# CORHEL-CME: An Interactive Tool For Modeling Solar Eruptions

Jon A. Linker  Tibor Torok  Cooper Downs  Ronald M. Caplan  Viacheslav Titov
Andres Reyes  Roberto Lionello  & Pete Riley  Predictive Science Inc., 9990
Mesa Rim Road Suite 170, San Diego, CA 92121
linkerj@predsci.com,tibor@predsci.com,cdowns<EMAIL_ADDRESS>titovv@predsci.com,areyes@predsci.com,lionel<EMAIL_ADDRESS>
###### Abstract

Coronal Mass Ejections (CMEs) are immense eruptions of plasma and magnetic
fields that are propelled outward from the Sun, sometimes with velocities
greater than 2000 km/s. They are responsible for some of the most severe space
weather at Earth, including geomagnetic storms and solar energetic particle
(SEP) events. We have developed CORHEL-CME, an interactive tool that allows
non-expert users to routinely model multiple CMEs in a realistic coronal and
heliospheric environment. The tool features a web-based user interface that
allows the user to select a time period of interest, and employs RBSL flux
ropes to create stable and unstable pre-eruptive configurations within a
background global magnetic field. The properties of these configurations can
first be explored in a zero-beta magnetohydrodynamic (MHD) model, followed by
complete CME simulations in thermodynamic MHD, with propagation out to 1 AU.
We describe design features of the interface and computations, including the
innovations required to efficiently compute results on practical timescales
with moderate computational resources. CORHEL-CME is now implemented at NASA’s
Community Coordinated Modeling Center (CCMC) using NASA Amazon Web Services
(AWS). It will be available to the public by the time this paper is published.

##  1 Introduction

Coronal Mass Ejections (CMEs) are magnificent displays of solar activity that
propel plasma and magnetic fields outward into the solar wind. In addition to
the inherent scientific interest of such phenomena, CMEs play a crucial role
in space weather at Earth (e.g., [1]). When the magnetic fields associated
with CMEs contain significant southward pointed fields (oppositely directed to
the magnetic field at the Earth’s magnetopause, i.e., negative $B_{z}$
fields), they can reconnect with the Earth’s magnetic field and peel away the
protective magnetopause, an essential trigger to geomagnetic activity.
Therefore, the magnetic structure of CMEs is of vital interest. The
compression regions and shock waves created by these events also play an
important role in the acceleration of solar energetic particles (SEPs) [2].

CMEs have been observed for over four decades, and while a basic picture of
their magnetic structure has emerged, fundamental questions remain. One of the
key reasons why these questions remain unresolved is that it is difficult to
measure the magnetic field in the corona. CMEs produce ancillary signatures in
a range of wavelengths that give important clues to the underlying structure
of CMEs, such as EUV waves [3], EUV dimming signatures [4], and the three-part
structure of CMEs observed in white light [5]. In order to reveal the physical
(e.g., magnetic) structure responsible for these observed features, a model
must include enough physics to reproduce observations for real events.
Thermodynamic magnetohydrodynamic (MHD) models of the solar corona, which
incorporate the relevant energy transport processes, have advanced to the
point that they can predict EUV/X-ray emission [6, 7, 8, 9, 10, 11, 12], as
well as scattering of white light observed in coronagraphs and at total solar
eclipses, for direct comparison with observations.

Figure 1: Framework diagram for CORHEL-CME.

Thermodynamic MHD models are now computed relatively routinely using the
Magnetohydrodynamic Algorithm outside a Sphere (MAS) code within the CORona-
HELiospheric (CORHEL) modeling suite. However, it is still very challenging to
model CME events, especially the most violent ones. The largest and fastest
CMEs usually originate in localized active regions (ARs) on the Sun. In
addition to modeling the global corona, the detailed structure of AR must also
be resolved. To allow non-expert users to simulate CMEs in a realistic coronal
and heliospheric environment, we have developed CORHEL-CME, which provides
MAS/CORHEL zero-beta and thermodynamic MHD simulations using observed
photospheric magnetic fields as boundary conditions. Fig. 1 shows a diagram of
the main steps in CORHEL-CME. In this paper, we describe the software
components of CORHEL-CME, the work flow for creating CME simulations, some of
the computational shortcuts employed to make the CME simulations feasible on
relatively modest computer hardware, and the standard diagnostics that are
output as part of a run. CORHEL-CME is now implemented at NASA’s Community
Coordinated Modeling Center (CCMC) using NASA Amazon Web Services (AWS).

##  2 Components of CORHEL-CME

Realistically simulating CMEs for a given event requires several key elements.
The background corona and solar wind must be simulated for the time period of
interest; this in turn requires solar magnetic data. This data is typically in
the form of full-Sun maps of the line-of-sight magnetic field measured in the
photosphere, from which the radial magnetic field is inferred. Major CME
events typically originate in solar active regions (ARs) where the magnetic
fields are highly energized (energy above the potential state), and so a pre-
eruptive configuration must be developed (also matching the photospheric field
measurements). To accomplish these tasks, CORHEL-CME employs the CORHEL
modeling suite (described next), and models of pre-eruptive configurations
based on analytical flux rope models (described in section 2.2.1). Parameter
specification for CORHEL-CME is made via a Web interface (section 2.3.

###  2.1 CORHEL

Advanced simulations of CMEs require a background model of the solar corona
and solar wind, applicable to the time of the event that is being studied.
CORHEL is a suite of coupled models and tools for describing the corona and
solar wind, developed by Predictive Science Inc. (PSI). It has been delivered
to the CCMC and AFRL, and solutions are available at PSI’s website
(www.predsci.com). The primary input data to CORHEL are synoptic maps of the
radial photospheric magnetic field, $B_{r}$, from observatories such as the
National Solar Observatories (NSO), SOLIS and GONG, and the Helioseismic and
Magnetic Imager (HMI) aboard the Solar Dynamics Observatory (SDO). An
essential aspect of CORHEL’s modeling approach is to divide the coronal and
heliospheric domains, and to use different codes and approximations in each
domain. The outer boundary for the coronal solutions is placed beyond the
Alfvén and sound speed critical points, at 20–30 solar radii ($R_{S}$). The
coronal solutions are used to drive the heliospheric solutions (extending to
$\gtrsim$ 230$R_{S}$), which can be computed in either the inertial frame or a
frame co-rotating with the Sun [13]. This approach has been applied to model
the propagation of observed CMEs from the low corona to 1 AU [11]. The MHD
runs use MAS—a high-performance code developed over several decades for
studying the structure and dynamics of the global corona and inner heliosphere
[14, 15, 16, 17, 18, 19, 8]. MAS can be run on massively parallel CPU systems
using MPI, and has been ported to GPUs using OpenACC [20] and Fortran standard
parallelism [21]. In thermodynamic MHD mode, MAS incorporates a realistic
energy equation; accounting for anisotropic thermal conduction, radiative
losses, and coronal heating. This allows plasma densities and temperatures to
be computed with sufficient accuracy to simulate extreme ultraviolet (EUV) and
X-ray emission observed from space [6]. The equations solved in MAS are shown
in Appendix A of [11]. The more recent versions of MAS employ a Wave-
Turbulence-Driven (WTD) approach for coronal heating and solar wind
acceleration; [22, 10] describe the additional equations solved. To rapidly
explore the stability and eruptive behavior of energized configurations,
CORHEL-CME also employs the simpler $\beta=0$ model (solution of the momentum
equation and Faraday’s law with plasma pressure = 0, e.g.,[23]), prior to
employing the more computationally intensive thermodynamic MHD model.

###  2.2 CME Modeling

MHD simulations are a powerful tool for studying CMEs. Versions of the MAS
code have been employed in state-of-the-art CME models for many years [23, 24,
25, 26, 27, 28, 13, 11, 29]. Such simulations are crucial for understanding
CME initiation and evolution. Traditionally they have been time-consuming to
develop and require significant expertise. The goal of CORHEL-CME is to
provide routine CME models that can be used by the broader scientific
community. The tool has been designed to be relatively straightforward to use,
but it also incorporates enough details to produce sufficiently accurate
results, including proper modeling of the pre-eruptive state and early
eruption phase. The latter is particularly important if the simulations are to
be used in modeling of Solar Particle Events (SPEs), since compressional
fronts, formed by the expansion of the CME, can accelerate particles at
heights as low as 1.3 $R_{\odot}$ in the corona [30]. A key component of CME
simulations is specifying a pre-eruptive configuration, described in the next
section. CORHEL-CME is designed to allow the user to first investigate
eruptions with the $\beta=0$ model, and, when the initial eruption properties
are satisfactory, perform full Sun-to-1 AU simulations of the CME with the
thermodynamic MHD model.

####  2.2.1 Pre-Eruptive Configurations

CORHEL-CME allows users to interactively perform routine simulations of CMEs
based on realistic pre-eruptive conditions. The TDM (Titov-Démoulin modified)
model [31] is an extension of the TD model [32], which has been widely used to
study pre-eruptive magnetic configurations (e.g., [33])

and as the initial condition for CME simulations (e.g., [34, 35, 36]). The TD
model is an approximately force-free equilibrium consisting of a toroidal MFR
(partly submerged below the photosphere) and a stabilizing (‘strapping’)
potential field produced by a fictitious pair of sub-photospheric monopoles.
The TDm model employs the same MFR geometry, but was designed such that the
MFR can be embedded into an arbitrary, locally bipolar potential field. This
facilitates the modeling of CMEs whose source regions have a relatively simple
magnetic structure [29]. An earlier version of our tool, CORHEL-MAS-TDM, is
available for runs on request at NASA CCMC
(https://ccmc.gsfc.nasa.gov/models/CORHEL~MAS-TDM~6.0) and provides $\beta=0$
modeling of CMEs in realistic solar magnetic fields. To overcome the
limitation of toroidal geometry, CORHEL-CME uses the RBSL (Regularized Biot-
Savart Laws) model [37, 38]. This model contains an MFR of arbitrary shape,
facilitating the modeling of complex CME source regions that have elongated
and curved polarity inversion lines (PILs). Fig. 2 shows a schematic of the
model and its application to the “Bastille Day” eruption that occurred on
7/14/2000. Both CORHEL-MAS-TDM and CORHEL-CME allow the user to insert the
MFRs into a given background field without altering the observed magnetogram.

###  2.3 Web Interface

CORHEL-CME is delivered as two software packages: One package is a GUI-based
web interface, the other is the CORHEL modeling suite. The web interface was
developed with a client-server model using Django/Python and JavaScript. It is
installed on a public-facing machine (a NASA AWS instance in the CCMC
implementation) and allows users to interact with the interface via a web
browser, eliminating the need for any local installation by users. The
interface hides many details of CME modeling, allowing users to focus on a few
free parameters that characterize pre-eruptive MFRs and their eruption. Once
all parameters are selected, CORHEL-CME creates two archive files for
download. One file contains all information needed to repeat or modify a
previous interface session; the other contains the files and scripts needed to
run a simulation with CORHEL/MAS. The optimum specifics of the simulation
(number of cores and grid points, computational mesh, run-time of the
simulation) are determined automatically.

The CORHEL modeling suite is installed on the computer where the MHD
computations are to be performed. In the CCMC implementation, the archive
files are automatically transferred to the computational node (at the present
time, an 8-GPU NASA AWS Instance). The CORHEL software contains scripts and
macros for automatic diagnostics, which are provided as a self-contained,
interactive html page upon completion of the run (section 5).

##  3 CORHEL-CME Workflow

Figure 2:  Main steps of modeling a CME with CORHEL-CME. See text for details.
Figure 3:  Navigation page of the CORHEL-CME web interface. The user is guided
through the process via various steps that include magnetogram and source-
region selection (Step 1), RBSL flux-rope configuration (Step 2), Set-up of
background models (Step 3) and final CME simulation (Step 4).

Fig. 2 visually illustrates the workflow for creating a CME simulation with
CORHEl-CME. The user is guided through these steps via a navigation page,
shown in Fig. 3. First (STEP 1), the user selects the date and time of the CME
to be modeled (Fig. 2(a)). The user is provided with choices for an
appropriate surface magnetic map for the relevant Carrington rotation. The
user can also select an existing coronal and heliospheric background solution
at this point, if such a solution is available in the database. In that case,
the surface map from that solution is loaded and used in the subsequent steps.
After the data source is selected, it is then automatically downloaded from
the corresponding data provider, and the user selects the location of the CME
source region (typically an active region) on the map, as shown in Fig. 2(b)).

