^{2p+1}\Pi_{\omega_{1},0}\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v]\Big{)}^{2p+1}$
(28) $\displaystyle=$ $\displaystyle
A(\rho,\omega,\alpha)\sin(t)\sin(x)+B(\rho,\omega,\alpha)\cos(t)\sin(x),$

where

$A(\rho,\omega,\alpha)=\omega^{2p+1}\int_{\mathbf{T}^{2}}\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v]\Big{)}^{2p+1}\sin(t)\sin(x)dtdx,$

and

$B(\rho,\omega,\alpha)=\omega^{2p+1}\int_{\mathbf{T}^{2}}\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v]\Big{)}^{2p+1}\cos(t)\sin(x)dtdx.$

Equating coefficients in (27) and (28) yields a pair of equations:

$\displaystyle-\omega\alpha\rho$
$\displaystyle=\omega^{2p+1}\int_{\mathbf{T}^{2}}\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v]\Big{)}^{2p+1}\sin(t)\sin(x)dtdx$
(29) $\displaystyle(m+1-\omega^{2})\rho$
$\displaystyle=\omega^{2p+1}\int_{\mathbf{T}^{2}}\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v]\Big{)}^{2p+1}\cos(t)\sin(x)dtdx.$
(30)

We divide both of these equations by $\rho$; we also divide (29) by $\omega$:

$\displaystyle-\alpha$
$\displaystyle=\frac{\omega^{2p}}{\rho}\int_{\mathbf{T}^{2}}\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v]\Big{)}^{2p+1}\sin(t)\sin(x)dtdx$
(31) $\displaystyle(m+1-\omega^{2})$
$\displaystyle=\frac{\omega^{2p+1}}{\rho}\int_{\mathbf{T}^{2}}\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v]\Big{)}^{2p+1}\cos(t)\sin(x)dtdx.$
(32)

Before proceeding further we observe that

$\Big{(}\partial_{t}[\rho\cos(t)\sin(x)+v]\Big{)}^{2p+1}=-\rho^{2p+1}\sin^{2p+1}(t)\sin^{2p+1}(x)+F(\rho,t,x),$

where

$F(\rho,t,x)=\sum_{j=1}^{2p+1}\binom{2p+1}{j}\Big{(}-\rho\sin(t)\sin(x)\Big{)}^{2p+1-j}(v_{t})^{j}.$
(33)

Returning to (31) and (32) we have

$\displaystyle-\alpha$
$\displaystyle=-\omega^{2p}\rho^{2p}\int_{\mathbf{T}^{2}}\sin^{2p+2}(t)\sin^{2p+2}(x)dtdx$
$\displaystyle\quad+\frac{\omega^{2p}}{\rho}\int_{\mathbf{T}^{2}}F(\rho,t,x)\sin(t)\sin(x)dtdx,$
and $\displaystyle m+1-\omega^{2}$
$\displaystyle=-\omega^{2p+1}\rho^{2p}\int_{\mathbf{T}^{2}}\sin^{2p+1}(t)\cos(t)\sin^{2p+2}(x)dtdx$
$\displaystyle\quad+\frac{\omega^{2p+1}}{\rho}\int_{\mathbf{T}^{2}}F(\rho,t,x)\cos(t)\sin(x)dtdx.$

Observe that

$\int_{\mathbf{T}^{2}}\sin^{2p+1}(t)\cos(t)\sin^{2p+2}(x)dtdx=0$

and let

$\theta=\theta(p)=\int_{\mathbf{T}^{2}}\sin^{2p+2}(t)\sin^{2p+2}(x)dtdx=\left(\binom{2p+1}{p}\frac{\pi}{2^{2p}}\right)^{2}.$

It now follows that the bifurcation equations can be written as

$\displaystyle\alpha$
$\displaystyle=\omega^{2p}\theta\rho^{2p}-\frac{\omega^{2p}}{\rho}\int_{\mathbf{T}^{2}}F(\rho,t,x)\sin(t)\sin(x)dtdx$
(34) $\displaystyle m+1-\omega^{2}$
$\displaystyle=\frac{\omega^{2p+1}}{\rho}\int_{\mathbf{T}^{2}}F(\rho,t,x)\cos(t)\sin(x)dtdx.$
(35)

We take the solution $v(\rho,\omega,\alpha)$ granted by Proposition 4.3 and
substitute it into the bifurcation equations (34), (35) and note that the
dependence on $v$ is through $F$.

