# Measurement of the Photon Detection Efficiency of Hamamatsu VUV4 SiPMs at
Cryogenic Temperature

R. Álvarez-Garrote  E. Calvo  A. Canto  J. I. Crespo-Anadón  C. Cuesta  A. de
la Torre Rojo  I. Gil-Botella  S. Manthey Corchado  I. Martín  C. Palomares
L. Pérez-Molina<EMAIL_ADDRESS>A. Verdugo de Osa

###### Abstract

Liquid argon time projection chambers (TPC) are widely used in neutrino
oscillation and dark matter experiments. Detection of scintillation light in
liquid argon TPC’s is challenging because of its short wavelength, in the VUV
range, and the cryogenic temperatures ($\sim$86 K) at which the sensors must
operate. Wavelength shifters (WLS) are typically needed to take advantage of
the high Photon Detection Efficiency (PDE) in the visible range of most of
photondetectors. The Hamamatsu VUV4 S13370–6075CN SiPMs can directly detect
VUV light without the use of WLS, which main benefit is an improved PDE at
these short wavelengths, but also the visible light from WLS. The manufacturer
(Hamamatsu Photonics K.K.) provides a complete characterization of these
devices at room temperature; however, previous studies have indicated a
decrease of the PDE at cryogenic temperature for VUV light. In this work, we
present the measurement of the PDE of VUV4 SiPMs at cryogenic temperature for
different wavelengths in the range [270, 570] nm. A dedicated measurement at
127 nm is also shown.

###### keywords:

Silicon photomultiplier (SiPM) , Photon detectors for UV , Cryogenic
temperature , Photon–detection efficiency (PDE) , DUNE , Noble liquid
detectors

††journal: Nuclear Physics A

[first]organization=Centro de Investigaciones, Energéticas, Medioambientales y
Tecnológicas (CIEMAT),addressline=Avda. Complutense 40, city=Madrid,
postcode=28040, country=Spain

##  1 Introduction

To optimize the energy resolution and sensitivity of the next generation of
liquid argon time proyection chamber (LArTPC) neutrino and direct dark matter
detection experiments it is essential to maximize the light collection process
[1, 2, 3, 4]. The main challenge relies on the detection of the LAr
scintillation photons whose peak emission wavelength is centered at 127 nm
[5]. These photons can be directly detected by VUV photodetectors or
indirectly after using a wavelength–shifter (WLS) typically around 420 nm. The
cryogenic environment is also a constraint on the choice of the photosensor.

As the large scale of the new generation detectors requires the use of large
area devices for light detection, these sensors can also be integrated in
light collection devices [6] or forming arrays [7] where WLS materials are
commonly used. Besides, the use of WLS dopants in LAr, like Xe, or the
installation of WLS coated reflective foils in the TPC walls is becoming a
general practice because of the dimensions of these giant detectors (several
meters) and the relative short scattering length in LAr for light of 127 nm (1
m). The inclusion of sensors sensitive to a large range of wavelengths is
interesting for detecting shifted and direct light with the same device. The
Hamamatsu VUV4 S13370 multi–pixel photon counting is a good candidate with a
photon detection efficiency (PDE) between 20% and 40% for wavelengths between
100 and 600 nm. In contrast to other SiPM models, for which it does not seem
to vary [8, 9], previous studies have shown a decrease in the PDE of these
SiPMs at cryogenic temperatures ($<$200 K) for 127 nm. In this paper, a
measurement of their PDE at cryogenic temperature (CT) for wavelengths ranging
from 127 to 570 nm is presented for the first time. The data could be of
particular interest for experiments requiring multi–wavelength detection.
Additionally, calibrated VUV4 SiPMs can be used in the measurement of the
conversion efficiency of wavelength shifters and for the calibration of
complex devices including wavelength shifting [10].

The paper is organized as follows. In section 2 we introduce the VUV4 SiPMs
characterized in this study. Then, in section 3 the employed methodology is
described, followed by the designed experimental setups in section 4. Finally,
in section 5 the results of the VUV4 SiPMs measurements are presented,
followed by concluding remarks in section 6.

##  2 Hamamatsu VUV4 SiPMs

The Hamamatsu (HPK) VUV4 SiPMs (series: S13370) are directly sensitive to the
LAr scintillation light of 127 nm, and, moreover, they are prepared to carry
out a stable performance at CT. Two VUV4 S13370–6075CN SiPMs have been studied
in this work. In Figure 1 we can see a picture of one SiPM and its dimensional
outline.

(a)

(b)

Figure 1: (a) Image of the S13370–6075CN SiPM model tested in this work. (b)
Dimensional outline [mm]

These SiPMs have a micro–cell pitch of 75 $\rm\mu$m, an effective
photosensitive area of $(6\times 6)$ mm2 with $70\%$ fill factor, and a
typical breakdown voltage at room temperature (RT) of $\sim$53 V. The devices
have a ceramic package and no window. They also contain a metal quenching
resistor to maintain its pulse shape at low temperatures. The main properties
provided by the manufacturer are summarized in Table 1.

Parameter | Value
---|---
Effective photosensitive area |  $(6\times 6)$ mm2
Size |  $(10\times 9)$ mm2
Pixel pitch |  $75$ $\rm\mu$m
Terminal capacitance |  $1.28$ nF
Fill factor | $70\%$
Crosstalk probability |  $5$$\%$
Gain | $5.8\cdot 10^{6}$
Dark counts (typ.) | 0.11 MHz
Breakdown voltage |  $(55\pm 0.2)$ V
Breakdown voltage [86.15 K] |  $(42\pm 0.2)$ V
Table 1: Key parameters of the HPK VUV4 SiPMs (series: S13370) provided by
Hamamatsu Photonics K.K. at an over–voltage of 4 V. RT is assumed unless
otherwise specified.

The SiPMs have been calibrated by the manufacturer at RT at over–voltage (OV)
of 4 V. The PDE for wavelengths from 120 nm to 900 nm measured by HPK is shown
in Figure 2.

Figure 2: Photon detection efficiency at RT and OV = 4 V from 120 nm to 900 nm
light measured and provided by Hamamatsu Photonics K.K.. The legend
corresponds to the serial numbers of the SiPM model tested (blue pentagons
#375 and orange stars #376).

These SiPMs have been considered by several experiments involving liquefied
noble gases [11, 12] and, therefore, there are studies in the literature where
key parameters and findings at CT are reported. Some of the them are of
particular interest to our study and are summarized as follows:

  * •

At a temperature of 163 K and 3.1 OV, the dark noise was measured to be (0.137
$\pm$ 0.002) Hz/mm2. The correlated avalanche noise (i.e., crosstalk and
afterpulses) is measured to be about 16%, three times larger than at RT (see
[11]).

  * •

The reflectivity of these devices in liquid Xe has been measured in [13]. The
specular reflectivity at 15∘ incidence of three samples of VUV4 SiPMs was
found to be around 29%. The same measurement in LAr gave similar results (see
[14]). This parameter can modify the effective PDE if it is not subtracted in
its measurement.

  * •

Finally, at an OV of 3.91 V and a wavelength of 127 nm in liquid argon at 91.2
K, the PDE was measured as 14.7${}^{+1.1}_{-2.4}$% (see [15]). There are not
measurements of the PDE at CT for wavelengths above the VUV range.

##  3 Methodology

###  3.1 PDE measurement for visible light

The PDE will be measured as the ratio of detected to incident photons. In this
case, the photons reflected in the SiPM surface are not subtracted and,
therefore, the measured PDE will depend on the incident angle. The measurement
of the PDE at CT for HPK VUV4 SiPM has been made using as reference the known
PDE at RT provided by the manufacturer using equation 1. The method is based
on the comparison of the light detected by the SiPM at different temperatures
when exposed to light of wavelengths in the range [270, 570] nm.

$\text{PDE}_{\text{CT}}=\frac{\text{Detected light (CT)}}{\text{Detected light
(RT)}}\cdot\text{PDE}_{\text{RT}}\quad.$ (1)

This methodology requires a controlled light source that emits a reproducible
number of photons. Due to the dependence of the SiPM gain with temperature, we
perform calibration measurements at the single photoelectron (SPE) level for
each studied temperature (298 K and 77 K). The SiPM is able to distinguish
signals from one, two or more photoelectrons (PE) and thus we can calculate
the gain by acquiring a charge spectrum. These measurements are always
performed at three different OV so the linear behavior of the gain can be
verified. The PDE is calculated using a higher intensity light regime (between
50–150 PEs) and for an over-voltage of 4 V, which is the OV recommended by the
manufacturer. The followed procedure was:

  1. 1.

Gain calibration measurements at RT for three different OV values.

  2. 2.

High-intensity light pulses at RT for different wavelengths between [270–570]
nm.

  3. 3.

Cool down of the system with liquid nitrogen (LN2).

  4. 4.

Gain calibration measurements at CT for three different OV values.

  5. 5.

High-intensity light pulses at CT for different wavelengths between [270–570]
nm.

###  3.2 PDE measurement for 127 nm light

The measurement at 127 nm requires the propagation of the light in vacuum from
the source, making the measurement much more complex. An alternative method
[16, 17] consists of the generation of 127 nm light directly in LAr with a
radioactive source and comparing the detected number of PE with the
theoretical expectation taking into account the ionizing particle, its energy,
and the LAr scintillation light–yield. This methodology requires the use of
simulations for calculating the solid angle of the sensor with respect to the
generated light. The effect of the LAr purity on the light–yield should also
be taken into account.

##  4 Experimental Setups

###  4.1 Setups for PDE measurement for visible light

Two different setups have been developed for this measurement, a tube at
vacuum equipped with a heat exchanger that will be our reference configuration
and another one, where the tube is filled with gas argon (GAr), used as
cross–check.

The read–out of the SiPM signal is the same in both cases and is shown in
Figure 3

Figure 3: Diagram of the electronic circuit. The dotted blue shaded side is
the front–end electronics, which will be at the same temperature than the SiPM
(left side); while the orange, where the digitization is included, will be
always at room temperature (right side).

The front–end electronics contains an amplifier, the voltage produced by the
SiPM at the input of the amplifier is the result of two 50 $\Omega$ resistors
connected in parallel. The voltage gain of the amplifier is 10 and the total
trans-impedance gain is $V_{\text{output}}/I_{\text{SiPM}}=250\ \Omega$. In
the right part of the diagram we can see that the ADC (model CAEN DT5725S
[18]) is connected to an amplifier (CAEN N979 model [18]). The signal output
is finally acquired with the ADC.

The trigger signal is provided by a pulse generator (model Aim-Tti 2511A) that
is synchronously sent to the ADC and the light source. The ADC provides the
digitized waveforms with 4 ns time and 14–bits resolution from the SiPM
signal.

Figure 4 shows an example average waveform of 20000 events corresponding to
270 nm LED light pulses obtained with the ADC at CT.