Next (STEP 2), the user designs an RBSL flux rope that is inserted into the
CME source region and provides the free magnetic energy required to power the
CME (see Fig. 2(c)). This step includes (i) the selection of a segment of the
polarity inversion line (PIL) along which the initial central section of the
flux-rope axis shall be placed at an arbitrary height above the solar surface,
(ii) selecting the locations of the rope footprints and its thickness, (iii)
further fine-tuning of the axis geometry. The user can modify the axis path,
guided by a side-view of the strapping-field contours, as shown in Fig. 2(d).
Once all these parameters are chosen, a downloadable tar file is produced that
contains all the necessary information and commands to run a zero-beta
simulation on a supercomputer. In the CCMC implementation, this file is
automatically transferred to the compute server to perform the run. The user
can then perform computationally inexpensive $\beta=0$ MHD runs with varying
field strengths to explore the stability of the MFR, to find the threshold for
eruption, and to explore the initial CME trajectory. These runs can typically
be completed in 10-30 minutes of wall clock time on multi-core or GPU systems.
As interacting CMEs are an important phenomena, CORHEL-CME allows a user to
implement multiple eruptions in multiple ARs, by proceeding through the steps
repetitively. Performing these types of calculations can greatly increase the
computational requirements, so the CCMC implementation presently limits this
capability to two events in a single run.

STEP 3 provides the capability to produce a background solution for the corona
and inner heliosphere (technically, these are two separate calculations).
These solutions are computed for individual Carrington rotations, and are
required for the Sun-to-1 AU CME simulation (Step 4). These background
calculations are intentionally performed with a relatively low spatial
resolution, and are later re-meshed to high resolution for CME runs (see
section 4). In the present version, the user can choose between two empirical
approaches for coronal heating (similar to [6]), which are briefly explained
in the corresponding interface tool tips. We expect this to be updated with
the WTD heating model (see section 2.1) in the near future. The heliospheric
background calculation is fully automated and requires no user input. After
the background run finishes, the background solution (both corona and
heliospheric) is added to the database, which allows subsequent users to
employ it in future calculations.

Finally (STEP 4), the user can perform a full Sun-to-1 AU CME run. This
requires a background solution for the Carrington rotation (corresponding to
the time when the CME occurred), to be present in the CORHEL-CME background-
solution database. The user will then select this solution during Step 1. If
no such solution is present, the user has to perform Step 3 first before
he/she can proceed further. Once a background solution is selected, the user
continues to Step 2 to either construct a new flux rope or to “confirm” a
previously constructed one (which can be loaded into the interface by
restoring the corresponding session), after which he/she can directly go to
Step 4 and start the CME simulation. No further input is required in Step 4.
Upon completion of each of these runs (zero-beta, background, CME), a tar file
that contains an extensive run report (see Section 5) is automatically
generated and can be downloaded by the user.

##  4 Efficiency Considerations

Figure 4: Flow chart showing the run pipeline for modeling a single Sun-to-1
AU CME in the corona and inner heliosphere. This corresponds to STEP 3 and
STEP 4 in Fig. 3.

An important goal for CORHEL-CME is to allow for real-world calculations with
moderate computational resources (presently implemented on an 8-GPU card NASA
AWS instance at CCMC). This requires optimizing the required steps in the
calculations. The sequence of MHD calculations performed, referred to as the
run pipeline, is depicted in Fig. 4.

Global coronal simulations require days of physical time to approximate the
corona, but only demand a few hours of computational time if ARs are resolved
coarsely. On the other hand, detailed models of the pre-event energized
magnetic structure in the AR require high resolution to account for specific
local features, where the magnetic field is very strong. The local physical
relaxation time in the AR is short (minutes to hours) because the Alfvén speed
is very high in these magnetically dominated regions, These considerations
have necessitated performing different steps of the CORHEl-CME pipeline with
different resolutions and flexibly transitioning computed solutions between
meshes of different resolution. We interpolate the simulation variables with
piece-wise monotonic splines (PCHIP) and analytic integral preserving
functions for the transition from one mesh to the other (a special procedure
is used to obtain the vector potential $\mathbf{A}$, see Appendix A). We refer
to this as re-meshing.

Figure 5:  Latitude-longitude map of $B_{r}$ boundary for low resolution
background simulation of the ambient corona around the time of the July 12,
2012 CME, together with the high resolution active region that would be
inserted and re-meshed for a CORHEL-CME simulation.

Re-meshing is used at several points in the run pipeline, as shown in Fig. 4
(a time t = 1 corresponds to 24 minutes). For example, we compute the
background simulation for the time period of interest at low resolution, to
minimize run time and data storage. If we performed the background simulation
at the resolution required to resolve important features of the AR magnetic
field, it greatly increases the computational requirements. Instead, we
interpolate our low-resolution simulation onto a higher-resolution grid, and
insert the resolved active region into this background solution, prior to
incorporating the RBSL flux rope (section 2.2.1). (The $\beta=0$ flux-rope
simulations are performed on the high resolution mesh.) This procedure is
depicted in Fig. 5. We have found that, in practice, the local perturbations
introduced by this insertion resolve rapidly and do not greatly perturb the
global coronal solution. There is minimal effects on the subsequent
propagation of the CME, as compared to a simulation that is performed entirely
at the higher resolution. We note that this re-meshing procedure differs from
adaptive-mesh refinement approaches, because the boundary condition is
modified; we are effectively solving a mathematically different problem from
the one we started with. We are relying on the physical effect that this
change is local to a region and does not greatly influence the global
solution. We re-mesh again to an intermediate resolution after the initial
eruption when the CME has strongly expanded, to compute the propagation of the
CME until it fully exits the outer domain boundary of the coronal calculation
at 30 $R_{\odot}$. The subsequent evolution is then followed in the
heliospheric calculation, which is also performed more efficiently as a
separate calculation.

##  5 Diagnostics Report

Numerical simulations of solar eruptions are typically difficult to interpret,
especially if they are performed with the thermodynamic MHD model. CORHEL-CME
incorporates the Psiweb Report Generator to create a self-contained
interactive web report (as an HTML page), for a quick look and assessment of
the MHD simulations. The report generator sets up working directories for each
report, gathers report parameters from configuration files, and launches PSI’s
post-processing tools written in Python, Bash, and Fortran. After all analyses
are performed for a given report, a self-contained web document is generated
using MkDocs (see https://www.mkdocs.org/). Plotly’s javascript library (see
https://plotly.com/javascript/) is used for three-dimensional (3D) field line
exploration, while the 2D slices and emission imagery pages are driven with
custom Javascript. The report is automatically produced and tarred for
download when a CORHEL-CME run has finished.

Figure 6:  Field lines displayed by the CORHEL-CME interactive visualization
tool, showing two flux ropes in the coronal simulation of the January 23, 2012
double-CME event. The plasma density (scaled by the distance $r^{2}$ from the
Sun) in the subsequent interplanetary simulation, shown in the equatorial
plane from an interactive animation in the run report.

Fig. 6 shows excerpts from a report created for a simulation of the double CME
event of January 23, 2012. The report provides a drop-down menu so users can
choose the set of variables animated on a given page (Fig. 6 (a).) Movies of
EUV and X-ray emission, as well as polarization brightness (pB) as would be
observed by different space-based instruments can be viewed from different
vantage points in the “emission” page (Fig. 6(b)). The field-line tool (Fig.
6(c)) allows users to plot different sets of field lines. The user can rotate
the plot, zoom into an area, and follow the evolution of the field lines via a
time slider. The helio report (Fig. 6(d)) provides an interactive animation of
various physical quantities in the equatorial plane (30–230 $R_{\odot}$). Time
series comparisons of the results with OMNI data can be obtained at the in
situ page.

After completion of a run, the full MHD data from the simulation can also be
obtained to perform more detailed analysis.

##  6 Summary

CORHEL-CME allows users to simulate CMEs for observed events, in coronal and
heliospheric environments modeled with observed photospheric magnetic fields.
A GUI interface allows users to develop pre-eruptive configurations, and
investigate stable flux ropes and eruptions in $\beta=0$ and full
thermodynamic MHD. After completion of a run, a downloadable diagnostics
report is created with quick-look, interactive visualizations. CORHEL-CME has
been designed to operate with moderate computational resources, to allow more
routine investigation of CME events. More detailed simulations can
subsequently be performed, if greater computational power is available. This
may be necessary for some types of scientific investigations. CORHEL-CME is
currently undergoing testing at NASA CCMC, implemented on NASA AWS. It will be
operational by the time of publication of this paper. In the future, we will
combine CORHEL-CME with the SPE Threat Assessment Tool (STAT, [39, 40]) to
allow users to simulate solar particle events.

## Appendix A

The MAS MHD code solves for Faraday’s law (advancement of the magnetic field),
using the vector potential $\mathbf{A}$ ($\mathbf{B}=\nabla\times\mathbf{A}$).
In MAS, differential operators are formed on staggered meshes to ensure that
standard vector identities (e.g $\nabla\cdot\nabla\times=0$) are preserved to
round-off error. Among other advantages, this enforces
$\nabla\cdot\mathbf{B}=0$ to machine precision in the calculations. In re-
meshing our solutions, the resampling of $\mathbf{A}$ from coarse to fine
meshes (or vice-versa), can produce unacceptable errors, even when higher-
order, smooth methods like PCHIP are used. This occurs because the current
density $\mathbf{J}=\nabla\times(\nabla\times\mathbf{B})$ (normalized units),
and the errors in $\mathbf{J}$ directly impact the balance of forces. To
obtain the re-meshed $\mathbf{A^{RME}}$ with sufficient accuracy, we perform
the following steps:

  1. 1.

Remesh the original boundary field $B_{r0}$ onto the new mesh, solve to obtain
the potential field $\mathbf{B^{pot0}}$.

  2. 2.

Remesh to obtain the original $\mathbf{B}$ in the domain: $\mathbf{B^{RM0}}$.

  3. 3.

Obtain the nonpotential part of the original $\mathbf{B}$ in the domain:
$\mathbf{B^{RME}}=\mathbf{B^{RM0}}-\mathbf{B^{pot0}}$.

  4. 4.

Obtain the re-meshed $\mathbf{J^{RME}}=\nabla\times\mathbf{B^{RME}}$

  5. 5.

Insert the new boundary field ($B_{r1}$) for the new mesh (e.g. insert the
high resolution $B_{r}$ at the boundary).

  6. 6.

Solve for the potential field to obtain $\mathbf{A^{pot1}}$ and
$\mathbf{B^{pot1}}$ (MAS employs a potential solver that finds the vector
potential for a $\nabla\times\mathbf{B}=0$ potential field).

  7. 7.

Utilize the following gauge choice for $\mathbf{A}$:

$\mathbf{A}=\left[\begin{array}[]{c}A_{r}\\\ A_{\theta}\\\
A_{\phi}\end{array}\right]=\nabla\times\left[\begin{array}[]{c}\psi\\\ 0\\\
0\end{array}\right]+\left[\begin{array}[]{c}A_{r}\\\ 0\\\
0\end{array}\right]=\left[\begin{array}[]{c}A_{r}\\\ \\\
\dfrac{1}{r}\left(\dfrac{1}{\sin\theta}\,\dfrac{\partial\psi}{\partial\phi}\right)\\\
\\\ -\dfrac{1}{r}\,\dfrac{\partial\psi}{\partial\theta}\end{array}\right].$
(1)

then

$\nabla^{2}_{\perp}A_{r}=-J^{{\rm
RME}}_{r};~{}~{}~{}\nabla^{2}_{\perp}\psi=-B^{{\rm RME}}_{r}.$ (2)

Equations (A.2) are series of 2D solves at each radial slice in the coronal
domain, that can be performed rapidly with preconditioned conjugate gradient
methods. This determines the nonpotential portion of $\mathbf{A}$
($\mathbf{A^{nonpot}}$). The desired
$\mathbf{A^{RME}}=\mathbf{A^{nonpot}}+\mathbf{A^{pot1}}$.

## Acknowledgments

This work was supported by AFOSR (contract FA9550-15-C-0001), the NASA SBIR
program (contract 80NSSC19C0193), the NASA Living With a Star Strategic
Capabilities program (grant 80NSSC22K0893), the NASA HTMS program (grant
80NSSC20K1274), the NSF PREEVENTS program (grant ICER1854790), the NASA HSOC
program (grant 80NSSC20K1285), NASA grant 80NSSC23K0258, and the Parker Solar
Probe WISPR contract NNG11EK11I to NRL (under subcontract N00173-19-C-2003 to
PSI). Computational resources were provided by the NASA High-End Computing
(HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames
Research Center and by the Expanse supercomputer at the San Diego
Supercomputing Center through the NSF ACCESS and XSEDE programs.

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# Poincaré compactification for semiflows of reaction-diffusion equations with
large diffusion and convection heating at the boundary

Leonardo Pires  State University of Ponta Grossa, Ponta Grossa PR, Brazil
<EMAIL_ADDRESS>
###### Abstract

In this paper, we study the Poincaré compactification of the limiting planar
semiflow of a coupled PDE-ODE system composed by a reaction-diffusion equation
with large diffusion coupled with an ODE by a boundary condition in a heating
transition region. The nonlinear sources are nondissipative polynomials. We
guarantee conditions to apply the Invariant Manifold Theorem in order to
reduce the dimension of the PDE and we prove that the compactified vector
fields are close in the $C^{1}$-norm. We give an example to present a
structurally stable problem without the global attractor.