####  4.2.1. Estimates on $F$

It will be worthwhile to have the following estimates related to the function
$F$ defined in (33):

######  Lemma 4.6.

Let $v=v(\rho,\omega,\alpha)$ denote the solution of (18) obtained in
Proposition 4.3. We have that

$\frac{|F(\rho,t,x)|}{\rho}\lesssim\rho^{2p+1}$

where the implicit constant only depends on $k_{0},m,p$.

###### Proof.

Using Proposition 4.3 we have that
$\|v\|_{V^{s}}\leq\rho^{2p+\frac{1}{2}}\leq\rho^{2}$

hence

$\displaystyle\left\|F(\rho,t,x)\right\|_{V^{s-1}}$
$\displaystyle=\left\|\sum_{j=1}^{2p+1}\binom{2p+1}{j}(-\rho\sin(t)\sin(x))^{2p+1-j}(v_{t})^{j}\right\|_{V^{s-1}}$
$\displaystyle\lesssim\sum_{j=1}^{2p+1}\binom{2p+1}{j}\rho^{2p+1-j}({\rho^{2}})^{j}=\sum_{j=1}^{2p+1}\binom{2p+1}{j}\rho^{2p+1+j}$
$\displaystyle\lesssim\rho^{2p+2}.$

Hence,

$\frac{\|F(\rho,t,x)\|_{V^{s-1}}}{\rho}\lesssim\rho^{2p+1}$

and the implicit constant depends only on $k_{0},p,m$. ∎

We are now equipped to solve the bifurcation equations (34) and (35).

######  Proposition 4.5.

Let $v=v(\rho,\omega,\alpha)$ denote the solution of (18) obtained in
Proposition 4.3. Let $\rho>0$ be sufficiently small; in particular we require
that $\rho$ satisfy the inequality (36) below. Then the bifurcation equation
(34) has a solution $\alpha(\omega,\rho)\gtrsim\rho^{2p}>0$, where the
implicit constant depends only on $k_{0},m,p$. Furthermore, for fixed $\rho>0$
sufficiently small, the map

$[W_{0},W_{1}]\ni\omega\mapsto\alpha(\omega,\rho)\in(0,\infty)$

is continuous. The bifurcation equation (35) has a solution
$\omega(\rho)\in[W_{0},W_{1}]$. Finally, we have that $\alpha\downarrow 0$ and
$\omega^{2}\to\omega_{1}^{2}=m+1$ as $\rho\downarrow 0$.

###### Proof.

Beginning with (34) we let

$\mathcal{B}(\alpha):=\omega^{2p}\theta\rho^{2p}-\frac{\omega^{2p}}{\rho}\int_{\mathbf{T}^{2}}F(\rho,t,x)\sin(t)\sin(x)dtdx.$

Using the fact that $\theta>0$ along with Lemma 4.6 we have that

$\mathcal{B}(\alpha)\leq|\mathcal{B}(\alpha)|\leq
W_{1}^{2p}\theta\rho^{2p}+C_{k_{0},m,p}W_{1}^{2p}\rho^{2p+1}.$

On the other hand,

$\displaystyle\mathcal{B}(\alpha)$ $\displaystyle\geq
W_{0}^{2p}\theta\rho^{2p}-C_{k_{0},m,p}W_{1}^{2p}\rho^{2p+1}$
$\displaystyle=\rho^{2p}(W_{0}^{2p}\theta-C_{k_{0},m,p}W_{1}^{2p}\rho)$
$\displaystyle\geq\frac{1}{2}W_{0}^{2p}\theta\rho^{2p},$

where the last inequality holds provided

$0<\rho\leq\frac{1}{2C_{k_{0},m,p}}\left(\frac{W_{0}}{W_{1}}\right)^{2p}\theta.$
(36)

Hence provided $\rho$ satisfies (36), we see that

$\mathcal{B}(\alpha)\in
I=\left[\frac{1}{2}W_{0}^{2p}\theta\rho^{2p},W_{1}^{2p}\theta\rho^{2p}+C_{k_{0},m,p}W_{1}^{2p}\rho^{2p+1}\right],$

for some constant $C_{k_{0},m,p}>0$ depending only on $k_{0},m,p$. We conclude
that $\mathcal{B}(I)\subset I$.