Figure 4: Average SiPM waveform obtained with the ADC at CT of 20000 pulses.

For measuring the PDE at different wavelengths we have made use of different
LEDs and lasers to illuminate the SiPM. Table 2 shows the manufacturers and
reference numbers, and the operational wavelengths of these light sources.

Type | Model |  $\lambda$ (nm)
---|---|---
LED | Roithner – UVR270-SA3P | 270
LED | Roithner – UVR280-SA3P | 280
LED | ThorLabs – LED315W | 317
LED | Roithner – XSL-355-3E-R6 | 355
LED | Roithner – LED385-33 | 385
Laser | PicoQuant – PDL828+LDH-P-C-405 | 405
Laser | PicoQuant – PDL828+LDH-P-C-420 | 420
LED | OSRAM – LB 543C-VAW-35 | 470
LED | Vishay – TLHG4200 | 570

Table 2: Information about manufacturers, reference number and wavelength of
the used LEDs and the setup where they are used. The references for the
manufacturers can be found in [19, 20, 21, 22, 23]

For the proposed measurement a very stable light source is needed. We have
performed dedicated measurements to ensure the repeatability of the amount of
light sent to the SiPM. A reference sensor has been illuminated with light
sources of different wavelengths in four separated data taking periods. The
mean value of the detected light in the four measurements for each wavelength
is shown in Figure 5. We can confirm the repeatability of the light intensity
as the deviations are smaller than 1% for all light sources except those of
270 and 385 nm, which are within 5%. These errors have been propagated to the
PDE calculation.

Figure 5: Mean number of detected PEs of the four individual measurements for
light sources of different wavelengths. All the points are represented with
error bars that represent the deviations observed in the measurements.

####  4.1.1 Vacuum heat exchanger setup

The main goal of the setup is to provide a controlled mechanism to change the
temperature of the photosensor without disrupting or altering the light
propagation medium to ensure that the same amount of light reaches the SiPM
independently of its temperature. The SiPM is cooled down with a heat
exchanger, the so–called cold–finger; to thermally insulate the SiPM attached
to the cold–finger, both the SiPM and the light source are placed in a tube at
vacuum. The technical design scheme of this setup can be seen in Figure 6(a).
The SiPM is placed at the end of a 1 m tube, where a $10^{-4}$ mbar vacuum is
created. The power supply and the output analog signal cables are connected to
the feed–through. To cool down the stainless steel tube, and thus the SiPM,
its lower part is introduced into a small vessel filled with LN2 as shown in
Figure 6(b).

(a)

(b)

Figure 6: (a) Technical scheme of the mechanical structure of the vacuum cold
finger setup. The vacuum is created in the whole stainless steel tube. The
corrugated part is placed in the scheme gap of the rigid tube to join both
parts in a flexible way. The heat exchanger is in the lower part of the tube,
where the SiPM is allocated, and it is eventually submerged in LN2. (b) The
lower part of the tube with the SiPM inside a vessel filled with LN2 to cool
down the SiPM.

The SiPM is facing an optical fiber, which runs from the feed–through and is
externally connected to the different light sources. The temperature sensor is
a PT100 manufactured by Heraeus [24] with a temperature range from 77 K to 933
K. It is located in the structure that supports the SiPM as can be seen in
Figure 7 (Left). The SiPM temperature is considered to be the one indicated by
the temperature sensor. The heat exchanger consists of a shielding wire mesh
wrapped around the SiPM holding bracket in contact with the tube (see Figure 7
(Right)). This part is submerged (once closed) in LN2 to cool down the SiPM by
thermal contact with the tube.

Figure 7: Left: Technical scheme of the mechanical structure of the vacuum
cold finger setup. The diffuser, the optical fiber, the SiPM and the
temperature sensor (PT100) are placed inside the tube. Right: Close–up view of
the mechanical implementation of the vacuum pipe containing the SiPM
photosensor and cold finger arrangement.

####  4.1.2 GAr vessel setup

A second setup have been designed to cross–check the results. In this case, we
have improved the temperature control and include the feasibility for
operating with GAr together with an 241Am source. This will allow us to
determine the PDE at 127 nm in the future. The determination of produced
scintillation light from excited gaseous argon is very challenging because of
its dependence on temperature and geometrical factors and requires further
development. However, the setup can also be used for the measurement of the
PDE for visible light. In fact, some eventual drawbacks of a non–vacuum
system, like the water condensation at CT, are avoided thanks to the filling
with GAr.

(a)

(b)

Figure 8: (a) Technical design of the GAr setup with dimensional information
in mm. (b) Zoom on the structure (yellow circle on the left) where the SiPM,
the PT100 sensors and the optical fiber are installed.

A schematic representation of the setup can be seen in Figure 8(a). It
consists of a 50 l vessel that holds a U–shaped tube through which GAr is
continuously circulating. In the center of the tube there is a holder where
the optical fiber is fixed facing the SiPM (see Figure 8(b)). A light diffuser
is used to homogenize the light reaching the SiPM from the fiber, reducing the
impact of the different refraction index for warm and cold GAr. As it is
closely allocated we can assume that the same amount of light is reaching the
SiPM at both RT and CT.

Once the setup is assembled, vacuum is created in the tube where the
photodetector was located, reaching levels of $10^{-4}$ mbar. Then the GAr is
allowed to circulate continuously through the tube. The gas is expelled to the
outside medium through a non–return valve that prevents contamination with
outside air. After taking data with several wavelengths light sources at RT,
the 50 l vessel is filled with LN2 to cool the U–tube, and therefore the SiPM
by thermal contact. Four thermal sensors located at different heights were
used to monitor the temperature change throughout the vessel. The required
measurements were successfully repeated at different cryogenic temperatures
and the results are described in section 5.

###  4.2 Setup for PDE measurement for 127 nm light

This measurement is done in the cryogenic setup prepared for the PDE
measurement of the DUNE X–ARAPUCA [17]. In this setup, we made use of a 300 l
cryogenic vessel with different concentric volumes, which schematic is shown
in Figure 9(a). There are two concentric internal volumes, a larger one (100
l), where the LN2 is introduced and a smaller one (18 l), where the VUV4 SiPMs
are located together with the X–ARAPUCA, and filled with GAr. In this 18 l
container, the GAr is liquefied by thermal contact with the LN2 of the
surrounding volume. This is achieved by controlling the pressure parameters
that regulate the temperature values necessary to carry out the liquefaction.
The GAr 99.9999% is liquefied with LN2 at 2.7 bar. To avoid outgassing we
perform successive vacuum cycles before introducing the optical and electrical
components that will be used to perform the measurements.

(a)

(b)

Figure 9: Setup scheme used for obtaining the absolute efficiency of the VUV4
SiPMs at 127 nm. (a) Cryogenic vessel with its concentric volumes where GAr is
liquified. (b) Picture of the black box holding the sensors together with the
241Am source with its corresponding diagram.

The 127 nm scintillating light is produced by an 241Am alpha source placed
together with two SiPMs in an opaque box, as shown in Figure 9(b). It emits
$\alpha$ particles with 5.485 MeV (84.45%) and 5.443 MeV (13.23%) energies
with an activity of (54.53 $\pm$ 0.82) Bq [25]. The $\alpha$ particles deposit
their energy inside the 4 cm sized black box, and the produced photons reach
the VUV4 SiPMs. The ADC acquisition is triggered by the coincidence of the two
SiPMs signal above a certain threshold. The setup includes an optical fiber
attached to a LED that allows the calibration of the SiPMs in the single PE
regime. In this case, the data taking is triggered by the pulse generator
synchronously with the LED.

##  5 Results

In the following section, the experimental results obtained with the setups
introduced in section 4 are presented. The PDE for visible light is determined
from the ratio of the light collected at CT and RT at a light regime for which
we have demonstrated a constant illumination of the sensors. Since the gain
and other operation parameters of the SiPMs depend on the temperature, the
estimation of the amount of PEs detected is crucial for a reliable measurement
of the PDE. First of all, the SiPM gain at CT and RT is computed from the
calibration runs. Then a dedicated study on the cross–talk at CT is presented
and finally, the results for the PDE are discussed.

###  5.1 Gain calibration

The gain calculation relies on the calibration of the sensor at the single
photo–electron level. While at CT we have a well–determined baseline and a
clear signal, at RT there is much more dark–count noise making the integration
of individual pulses more complicated. Therefore the data sample is filtered
to get events with a single pulse and a clean baseline in the integration
window.

Experimentally, the gain can be obtained from the charge spectra after
integrating the individual pulses. These spectra are fitted to N Gaussians,
corresponding to the electronic noise (pedestal) and the charge deposited by
1… N-1 PE. This is presented in Figure 10 for the lowest OV and at RT.

Figure 10: Example charge spectrum corresponding to a calibration at OV $=3$ V
at RT with a three Gaussian fit.

The distance between consecutive maximums of the Gaussians is the gain of the
system.The error accounts for the dispersion of the measurements for different
calibration runs. Figure 11 shows these values graphically for both
temperature regimes and their respective linear fit.

Figure 11: Mean gain values obtained at different temperatures versus
overvoltage and their respective linear fit. For the same OV, the SiPM gain at
CT (blue circles) is higher than at RT (red triangles), as expected.

###  5.2 Cross–talk determination

The number of detected PEs detected by the SiPM should be corrected by the
cross–talk probability (that is the probability that an avalanching microcell
causes an avalanche in a second microcell). It is known that the cross–talk
probability changes with temperature [11, 26]. So, it is critical to correctly
account for this effect in the PDE determination.

We have performed a dedicated analysis to extract the cross–talk value for the
VUV4 SiPMs (series: S13370) at CT. To describe the joint effect of cross–talk
and afterpulses in the PE distribution we have made use of a composite
Poissonian as described in [27, 28]. Figure 12 represents the different
predicted distributions for the normalized charge histograms depending on the
cross–talk value and how it compares with a pure Poisson distribution.

Figure 12: Compound Poisson distribution function for $p=$ [0.05, 0.15, 0.25,
0.35] being $p=0$ the pure Poisson distribution.

For computing the cross–talk probability ($P_{CT}$) at CT we have taken a
calibration run with enough light so that we can represent the plot shown in
Figure 13 and perform a Vinogradov fit to the mentioned composite Poissonian.

(a)

(b)

Figure 13: (a) Charge histogram for a calibration run at OV $=4$ V. (b)
Normalized density charge histogram fitted with a Vinogradov fit to extract
the cross–talk value at CT.

The $P_{CT}$ computed for this SiPM at CT and an OV 4V is $(14.84\pm 0.24)\%$
which is in perfect agreement with the presented value in [15]. At RT we use
the value measured by HPK of $5\%$ because after filtering the signal, the
resulting charge histogram does not have enough statistics for performing a
Vinogradov fit. From the probability value we compute the correction factor as
$f_{CT}=1/(1+K_{dup})$ where $K_{dup}=P_{CT}/(1-P_{CT})$ [27].