††journal: …

Keywords: Poincaré compactification, reaction-diffusion equations, large
diffusion, convection heating at the boundary, structural stability.

MSC 2020 Classification: 34D30, 34D45, 37D15, 34C45, 35K57, 35K67.

##  1 Introduction

Reaction-diffusion equations with large diffusion and convection heating at
the boundary consist of a $\varepsilon$ parameter family of PDE-ODE coupled by
one part of the boundary and vector fields. When the parameter $\varepsilon\to
0$ the limiting problem becomes a planar ODE. In [10] was shown that the
semiflow of the PDE-ODE is a $C^{1}$ perturbation of the limiting planar
semiflow and thus, under suitable conditions, the limiting problem is
structurally stable in the sense that, for small value of the parameter
$\varepsilon$ there is a topological equivalence between the phase spaces of
the PDE-ODE and the planar ODE. The technique used in [10] was the existence
of an invariant manifold in order to reduce the problem to a two-dimensional
space and, after that, the existence of the global attractor allows to compare
the semiflows in a compact set containing all the dynamics of the problem. In
this context, the semiflows behave as Morse-Smale semiflows in a two-
dimensional compact manifold where structural stability is well-known by
results of [9].

The existence of the global attractor is essential in [10]. However, there are
some interesting situations in which the existence of the global attractor
does not take place. In this paper, we address problems whose source are
nondissipative polynomials and thus, there is no compact set attracting all
bounded sets. We used the invariant manifold technique to reduce the PDE-ODE
to a two-dimensional space identified with $\mathbb{R}^{2}$. Since the problem
now is acting in the plane we perform a Poincaré compactification in order to
compare the semiflows. This way, we are able here to obtain the structural
stability for some problems that do not fit in the assumptions of [10] due to
the absence of the global attractor.

Let $\Omega$ be a cylindrical solid body placed along the $x$-axis between
$x=0$ and $x=1$ and, let $T$ be a cylinder-shaped tank of the same radius as
$\Omega$ along the $x$-axis between $x=-1$ and $x=0$ (Figure 1). The area of
contact between $\Omega$ and $T$ at $x=0$ is denoted by $\Gamma_{0}$. The tank
$T$ is filled with liquid well stirred throughout the process. At $x=-1$
through one pipe, a liquid is entering the tank at a constant flow rate, and,
through another pipe, a liquid of temperature $v(t)$ is leaving the tank at
the same entering flow rate.

$\Gamma_{0}$$-1$$0$$1$$\Omega$$T$liquidsolid$u^{\varepsilon}(x,t)$$v(t)$
Figure 1: Domain

This type of problem was considered in [6] where it is well-known that the
symmetry of the cylindrical solid domain allows us to consider the unknown
temperature $u^{\varepsilon}(x,t)$ inside $\Omega$ depends on one spatial
variable $x\in(0,1)$. Thus, the coupled system to describe the heat transfer
through $\Gamma_{0}$ is given by the following equations

$\begin{cases}u_{t}^{\varepsilon}-(a_{\varepsilon}(x)u_{x}^{\varepsilon})_{x}+\lambda
u^{\varepsilon}=f_{1}(u^{\varepsilon}),\quad&\hbox{in
}(0,1)\times(0,\infty),\\\ u^{\varepsilon}_{x}(1,t)=0,&\hbox{in
}(0,\infty),\\\ -a_{\varepsilon}(0)u^{\varepsilon}_{x}(0,t)=g_{1}(v),&\hbox{in
}(0,\infty),\\\ \dot{v}+\beta v=f_{2}(u^{\varepsilon}(0,t))+g_{2}(v),&\hbox{in
}(0,\infty),\\\ v(0)=v_{0}=u^{\varepsilon}(x,0),&\hbox{in }(0,1),\end{cases}$
(1.1)

where $\varepsilon\in(0,\varepsilon_{0}]$, is a parameter, $\lambda$ and
$\beta$ are positive constants and the nonlinerities $f_{1},f_{2},g_{1}$ and
$g_{2}$ are polynomials such that, the maximum $d$ between the degrees of
$f_{1},f_{2},g_{1},g_{2}$ satisfies $d-2\geq 0$ and $d-1$ is greater than or
equal to degree of $g_{1}$ and $f_{1}$. As in [10] we assume that the
diffusion coefficients $a_{\varepsilon}\in C^{1}([0,1],[m_{0},\infty))$, for
some $m_{0}>0$, satisfy

$\tau(\varepsilon):=\min_{x\in[0,1]}a_{\varepsilon}(x)\to\infty\quad\hbox{as}\quad\varepsilon\to
0.$ (1.2)

The convergence (1.2) is known as large diffusion and was widely studied in
the works [1, 2, 7, 8, 11]. In order to describe our results and provide
motivation for the limiting equation $\varepsilon\to 0$, we integrate the PDE
in (1.1) to obtain

$\int_{0}^{1}u^{\varepsilon}_{t}\,dx-[a_{\varepsilon}(1)u^{\varepsilon}_{x}(1,t)-a_{\varepsilon}(0)u^{\varepsilon}_{x}(0,t)]+\int_{0}^{1}\lambda
u^{\varepsilon}\,dx=\int_{0}^{1}f_{1}(u^{\varepsilon})\,dx.$

By boundary conditions, we obtain

$\int_{0}^{1}u^{\varepsilon}_{t}\,dx+\int_{0}^{1}\lambda
u^{\varepsilon}\,dx=\int_{0}^{1}f_{1}(u^{\varepsilon})\,dx+g_{1}(v).$

Now, we use the well-known fact of [7, 8] that the large diffusion implies
spatial homogenization, that is, we must have $u^{\varepsilon}(x,t)$ converges
as $\varepsilon\to 0$ to a function $u(t)$ independent of the spatial
variable. Thus, taking $\varepsilon\to 0$ we obtain

$\dot{u}+\lambda u=f_{1}(u)+g_{1}(v).$

Adding the ODE of (1.1), we obtain in the limiting process the following
planar ODE

$\begin{cases}\dot{u}=-\lambda u+f_{1}(u)+g_{1}(v),\\\ \dot{v}=-\beta
v+f_{2}(u)+g_{2}(v).\end{cases}$ (1.3)

The objective of this paper is to show that for $\varepsilon$ sufficiently
small, the phase spaces of (1.1) and (1.3) are topologically conjugated. In
doing this, we project the semiflow of (1.1) in a two-dimensional space
$Y_{\varepsilon}$ identified with $\mathbb{R}^{2}$. We are interested in the
case of nondissipative polynomials sources, thus we do not have a compact set
in $\mathbb{R}^{2}$ to compare the semiflows satisfactorily. Hence, we perform
a compactification of the semiflows to the Poincaré disk to show the $C^{1}$
convergence of the compactified vector fields. In particular, when we choose
polynomials sources providing Morse-Smale semiflows, we obtain a structurally
stable problem.

In Section 2, we present the functional setting to deal with (1.1) and (1.3)
and we guarantee all requirements to apply the Invariant Manifold Theorem. In
Section 3, we perform a Poincaré compactification of the semiflows and we
prove the $C^{1}$ closeness of the compactified vector fields. In Section 4,
we give an example to show how our results work.

##  2 Invariant Manifold Theorem

We start writing (1.1) and (1.3) abstractly as an evolution equation in a
suitable Banach space. For this, we denote $L^{2}_{c}$ as the set of constant
functions in the usual space $L^{2}(0,1)$ and the Sobolev space $H^{1}(0,1)$.
Our phase spaces are the products $H^{1}(0,1)\times L^{2}_{c}$ and
$L^{2}(0,1)\times L^{2}_{c}$ which are closed subsets of $L^{2}(0,1)\times
L^{2}(0,1)$ with product norm.

We define for $\varepsilon\in(0,\varepsilon_{0}]$ the operator
$A_{\varepsilon}:D(A_{\varepsilon})\subset L^{2}(0,1)\times L^{2}(0,1)\to
L^{2}(0,1)\times L^{2}(0,1)$ by

$\begin{cases}D(A_{\varepsilon})=\\{(u,v)\in H^{1}(0,1)\times
L^{2}_{c}:(a_{\varepsilon}u_{x})_{x}\in
L^{2}(0,1),\,u_{x}(1)=0,\,a_{\varepsilon}(0)u_{x}(0)=0\\},\\\
A_{\varepsilon}(u,v)=(-(a_{\varepsilon}u_{x})_{x}+\lambda u,\beta
v)\end{cases}$ (2.1)

and, for $\varepsilon=0$, we define the operator $A_{0}:L^{2}_{c}\times
L_{c}^{2}\to L^{2}_{c}\times L_{c}^{2}$ by $A_{0}(u,v)=(\lambda u,\beta v)$.

If we define $F:H^{1}(0,1)\times L^{2}_{c}\to L^{2}(0,1)\times L^{2}_{c}$ by
$F(u,v)=(f_{1}(u)+g_{1}(v),f_{2}(u(0))+g_{2}(v))$, where we have used the same
notation for $f_{1},f_{2},g_{1},g_{2}$ and its Nemistky functionals, then
(1.1) and (1.3) can be written as

$\begin{cases}U_{t}+A_{\varepsilon}U=F(U),\\\
U(0)=U_{0},\,\,\,\varepsilon\in[0,\varepsilon_{0}],\end{cases}$ (2.2)

where we are denoting $U=(u,v)$ a point of $H^{1}(0,1)\times L^{2}_{c}$.

Notice that when $\varepsilon=0$, $F$ acts in $L^{2}_{c}\times L_{c}^{2}$ into
itself.

We have $A_{\varepsilon}$ a positive self-adjoint operator with compact
resolvent, $\varepsilon\in(0,\varepsilon_{0}]$. Hence, the spectrum of
$A_{\varepsilon}$ can be written in the following form

$0<\lambda_{1}^{\varepsilon}<\lambda_{2}^{\varepsilon}<\lambda_{3}^{\varepsilon}<\lambda_{4}^{\varepsilon}<...$

Denote
$\varphi_{1}^{\varepsilon},\varphi_{2}^{\varepsilon},\varphi_{3}^{\varepsilon},...$
the corresponding eigenfunctions.

Next, we summarize some spectral properties of $A_{\varepsilon}$ whose proof
can be found in [10] or alternatively in [13, 14].

######  Theorem 2.1.

Let $A_{\varepsilon}$ be the operator defined in (2.1). Then it is valid for
the following properties.

  * (i)

$\lambda_{1}^{\varepsilon}=\beta$ and $\varphi^{\varepsilon}_{1}=(0,1)$.

  * (ii)

$\lambda_{2}^{\varepsilon}\to\lambda$ as $\varepsilon\to 0$ and
$\varphi^{\varepsilon}_{2}\to(1,0)$ as $\varepsilon\to 0$.

  * (iii)

$\lambda_{k}^{\varepsilon}\to\infty$ as $\varepsilon\to 0$ for $k\geq 3$.

Now, we introduce some notations to state the Invariant Manifold Theorem. We
denote $Y_{\varepsilon}=span[(0,1),\varphi_{2}^{\varepsilon}]$ the space
generated by the first two eigenfunctions of $A_{\varepsilon}$ and we denote
$L_{\varepsilon}$ the spectral projection of $L^{2}(0,1)\times L^{2}_{c}$ onto
$Y^{\varepsilon}$. For each $\varepsilon\in(0,\varepsilon_{0}]$, we decompose
$H^{1}(0,1)\times L_{c}^{2}=Y_{\varepsilon}\oplus Y^{\perp}_{\varepsilon}$,
and we denote $A_{\varepsilon}^{+}=A_{\varepsilon}|_{Y_{\varepsilon}}$ and
$A_{\varepsilon}^{-}=A_{\varepsilon}|_{Y_{\varepsilon}^{\perp}}$. These
decompositions allow us to rewrite (2.2) as the following coupled equation

$\begin{cases}V_{t}^{\varepsilon}+A_{\varepsilon}^{+}V^{\varepsilon}=L_{\varepsilon}F(V^{\varepsilon}+Z^{\varepsilon}):=H_{\varepsilon}(V^{\varepsilon},Z^{\varepsilon}),\\\
Z_{t}^{\varepsilon}+A_{\varepsilon}^{-}Z^{\varepsilon}=(I-L_{\varepsilon})F(V^{\varepsilon}+Z^{\varepsilon}):=G_{\varepsilon}(V^{\varepsilon},Z^{\varepsilon}).\end{cases}$
(2.3)

The Invariant Manifold Theorem whose proof can be found in [1, 12] and [15]
states as follows.