######  Remark 2.

Assumption 2 implies that the domain of $\mathcal{B}$ is
$[\frac{1}{4}W_{0}^{2p}\theta\rho^{2p},\infty)\supset I$. Had we changed the
exponent in $\rho$ in Assumption 2, we would not necessarily be able to
conclude that $I$ is contained in the domain of $\mathcal{B}$ and hence would
not get a fixed point from these estimates.

We will show that actually $\mathcal{B}$ is a contraction on $I$. To see this
by Proposition 4.4 we have (using that in particular
$\|v(\alpha)\|_{V^{s}}\leq\rho^{2}$)

$\displaystyle\left|\partial_{\alpha}\mathcal{B(\alpha)}\right|\lesssim\frac{1}{\rho}\sum_{j=1}^{2p+1}\binom{2p+1}{j}\left\|(-\rho\sin(t)\sin(x))^{2p+1-j}j(v(\alpha)_{t})^{j-1}\partial_{\alpha}v(\alpha)_{t}\right\|_{V^{s-1}}$
$\displaystyle\lesssim\frac{1}{\rho}\sum_{j=1}^{2p+1}\binom{2p+1}{j}\rho^{2p+1-j}j(\rho^{2})^{j-1}\rho^{\frac{1}{2}}\lesssim\rho^{2p-\frac{1}{2}},$

so it follows that $\left|\partial_{\alpha}\mathcal{B(\alpha)}\right|$ can be
made smaller than $\frac{1}{2}$ as long as $\rho\in(0,1)$ is small enough
depending only on $p,m,k_{0}$ meaning $\mathcal{B}$ is a contraction. This
grants us a unique fixed point for $\mathcal{B}$ on $I$. Finally we insert the
solution $\alpha\in I$ of (34) into the bifurcation equation (35). For
convenience we will solve the latter for $\omega^{2}$ so we define

$\mathcal{G}(\xi):=m+1-\frac{\xi^{\frac{2p+1}{2}}}{\rho}\int_{\mathbf{T}^{2}}F(\rho,t,x)\cos(t)\sin(x)dtdx,$

where $\xi=\omega^{2}$. As above we note that if $\xi\in[W_{0}^{2},W_{1}^{2}]$
then using 4.6

$\mathcal{G}(\xi)\leq|\mathcal{G}(\xi)|\leq
m+1+D_{k_{0},m,p}W_{1}^{2p+1}\rho^{2p+1}\leq W_{1}^{2}$

if

$0<\rho^{2p+1}\lesssim\frac{W_{1}^{2}-(m+1)}{W_{1}^{2p+1}},$ (37)

where we note that the right hand side of the above inequality is positive
since $W_{1}^{2}>m+1$. Furthermore,

$\mathcal{G}(\xi)\geq m+1-D_{k_{0},m,p}W_{1}^{2p+1}\rho^{2p+1}\geq W_{0}^{2}$

if

$0<\rho^{2p+1}\leq\frac{m+1-W_{0}^{2}}{D_{k_{0},m,p}W_{1}^{2p+1}},$ (38)

for some constant $D_{k_{0},m,p}>0$. Note again that the right hand side above
is positive since $m+1>W_{0}^{2}$. The continuity of $\mathcal{G}$ on
$[W_{0}^{2},W_{1}^{2}]$ follows from the fact that $\alpha(\rho,\omega)$ is
continuous in $\omega$, which is a consequence of the fact that $\mathcal{B}$
is a contraction (see Corollary 5.2.2 of [7]), and we conclude the existence a
fixed point of $\mathcal{G}$ and hence a solution
$\omega(\rho)=\sqrt{\xi}\in[W_{0},W_{1}]$ of (35) by the Brouwer fixed point
theorem.

The final statement concerning the limits is clear for the bounds we obtained
here and their dependence on $\rho$. ∎

This completes the proof of Theorem 2 and hence also Theorem 1. We note that
the statement concerning $C^{k_{0}}$ regularity of the solutions follows from
Sobolev embedding since, in particular, $s>k_{0}+1$.

##  5\. Acknowledgments

The authors would like to thank the anonymous referee for a careful reading
and helpful tips to improve the exposition.

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