###  5.3 Cryogenic Photon Detection Efficiency computation

####  5.3.1 PDE measurement for visible light

The amount of light detected by the SiPM at a given overvoltage is measured in
terms of PEs, and is determined as the integrated charge in a 3 $\mu$s
waveform divided by the SiPM gain at the corresponding temperature for this
overvoltage.

The ratio of the number of PEs measured at RT and CT for the same light
intensity at a given wavelength is used to determine the PDE at CT as shown in
equation 1. The obtained value should be corrected by the increase of the
cross–talk at CT.

The PDE values obtained in the visible wavelength range are tabulated in Table
3 and shown in Figure 14 for one SiPM.

$\lambda$ (nm) | PDE${}_{\text{RT}}$ ($\%$) | PDE${}_{\text{CT}}$ ($\%$) | Ratio
---|---|---|---
270 | 19.10 | 8.05 ± 0.36 | 0.42 ± 0.02
280 | 21.00 | 8.66 ± 0.25 | 0.41 ± 0.01
317 | 26.45 | 10.67 ± 0.41 | 0.40 ± 0.02
355 | 25.05 | 11.00 ± 0.49 | 0.44 ± 0.02
385 | 29.35 | 15.52 ± 0.83 | 0.53 ± 0.03
405 | 32.10 | 17.27 ± 0.36 | 0.54 ± 0.01
420 | 33.50 | 21.02 ± 0.44 | 0.63 ± 0.01
470 | 34.80 | 27.76 ± 0.53 | 0.80 ± 0.02
570 | 28.40 | 20.11 ± 0.51 | 0.71 ± 0.02
Table 3: PDE for both temperature regimes. The RT one is given by the
manufacturer, while the CT one is the obtained after the analysis described in
this document. Figure 14: PDEs values at RT (grey diamonds) and at CT for
vacuum setup (green circles).

We can see that the PDE at CT shows a clear decrease compared to its value at
RT. Not only that, but there is a dependence with the wavelength that is
presented in Figure 15.

Figure 15: Ratio between the PE measured at CT and RT for the studied range of
wavelengths.

We can clearly notice a minimum at $\sim$300 nm and a maximum at $\sim$500 nm.

The same procedure is followed for the GAr setup, where in order to ensure
that the light reaching the SiPMs is independent of the temperature of the
setup, we allocate the optical fiber close enough to the sensors and
additionally we use a diffuser so that the light is emitted homogeneously.

Additionally, as we have increased the number of temperature sensors from one
to four, we have a better determination of the setup temperature. This allows
us to do measurements at intermediate temperatures during cooling. In Figure
16 we can see the results obtained in GAr for two cryogenic temperatures, 128
K and 86 K.

Figure 16: PDEs values at RT (grey diamonds), for GAr setup at 128 K (pink
down triangles) and at 86 K (orange squares) and for vacuum setup at 77 K
(green circles).

We can clearly notice the same decrease of PDE with temperature, validating
our results at 77 K. The measurement at 128 K gives additional information on
the temperature–dependence of the PDE.

####  5.3.2 PDE measurement for 127 nm light

As introduced in section 3, we have performed another measurement by
submerging the SiPMs in LAr together with an 241Am source to measure the PDE
at CT and at 127 nm. For this computation, we compare the light collected by
the SiPM with the expected number of PEs taking into account the position of
the SiPMs relative to the source and the scintillation light–yield of an
$\alpha$ particle of 5.48 MeV in LAr. To determine the fraction of the
produced scintillation light that hits the SiPM area, a detailed simulation of
the 4 cm side black box, including the dimensions of the source ($\oslash=20$
mm) and a realistic simulation of the sensor holders, is made. Figure 17
features a view of the simulated elements and the trajectories of the
isotropic generated photons.

Figure 17: Simulation of optical photons from an extense source to study their
geometrical distribution and the number reaching the VUV4 SiPMs with GEANT4
[29].

The number of generated scintillation photons is determined considering a
light–yield of 51000 photons/MeV [30] and a quenching factor of 0.70 for
$\alpha$ particles [31]. The increase of the cross–talk at CT is corrected as
in the measurement for visible light. Finally, the quenching of the
scintillation light production by impurities in the LAr that affects the
triplet emission with a longer decay time is computed by following the next
equation [32]:

$f^{-1}_{\text{purity}}\equiv
A_{\text{triplet}}\frac{\tau^{\text{exp}}_{\text{{slow}}}}{\tau^{\text{th}}_{\text{{slow}}}}+A_{\text{singlet}}=(0.94\pm
0.02)\quad,$ (2)

where $\rm\tau^{th}_{\text{slow}}=1.41\ \mu$s, $\rm A_{triplet}=0.23$, $\rm
A_{singlet}=0.77$ are the theoretical parameters describing an ideal
scintillation profile and are obtained from the literature [17, 33]. The
measured $\tau^{\text{exp}}_{\text{slow}}$ is obtained by fitting the
deconvoluted scintillation profile of the SiPMs as shown in Figure 18.

Figure 18: Fit of the deconvoluted scintillation profile for the VUV4 SiPM.

As introduced in section 4, we have made use of two VUV4 SiPMs that have been
calibrated as in section 5.1. For obtaining the charge associated to the LAr
scintillation photons we acquire signals from a coincident trigger, resulting
in an averaged charge of $(55.00\pm 1.99)$ PE. The expected number of photons
in each SiPM after applying the purity factor is $495$ PE. With the cross–talk
correction, the estimated PDE at CT for the LAr scintillation light is:

$\text{PDE}_{\text{127 nm}}=(12.32\pm 1.09)\%\quad.$

The PDE has been computed individually for the two SiPMs and the shown result
is the mean value. We can see that the absolute measurement for 127 nm shows a
decrease in PDE at CT. Moreover, this value can be compared with the recently
published value in [15], $(14.7^{+1.1}_{-2.4})$%. In this study they have
subtracted the reflected photons ($24\%$) from the contribution of photons
hitting the sensor, contrary to our method. The result is compatible within
errors with our current result if the same definition is applied. The main
uncertainty of our measurement is $8.3\%$ coming from the geometrical
tolerance of our setup, i.e., the uncertainty on its position, followed by a
$1.9\%$ uncertainty due to the purity correction.

##  6 Conclusions

In this work, we have presented the measurement of the absolute PDE of HPK
VUV4 S13370–6075CN SiPMs at CT for different wavelengths in the range [270,
570] nm and at 127 nm with a detailed description of the setups developed for
this measurement. The results are summarized in Figure 19.

Figure 19: PDEs values at RT (grey diamonds), at 128 K for GAr setup (pink
down triangles), at 86 K for GAr setup (orange squares) and at 86 K for LAr
setup (purple up triangle) and at 77 K for vacuum setup (green circles).

We can clearly notice that for the three presented setups we obtain compatible
results. The most remarkable conclusion is the decrease of the PDE for these
SiPMs when operating at CT. The PDE reduction at CT depends on the wavelength,
being maximal ($\sim$60%) at lower wavelengths ($<$350 nm) and about 20% for
wavelengths around 450 nm. The PDE has been measured at three different
cryogenic temperatures: 77 K, 86 K and 128 K. The sensitivity of the
measurements allows to observe a higher PDE at 128 K. Besides that, this study
confirms the value of the PDE at 127 nm presented in previous works.

## Acknowledgments

The present research has been supported and partially funded by European Union
NextGenerationEU/PRTR, European Union’s Horizon 2020 Research and Innovation
programme under Grant Agreement No. 101004761, by
MCIN/AEI/10.13039/501100011033 under Grants No. PID2019–104676GB–C31,
RYC2021–031667–I, PRE2019–090468 and PRE2020–094863 of Spain.

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# On existence and uniqueness of solutions of thermal phase field models with
a general class of nonlinearities

###### Abstract.

We prove the existence and uniqueness of solutions for a family of nonlinear
parabolic systems related to phase field models taking in account variations
of temperature and the possibility of a general class of nonlinearities. The
present results generalizes in certain aspects the already published ones in
the literature.

###### Key words and phrases:

Phase field model, Nonlinear parabolic systems, Leray- Schauder degree.

###### 2010 Mathematics Subject Classification:

Primary 35K61; Secundary 35Q99.

The first author was partially supported by FAPESP, Brazil, grant
2013/22328-8.

The second author was partially supported by CNPq, Brazil, grant
307833/2008-9.

The third author was partially supported by FAPESP, Brazil, grants
2010/10087-8, 2012/15379-2.

A. L. A. de Araujo
Universidade Federal de Viçosa, CCE, Departamento de Matemática, Viçosa, MG,
Brasil
E-mail<EMAIL_ADDRESS>
J.L. Boldrini
Universidade Estadual de Campinas, IMECC, Departamento de Matemática,
Campinas, SP, Brasil
E-mail<EMAIL_ADDRESS>
B.M.R. Calsavara
Universidade Estadual de Campinas, IMECC, Departamento de Matemática,
Campinas, SP, Brasil
E-mail<EMAIL_ADDRESS>
##  1\. Introduction

Let $\Omega\subset\mathrm{I\\!R\\!}^{N}$ be a bounded domain and $T>0$ a
finite constant. We denote $Q=\Omega\times(0,T)$ and
$\sum=\partial\Omega\times(0,T)$ and consider the following nonlinear
parabolic system

$\left\\{\begin{array}[]{lll}\displaystyle u_{t}+l\phi_{t}=\Delta
u+f(x,t)&\textup{in}&Q,\\\
\displaystyle\phi_{t}=\Delta\phi+F(x,t,\phi)+u&\textup{in}&Q,\\\
\displaystyle\partial
u/\partial\nu=\partial\phi/\partial\nu=0&\textup{on}&\sum,\\\ \displaystyle
u(x,0)=u_{0}(x),\ \phi(x,0)=\phi_{0}(x)&\textup{for
all}&x\in\Omega,\end{array}\right.$ (1)

that may be used to model the solidification/melting process of pure materials
in the region $\Omega$ when one takes in consideration the variations of
temperature. Here the unknowns are the functions $u$ and $\phi$, related
respectively to the temperature distribution of the material and the phase
function, called phase-field, used to distinguish between the liquid and solid
phases of the material. $F(x,t,\phi)$ in the phase-field equation is related
to the derivative with respect to $\phi$ of the atomic interaction potential
of the material being considered; $f(x,t)$ is related to the density of heat
sources or sinks, and the constant $l>0$ is related to the latent heat. Since
they are not relevant for the purposes the discussion in this article, to
simplify the notation, we assumed values the other material constants to be
one; our results would exactly the same for other values of these constants.

We remark that the nonlinearity in (1) is given by $F(x,t,\phi)$ and that to
model different realistic situations one must consider different types of such
nonlinearities. Thus, it is important to understand the behavior of such
systems for several classes of $F(x,t,\phi)$, and in particular, to know
results on existence of solutions, their regularity, uniqueness, asymptotic
behavior, and so on.