######  Theorem 2.2.

For $\varepsilon_{0}>0$ sufficiently small, there is an invariant manifold
$\mathcal{M}_{\varepsilon}$ for (2.2) given by

$\mathcal{M}_{\varepsilon}=\\{U^{\varepsilon}\in H^{1}(0,1)\times
L^{2}_{c}\,;\,U^{\varepsilon}=L_{\varepsilon}U^{\varepsilon}+s_{*}^{\varepsilon}(L_{\varepsilon}U^{\varepsilon})\\},\quad\varepsilon\in(0,\varepsilon_{0}],$

where $s_{\ast}^{\varepsilon}:Y_{\varepsilon}\to Y_{\perp}^{\varepsilon}$ is a
continuously differentiable map satisfying

$\sup_{\tilde{V}^{\varepsilon}\in
Y_{\varepsilon}}\|s_{\ast}^{\varepsilon}(\tilde{V}^{\varepsilon})\|_{H^{1}(0,1)}+\sup_{\tilde{V}^{\varepsilon}\in
Y_{\varepsilon}}\|\dot{s}_{\ast}^{\varepsilon}(\tilde{V}^{\varepsilon})\|_{\mathcal{L}(Y_{\varepsilon},H^{1}(0,1))}\to
0,\quad\hbox{as }\varepsilon\to 0.$

The invariant manifold $\mathcal{M}_{\varepsilon}$ is exponentially attracting
and the flow of $U_{0}^{\varepsilon}\in\mathcal{M}_{\varepsilon}$ is given by

$\tilde{T}_{\varepsilon}(t)U_{0}^{\varepsilon}=V^{\varepsilon}(t)+s_{\ast}^{\varepsilon}(V^{\varepsilon}(t)),\quad
t\geq 0,$

where $V^{\varepsilon}(t)$ satisfies

$\begin{cases}{V_{t}^{\varepsilon}}+A_{\varepsilon}^{+}V^{\varepsilon}=H_{\varepsilon}(V^{\varepsilon},s_{\ast}^{\varepsilon}(V^{\varepsilon})),\quad
t\geq 0,\\\ V^{\varepsilon}(0)=L_{\varepsilon}U_{0}^{\varepsilon}\in
Y_{\varepsilon}.\end{cases}$ (2.4)

Notice that the equation (2.4) is an ODE in $Y_{\varepsilon}$ that describes
the behavior of the PDE-ODE (1.1). Moreover, $Y^{\varepsilon}$ is a two-
dimensional space generated by eigenfunctions $(1,0)$ and
$\varphi^{\varepsilon}_{2}$. In what follows, we identify $Y^{\varepsilon}$
with $\mathbb{R}^{2}$ and write (2.4) as a planar ODE.

In $L^{2}(0,1)\times L_{c}^{2}$ we consider the following inner product

$\left\langle(u,v),(\varphi,\psi)\right\rangle=\int_{0}^{1}u\varphi\,dx+\psi
v.$ (2.5)

Recall that we are denoting
$Y_{\varepsilon}=span[(0,1),\varphi_{2}^{\varepsilon}]$. Since
$\varphi_{2}^{\varepsilon}\in H^{1}(0,1)\times L_{c}^{2}$ and is orthogonal to
$(0,1)$, we have $\varphi_{2}^{\varepsilon}=(\varphi^{\varepsilon},0)$, for
some $\varphi^{\varepsilon}\in H^{1}(0,1)$. Hence, we can write
$Y_{\varepsilon}=span[(0,1),(\varphi^{\varepsilon},0)]$, where
$\varphi^{\varepsilon}\to 1$ as $\varepsilon\to 0$ according with Theorem 2.1.

The decomposition $H^{1}(0,1)\times L_{c}^{2}=Y_{\varepsilon}\oplus
Y^{\perp}_{\varepsilon}$, allows us write a solution
$(w^{\varepsilon}(x,t),v(t))\in H^{1}(0,1)\times L_{c}^{2}$ of the PDE-ODE
(1.1) as,

$(w^{\varepsilon}(x,t),v(t))=w_{1}^{\varepsilon}(t)(0,1)+u^{\varepsilon}(t)(\varphi^{\varepsilon}(x),0)+(w_{\perp}^{\varepsilon}(x,t),0),$
(2.6)

where $(w_{\perp}^{\varepsilon}(x,t),0)\in Y_{\varepsilon}^{\perp}$.

Taking the inner product (2.5) of (2.6) with $(0,1)$, we obtain

$\left\langle(w^{\varepsilon},v),(0,1)\right\rangle=\left\langle
w_{1}^{\varepsilon}(0,1),(0,1)\right\rangle+\left\langle
u^{\varepsilon}(\varphi^{\varepsilon},0),(0,1)\right\rangle+\left\langle(w_{\perp}^{\varepsilon},0),(0,1)\right\rangle$

which implies $w^{\varepsilon}_{1}=v$.

Besides that, if we take the inner product (2.5) of (2.6) with
$(\varphi^{\varepsilon},0)$, we obtain

$\left\langle(w^{\varepsilon},v),(\varphi^{\varepsilon},0)\right\rangle=\left\langle
w_{1}(0,1),(\varphi^{\varepsilon},0)\right\rangle+\left\langle
u^{\varepsilon}(\varphi^{\varepsilon},0),(\varphi^{\varepsilon},0)\right\rangle+\left\langle(w_{\perp}^{\varepsilon},0),(\varphi^{\varepsilon},0)\right\rangle$

which implies
$\int_{0}^{1}w^{\varepsilon}\varphi^{\varepsilon}\,dx=u^{\varepsilon}$, where
we have used that $\|\varphi^{\varepsilon}\|_{L^{2}(0,1)}=1$ and
$w^{\varepsilon}_{\perp}\perp\varphi^{\varepsilon}$ in $L^{2}(0,1)$.

It follows from PDE-ODE (1.1) that

$\begin{cases}\dot{u}^{\varepsilon}(t)=\int_{0}^{1}(a_{\varepsilon}(x)w^{\varepsilon}_{x}(x,t))_{x}\varphi^{\varepsilon}(x)\,dx-\int_{0}^{1}\lambda
w^{\varepsilon}(x,t)\varphi^{\varepsilon}(x)\,dx+\int_{0}^{1}f_{1}(w^{\varepsilon}(x,t))\varphi^{\varepsilon}(x)\,dx,\\\
\dot{v}(t)=-\beta v(t)+f_{2}(w^{\varepsilon}(0,t))+g_{2}(v(t)).\end{cases}$

The boundary conditions and the fact that $(\varphi^{\varepsilon},0)\in
D(A_{\varepsilon})$ imply that

$\displaystyle\int_{0}^{1}(a_{\varepsilon}w_{x}^{\varepsilon})_{x}\varphi^{\varepsilon}\,dx$
$\displaystyle=-a_{\varepsilon}(0)w^{\varepsilon}_{x}(0)\varphi^{\varepsilon}(0)-\int_{0}^{1}a_{\varepsilon}w_{x}^{\varepsilon}\varphi_{x}^{\varepsilon}\,dx$
$\displaystyle=-a_{\varepsilon}(0)w^{\varepsilon}_{x}(0)\varphi^{\varepsilon}(0)+a_{\varepsilon}(0)\varphi^{\varepsilon}_{x}(0)w^{\varepsilon}(0)+\int_{0}^{1}(a_{\varepsilon}\varphi_{x}^{\varepsilon})_{x}w^{\varepsilon}\,dx$
$\displaystyle=g_{1}(v)\varphi^{\varepsilon}(0)+\int_{0}^{1}(a_{\varepsilon}\varphi_{x}^{\varepsilon})_{x}w^{\varepsilon}\,dx.$

Since $(\varphi^{\varepsilon},0)$ is an eigenfunction associated with
$\lambda_{2}^{\varepsilon}$ and
$w^{\varepsilon}=u^{\varepsilon}\varphi^{\varepsilon}+w^{\varepsilon}_{\perp}$,
we have

$\begin{cases}\dot{u}^{\varepsilon}=-\lambda_{2}^{\varepsilon}u^{\varepsilon}+\int_{0}^{1}f_{1}(u^{\varepsilon}\varphi^{\varepsilon}+w_{\perp}^{\varepsilon})\varphi^{\varepsilon}\,dx+g_{1}(v)\varphi^{\varepsilon}(0),\\\
\dot{v}=-\beta
v+f_{2}(u^{\varepsilon}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot))+g_{2}(v).\end{cases}$
(2.7)

The equation (2.7) is the version of the equation (2.4) to $\mathbb{R}^{2}$.

The next result ensures that the behavior of the PDE-ODE (1.1) is dictated by
the ODE (2.7).

######  Theorem 2.3.

Let $(w^{\varepsilon},v)$ be a solution of (1.1) and $w^{\varepsilon}_{\perp}$
be as in (2.6). Then it is valid the following convergence

$\sup_{t\in(0,1)}\|w^{\varepsilon}_{\perp}(x,t)\|_{H^{1}(0,1)}\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

###### Proof.

By (2.6) we have that
$w^{\varepsilon}=u^{\varepsilon}\varphi^{\varepsilon}+w^{\varepsilon}_{\perp}$.
Since $w^{\varepsilon}$ is a solution of (1.1) and $u^{\varepsilon}$ satisfies
(2.7), we have

$\displaystyle(w^{\varepsilon}_{\perp})_{t}$
$\displaystyle=w^{\varepsilon}_{t}-\dot{u}^{\varepsilon}\varphi^{\varepsilon}$
$\displaystyle=(a_{\varepsilon}w^{\varepsilon}_{x})_{x}-\lambda
w^{\varepsilon}+f_{1}(w^{\varepsilon})+\lambda_{2}^{\varepsilon}u^{\varepsilon}\varphi^{\varepsilon}-\int_{0}^{1}f_{1}(u^{\varepsilon}\varphi^{\varepsilon}+w_{\perp}^{\varepsilon})\varphi^{\varepsilon}\,dy\varphi^{\varepsilon}-g_{1}(v)\varphi^{\varepsilon}(0)\varphi^{\varepsilon}$
$\displaystyle=(a_{\varepsilon}(w_{\perp}^{\varepsilon})_{x})_{x}-\lambda
w_{\perp}^{\varepsilon}+f_{1}(u^{\varepsilon}\varphi^{\varepsilon}+w^{\varepsilon}_{\perp})-\int_{0}^{1}f_{1}(u^{\varepsilon}\varphi^{\varepsilon}+w_{\perp}^{\varepsilon})\varphi^{\varepsilon}\,dy\varphi^{\varepsilon}-g_{1}(v)\varphi^{\varepsilon}(0)\varphi^{\varepsilon},$

whew we have used
$(a_{\varepsilon}\varphi^{\varepsilon}_{x})_{x}-\lambda\varphi^{\varepsilon}+\lambda^{\varepsilon}_{2}\varphi^{\varepsilon}=0$.

Hence, $w^{\varepsilon}_{\perp}$ is a solution of the following parabolic
equation

$\begin{cases}(w^{\varepsilon}_{\perp})_{t}-(a_{\varepsilon}(w^{\varepsilon}_{\perp})_{x})_{x}+\lambda
w^{\varepsilon}_{\perp}=R_{\varepsilon}(u^{\varepsilon},v,w_{\perp}^{\varepsilon}),\quad&\hbox{in
}(0,1)\times(0,\infty),\\\ (w^{\varepsilon}_{\perp})_{x}(1)=0,&\hbox{in
}(0,\infty),\\\
-a_{\varepsilon}(0)(w^{\varepsilon}_{\perp})_{x}(0)=g_{1}(v),&\hbox{in
}(0,\infty),\\\ w^{\varepsilon}_{\perp}(0)=w_{0},&\hbox{in }(0,1),\end{cases}$
(2.8)

where

$R_{\varepsilon}(u^{\varepsilon},v,w_{\perp}^{\varepsilon})=f_{1}(u^{\varepsilon}\varphi^{\varepsilon}+w^{\varepsilon}_{\perp})-\int_{0}^{1}f_{1}(u^{\varepsilon}\varphi^{\varepsilon}+w_{\perp}^{\varepsilon})\varphi^{\varepsilon}\,dy\varphi^{\varepsilon}-g_{1}(v)\varphi^{\varepsilon}(0)\varphi^{\varepsilon}.$

We define the operator $B_{\varepsilon}:D(B_{\varepsilon})\subset
L^{2}(0,1)\to L^{2}(0,1)$ by $D(B_{\varepsilon})=\\{u\in
H^{1}(0,1):(a_{\varepsilon}u_{x})_{x}\in
L^{2}(0,1),\,u_{x}(1)=0,\,a_{\varepsilon}(0)u_{x}(0)=0\\}$ and
$B_{\varepsilon}u=-(a_{\varepsilon}u_{x})_{x}+\lambda u$. Since
$w_{\perp}^{\varepsilon}$ is a solution of (2.8), it must satisfies the
following variation of constants formula,

$w_{\perp}^{\varepsilon}(t)=e^{-B_{\varepsilon}t}w_{0}+\int_{0}^{t}e^{-B_{\varepsilon}(t-s)}R_{\varepsilon}(u^{\varepsilon},v,w_{\perp}^{\varepsilon})\,ds.$

As in [7, eq. (2.7)], we can take constants $C_{1}>0$ and $\mu>0$ independent
of $\varepsilon$, such that the linear semiflow
$\\{e^{-B_{\varepsilon}t}\\}_{t\geq 0}$ restrict to $Y_{\perp}^{\varepsilon}$
satisfy the properties

$\begin{cases}\|e^{-B_{\varepsilon}t}w_{0}\|_{H^{1}(0,1)}\leq
C_{1}t^{-\frac{1}{2}}e^{-\mu\tau(\varepsilon)t}\|w_{0}\|_{L^{2}(0,1)},\\\
\|e^{-B_{\varepsilon}t}w_{0}\|_{H^{1}(0,1)}\leq
C_{1}e^{-\mu\tau(\varepsilon)t}\|w_{0}\|_{H^{1}(0,1)},\end{cases}$ (2.9)

where $\tau(\varepsilon)$ is given by (1.2).

Moreover, we can found constants $K_{1},K_{2}$ independents of $\varepsilon$
such that

$\displaystyle\|R_{\varepsilon}(u^{\varepsilon},v,w_{\perp}^{\varepsilon})\|_{L^{2}(0,1)}$
$\displaystyle\leq|\int_{0}^{1}f_{1}(u^{\varepsilon}\varphi^{\varepsilon}(x)+w^{\varepsilon}_{\perp}(x))-f_{1}(u^{\varepsilon}\varphi^{\varepsilon}(y)+w_{\perp}^{\varepsilon}(y))\varphi^{\varepsilon}(y)|\,dy+|g_{1}(v)\varphi^{\varepsilon}(0)|$
$\displaystyle\leq\int_{0}^{1}|f_{1}(u^{\varepsilon}\varphi^{\varepsilon}(x)+w^{\varepsilon}_{\perp}(x))||1-\varphi^{\varepsilon}(y)|\,dy$
$\displaystyle+\int_{0}^{1}|f_{1}(u^{\varepsilon}\varphi^{\varepsilon}(x)+w_{\perp}^{\varepsilon}(x))-f_{1}(u^{\varepsilon}\varphi^{\varepsilon}(y)+w_{\perp}^{\varepsilon}(y))||\varphi^{\varepsilon}(y)|\,dy+|g_{1}(v)\varphi^{\varepsilon}(0)|$
$\displaystyle\leq o(1)+K_{2}+K_{1}\|w^{\varepsilon}_{\perp}\|_{H^{1}(0,1)},$

where $o(1)$ denotes a function that goes to zero as $\varepsilon\to 0$ and,
we have used Theorem 2.1 and the continuity of $f_{1}$ and $g_{1}$ to
guarantee the boundness constants.

Hence, by (2.9), we have

$\displaystyle\|w_{\perp}^{\varepsilon}(t)\|_{H^{1}(0,1)}$ $\displaystyle\leq
C_{1}e^{-\mu\tau(\varepsilon)t}\|w_{0}\|_{H^{1}(0,1)}+\int_{0}^{t}e^{-\mu\tau(\varepsilon)(t-s)}(t-s)^{-\frac{1}{2}}\|R_{\varepsilon}(u^{\varepsilon},v,w_{\perp}^{\varepsilon})\|_{H^{1}(0,1)}\,ds$
$\displaystyle\leq
C_{1}e^{-\mu\tau(\varepsilon)t}\|w_{0}\|_{H^{1}(0,1)}+(o(1)+K_{2})\int_{0}^{t}(t-s)^{\frac{1}{2}}e^{-\mu\tau(\varepsilon)(t-s)}\,ds$
$\displaystyle+K_{1}\int_{0}^{t}(t-s)^{\frac{1}{2}}e^{-\mu\tau(\varepsilon)(t-s)}\|w^{\varepsilon}_{\perp}(s)\|_{H^{1}(0,1)}\,ds.$

The result follows applying the Gronwall inequality.

∎

For each $\varepsilon\in[0,\varepsilon_{0}]$, we define the planar vector
field $X_{\varepsilon}(u,v)=(P_{\varepsilon}(u,v),Q_{\varepsilon}(u,v))$,
where

$\begin{cases}P_{0}(u,v)=-\lambda u+f_{1}(u)+g_{1}(v),\\\ Q_{0}(u,v)=-\beta
v+f_{2}(u)+g_{2}(v),\end{cases}$ (2.10)

and

$\begin{cases}P_{\varepsilon}(u,v)=-\lambda_{2}^{\varepsilon}u+\int_{0}^{1}f_{1}(u\varphi^{\varepsilon}+\theta^{\varepsilon}(u\varphi^{\varepsilon}))\varphi^{\varepsilon}\,dx+g_{1}(v)\varphi^{\varepsilon}(0),\\\
Q_{\varepsilon}(u,v)=-\beta
v+f_{2}(u\varphi^{\varepsilon}(0)+\theta^{\varepsilon}(u\varphi^{\varepsilon}(0)))+g_{2}(v).\end{cases}$
(2.11)

Since we do not have a compact set containing all dynamics of the problem,
taking the $C^{1}$ norm $\|X_{\varepsilon}-X_{0}\|_{1}$ in compact sets does
not produce suitable comparison due to the possibility of the dynamics at
infinite. Taking the $C^{1}$ norm $\|X_{\varepsilon}-X_{0}\|_{1}$ in all
$\mathbb{R}^{2}$ is not a good strategy due to the unboundedness of the
polynomial sources. In the next section, we will find a way to overcome this
issue.

##  3 Poincaré compactification

In this section, we perform a Poincaré compactification for semiflows of (1.1)
and (1.3) and we prove the $C^{1}$ closeness of the respective compactified
vector fields. Our main reference for Poincaré compactification is [5] we also
refer to the works [3, 4].

Let $M\subset\mathbb{R}^{3}$ be a compact differentiable two-dimensional
manifold. Consider an open cover $M=V_{1}\cup...\cup V_{k}$ with $V_{i}$
contained in a local chart $(\phi_{i},U_{i})$ such that $\phi_{i}(U_{i})=B(2)$
and $\phi_{i}(V_{i})=B(1)$, where $B(1)$ and $B(2)$ are open balls centered in
the origin of $\mathbb{R}^{2}$ of radius $1$ and $2$, respectively. The
$C^{1}$-norm of a vector field $X$ in $M$ is defined by

$\|X\|_{1}=\max_{1\leq i\leq k}\sup_{(x,y)\in
B(1)}\\{\|X(\phi^{-1}_{i}(x,y))\|_{\mathbb{R}^{2}},\|X^{\prime}(\phi^{-1}_{i}(x,y))\|_{\mathcal{L}(\mathbb{R}^{2})}\\}.$

Let
$X_{\varepsilon}(u^{\varepsilon},v)=(P_{\varepsilon}(u^{\varepsilon},v),Q_{\varepsilon}(u^{\varepsilon},v))$,
$\varepsilon\in[0,\varepsilon_{0}]$, be the polynomial vector field defined in
(2.10) and (2.11) and denote $d$ the maximum degree of $P_{\varepsilon}$ and
$Q_{\varepsilon}$.

We embedded $\mathbb{R}^{2}$ into $\mathbb{R}^{3}$ as the plane
$\\{(u,v,1):(u,v)\in\mathbb{R}^{2}\\}$ tangent to $\mathbb{S}^{2}$ at point
$(0,0,1)$. We decompose
$\mathbb{S}^{2}=\mathbb{H}^{+}\cup\mathbb{S}^{1}\cup\mathbb{H}^{-}$, whee
$\mathbb{H}^{+}$ is the northern hemisphere, $\mathbb{H}^{-}$ is the southern
hemisphere and $\mathbb{S}^{1}$ is the equator.

The compactification of the vector field
$X_{\varepsilon}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ to the sphere
$\mathbb{S}^{2}$ is given by central projections
$f^{+},f^{-}:\mathbb{R}^{2}\to\mathbb{S}^{2}$, $f^{-}=-f^{+}$, where

$f^{+}(u,v)=\Big{(}\frac{u}{\Delta(u,v)},\frac{v}{\Delta(u,v)},\frac{1}{\Delta(u,v)}\Big{)},\quad\Delta(u,v)=\sqrt{u^{2}+v^{2}+1}.$

In $\mathbb{S}^{2}$ we consider the usual six local charts
$\mathbb{S}^{2}=U_{1}\cup U_{2}\cup...\cup U_{6}$ given by

$U_{1}=\\{(y_{1},y_{2},y_{3})\in\mathbb{S}^{2}:y_{1}>0\\},U_{2}=\\{(y_{1},y_{2},y_{3})\in\mathbb{S}^{2}:y_{2}>0\\},U_{3}=\\{(y_{1},y_{2},y_{3})\in\mathbb{S}^{2}:y_{3}>0\\}$
$U_{4}=\\{(y_{1},y_{2},y_{3})\in\mathbb{S}^{2}:y_{1}<0\\},U_{5}=\\{(y_{1},y_{2},y_{3})\in\mathbb{S}^{2}:y_{2}<0\\},U_{6}=\\{(y_{1},y_{2},y_{3})\in\mathbb{S}^{2}:y_{3}<0\\}.$

and local maps are given by

$\phi_{1}(u,v,w)=\Big{(}\frac{v}{u},\frac{w}{u}\Big{)},\phi_{2}(u,v,w)=\Big{(}\frac{u}{v},\frac{w}{v}\Big{)},\phi_{3}(u,v,w)=\Big{(}\frac{u}{w},\frac{v}{w}\Big{)}$
$\phi_{4}(u,v,w)=\Big{(}-\frac{v}{u},-\frac{w}{u}\Big{)},\phi_{5}(u,v,w)=\Big{(}-\frac{u}{v},-\frac{w}{v}\Big{)},\phi_{5}(u,v,w)=\Big{(}-\frac{u}{w},-\frac{v}{w}\Big{)}.$

In order to study $X_{\varepsilon}$ in $\mathbb{R}^{2}$ including its behavior
near infinity it is enough working on the Poincaré disk
$\mathbb{D}^{2}=\mathbb{H}^{+}\cup\mathbb{S}^{1}$. The compactification
$\bar{X}_{\varepsilon}:\mathbb{D}^{2}\to\mathbb{D}^{2}$ of the vector field
$X_{\varepsilon}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is given by

$\bar{X}_{\varepsilon}(Y)=\rho(Y)Df^{+}(X)X_{\varepsilon}(X),\quad
Y=F^{+}(X),$ (3.1)

where $\rho(Y)$ is a bounded correction therm.

According with [5, Chapter 5], the expression (3.1) in the charts
$(\phi_{i},U_{i})$, $i=1,...,6$ is given as follows.

In the chart $(\phi_{1},U_{1})$ make the change of variable $(u,v)=(1/x,x/y)$
to obtain

$\begin{cases}\dot{x}=y^{d}\Big{[}-xP_{\varepsilon}\Big{(}\frac{1}{y},\frac{x}{y}\Big{)}+Q_{\varepsilon}\Big{(}\frac{1}{y},\frac{x}{y}\Big{)}\Big{]},\\\
\dot{y}=-y^{d+1}P_{\varepsilon}\Big{(}\frac{1}{y},\frac{x}{y}\Big{)}.\end{cases}$
(3.2)

In the chart $(\phi_{2},U_{2})$ make the change of variable $(u,v)=(x/y,1/x)$
to obtain

$\begin{cases}\dot{x}=y^{d}\Big{[}P_{\varepsilon}\Big{(}\frac{x}{y},\frac{1}{y}\Big{)}-xQ_{\varepsilon}\Big{(}\frac{x}{y},\frac{1}{y}\Big{)}\Big{]},\\\
\dot{y}=-y^{d+1}Q_{\varepsilon}\Big{(}\frac{x}{y},\frac{1}{y}\Big{)}.\end{cases}$
(3.3)

In the chart $(\phi_{3},U_{3})$ make the change of variable $(u,v)=(x,y)$ to
obtain

$\begin{cases}\dot{x}=P_{\varepsilon}(x,y),\\\
\dot{y}=Q_{\varepsilon}(x,y).\end{cases}$ (3.4)

Moreover, the expression for $\bar{X}_{\varepsilon}$ in the charts
$(\phi_{4},U_{4})$, $(\phi_{5},U_{5})$ and $(\phi_{6},U_{6})$ is the same as
for $(\phi_{1},U_{1})$, $(\phi_{2},U_{2})$ and $(\phi_{3},U_{3})$ multiplied
by $(-1)^{d}$, respectively. Therefore, all calculations can be done in the
first three charts.