The first rigorous mathematical analysis for a phase-field model as in (1) was
done by Caginalp in [2] by considering the nonlinearity coming from the
classical two-wells potential: $F(\phi)=(\phi-\phi^{3})/2$. Along the time
other types of non-linearity were considered, among others are the works of
Hoffman and Jiang [5], Bates and Zheng [1], and Moroşanu, Motreanu [9, 10] and
Cârja et all [3] (this last paper considers the case of non homogeneous
boundary conditions).

Our objective in this article is to present a result on existence, uniqueness
and regularity of solutions of system (1) for a class of nonlinearities
$F(x,t,\phi)$ that includes the one derived from the two-wells potential in
the usual tridimensional case and is related to the class considered in
Moroşanu and Motreanu [9, 10]. It is a bit difficult to compare in general
terms ours and Moroşanu and Motreanu’s classes, but, as we will show, at least
for the usual and important subclass of autonomous and homogeneous
nonlinearities, that is, nonlinearities that depend only on $\phi$, our
subclass is strictly larger than the corresponding subclass in [10]. Details
of these comparisons and further commentaries are presented in Section 3.

As for the techniques we use to prove our results, similarly as in the works
of Hoffman and Jiang [5] and Moroşanu and Motreanu [9, 10], at certain point
we use the Leray-Schauder fixed point theorem.

The paper is organized as follows. In Section $2$ we fix the notations, and
recall certain concepts and results that will use later on; in Section $3$ we
explicitly state our technical assumptions and our main result concerning
existence, regularity and uniqueness of solutions; we also discuss with some
detail the relations between ours and the results in [9, 10]; in Section $4$,
we analyze an auxiliary problem related to the phase-field equation in (1). We
remark that the same auxiliary problem was already treated in [9, 10], but we
have to reconsider it because in the present work we need some new specific
arguments. Finally, Section $5$ is dedicated to the proof of our main result.

##  2\. Preliminaries

Let $\Omega\subset\mathrm{I\\!R\\!}^{N}$ be an open and bounded domain with a
sufficiently smooth boundary, $\partial\Omega\in C^{3}$ and
$Q=\Omega\times(0,T)$ the space-time cylinder with lateral surface
$\sum=\partial\Omega\times(0,T)$. For $t\in(0,T]$, we denote
$Q_{t}=\Omega\times(0,t)$.

Given $X,Y$ Banach spaces, we denote by $X\hookrightarrow Y$ the continuous
immersion of $X$ in $Y$ and by $\textstyle{\displaystyle
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$ the
compact immersion of $X$ in $Y$.

Next, to ease the references, we state the following embedding result for
Sobolev spaces of type $W^{r,s}_{p}(Q)$, which is a particular case of Lemma
$3.3$ in Ladyzhenskaya et all [6, pp. 80], obtained by taking $l=1$ and
$r=s=0$ in there.

######  Lemma 2.1.

Let $\Omega$ a domain of $\mathrm{I\\!R\\!}^{N}$ with boundary
$\partial\Omega$ (at least satisfying the cone property). Then for any
function $u\in W^{2,1}_{p}(Q)$ we also have $u\in L^{q}(Q)$, and it is valid
the following inequality

$\displaystyle\displaystyle\|u\|_{L^{q}(Q)}\leq C\|u\|_{W^{2,1}_{p}(Q)},$

provided that: $q=\infty$ if $p>\frac{N+2}{2}$; $q\geq 1$ if $p=\frac{N+2}{2}$
and $q=\frac{p(N+2)}{N+2-2p}$ if $p<\frac{N+2}{2}$. The constant $C$ depends
only on $T,p,q,N$ and $\Omega$.

##  3\. Main result, comparisons and commentaries

For the rest of this article we use the following technical hypotheses:

$(H_{0})$:

$f$ (or $g)\in L^{p}(Q)$ with $p\geq 2$, $\phi_{0}$ (or $u_{0})\in
W^{2-2/p}_{p}(\Omega)$ such that
$\displaystyle\partial\phi_{0}/\partial\nu=0$ (or $\displaystyle\partial
u_{0}/\partial\nu=0$) on $\sum=\partial\Omega\times(0,T)$.

$(H_{1})$:

There is a constant $a_{0}\in\mathrm{I\\!R\\!}$ such that for any $(x,t)\in Q$
and $z_{1},z_{2}\in\mathrm{I\\!R\\!}$, we have

$\displaystyle(F(x,t,z_{1})-F(x,t,z_{2}))(z_{1}-z_{2})\leq
a_{0}(z_{1}-z_{2})^{2}.$

$(H_{2})$:

There is a function
$G:Q\times\mathrm{I\\!R\\!}^{2}\rightarrow\mathrm{I\\!R\\!}$ such that for any
$(x,t)\in Q$ and $z_{1},z_{2}\in\mathrm{I\\!R\\!}$, there holds

$(F(x,t,z_{1})-F(x,t,z_{2}))^{2}\leq G(x,t,z_{1},z_{2})(z_{1}-z_{2})^{2},$

where $G(x,t,z_{1},z_{2})\leq c_{0}(1+|z_{1}|^{2r-2}+|z_{2}|^{2r-2})$ with
constants $c_{0}$ and $r\geq 1$.

$(H_{3})$:

For $N\in IN-\\{0\\}$, the allowed values of the parameter $r$ in the previous
hypothesis are either any $r\geq 1$ when $p\geq(N+2)/2$ or $1\leq
r<\frac{N+2}{N+2-2p}$ when $p<(N+2)/2$.

$(H_{4})$:

$F:Q\times\mathrm{I\\!R\\!}\rightarrow\mathrm{I\\!R\\!}$ is a Caratheodory
function, i. e., $F(.,.,z)$ is measurable on $Q$, $\forall
z\in\mathrm{I\\!R\\!}$, and $F(x,t,.)\in
C(\mathrm{I\\!R\\!},\mathrm{I\\!R\\!})$, $\forall(x,t)\in Q$, $F(.,.,0)\in
L^{\infty}(Q)$.

For future use in the comparison with the results of [9], [10], and also to
obtain better regularity results for the solutions, we state below the
following results.

######  Lemma 3.1.

Assumption ($H_{2}$) and ($H_{4}$) implies that $F$ fulfills the growth
condition

$|F(x,t,z)|\leq a(1+|z|^{r}),\forall(x,t,z)\in Q\times\mathrm{I\\!R\\!},$ (2)

where $a>0$ is a constant.

Proof: By setting $z_{1}=z$ and $z_{2}=0$ in $(H_{2})$, we get

$\begin{array}[]{lll}|F(x,t,z)|&\leq&|F(x,t,0)|+G^{1/2}(x,t,z,0)|z|\\\
&\leq&|F(x,t,0)|+c_{0}^{1/2}(1+|z|^{2r-2})^{1/2}|z|,\ \forall\
z\in\mathrm{I\\!R\\!}.\end{array}$

Since $F(.,.,0)\in L^{\infty}(Q)$, relation (2) follows.

######  Lemma 3.2.

Assumption ($H_{1}$) and ($H_{4}$) implies that for some constant $d_{0}>0$,

$F(x,t,z)z\leq d_{0}(1+z^{2}),\quad\forall(x,t)\in Q,\,z\in\mathrm{I\\!R\\!}.$
(3)

Proof: By setting $z_{1}=z$ and $z_{2}=0$ in $(H_{1})$, we get $F(x,t,z)z\leq
F(x,t,0)z+a_{0}z^{2}$, $\forall\ z\in\mathrm{I\\!R\\!}$ . Since $F(.,.,0)\in
L^{\infty}(Q)$, we have
$F(x,t,z)z\leq\|F(\cdot,\cdot,0)\|^{2}_{L^{\infty}(Q)}/2+z^{2}/2+a_{0}z^{2}$,
$\forall\ z\in\mathrm{I\\!R\\!}$, and the result follows with
$d_{0}=\max\left\\{\|F(\cdot,\cdot,0)\|^{2}_{L^{\infty}(Q)}/2,1/2+a_{0}\right\\}$.

The main purpose of the present article is to prove the following result

######  Theorem 3.1.

Under assumptions $(H_{0})-(H_{4})$. Then the problem (1) has a unique
solution $(u,\phi)\in W^{2,1}_{p}(Q)\times W^{2,1}_{p}(Q)$. The solution
$(u,\phi)$ satisfies

$\|u\|_{W^{2,1}_{p}(Q)}+\|\phi\|_{W^{2,1}_{p}(Q)}\leq
C(1+\|\phi_{0}\|_{W^{2-2/p}_{p}(\Omega)}+\|u_{0}\|_{W^{2-2/p}_{p}(\Omega)}+\|f\|_{L^{p}(Q)}),$
(4)

where the constants $C$ depends only $|\Omega|,\alpha,T,p,r,c_{0},a,d_{0}$.

The proof of this result will be given in Section $5$. But before proceeding
with the preparation for such proof, in the following we will compare ours and
the corresponding results in the articles by Moroşanu and Motreanu [9], [10];
just concerning the class of nonlinearities being considered, we also briefly
compare ours with the one in the article by Cârja et all [3].

In the paper of Moroşanu and Motreanu [9], the authors used the following
assumptions about the non-linearity $F$:

$(M_{1})$:

There is a constant $a_{0}\in\mathrm{I\\!R\\!}$ such that for any $(x,t)\in Q$
and $z_{1},z_{2}\in\mathrm{I\\!R\\!}$ there holds

$(F(x,t,z_{1})-F(x,t,z_{2}))(z_{1}-z_{2})\leq a_{0}(z_{1}-z_{2})^{2}.$

$(M_{2})$:

There is a function
$\overline{F}:Q\times\mathrm{I\\!R\\!}^{2}\rightarrow\mathrm{I\\!R\\!}$ such
that for any $(x,t)\in Q$ and $z_{1},z_{2}\in\mathrm{I\\!R\\!}$, we have

$(F(x,t,z_{1})-F(x,t,z_{2}))^{2}\leq\overline{F}(x,t,z_{1},z_{2})(z_{1}-z_{2})^{2},$

where $\overline{F}(x,t,z_{1},z_{2})\leq
c_{0}(1+|z_{1}|^{2r-2}+|z_{2}|^{2r-2})$, with some constants $c_{0}$ and
$r\geq 1$.