######  Theorem 3.1.

Let $\bar{X}_{\varepsilon}$ and $\bar{X}_{0}$ be the respective
compactification to the Poncaré disk $\mathbb{D}^{2}$ of the polynomial vector
fields $X_{\varepsilon}$ and $X_{0}$ defined in (2.10) and (2.11). Then it is
valid the following convergence

$\|\bar{X}_{\varepsilon}-\bar{X}_{0}\|_{1}\to
0\quad\hbox{as}\quad\varepsilon\to 0.$ (3.5)

###### Proof.

Recall that

$\begin{cases}P_{\varepsilon}(u,v)=-\lambda_{2}^{\varepsilon}u+\int_{0}^{1}f_{1}(u\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot))\varphi^{\varepsilon}(r)\,dr+g_{1}(v)\varphi^{\varepsilon}(0),\\\
Q_{\varepsilon}(u,v)=-\beta
v+f_{2}(u\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot))+g_{2}(v),\end{cases}$

and

$\begin{cases}P_{0}(u,v)=-\lambda u+f_{1}(u)+g_{1}(v),\\\ Q_{0}(u,v)=-\beta
v+f_{2}(u)+g_{2}(v).\end{cases}$

Step 1. $C^{0}$-convergence.

In the chart $(\phi_{1},U_{1})$, $P_{\varepsilon}$ and $P_{0}$, read as

$\bar{P}_{\varepsilon}(x,y)=y^{d}\Big{[}-x\Big{[}-\lambda_{2}^{\varepsilon}\frac{1}{y}+\int_{0}^{1}f_{1}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)\,dr+g_{1}\Big{(}\frac{x}{y}\Big{)}\varphi^{\varepsilon}(0)\Big{]}\\\
+\Big{(}-\beta\frac{x}{y}+f_{2}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}+g_{2}\Big{(}\frac{x}{y}\Big{)}\Big{)}\Big{]}$

and

$\bar{P}_{0}(x,y)=y^{d}\Big{[}-x\Big{[}-\lambda\frac{1}{y}+f_{1}\Big{(}\frac{1}{y}\Big{)}+g_{1}\Big{(}\frac{x}{y}\Big{)}\Big{]}+\Big{(}-\beta\frac{x}{y}+f_{2}\Big{(}\frac{1}{y}\Big{)}+g_{2}\Big{(}\frac{x}{y}\Big{)}\Big{)}\Big{]}.$

Thus,

$\displaystyle|\bar{P}_{\varepsilon}(x,y)-\bar{P}_{0}(x,y)|$
$\displaystyle\leq|\lambda-\lambda_{2}^{\varepsilon}||xy^{d-1}|+\int_{0}^{1}|xy^{d}||f_{1}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)-f_{1}\Big{(}\frac{1}{y}\Big{)}|\,dr$
$\displaystyle+|xy^{d}g_{1}\Big{(}\frac{x}{y}\Big{)}||\varphi^{\varepsilon}(0)-1|+|y^{d}||f_{2}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}-f_{2}\Big{(}\frac{1}{y}\Big{)}|.$

But,

$\displaystyle\int_{0}^{1}$
$\displaystyle|xy^{d}||f_{1}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)-f_{1}\Big{(}\frac{1}{y}\Big{)}|\,dr\leq\int_{0}^{1}|xy^{d}||f_{1}\Big{(}\frac{1}{y}\Big{)}|1-\varphi^{\varepsilon}(r)|\,dr$
$\displaystyle+\int_{0}^{1}K_{1}|xy^{d}||\frac{1}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)-\frac{1}{y}|\,dr$
$\displaystyle\leq\int_{0}^{1}|xy^{d}||f_{1}\Big{(}\frac{1}{y}\Big{)}|1-\varphi^{\varepsilon}(r)|\,dr+\int_{0}^{1}K_{1}|xy^{d-1}||\varphi^{\varepsilon}(r)-1|+K_{1}|xy^{d}||w_{\perp}^{\varepsilon}(r,\cdot)|\,dr,$

where $K_{1}$ depends on the Lipschitz constant of $f_{1}$ and the boundness
of $\varphi^{\varepsilon}$ on $B(1)$.

In a similar way,

$|y^{d}||f_{2}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}-f_{2}\Big{(}\frac{1}{y}\Big{)}|\leq
K_{2}|y^{d-1}||\varphi^{\varepsilon}(0)-1|+K_{2}|y^{d}||w_{\perp}^{\varepsilon}(0,\cdot)|,$

where $K_{2}$ denotes the Lipschitz constant of $f_{2}$ on $B(1)$.

Since $d$ is greater than or equal degree of $g_{1}$ and $f_{1}$, we have that
$|xy^{d}f_{1}(1/y)|$ and $|xy^{d}g_{1}(x/y)|$ are bounded in $B(1)$. Moreover,
$d-1\geq 0$ and, by Theorems 2.1 and 2.3,
$\lambda_{2}^{\varepsilon}\to\lambda$, $\varphi^{\varepsilon}\to 1$ and
$w^{\varepsilon}_{\perp}\to 0$ as $\varepsilon\to 0$. Putting all estimates
together, we obtain

$\sup_{(x,y)\in B(1)}|\bar{P}_{\varepsilon}(x,y)-\bar{P}_{0}(x,y)|\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

Moreover, in the chart $(\phi_{1},U_{1})$, $Q_{\varepsilon}$ and $Q_{0}$, read
as

$\bar{Q}_{\varepsilon}(x,y)=-y^{d+1}\Big{[}-\lambda_{2}^{\varepsilon}\frac{1}{y}+\int_{0}^{1}f_{1}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)\,dr+g_{1}\Big{(}\frac{x}{y}\Big{)}\varphi^{\varepsilon}(0)\Big{]}$

and

$\bar{Q}_{0}(x,y)=-y^{d+1}\Big{[}-\lambda\frac{1}{y}+f_{1}\Big{(}\frac{1}{y}\Big{)}+g_{1}\Big{(}\frac{x}{y}\Big{)}\Big{]}.$

Thus,

$\displaystyle|\bar{Q}_{\varepsilon}(x,y)-\bar{Q}_{0}(x,y)|$
$\displaystyle\leq|\lambda-\lambda_{2}^{\varepsilon}||y^{d}|+\int_{0}^{1}|y^{d+1}||f_{1}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)-f_{1}\Big{(}\frac{1}{y}\Big{)}|\,dr$
$\displaystyle+|y^{d+1}g_{1}\Big{(}\frac{x}{y}\Big{)}||\varphi^{\varepsilon}(0)-1|.$

But

$\displaystyle\int_{0}^{1}$
$\displaystyle|y^{d+1}||f_{1}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)-f_{1}\Big{(}\frac{1}{y}\Big{)}|\,dr\leq\int_{0}^{1}|y^{d+1}||f_{1}\Big{(}\frac{1}{y}\Big{)}|1-\varphi^{\varepsilon}(r)|\,dr$
$\displaystyle+\int_{0}^{1}K_{1}|y^{d+1}||\frac{1}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)-\frac{1}{y}|\,dr$
$\displaystyle\leq\int_{0}^{1}|y^{d+1}||f_{1}\Big{(}\frac{1}{y}\Big{)}|1-\varphi^{\varepsilon}(r)|\,dr+\int_{0}^{1}K_{1}|y^{d}||\varphi^{\varepsilon}(r)-1|+K_{1}|y^{d+1}||w_{\perp}^{\varepsilon}(r,\cdot)|\,dr,$

where $K_{1}$ depends on the Lipschitz constant of $f_{1}$ and the boundness
of $\varphi^{\varepsilon}$ on $B(1)$.

Since $d$ is greater than or equal degree of $f_{1}$ and $g_{1}$, we have that
$|y^{d+1}f_{1}(1/y)|$ and $|y^{d+1}g_{1}(x/y)|$ are bounded in $B(1)$. By
convergence properties $\lambda_{2}^{\varepsilon}\to\lambda$,
$\varphi^{\varepsilon}\to 1$ and $w^{\varepsilon}_{\perp}\to 0$ as
$\varepsilon\to 0$, we obtain

$\sup_{(x,y)\in B(1)}|\bar{Q}_{\varepsilon}(x,y)-\bar{Q}_{0}(x,y)|\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

In the chart $(\phi_{2},U_{2})$, $P_{\varepsilon}$ and $P_{0}$ reads as

$\bar{P}_{\varepsilon}(x,y)=y^{d}\Big{[}-\lambda_{2}^{\varepsilon}\frac{x}{y}+\int_{0}^{1}f_{1}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)\,dr+g_{1}\Big{(}\frac{1}{y}\Big{)}\varphi^{\varepsilon}(0)\\\
-x\Big{(}-\beta\frac{1}{y}+f_{2}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}+g_{2}\Big{(}\frac{1}{y}\Big{)}\Big{)}\Big{]}$

and

$\bar{P}_{0}(x,y)=y^{d}\Big{[}-\lambda\frac{x}{y}+f_{1}\Big{(}\frac{x}{y}\Big{)}+g_{1}\Big{(}\frac{1}{y}\Big{)}-x\Big{(}-\beta\frac{1}{y}+f_{2}\Big{(}\frac{x}{y}\Big{)}+g_{2}\Big{(}\frac{1}{y}\Big{)}\Big{)}\Big{]}.$

Thus,

$\displaystyle|\bar{P}_{\varepsilon}(x,y)-\bar{P}_{0}(x,y)|$
$\displaystyle\leq|\lambda-\lambda_{2}^{\varepsilon}||xy^{d-1}|+\int_{0}^{1}|y^{d}||f_{1}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)-f_{1}\Big{(}\frac{x}{y}\Big{)}|\,dr$
$\displaystyle+|y^{d}g_{1}\Big{(}\frac{1}{y}\Big{)}||\varphi^{\varepsilon}(0)-1|+|xy^{d}||f_{2}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}-f_{2}\Big{(}\frac{x}{y}\Big{)}|$

But

$\displaystyle\int_{0}^{1}$
$\displaystyle|y^{d}||f_{1}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)-f_{1}\Big{(}\frac{x}{y}\Big{)}|\,dr\leq\int_{0}^{1}|y^{d}||f_{1}\Big{(}\frac{x}{y}\Big{)}|1-\varphi^{\varepsilon}(r)|\,dr$
$\displaystyle+\int_{0}^{1}K_{1}|y^{d}||\frac{x}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)-\frac{x}{y}|\,dr$
$\displaystyle\leq\int_{0}^{1}|y^{d}||f_{1}\Big{(}\frac{x}{y}\Big{)}|1-\varphi^{\varepsilon}(r)|\,dr+\int_{0}^{1}K_{1}|xy^{d-1}||\varphi^{\varepsilon}(r)-1|+K_{1}|y^{d}||w_{\perp}^{\varepsilon}(r,\cdot)|\,dr$

and

$\displaystyle|xy^{d}||f_{2}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}-f_{2}\Big{(}\frac{x}{y}\Big{)}|\leq
K_{2}|x^{2}y^{d-1}||\varphi^{\varepsilon}(0)-1|+K_{2}|xy^{d}||w_{\perp}^{\varepsilon}(0,\cdot)|,$

where $K_{2}$ denotes the Lipschitz constant of $f_{2}$ on $B(1)$.