$(M_{3})$:

There exist functions $k$ and $h$ such that $F(x,t,z)=k(x,t,z)-h(z)$,
$\forall(x,t)\in Q$, $z\in\mathrm{I\\!R\\!}$, where
$k:Q\times\mathrm{I\\!R\\!}\rightarrow\mathrm{I\\!R\\!}$ is a Caratheodory
function, i. e., $k(.,.,z)$ is measurable on $Q$, $\forall
z\in\mathrm{I\\!R\\!}$, and $k(x,t,.)\in
C(\mathrm{I\\!R\\!},\mathrm{I\\!R\\!})$, $\forall(x,t)\in Q$, $F(.,.,0)\in
L^{\infty}(Q)$ and $h\in C^{1}(\mathrm{I\\!R\\!},\mathrm{I\\!R\\!})$. In
addition, the function $k$ and $h$ verify the assumptions

$(i)$:

$k(x,t,z)^{2}\leq a_{1}h(z)z+a_{2}(1+z^{2})$, $\forall(x,t,z)\in
Q\times\mathrm{I\\!R\\!}$, for some constants $a_{1}>0$, $a_{2}>0$,

$(ii)$:

$-b_{0}\leq h^{\prime}(z)\leq b_{1}(1+|z|^{r-1})$, $\forall
z\in\mathrm{I\\!R\\!}$ and $1\leq r<(N+2)/N-2$ if $N>2$, for some constants
$b_{0},b_{1}>0$.

With assumption $(M_{2})$ the authors proved (2), i.e,

$|k(x,t,z)-h(z)|\leq a(1+|z|^{r}),\quad\forall\ (x,t,z)\in
Q\times\mathrm{I\\!R\\!},\mbox{ where }a>0\mbox{ is a constant. }$

######  Lemma 3.3.

The condition $(M_{3})(i)$ implies the relation

$\displaystyle F(x,t,z)z\leq d_{0}(1+z^{2}),\,\,\,\,\forall(x,t)\in
Q,\,z\in\mathrm{I\\!R\\!},$

where $d_{0}$ is a positive constant. That is, the above condition
$(M_{3})(i)$ implies (3).

Proof: The condition $(M_{3})(i)$ and Young’s inequality show that

$\displaystyle\begin{array}[]{lll}\displaystyle
k(x,t,z)z=\left(\frac{\sqrt{2}}{\sqrt{a_{1}}}k(x,t,z)\right)\left(\frac{\sqrt{a_{1}}}{\sqrt{2}}z\right)&\leq&\displaystyle\frac{1}{a_{1}}k(x,t,z)^{2}+\frac{a_{1}}{4}z^{2}\\\
\displaystyle&\leq&h(z)z+d_{0}(1+z^{2}),\end{array}$

for all $(x,t)\in Q$ and $z\in\mathrm{I\\!R\\!}$.

Hence

$(k(x,t,z)-h(z))z\leq d_{0}(1+z^{2}),\,\,\,\,\,\forall(x,t)\in
Q,\,z\in\mathrm{I\\!R\\!},$

and so (3) is established.

In the paper of Moroşanu and Motreanu [10], the authors used on non-linearity
$F$ the assumptions (2), $(M_{1})$, $(M_{2})$ and

$(M_{4})$:

$F(x,t,z)|z|^{pr-r-1}z\leq\alpha(1+|z|^{pr-1})-\beta|z|^{pr}$, for constants
$\alpha,\beta>0$ and $r\geq 1$ provided

$r<\frac{N+2}{N+2-2p},\;\mbox{ if }\;N+2-2p>0.$

######  Remark 3.1.

It follows from Lemma 3.3, $(M_{1})$ and $(M_{2})$ that the nonlinearity,
$F(x,t,z)$ considered in of Moroşanu and Motreanu [9] satisfies our hypotheses
$(H_{1})-(H_{4})$.

In the same way, one can prove that the class of nonlinearity considered in
Moroşanu [10] also satisfies our hypotheses $(H_{1})-(H_{4})$.

As for the paper of Cârja et all [3], the authors consider an autonomous non-
linearity $F$ of form

$F(\phi)=f(\phi)-a_{s}|\phi|^{s-1}\phi,\,\,\forall\,\phi\in\mathbb{R},$

with $a_{s}>0$ and $s\geq 3$ satisfying

$s<\frac{N+2}{N+2-2p},\;\mbox{ if }\;N+2-2p>0,$

while $f(\phi)\in C^{1}(\mathbb{R})$ fulfills, for constants $b_{1},b_{2}>0$,
the following properties:

$|f^{\prime}(\phi)|\leq b_{1}(1+|\phi|^{s-2}),\,\,\forall\,\phi\in\mathbb{R}$

and

$(f(\phi_{1})-f(\phi_{2}))(\phi_{1}-\phi_{2})\leq
b_{2}(\phi_{1}-\phi_{2})^{2},\,\,\forall\,\phi_{1},\phi_{2}\in\mathbb{R}.$

In [3, Lemma 1.1] it is proved that $F$ satisfies $(M_{1})$, $(M_{2})$ and
$(M_{4})$ with $r=s$. Therefore, by just comparing the allowed class of
nonlinearities, since [3] is also concerned with non homogeneous boundary
conditions, the nonlinearities in [3] also satisfies the hypotheses of the
present work.

The previous remarks raise the possibility that our results generalize the
ones of [3], [9] and [10]. This in fact is so in the sense to be described in
the following.

The results in the present work improve the results of Moroşanu and Motreanu
[9] in the sense that we have a larger range for the allowed values of the
parameter $r$ appearing in hypothesis $(H_{2})$. In [9], it is required that
$1\leq r<(N+2)/(N-2)$, when $N\geq 3$, whereas here, according to $(H_{3})$ we
improves to either any $r\geq 1$ when $p\geq(N+2)/2$ or $1\leq
r<(N+2)/(N+2-2p)$ when $p<(N+2)/2$. Observe that this last condition is
exactly the same as in Moroşanu Motreanu [10], but in that paper a further
restriction on the nonlinearity is required (see $(M_{4})$). In fact, in the
following we will give an example of a function satisfying our hypotheses of
nonlinearity but not all the assumptions of $(M_{4})$ in [10]; this means
that, at least in the autonomous case, our class of nonlinearities is also
strictly larger than the one in [10].

######  Example 3.1.

Fix $p>2$ and consider $r_{1},r_{2}$ satisfying

$\begin{array}[]{l}\displaystyle 1\leq r_{2}<r_{1}<\frac{N+2}{N+2-2p},\mbox{
with }N+2-2p>0,\\\ \displaystyle r_{1}<r.\end{array}$ (6)

Then $F:\mathrm{I\\!R\\!}\rightarrow\mathrm{I\\!R\\!}$ defined by

$\displaystyle F(z)=|z|^{r_{2}-1}z-|z|^{r_{1}-1}z,\ \ \forall\
z\in\mathrm{I\\!R\\!},$ (7)

satisfies the conditions of the present article but does not satisfies
$(M_{4})$.

To check the previous claim, we start by showing that the function $F$ defined
by (7) satisfies our hypothesis ($H_{1}$), ($H_{2}$) and ($H_{4}$) when

$r\geq r_{1}.$

From (6), (7) and Mean Value Theorem we have: $\exists t_{0}\in(0,1)$ such
that

$\begin{array}[]{l}(F(z_{1})-F(z_{2}))(z_{1}-z_{2})\\\
=(r_{2}|(1-t_{0})z_{1}+t_{0}z_{2}|^{r_{2}-1}-r_{1}|(1-t_{0})z_{1}+t_{0}z_{2}|^{r_{1}-1})(z_{1}-z_{2})^{2},\end{array}$

$\forall\ z_{1}$, $\ z_{2}\ \in\mathrm{I\\!R\\!}$.

But, the function $g(t)=r_{2}|t|^{r_{2}-1}-r_{1}|t|^{r_{1}-1}$ is such that

$g(t)\leq 0,\ \textup{if}\
|t|\geq\left(\frac{r_{2}}{r_{1}}\right)^{\frac{1}{r_{1}-r_{2}}}$

and

$g(t)\leq
r_{2}\left(\frac{r_{2}}{r_{1}}\right)^{\frac{r_{2}-1}{r_{1}-r_{2}}}+r_{1}\left(\frac{r_{2}}{r_{1}}\right)^{\frac{r_{1}-1}{r_{1}-r_{2}}},\
\textup{if}\ |t|<\left(\frac{r_{2}}{r_{1}}\right)^{\frac{1}{r_{1}-r_{2}}}.$

So there exists a constant $a_{0}>0$ such that

$g(t)\leq a_{0},\forall\ t\in\mathrm{I\\!R\\!}.$

Therefore

$\begin{array}[]{l}(F(z_{1})-F(z_{2}))(z_{1}-z_{2})\leq
a_{0}(z_{1}-z_{2})^{2},\forall z_{1},\ z_{2}\
\in\mathrm{I\\!R\\!},\end{array}$

which ensures that our hypothesis ($H_{1}$) is satisfied.

From (7) and Mean Value Theorem we have

$\begin{array}[]{lll}|F(z_{1})-F(z_{2})|&\leq&\left||z_{1}|^{r_{2}-1}z_{1}-|z_{2}|^{r_{2}-1}z_{2}\right|+\left||z_{1}|^{r_{1}-1}z_{1}-|z_{2}|^{r_{1}-1}z_{2}\right|\\\
&\leq&G(z_{1},z_{2})|z_{1}-z_{2}|,\ \ \forall z_{1},\ z_{2}\
\in\mathrm{I\\!R\\!},\end{array}$

where

$\begin{array}[]{l}\displaystyle G(z_{1},z_{2})=r_{2}\sup_{0\leq t\leq
1}|(1-t)z_{1}+tz_{2}|^{r_{2}-1}+r_{1}\displaystyle\sup_{0\leq t\leq
1}|(1-t)z_{1}+tz_{2}|^{r_{1}-1}\\\ \displaystyle\phantom{G(z_{1},z_{2})}\leq
r_{2}(|z_{1}|+|z_{2}|)^{r_{2}-1}+r_{1}(|z_{1}|+|z_{2}|)^{r_{1}-1}\\\
\displaystyle\phantom{G(z_{1},z_{2})}\leq
a(1+|z_{1}|^{r_{1}-1}+|z_{2}|^{r_{1}-1})\\\
\displaystyle\phantom{G(z_{1},z_{2})}\leq
a(1+|z_{1}|^{r-1}+|z_{2}|^{r-1}),\qquad\qquad\qquad\forall z_{1},\ z_{2}\
\in\mathrm{I\\!R\\!},\end{array}$

for some constant $a>0$; which ensures that our hypothesis ($H_{2}$) is
satisfied with $F(x,t,z)=F(z)$.

Moreover, from (6) and (7) we have constants $c_{1},c_{2}>0$ such that

$|F(z)|\leq|z|^{r_{2}}+|z|^{r_{1}}\leq c_{1}(1+|z|^{r_{1}}),\ \ \forall z\
\in\mathrm{I\\!R\\!},$

and

$F(z)z=|z|^{r_{2}+1}-|z|^{r_{1}+1}=|z|(|z|^{r_{2}}-|z|^{r_{1}})\leq
c_{2}(1+z^{2}),\ \ \forall z\ \in\mathrm{I\\!R\\!},$

because, $|z|^{r_{2}}-|z|^{r_{1}}\leq 1$ if $|z|\leq 1$ and by (6),
$|z|^{r_{2}}-|z|^{r_{1}}\leq 0$ if $|z|>1$, because $1\leq r_{2}<r_{1}$. The
last two inequalities ensures that our hypothesis ($H_{4}$) is satisfied with
$F(x,t,z)=F(z)$.

Next, we will show that $F$ defined in (7) does not satisfy the assumption
$(M_{4})$ in Moroşanu and Motreanu [10] (same as hypothesis $(H_{0})$ in Cârja
et all [3]), for any $r_{1}<r<(N+2)/(N+2-2p)$.

If $1\leq r_{2}<r_{1}<r$, we will show that $F$ defined in (7) does not
satisfy the inequality $(M_{4})$.

Indeed, suppose there is $\alpha,\beta$ such that $F$ satisfies $(M_{4})$ for
all $z\in\mathrm{I\\!R\\!}$. On other hand, we have that

$\begin{array}[]{l}\displaystyle
F(z)|z|^{pr-r-1}z=\left(|z|^{r_{2}-1}z-|z|^{r_{1}-1}z\right)|z|^{pr-r-1}z\\\
\displaystyle\phantom{F(z)|z|^{pr-r-1}z}=|z|^{pr+r_{2}-r}-|z|^{pr+r_{2}-r}\end{array}$

Therefore,

$|z|^{pr+r_{2}-r}-|z|^{pr+r_{1}-r}\leq\alpha(1+|z|^{pr-1})-\beta|z|^{pr},$

which can be written as

$|z|^{pr+r_{2}-r}+\beta|z|^{pr}\leq\alpha(1+|z|^{pr-1})+|z|^{pr+r_{1}-r}.$

Dividing this last inequality by $|z|^{pr}$, with $z\neq 0$, we obtain that

$|z|^{r_{2}-r}+\beta\leq\alpha\left[\frac{1}{|z|^{pr}}+\frac{1}{|z|}\right]+|z|^{r_{1}-r},$

for all $z\in\mathrm{I\\!R\\!}-\\{0\\}$. This is a contradiction because,

$\lim_{|z|\rightarrow+\infty}|z|^{r_{2}-r}+\beta=\beta>0$

and

$\lim_{|z|\rightarrow+\infty}\alpha\left[\frac{1}{|z|^{pr}}+\frac{1}{|z|}\right]+|z|^{r_{1}-r}=0,$

because $1\leq r_{2}<r_{1}<r$.

######  Remark 3.2.

If we restrict ourselves to the subclass of autonomous nonlinearities, i.e.,
nonlinearities of form $F(x,t,z)=F(z)$, our last example shows that our
hypotheses $(H_{1})-(H_{4})$ are less restrictive than the hypotheses
$(M_{1})-(M_{4})$ in Moroşanu and Motreanu [10]. By [3, Lemma 1.1] our
hypotheses $(H_{1})-(H_{4})$ are less restrictive than the hypotheses [3].

######  Remark 3.3.

By taking $N=3$, $k(x,t,z)=a(x,t)z+b(x,t)z^{2}$, with $a,b\in L^{\infty}(Q)$
and $h(z)=z^{3}$, problem (1) generalizes the parabolic problem studied by
Hoffman and Jiang in [5].

Finally, we will need the following result whose proof can be found in
Ladyzhenskaya, [6, p. $341$].

######  Proposition 3.1.

Assume that $f\in L^{p}(Q)$ and $u_{0}\in W^{2-2/p}_{p}(\Omega)$. Then the
linear problem

$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle u_{t}-\Delta
u=f(x,t)&\textup{in }Q:=\Omega\times(0,T),\\\ \\\ \displaystyle\partial
u/\partial\nu=0&\textup{on }\sum=\partial\Omega\times(0,T),\\\ \\\
\displaystyle u(x,0)=u_{0}(x)&\textup{for all }x\in\Omega,\end{array}\right.$

has a unique solution $u\in W^{2,1}_{p}(Q)$ satisfying the estimate

$\|u\|_{W^{2,1}_{p}(Q)}\leq
C\left(\|u_{0}\|_{W^{2-2/p}_{p}(\Omega)}+\|f\|_{L^{p}(Q)}\right),$ (9)

##  4\. An auxiliary problem

We will need results concerning the following auxiliary nonlinear parabolic
boundary value problem:

$\left\\{\begin{array}[]{lll}\displaystyle\phi_{t}-\Delta\phi=F(x,t,\phi)+g(x,t)&\textup{if}&(x,t)\in
Q,\\\
\displaystyle\partial\phi/\partial\nu=0&\textup{on}&\sum=\partial\Omega\times(0,T),\\\
\displaystyle\phi(x,0)=\phi_{0}(x)&x\in\Omega.&\end{array}\right.$ (10)

Let $q$ give by

$q:=\left\\{\begin{array}[]{lll}\textup{any number}\ \geq
pr&\textup{if}&p\geq\frac{N+2}{2}.\\\ \displaystyle\textup{any number in}\
\left[pr,\frac{p(N+2)}{N+2-2p}\right)&\textup{if}&p<\frac{N+2}{2}.\end{array}\right.$
(11)

Notice that (11) makes sense by $(H_{3})$.

For this auxiliary problem, we have the following existence and regularity
result.

######  Proposition 4.1.

Assume that conditions $(H_{0})-(H_{4})$ hold. Then problem (10) admits a
unique solution $\phi\in W^{2,1}_{p}(Q)$ satisfying the estimate

$\|\phi\|_{W^{2,1}_{p}(Q)}\leq
C(1+\|\phi_{0}\|_{W^{2-2/p}_{p}(\Omega)}+\|g\|_{L^{p}(Q)}),$ (12)

where $C$ is a constant which depends only
$|\Omega|,T,p,r,a,a_{0},a_{1},a_{2},b_{0}$.

If $\phi_{1},\phi_{2}$ are solutions of (10) corresponding to
$\phi_{01},\phi_{02}\in W^{2-2/p}_{p}(\Omega)$ (instead of $\phi_{0}$) and
$g_{1},g_{2}$ (in place $g$), respectively, with
$\|\phi_{1}\|_{W^{2,1}_{p}(Q)},\|\phi_{2}\|_{W^{2,1}_{p}(Q)}\leq M$, then

$\|\phi_{1}-\phi_{2}\|_{W^{2,1}_{p}(Q)}\leq
C\left(\|\phi_{01}-\phi_{02}\|_{W^{2-2/p}_{p}(\Omega)}+\|g_{1}-g_{2}\|_{L^{p}(Q)}\right),$
(13)

where $C$ is a constant which depends only
$|\Omega|,T,M,p,r,a,a_{0},a_{1},a_{2},b_{0}$.

###  4.1. Preparatory results

To prove Proposition 4.1, we will apply the Leray-Schauder’s fixed point
theorem. For this, let us define the nonlinear operator

$\begin{array}[]{rccl}L:&L^{pr}(Q)\times[0,1]&\rightarrow&L^{pr}(Q)\\\
&(w,\lambda)&\rightarrow&L(w,\lambda)=\phi,\end{array}$ (14)

where $\phi$ is the solution of the linear problem

$\left\\{\begin{array}[]{ll}\displaystyle\phi_{t}-\Delta\phi=\lambda(F(x,t,w)+g(x,t))&\textup{in
}Q,\\\ \\\ \displaystyle\partial\phi/\partial\nu=0&\textup{on }\sum,\\\ \\\
\displaystyle\phi(x,0)=\phi_{0}(x)&\mbox{for all
}x\in\Omega.\end{array}\right.$ (15)

First of all, we have to check that $L$ is well defined. In fact, according to
(2) we have $F(\cdot,\cdot,w(\cdot))\in L^{p}(Q)$, $\forall w\in L^{pr}(Q)$.
Then $F(\cdot,\cdot,w(\cdot))+g\in L^{p}(Q)$. Since
$\lambda(F(\cdot,\cdot,w(\cdot))+g)\in L^{p}(Q)$, by the $L^{p}$-theory (see
Ladyzhenskaya, [6, pp. 341]), we have that there exists a unique solution
$\phi\in W^{2,1}_{p}(Q)$ of problem (15). It follows from ($H_{3}$), (11) and
Lemma 2.1 that $W^{2,1}_{p}(Q)\hookrightarrow L^{pr}(Q)$, and thus $L$ is well
defined.

Next, we prove that $L$ has suitable properties.

######  Lemma 4.1.

The mapping $L:L^{pr}(Q)\times[0,1]\rightarrow L^{pr}(Q)$ defined in (14) has
the following properties:

(i):

$L(\cdot,\lambda):L^{pr}(Q)\rightarrow L^{pr}(Q)$ is compact for every
$\lambda\in[0,1]$, i.e., it is continuous and maps bounded sets into
relatively compacts sets.

(ii):

For every $\epsilon>0$ and every bounded set $A\subset L^{pr}(Q)$ there exists
$\delta>0$ such that

$\|L(w,\lambda_{1})-L(w,\lambda_{2})\|_{L^{pr}(Q)}<\epsilon,$

whenever $w\in A$ and $|\lambda_{1}-\lambda_{2}|<\delta$.

Proof: We start with the proof of (i). To prove the continuity of
$L(\cdot,\lambda)$, let us consider $w_{1},w_{2}\in L^{pr}(Q)$ and
$\phi_{1}=L(w_{1},\lambda)$, $\phi_{2}=L(w_{2},\lambda)$ the corresponding
solutions. From (14) and (15), we obtain

$\left\\{\begin{array}[]{ll}(\phi_{1}-\phi_{2})_{t}-\Delta(\phi_{1}-\phi_{2})=\lambda(F(x,t,w_{1})-F(x,t,w_{2}))&\textup{in
}Q,\\\ \\\ \partial(\phi_{1}-\phi_{2})/\partial\nu=0&\textup{on }\sum,\\\ \\\
(\phi_{1}-\phi_{2})(x,0)=0&\mbox{for all }x\in\Omega.\end{array}\right.$

If $w$ is in ${L^{pr}(Q)}$, from Lemma 3.1, $F(\cdot,\cdot,w)$ is in
$L^{p}(Q)$. As above, the $L^{p}$-theory implies the estimate

$\|\phi_{1}-\phi_{2}\|_{W^{2,1}_{p}(Q)}\leq
C\|F(x,t,w_{1})-F(x,t,w_{2})\|_{L^{p}(Q)}.$

Thus the operator $L(\cdot,\lambda):L^{pr}(Q)\rightarrow W^{2,1}_{p}(Q)$,
given by (14), is continuous. By the compact embedding
$W^{2,1}_{p}(Q)\hookrightarrow L^{pr}(Q)$ (see Lions [8, pp. 21]), we conclude
that $L(\cdot,\lambda):L^{pr}(Q)\rightarrow L^{pr}(Q)$ is continuous and
compact for each fixed $\lambda\in[0,1]$.