Since $d$ is greater than or equal degree of $g_{1}$ and $f_{1}$, we have that
$|y^{d}f_{1}(x/y)|$ and $|y^{d}g_{1}(1/y)|$ are bounded in $B(1)$. By
convergence properties $\lambda_{2}^{\varepsilon}\to\lambda$,
$\varphi^{\varepsilon}\to 1$ and $w^{\varepsilon}_{\perp}\to 0$ as
$\varepsilon\to 0$, we obtain

$\sup_{(x,y)\in B(1)}|\bar{P}_{\varepsilon}(x,y)-\bar{P}_{0}(x,y)|\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

Moreover, in the chart $(\phi_{2},U_{2})$, $Q_{\varepsilon}$ and $Q_{0}$ reads
as

$\bar{Q}_{\varepsilon}(x,y)=-y^{d+1}\Big{[}-\beta\frac{1}{y}+f_{2}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}+g_{2}\Big{(}\frac{1}{y}\Big{)}\Big{]}$

and

$\bar{Q}_{0}(x,y)=-y^{d+1}\Big{[}-\beta\frac{1}{y}+f_{2}\Big{(}\frac{x}{y}\Big{)}+g_{2}\Big{(}\frac{1}{y}\Big{)}\Big{]}.$

Thus,

$\displaystyle|\bar{Q}_{\varepsilon}(x,y)-\bar{Q}_{0}(x,y)|$
$\displaystyle\leq|y^{d+1}||f_{2}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}-f_{2}\Big{(}\frac{x}{y}\Big{)}|$
$\displaystyle\leq
K_{2}|xy^{d}||\varphi^{\varepsilon}(0)-1|+K_{2}|y^{d+1}||w_{\perp}^{\varepsilon}(0,\cdot)|.$

By convergence properties $\varphi^{\varepsilon}\to 1$ and
$w^{\varepsilon}_{\perp}\to 0$ as $\varepsilon\to 0$, we obtain

$\sup_{(x,y)\in B(1)}|\bar{Q}_{\varepsilon}(x,y)-\bar{Q}_{0}(x,y)|\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

In the chart $(\phi_{3},U_{3})$, $P_{\varepsilon}$ and $P_{0}$ reads as

$\bar{P}_{\varepsilon}(x,y)=-\lambda_{2}^{\varepsilon}x+\int_{0}^{1}f_{1}(x\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot))\varphi^{\varepsilon}(r)\,dr+g_{1}(y)\varphi^{\varepsilon}(0)$

and

$\bar{P}_{0}(x,y)=-\lambda x+f_{1}(x)+g_{1}(y).$

Thus,

$\displaystyle|\bar{P}_{\varepsilon}(x,y)-\bar{P}_{0}(x,y)|$
$\displaystyle\leq\int_{0}^{1}|f_{1}(x\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot))\varphi^{\varepsilon}(r)-f_{1}(x)|\,dr$
$\displaystyle\leq\int_{0}^{1}|f_{1}(x)||1-\varphi^{\varepsilon}(r)|\,dr+\int_{0}^{1}K_{1}|x\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)-x|\,dr$
$\displaystyle\leq\int_{0}^{1}K_{1}|x||\varphi^{\varepsilon}(r)-1|+K_{1}|w_{\perp}^{\varepsilon}(r,\cdot)|\,dr.$

By convergence properties $\varphi^{\varepsilon}\to 1$ and
$w^{\varepsilon}_{\perp}\to 0$ as $\varepsilon\to 0$, we obtain

$\sup_{(x,y)\in B(1)}|\bar{P}_{\varepsilon}(x,y)-\bar{P}_{0}(x,y)|\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

Moreover, in the chart $(\phi_{3},U_{3})$, $Q_{\varepsilon}$ and $Q_{0}$ reads
as

$\bar{Q}_{\varepsilon}(x,y)=-\beta
y+f_{2}(x\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot))+g_{2}(y),$

and

$\bar{Q}_{0}(x,y)=-\beta y+f_{2}(x)+g_{2}(y).$

Thus

$\displaystyle|\bar{Q}_{\varepsilon}(x,y)-\bar{Q}_{0}(x,y)|$
$\displaystyle\leq|f_{2}(x\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot))-f_{2}(x)|$
$\displaystyle\leq
K_{2}|x||\varphi^{\varepsilon}(0)-1|+K_{2}|w_{\perp}^{\varepsilon}(0,\cdot)|.$

By convergence properties $\varphi^{\varepsilon}\to 1$ and
$w^{\varepsilon}_{\perp}\to 0$ as $\varepsilon\to 0$, we obtain

$\sup_{(x,y)\in B(1)}|\bar{Q}_{\varepsilon}(x,y)-\bar{Q}_{0}(x,y)|\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

Step 2. $C^{1}$-convergence.

In the chart $(\phi_{1},U_{1})$, the partial derivatives of
$\bar{P}_{\varepsilon}$ and $\bar{P}_{0}$ with respect to $x$ reads as

$\displaystyle\frac{\partial\bar{P}_{\varepsilon}}{\partial x}(x,y)$
$\displaystyle=y^{d-1}\lambda_{2}^{\varepsilon}-y^{d}\int_{0}^{1}f_{1}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)\,dr$
$\displaystyle-y^{d}g_{1}\Big{(}\frac{x}{y}\Big{)}\varphi^{\varepsilon}(0)-xy^{d-1}g^{\prime}_{1}\Big{(}\frac{x}{y}\Big{)}\varphi^{\varepsilon}(0)-\beta
y^{d-1}+y^{d-1}g^{\prime}_{2}\Big{(}\frac{x}{y}\Big{)}$

and

$\displaystyle\frac{\partial\bar{P}_{0}}{\partial x}(x,y)$
$\displaystyle=y^{d-1}\lambda-y^{d}f_{1}\Big{(}\frac{1}{y}\Big{)}$
$\displaystyle-y^{d}g_{1}\Big{(}\frac{x}{y}\Big{)}-xy^{d-1}g^{\prime}_{1}\Big{(}\frac{x}{y}\Big{)}-\beta
y^{d-1}+y^{d-1}g^{\prime}_{2}\Big{(}\frac{x}{y}\Big{)}.$

Thus,

$\displaystyle\Big{|}\frac{\partial\bar{P}_{\varepsilon}}{\partial
x}(x,y)-\frac{\partial\bar{P}_{0}}{\partial x}(x,y)\Big{|}$
$\displaystyle\leq|y^{d-1}|\lambda_{2}^{\varepsilon}-\lambda|+|y^{d}|\int_{0}^{1}|f_{1}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)-f_{1}\Big{(}\frac{1}{y}\Big{)}|\,dr$
$\displaystyle+|y^{d}g_{1}\Big{(}\frac{x}{y}\Big{)}||\varphi^{\varepsilon}(0)-1|+|xy^{d-1}g^{\prime}_{1}\Big{(}\frac{x}{y}\Big{)}||\varphi^{\varepsilon}(0)-1|.$

Since $d$ is greater than or equal degree of $g_{1}$, we have that
$|y^{d}g_{1}(x/y)|$ and $|xy^{d-1}g^{\prime}_{1}(x/y)|$ are bounded in $B(1)$
The same boundness is true for $|y^{d}f_{1}(1/y)|$. By convergence properties
$\lambda_{2}^{\varepsilon}\to\lambda$, $\varphi^{\varepsilon}\to 1$ and
$w_{\perp}^{\varepsilon}\to 0$ as $\varepsilon\to 0$, we obtain

$\sup_{(x,y)\in B(1)}\Big{|}\frac{\partial\bar{P}_{\varepsilon}}{\partial
x}(x,y)-\frac{\partial\bar{P}_{0}}{\partial x}(x,y)\Big{|}\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

In the chart $(\phi_{1},U_{1})$, the partial derivatives of
$\bar{P}_{\varepsilon}$ and $\bar{P}_{0}$ with respect to $y$ reads as

$\displaystyle\frac{\partial\bar{P}_{\varepsilon}}{\partial y}(x,y)$
$\displaystyle=x(d-1)y^{d-2}\lambda_{2}^{\varepsilon}-xdy^{d-1}\int_{0}^{1}f_{1}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)\,dr$
$\displaystyle+xy^{d-2}\int_{0}^{1}f^{\prime}_{1}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)\,dr$
$\displaystyle-
xdy^{d-1}g_{1}\Big{(}\frac{x}{y}\Big{)}\varphi^{\varepsilon}(0)+x^{2}y^{d-2}g^{\prime}_{1}\Big{(}\frac{x}{y}\Big{)}\varphi^{\varepsilon}(0)$
$\displaystyle-\beta
x(d-1)y^{d-2}+dy^{d-1}f_{2}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}-y^{d-2}f^{\prime}_{2}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}$
$\displaystyle+dy^{d-1}g_{2}\Big{(}\frac{x}{y}\Big{)}-xy^{d-2}g^{\prime}_{2}\Big{(}\frac{x}{y}\Big{)}$

and

$\displaystyle\frac{\partial\bar{P}_{0}}{\partial y}(x,y)$
$\displaystyle=x(d-1)y^{d-2}\lambda-xdy^{d-1}f_{1}\Big{(}\frac{1}{y}\Big{)}$
$\displaystyle+xy^{d-2}f^{\prime}_{1}\Big{(}\frac{1}{y}\Big{)}$
$\displaystyle-
xdy^{d-1}g_{1}\Big{(}\frac{x}{y}\Big{)}+x^{2}dy^{d-2}g^{\prime}_{1}\Big{(}\frac{x}{y}\Big{)}$
$\displaystyle-\beta
x(d-1)y^{d-2}+dy^{d-1}f_{2}\Big{(}\frac{1}{y}\Big{)}-y^{d-2}f^{\prime}_{2}\Big{(}\frac{1}{y}\Big{)}$
$\displaystyle+dy^{d-1}g_{2}\Big{(}\frac{x}{y}\Big{)}-xy^{d-2}g^{\prime}_{2}\Big{(}\frac{x}{y}\Big{)}.$

Since $d-2\geq 0$ and $d-1$ is greater than or equal degree of $g_{1}$ and
$f_{1}$, as in the previous estimates, we obtain

$\sup_{(x,y)\in B(1)}\Big{|}\frac{\partial\bar{P}_{\varepsilon}}{\partial
y}(x,y)-\frac{\partial\bar{P}_{0}}{\partial y}(x,y)\Big{|}\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

Moreover, in the chart $(\phi_{1},U_{1})$, the partial derivatives of
$\bar{Q}_{\varepsilon}$ and $\bar{Q}_{0}$ with respect to $x$ reads as

$\displaystyle\frac{\partial\bar{Q}_{\varepsilon}}{\partial x}(x,y)$
$\displaystyle=-y^{d}g^{\prime}_{1}\Big{(}\frac{x}{y}\Big{)}\varphi^{\varepsilon}(0)$

and

$\displaystyle\frac{\partial\bar{Q}_{0}}{\partial x}(x,y)$
$\displaystyle=-y^{d}g^{\prime}_{1}\Big{(}\frac{x}{y}\Big{)}.$

Thus,

$\sup_{(x,y)\in B(1)}\Big{|}\frac{\partial\bar{Q}_{\varepsilon}}{\partial
x}(x,y)-\frac{\partial\bar{Q}_{0}}{\partial x}(x,y)\Big{|}\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

In the chart $(\phi_{1},U_{1})$, the partial derivatives of
$\bar{Q}_{\varepsilon}$ and $\bar{Q}_{0}$ with respect to $y$ reads as

$\displaystyle\frac{\partial\bar{Q}_{\varepsilon}}{\partial y}(x,y)$
$\displaystyle=d\lambda_{2}^{\varepsilon}y^{d-1}-(d+1)y^{d}\int_{0}^{1}f_{1}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)\,dr$
$\displaystyle+y^{d-1}\int_{0}^{1}f^{\prime}_{1}\Big{(}\frac{1}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)\,dr$
$\displaystyle-(d+1)y^{d}g_{1}\Big{(}\frac{x}{y}\Big{)}\varphi^{\varepsilon}(0)+xy^{d-1}g^{\prime}_{1}\Big{(}\frac{x}{y}\Big{)}\varphi^{\varepsilon}(0)$

and

$\displaystyle\frac{\partial\bar{Q}_{0}}{\partial y}(x,y)$
$\displaystyle=d\lambda y^{d-1}-(d+1)y^{d}f_{1}\Big{(}\frac{1}{y}\Big{)}$
$\displaystyle+y^{d-1}f^{\prime}_{1}\Big{(}\frac{1}{y}\Big{)}$
$\displaystyle-(d+1)y^{d}g_{1}\Big{(}\frac{x}{y}\Big{)}+xy^{d-1}g^{\prime}_{1}\Big{(}\frac{x}{y}\Big{)}.$

Since $d-1\geq 0$ and $d$ is greater than degree of $g_{1}$ and $f_{1}$, as in
the previous estimates, we obtain

$\sup_{(x,y)\in B(1)}\Big{|}\frac{\partial\bar{Q}_{\varepsilon}}{\partial
y}(x,y)-\frac{\partial\bar{Q}_{0}}{\partial y}(x,y)\Big{|}\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

In the chart $(\phi_{2},U_{2})$, the partial derivatives of
$\bar{P}_{\varepsilon}$ and $\bar{P}_{0}$ with respect to $x$ reads as

$\displaystyle\frac{\partial\bar{P}_{\varepsilon}}{\partial x}(x,y)$
$\displaystyle=-y^{d-1}\lambda_{2}^{\varepsilon}+y^{d-1}\int_{0}^{1}f^{\prime}_{1}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)\,dr$
$\displaystyle+\beta
y^{d-1}-y^{d}f_{2}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}-xy^{d-1}f^{\prime}_{2}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}$
$\displaystyle-y^{d}g_{2}\Big{(}\frac{1}{y}\Big{)}$

and

$\displaystyle\frac{\partial\bar{P}_{0}}{\partial x}(x,y)$
$\displaystyle=-y^{d-1}\lambda+y^{d-1}f^{\prime}_{1}\Big{(}\frac{x}{y}\Big{)}$
$\displaystyle+\beta
y^{d-1}-y^{d}f_{2}\Big{(}\frac{x}{y}\Big{)}-xy^{d-1}f^{\prime}_{2}\Big{(}\frac{x}{y}\Big{)}$
$\displaystyle-y^{d}g_{2}\Big{(}\frac{1}{y}\Big{)}$