Next, we prove (ii). Let us fix a bounded set $A\subset L^{pr}(Q)$ and
consider $(\phi_{1},\lambda_{1})$, $(\phi_{2},\lambda_{2})\in
L^{pr}(Q)\times[0,1]$ the corresponding solutions of (15), where we take any
$w\in A$. We have From (14) and (15)

$\left\\{\begin{array}[]{ll}(\phi_{1}-\phi_{2})_{t}-\Delta(\phi_{1}-\phi_{2})=(\lambda_{1}-\lambda_{2})(F(x,t,w)+g(x,t))&\textup{in
}Q,\\\ \\\ \partial(\phi_{1}-\phi_{2})/\partial\nu=0&\textup{on }\sum,\\\ \\\
(\phi_{1}-\phi_{2})(x,0)=0&\mbox{for all }x\in\Omega.\end{array}\right.$

The $L^{q}$-theory (see Ladyzhenskaya, [[6], pp. 341]) provides the estimate

$\|\phi_{1}-\phi_{2}\|_{W^{2,1}_{p}(Q)}\leq
C|\lambda_{1}-\lambda_{2}|(\|F(x,t,w)\|_{L^{p}(Q)}+\|g\|_{L^{p}(Q)}).$

From Lemma 3.1 and (11), we have that

$\|\phi_{1}-\phi_{2}\|_{W^{2,1}_{p}(Q)}\leq
C(A,\|g\|_{L^{p}(Q)})|\lambda_{1}-\lambda_{2}|,$

where $C(A,\|g\|_{L^{p}(Q)})$ is a bounded constant because $A$ is a bounded
set.

The proof of (ii) then follows from Lemma 2.1, (11) and the last inequality by
taking $\displaystyle\delta=\epsilon/2C(A,\|g\|_{L^{p}(Q)})$ for each fixed
$\epsilon>0$.

In the next result, we prove estimates for any possible fixed points of
$L(\cdot,\lambda)$.

######  Lemma 4.2.

Suppose that assumptions $(H_{0})-(H_{4})$ are satisfied. Then, there exists a
number $\rho>0$, such that any fixed point $\phi\in L^{pr}(Q)$ of
$T(\cdot,\lambda)$ for any $\lambda\in[0,1]$, i. e., $T(\phi,\lambda)=\phi$
for some $\lambda\in[0,1]$, satisfies

$\|\phi\|_{L^{pr}(Q)}<\rho;$ (16)

Proof: Let $\phi\in L^{pr}(Q)$ a fixed point of $T(\cdot,\lambda)$ for some
$\lambda\in[0,1]$. Then, $\phi$ solves the problem

$\left\\{\begin{array}[]{ll}\displaystyle\phi_{t}-\Delta\phi=\lambda(F(x,t,\phi)+g(x,t))&\textup{in
}Q,\\\ \\\ \displaystyle\partial\phi/\partial\nu=0&\textup{on }\sum,\\\ \\\
\displaystyle\phi(x,0)=\phi_{0}(x)&\mbox{for all
}x\in\Omega.\end{array}\right.$ (17)

Denote

$Q_{t}=\Omega\times(0,t),\ t\in(0,T].$

By multiplying first equation in (10) by $\phi$, integrating over $Q_{t}$,
using Lemma 3.3, Green’s formula and Young’s inequality, we obtain

$\begin{array}[]{c}\displaystyle\frac{1}{2}\int_{\Omega}\phi^{2}(t)dx+\int_{Q_{t}}|\nabla\phi|^{2}dxds\\\
\displaystyle\leq\frac{1}{2}\left(\|\phi_{0}\|^{2}_{L^{2}(\Omega)}+\|g\|^{2}_{L^{2}(Q)}\right)+\frac{1}{2}\int_{Q_{t}}\phi^{2}dxds+\int_{Q_{t}}F(x,s,\phi)\phi
dxds.\end{array}$

In view of (3), we get further

$\frac{1}{2}\int_{\Omega}\phi^{2}(t)dx+\int_{Q_{t}}|\nabla\phi|^{2}dxds\leq
C_{1}\left(1+\|\phi_{0}\|^{2}_{L^{p}(\Omega)}+\|g\|^{2}_{L^{p}(Q)}\right)+C_{2}\int_{Q_{t}}\phi^{2}dxds$

where $C_{1}$ and $C_{2}$ denote positive constants. By Gronwall’s inequality,
we arrive at

$\displaystyle\frac{1}{2}\int_{\Omega}\phi^{2}(t)dx+\int_{Q_{t}}|\nabla\phi|^{2}dxds\leq
C_{0}\left(1+\|\phi_{0}\|^{2}_{W^{2-2/p}_{p}(\Omega)}+\|g\|^{2}_{L^{p}(Q)}\right),\
\forall\ t\in(0,T].$ (18)

As $F(\cdot,\cdot,\phi)+g\in L^{p}(Q)$. By the $L^{p}$-theory (see
Ladyzhenskaya, [6, pp. 341],) we have that

$\displaystyle\|\phi\|_{W^{2,1}_{p}(Q)}\leq
C\left(1+\|\phi_{0}\|_{W^{2-\frac{2}{p}}_{p}(\Omega)}+\|F(\cdot,\cdot,\phi)\|_{L^{p}(Q)}+\|g\|_{L^{p}(Q)}\right).$

By using $q\geq pr$ and the convexity of $|\cdot|^{p/r}$ and Lemma 3.1, we get

$\|F(\cdot,\cdot,\phi)\|_{L^{p}(Q)}\leq
C\|F(\cdot,\cdot,\phi)\|_{L^{q/r}(Q)}\leq C(1+\|\phi\|_{L^{q}(Q)})$

Next, by (11) and Lemma 2.1, we have the sequence of imbeddings

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
19.83612pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\displaystyle
W^{2,1}_{p}(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
19.83612pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
43.83612pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
43.83612pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{L^{q}(Q)}$}}}}}}}\ignorespaces}}}}\ignorespaces\hookrightarrow
L^{2}(Q),$

where the first immersion is compact by Lions [8, pp. 21, Remark 2.3]. The
interpolation inequality yields that for all $\epsilon>0$, there exists
$C(\epsilon)>0$ such that

$\displaystyle\|v\|_{L^{q}(Q)}\leq\epsilon\|v\|_{W^{2,1}_{p}(Q)}+C(\epsilon)\|v\|_{L^{2}(Q)},\
\ \forall v\in W^{2,1}_{p}(Q).$

From (18) and of the two last inequalities, we derive that

$\displaystyle(1-\epsilon C)\|\phi\|_{W^{2,1}_{p}(Q)}\leq
C\left(1+\|\phi_{0}\|_{W^{2-\frac{2}{p}}_{p}(\Omega)}+\|\phi_{0}\|_{W^{1}_{2}(\Omega)}+\|g\|_{L^{p}(Q)}\right),$

where $\epsilon>0$ is small enough such that $1-\epsilon C>0$. So we get

$\begin{array}[]{lll}\|\phi\|_{W^{2,1}_{p}(Q)}&\leq&C\left(1+\|\phi_{0}\|_{W^{2-\frac{2}{p}}_{p}(\Omega)}+\|g\|_{L^{p}(Q)}\right).\end{array}$
(19)

From (11), Lemma 2.1 and (19), we have

$\displaystyle\|\phi\|_{{L^{pr}(Q)}}\leq
C\left(1+\|\phi_{0}\|_{W^{2-\frac{2}{p}}_{p}(\Omega)}+\|g\|_{L^{p}(Q)}\right),$
(20)

which leads to the conclusion that the claim in (16) holds true.

###  4.2. Proof of Proposition 4.1

From Lemma (4.2), we know the existence of a number $\rho>0$ which satisfies
the property stated in (16) and we have that the linear heat equation,
$\phi-T(\phi,0)=0$, admits a unique solution, i.e., the unique solution of

$\left\\{\begin{array}[]{ll}\displaystyle\phi_{t}-\Delta\phi=0&\;\;\;\;\;\;\;\mbox{in
}Q,\\\ \displaystyle\partial\phi/\partial\nu=0&\;\;\;\;\;\;\;\mbox{on
}\sum,\\\ \phi(x,0)=\phi_{0}(x)&\;\;\;\;\;\;\;\mbox{for all
}x\in\Omega.\end{array}\right.$

It follows from Leray-Schauder’s fixed point theorem (see Friedman [4, pp.
189, Theorem 3]) that the problem

$\phi-T(\phi,1)=0,$ (21)

has a solution $\phi$.

It follows from (14) that $\phi$ is a solution of the problem (10); from Lemma
4.2 and (19) we have $\phi\in W^{2,1}_{p}(Q)$ and

$\|\phi\|_{W^{2,1}_{p}(Q)}\leq
C_{1}\left(1+\|\phi_{0}\|_{W^{2-\frac{2}{p}}_{p}(\Omega)}+\|g\|_{L^{p}(Q)}\right).$

Since by hypothesis, $\phi_{1},\phi_{2}\in W^{2,1}_{p}(Q)$ solve (10),
respectively with $\phi_{01}$, $\phi_{02}$ and $g_{1}$, $g_{2}$ ,
$\phi_{1}-\phi_{2}\in W^{2,1}_{p}(Q)$ and satisfies

$\left\\{\begin{array}[]{ll}(\phi_{1}-\phi_{2})_{t}-\Delta(\phi_{1}-\phi_{2})=F(x,t,\phi_{1})-F(x,t,\phi_{2})+g_{1}-g_{2}&\textup{in
}Q,\\\ \\\ \partial(\phi_{1}-\phi_{2})/\partial\nu=0&\textup{on }\sum,\\\ \\\
(\phi_{1}-\phi_{2})(x,0)=\phi_{01}-\phi_{02}&\mbox{for all
}x\in\Omega.\end{array}\right.$ (22)

By multiplying the first equation in (10) by
$|\phi_{1}-\phi_{2}|^{p-2}(\phi_{1}-\phi_{2})$, integrating over $Q_{t}$, and
using Green’s theorem, we obtain

$\begin{array}[]{c}\displaystyle\frac{1}{p}\int_{\Omega}|\phi_{1}-\phi_{2}|^{p}(t)dx+(p-1)\int_{Q_{t}}|\nabla(\phi_{1}-\phi_{2})|^{2}|\phi_{1}-\phi_{2}|^{p-2}dxds\\\
\displaystyle\leq\frac{1}{p}\int_{\Omega}|\phi_{01}-\phi_{02}|^{p}dx+\frac{2^{p}}{p}\int_{Q}|g_{1}-g_{2}|^{p}dxds+\frac{p-1}{p}\frac{1}{2^{\frac{p}{p-1}}}\int_{Q_{t}}|\phi_{1}-\phi_{2}|^{p}dxds\\\
\displaystyle+\int_{Q_{t}}\left(F(x,s,\phi_{1})-F(x,s,\phi_{2})\right)|\phi_{1}-\phi_{2}|^{p-2}(\phi_{1}-\phi_{2})dxds.\end{array}$