Since $d-1\geq 0$, as in the previous estimates, we obtain

$\sup_{(x,y)\in B(1)}\Big{|}\frac{\partial\bar{P}_{\varepsilon}}{\partial
x}(x,y)-\frac{\partial\bar{P}_{0}}{\partial x}(x,y)\Big{|}\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

In the chart $(\phi_{2},U_{2})$, the partial derivatives of
$\bar{P}_{\varepsilon}$ and $\bar{P}_{0}$ with respect to $y$ reads as

$\displaystyle\frac{\partial\bar{P}_{\varepsilon}}{\partial y}(x,y)$
$\displaystyle=-x(d-1)y^{d-2}\lambda_{2}^{\varepsilon}+dy^{d-1}\int_{0}^{1}f_{1}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)\,dr$
$\displaystyle-
xy^{d-2}\int_{0}^{1}f^{\prime}_{1}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot)\Big{)}\varphi^{\varepsilon}(r)\,dr$
$\displaystyle+dy^{d-1}g_{1}\Big{(}\frac{1}{y}\Big{)}\varphi^{\varepsilon}(0)-y^{d-2}g^{\prime}_{1}\Big{(}\frac{1}{y}\Big{)}\varphi^{\varepsilon}(0)$
$\displaystyle+\beta
x(d-1)y^{d-2}-dxy^{d-1}f_{2}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}-x^{2}y^{d-2}f^{\prime}_{2}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}$
$\displaystyle-
dxy^{d-1}g_{2}\Big{(}\frac{1}{y}\Big{)}+xy^{d-2}g^{\prime}_{2}\Big{(}\frac{1}{y}\Big{)}$

and

$\displaystyle\frac{\partial\bar{P}_{0}}{\partial y}(x,y)$
$\displaystyle=-(d-1)y^{d-2}\lambda+dy^{d-1}f_{1}\Big{(}\frac{x}{y}\Big{)}$
$\displaystyle-x^{2}y^{d-2}f^{\prime}_{1}\Big{(}\frac{x}{y}\Big{)}$
$\displaystyle+dy^{d-1}g_{1}\Big{(}\frac{1}{y}\Big{)}-y^{d-2}g^{\prime}_{1}\Big{(}\frac{1}{y}\Big{)}$
$\displaystyle+\beta
x(d-1)y^{d-2}-xdy^{d-1}f_{2}\Big{(}\frac{x}{y}\Big{)}-x^{2}y^{d-2}f^{\prime}_{2}\Big{(}\frac{x}{y}\Big{)}$
$\displaystyle-
xdy^{d-1}g_{2}\Big{(}\frac{1}{y}\Big{)}+xy^{d-2}g^{\prime}_{2}\Big{(}\frac{1}{y}\Big{)}.$

Since $d-2\geq 0$ and $d-1$ is greater than or equal degree of $g_{1}$ and
$f_{1}$, as in the previous estimates, we obtain

$\sup_{(x,y)\in B(1)}\Big{|}\frac{\partial\bar{P}_{\varepsilon}}{\partial
y}(x,y)-\frac{\partial\bar{P}_{0}}{\partial y}(x,y)\Big{|}\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

Moreover, in the chart $(\phi_{2},U_{2})$, the partial derivatives of
$\bar{Q}_{\varepsilon}$ and $\bar{Q}_{0}$ with respect to $x$ reads as

$\displaystyle\frac{\partial\bar{Q}_{\varepsilon}}{\partial x}(x,y)$
$\displaystyle=xy^{d-1}f^{\prime}_{2}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}$

and

$\displaystyle\frac{\partial\bar{Q}_{0}}{\partial x}(x,y)$
$\displaystyle=xy^{d-1}f^{\prime}_{2}\Big{(}\frac{x}{y}\Big{)}.$

Thus,

$\sup_{(x,y)\in B(1)}\Big{|}\frac{\partial\bar{Q}_{\varepsilon}}{\partial
x}(x,y)-\frac{\partial\bar{Q}_{0}}{\partial x}(x,y)\Big{|}\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

In the chart $(\phi_{2},U_{2})$, the partial derivatives of
$\bar{Q}_{\varepsilon}$ and $\bar{Q}_{0}$ with respect to $y$ reads as

$\displaystyle\frac{\partial\bar{Q}_{\varepsilon}}{\partial y}(x,y)$
$\displaystyle=\beta
dy^{d-1}-(d+1)y^{d}f_{2}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}+xy^{d-1}f^{\prime}_{2}\Big{(}\frac{x}{y}\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot)\Big{)}$
$\displaystyle-(d+1)y^{d}g_{2}\Big{(}\frac{1}{y}\Big{)}+y^{d-1}g^{\prime}_{2}\Big{(}\frac{1}{y}\Big{)}$

and

$\displaystyle\frac{\partial\bar{Q}_{0}}{\partial y}(x,y)$
$\displaystyle=\beta
dy^{d-1}-(d+1)y^{d}f_{2}\Big{(}\frac{x}{y}\Big{)}+xy^{d-1}f^{\prime}_{2}\Big{(}\frac{x}{y}\Big{)}$
$\displaystyle-(d+1)y^{d}g_{2}\Big{(}\frac{1}{y}\Big{)}+y^{d-1}g^{\prime}_{2}\Big{(}\frac{1}{y}\Big{)}.$

Since $d-1\geq 0$, as in the previous estimates, we obtain

$\sup_{(x,y)\in B(1)}\Big{|}\frac{\partial\bar{Q}_{\varepsilon}}{\partial
y}(x,y)-\frac{\partial\bar{Q}_{0}}{\partial y}(x,y)\Big{|}\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

Finally, In the chart $(\phi_{3},U_{3})$, the partial derivatives of
$\bar{P}_{\varepsilon}$ and $\bar{P}_{0}$ with respect to $x$ reads as

$\displaystyle\frac{\partial\bar{P}_{\varepsilon}}{\partial x}(x,y)$
$\displaystyle=-\lambda_{2}^{\varepsilon}+\int_{0}^{1}f^{\prime}_{1}(x\varphi^{\varepsilon}(r)+w_{\perp}^{\varepsilon}(r,\cdot))\varphi^{\varepsilon}(r)\,dr$

and

$\displaystyle\frac{\partial\bar{P}_{0}}{\partial x}(x,y)$
$\displaystyle=-\lambda+f^{\prime}_{1}(x)$

As in the previous estimates, we obtain

$\sup_{(x,y)\in B(1)}\Big{|}\frac{\partial\bar{P}_{\varepsilon}}{\partial
x}(x,y)-\frac{\partial\bar{P}_{0}}{\partial x}(x,y)\Big{|}\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

In the chart $(\phi_{3},U_{3})$, the partial derivatives of
$\bar{P}_{\varepsilon}$ and $\bar{P}_{0}$ with respect to $y$ reads as

$\displaystyle\frac{\partial\bar{P}_{\varepsilon}}{\partial y}(x,y)$
$\displaystyle=g^{\prime}_{1}(y)\varphi^{\varepsilon}(0)$

and

$\displaystyle\frac{\partial\bar{P}_{0}}{\partial y}(x,y)$
$\displaystyle=g^{\prime}_{1}(y)$

As in the previous estimates, we obtain

$\sup_{(x,y)\in B(1)}\Big{|}\frac{\partial\bar{P}_{\varepsilon}}{\partial
y}(x,y)-\frac{\partial\bar{P}_{0}}{\partial y}(x,y)\Big{|}\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

Moreover, in the chart $(\phi_{3},U_{3})$, the partial derivatives of
$\bar{Q}_{\varepsilon}$ and $\bar{Q}_{0}$ with respect to $x$ reads as

$\displaystyle\frac{\partial\bar{Q}_{\varepsilon}}{\partial x}(x,y)$
$\displaystyle=f^{\prime}_{2}(x\varphi^{\varepsilon}(0)+w_{\perp}^{\varepsilon}(0,\cdot))$

and

$\displaystyle\frac{\partial\bar{Q}_{0}}{\partial x}(x,y)$
$\displaystyle=f^{\prime}_{2}(x).$

Thus,

$\sup_{(x,y)\in B(1)}\Big{|}\frac{\partial\bar{Q}_{\varepsilon}}{\partial
x}(x,y)-\frac{\partial\bar{Q}_{0}}{\partial x}(x,y)\Big{|}\to
0\quad\hbox{as}\quad\varepsilon\to 0.$

In the chart $(\phi_{3},U_{3})$, the partial derivatives of
$\bar{Q}_{\varepsilon}$ and $\bar{Q}_{0}$ with respect to $y$ reads as

$\displaystyle\frac{\partial\bar{Q}_{\varepsilon}}{\partial y}(x,y)$
$\displaystyle=-\beta+g^{\prime}_{2}(y)$

and

$\displaystyle\frac{\partial\bar{Q}_{0}}{\partial y}(x,y)$
$\displaystyle=-\beta+g^{\prime}_{2}(y)$

Thus,

$\Big{|}\frac{\partial\bar{Q}_{\varepsilon}}{\partial
y}(x,y)-\frac{\partial\bar{Q}_{0}}{\partial y}(x,y)\Big{|}=0.$

Putting all convergence together we obtain (3.5). ∎

######  Remark 3.2.

Notice that, if $g_{1}(0)=0$ then we have

$|y^{d-1}g\Big{(}\frac{x}{y}\Big{)}||\varphi^{\varepsilon}(0)-1|\leq
K_{2}|y^{d}||\varphi^{\varepsilon}(0)-1|\to 0\quad\hbox{as}\quad\varepsilon\to
0,$

where $K_{2}$ denotes the Lipschitz constant of $g_{1}$ o $B(1)$. Hence, we
can remove the assumption $d-1$ greater than or equal degree of $g_{1}$ in
this case.

A similar remark is valid when $f_{1}(0)=0$.

##  4 Example

Taking $\lambda=1$, $f_{1}(u)=-u^{2}$, $g_{1}(v)=v^{2}$, $\beta=1$,
$f_{2}=u^{2}$ and $g_{2}=v^{2}+2v$, the ODE (1.3) takes the form

$\begin{cases}\dot{u}=-u^{2}+v^{2}-u,\\\ \dot{v}=v^{2}+u^{2}+v.\end{cases}$
(4.1)

In what follows we will determine the phase portrait of (4.1) on the Poincaré
disk. We have

$\begin{cases}P_{0}(u,v)=-u^{2}+v^{2}-u,\\\
Q_{0}(u,v)=v^{2}+u^{2}+v.\end{cases}$

The vector field $X_{0}(u,v)=(P(u,v),Q(u,v))$ has two hyperbolic equilibria,
the origin which is a saddle and $X^{\ast}\approx(0.419643,-0.771845)$ which
is a stable focus.

In the chart $(\varphi_{1},U_{1})$ the vector field $X_{0}(u,v)$ has the
expression

$\begin{cases}\dot{x}=-x^{3}-x^{2}-x-2xy+1,\\\
\dot{y}=-y^{2}-y-x^{2}y.\end{cases}$

Therefore, on $U_{1}$ there is a unique infinite hyperbolic equilibrium
$X^{\infty}\approx(0.543689,0)$ which is a stable node. Since $d=2$, the
diametrically opposite infinite hyperbolic equilibrium is an unstable node.

In the chart $(\varphi_{2},U_{2})$ the vector field $X(u,v)$ has the
expression

$\begin{cases}\dot{x}=-x^{3}+x^{2}+x+2xy+1,\\\
\dot{y}=y^{2}+y-x^{2}y.\end{cases}$

Therefore, on $U_{2}$ there is a unique infinite hyperbolic equilibrium
$X^{\infty}\approx(1.83929,0)$ which is a stable node. Since $d=2$, the
diametrically opposite infinite hyperbolic equilibrium is an unstable node.

Thus, the compactified vector field $\bar{X}_{0}$ has only hyperbolic
equilibria and, there are no saddle-connections in
$\mathbb{D}^{2}\setminus\mathbb{S}^{1}$. According with [9, Chapter 4],
$\bar{X}_{0}$ is a Morse-Smale vector field. Since this kind of vector field
is structurally stable and, by Theorem 3.1, $\bar{X}_{\varepsilon}$ is close
$\bar{X}_{0}$ in the $C^{1}-$norm, for small $\varepsilon$ the phase portrait
of the PDE-ODE (1.1) has the same behavior of (1.3) for the above choices of
$\lambda,\beta,f_{1},f_{2},g_{1}$ and $g_{2}$.

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