Due to assumption ($H_{1}$) and Gronwall’s inequality, it results

$\displaystyle\|\phi_{1}-\phi_{2}\|_{L^{p}(\Omega)}^{p}\leq
C(T,r,p)\left(\|\phi_{01}-\phi_{02}\|_{L^{p}(\Omega)}^{p}+\|g_{1}-g_{2}\|_{L^{p}(Q)}^{p}\right)$
(23)

According to (11) and Lemma 2.1, we have $\phi_{1},\phi_{2}\in L^{pr}(Q)$,
which in conjunction with relation (2) yields that
$F(\cdot,\cdot,\phi_{1})-F(\cdot,\cdot,\phi_{2})\in L^{p}(Q)$. By applying
$L_{p}-$theory to the problem (22) and by using $(H_{0})$, we get the estimate

$\begin{array}[]{l}\displaystyle\|\phi_{1}-\phi_{2}\|_{W^{2,1}_{p}(Q)}^{p}\leq
C(|\Omega|,T,N)\\\
\displaystyle\times\left(\|\phi_{01}-\phi_{02}\|_{W^{2-2/p}_{p}(\Omega)}^{p}+\|F(\cdot,\cdot,\phi_{1})-F(\cdot,\cdot,\phi_{2})\|_{L^{p}(Q)}^{p}+\|g_{1}-g_{2}\|_{L^{p}(Q)}^{p}\right)\end{array}$
(24)

The inequality $q>pr$ allow us to fix a number $m$ such that

$\displaystyle 2\leq p\leq\frac{qp}{q+p-pr}\leq pr<m\leq q.$ (25)

Consequently, the next sequence of embeddings holds

$\displaystyle W^{2,1}_{p}(Q)\subset L^{q}(Q)\subset L^{m}(Q)\subset
L^{pr}(Q)\subset L^{p}(Q)\subset L^{2}(Q).$ (26)

From $(H_{2})$, (25) and H$\ddot{o}$lder’s inequality, with
$\frac{1}{q_{1}}+\frac{1}{q_{2}}=1$, where $q_{1}=p/m,q_{2}=m/(m-p)$, we
conclude that

$\begin{array}[]{l}\displaystyle\|F(\cdot,\cdot,\phi_{1})-F(\cdot,\cdot,\phi_{2})\|_{L^{p}(Q)}\leq\left\|G(x,t,\phi_{1},\phi_{2})^{1/2}|\phi_{1}-\phi_{2}|\right\|_{L^{p}(Q)}\\\
\hskip
85.35826pt\displaystyle=\left(\int_{Q}G(x,t,\phi_{1},\phi_{2})^{p/2}|\phi_{1}-\phi_{2}|^{p}dxdt\right)^{\frac{1}{p}}\\\
\hskip
85.35826pt\displaystyle\leq\left(\int_{Q}G(x,t,\phi_{1},\phi_{2})^{n_{0}/2}dxdt\right)^{\frac{1}{n_{0}}}\|\phi_{1}-\phi_{2}\|_{L^{m}(Q)},\end{array}$
(27)

where $n_{0}=mp/(m-p)$. We remark that the previous computations make sense
because $G(x,t,\phi_{1},\phi_{2})^{n_{0}/2}\in L^{1}(Q)$. Indeed, taking into
account the growth condition in $(H_{2})$, $G(x,t,\phi_{1},\phi_{2})\in
L^{\frac{q}{2(r-1)}}(Q)$, whenever $\phi_{1},\phi_{2}\in L^{q}(Q)$, and by
(25) it is true that

$\displaystyle\frac{q}{r-1}>n_{0}>2.$ (28)

Then, the inequalities in (28) lead to the claim above. Combining (24) and
(27), we arrive at

$\begin{array}[]{l}\displaystyle\|\phi_{1}-\phi_{2}\|_{W^{2,1}_{p}(Q)}^{p}\leq
C(|\Omega|,T,N)\left(\|\phi_{01}-\phi_{02}\|_{W^{2-2/p}_{p}(\Omega)}^{p}\right.\\\
\hskip
56.9055pt\displaystyle\left.+\|G(x,t,\phi_{1},\phi_{2})\|_{L^{\frac{n_{0}}{2}}(Q)}^{1/2}\|\phi_{1}-\phi_{2}\|_{L^{m}(Q)}+\|g_{1}-g_{2}\|_{L^{p}(Q)}^{p}\right)\end{array}$
(29)

In addition, we have for any $(x,t)\in Q$ that

$\begin{array}[]{l}\displaystyle\left(1+|\phi_{1}(x,t)|^{2r-2}+|\phi_{2}(x,t)|^{2r-2}\right)^{n_{0}/2}\\\
\hskip 56.9055pt\displaystyle\leq
C(p,r)(1+|\phi_{1}(x,t)|^{n_{0}(r-1)}+|\phi_{2}(x,t)|^{n_{0}(r-1)}).\end{array}$

This last relation, inequality (28) and estimate (29) then imply

$\begin{array}[]{l}\displaystyle\|\phi_{1}-\phi_{2}\|_{W^{2,1}_{p}(Q)}\\\
\displaystyle\leq
C(|\Omega|,T,N)\left[\|\phi_{01}-\phi_{02}\|_{W^{2-2/p}_{p}(\Omega)}+\|g_{1}-g_{2}\|_{L^{p}(Q)}\right.\\\
\hskip
14.22636pt\displaystyle\left.+C(|\Omega|,T,r,p,b_{0})\left(1+\|\phi_{1}\|_{L^{n_{0}(r-1)}}^{r-1}+\|\phi_{2}\|_{L^{n_{0}(r-1)}}^{r-1}\right)\|\phi_{1}-\phi_{2}\|_{L^{m}(Q)}\right]\vspace{0.1cm}\\\
\displaystyle\leq
C(|\Omega|,T,N,p,r,b_{0})\left[\|\phi_{01}-\phi_{02}\|_{W^{2-2/p}_{p}(\Omega)}+\|g_{1}-g_{2}\|_{L^{p}(Q)}\right.\\\
\hskip
14.22636pt\displaystyle\left.+\left(1+\|\phi_{1}\|_{L^{n_{0}(r-1)}}^{r-1}+\|\phi_{2}\|_{L^{n_{0}(r-1)}}^{r-1}\right)\|\phi_{1}-\phi_{2}\|_{L^{m}(Q)}\right]\vspace{0.1cm}\\\
\displaystyle\leq C(|\Omega|,T,N,p,r,b_{0})(1+2M^{r-1})\\\ \hskip
14.22636pt\displaystyle\times\left[\|\phi_{01}-\phi_{02}\|_{W^{2-2/p}_{p}(\Omega)}+\|g_{1}-g_{2}\|_{L^{p}(Q)}+\|\phi_{1}-\phi_{2}\|_{L^{m}(Q)}\right]\end{array}$
(30)

Due to the embeddings in (26), the interpolation inequality (see Lions [7, pp.
58, Lemma 5.1]) yields that for all $\epsilon>0$, there exists $C(\epsilon)>0$
such that

$\displaystyle\|\phi\|_{L^{m}(Q)}\leq\epsilon\|\phi\|_{W^{2,1}_{p}(Q)}+\|\phi\|_{L^{p}(Q)},\forall\phi\in
W^{2,1}_{p}(Q).$ (31)

Thus, from (23), (30) and (31), we get that

$\begin{array}[]{c}\displaystyle(1-\epsilon
C(|\Omega|,T,N,p,r,b_{0}))\|\phi_{1}-\phi_{2}\|_{W^{2,1}_{p}(Q)}\\\
\displaystyle\leq
C(|\Omega|,T,N,p,r,a_{0},b_{0})\left[\|\phi_{01}-\phi_{02}\|_{W^{2-2/p}_{p}(\Omega)}+\|g_{1}-g_{2}\|_{L^{p}(Q)}\right].\end{array}$
(32)

Next, by taking $\epsilon>0$ such that $1-\epsilon
C(|\Omega|,T,N,p,r,b_{0})>0$, the last inequality implies the estimate (13),
completing the proof of Proposition 4.1.

##  5\. Proof of the main result

###  5.1. Preparatory results

To prove Theorem 3.1, we will again apply the Leray-Schauder’s fixed point
theorem. Consider the nonlinear operator $\mathcal{L}$ defined by

$\begin{array}[]{rccl}\mathcal{L}:&L^{p}(Q)\times[0,1]&\rightarrow&L^{p}(Q)\\\
&(g,\lambda)&\rightarrow&\mathcal{L}(g,\lambda)=u,\end{array}$ (33)

where $u$ is the unique solution of the following linear parabolic boundary
value problem

$\left\\{\begin{array}[]{ll}\displaystyle u_{t}-\Delta
u=\lambda(-l\phi_{t}+f)&\mbox{ in }Q,\\\ \displaystyle\partial
u/\partial\nu=0&\mbox{ on}\sum,\\\ \displaystyle
u(x,0)=\lambda{u}_{0}(x)&\mbox{ for all }x\in\Omega,\end{array}\right.$ (34)

with $\phi$ the unique solution of the following semilinear parabolic boundary
value problem (see Proposition 4.1)

$\left\\{\begin{array}[]{ll}\displaystyle\phi_{t}-\Delta\phi=F(x,t,\phi)+g&\mbox{
in }Q,\\\ \displaystyle\partial\phi/\partial\nu=0&\mbox{ on }\sum,\\\
\displaystyle\phi(x,0)={\varphi}_{0}(x)&\mbox{ for all
}x\in\Omega.\end{array}\right.$ (35)

First of all, we observe that the operator $\mathcal{L}$ is well defined. In
fact, according to Proposition 4.1, since $g\in L^{p}(Q)$, there is only one
solution $\phi\in W^{2,1}_{p}(Q)$ of (35). And since, $-l\phi_{t}+f\in
L^{p}(Q)$, it follows from the Proposition 3.1, that (34) has a unique
solution $u\in W^{2,1}_{p}(Q)$. Therefore, $\mathcal{L}(g,\lambda)=u\in
W^{2,1}_{p}(Q)\subset L^{p}(Q)$, and $\mathcal{L}$ is well defined.

######  Lemma 5.1.

The mapping $\mathcal{L}:L^{p}(Q)\times[0,1]\rightarrow L^{p}(Q)$ has the
following properties:

(i):

$\mathcal{L}(\cdot,\lambda):L^{p}(Q)\rightarrow L^{p}(Q)$ is compact for every
$\lambda\in[0,1]$, i. e., it is continuous and maps bounded sets into
relatively compacts sets.

(ii):

For every $\epsilon>0$ and every bounded set $A\subset L^{p}(Q)$ there exists
$\delta>0$ such that, whenever $g\in A$ and
$|\lambda_{1}-\lambda_{2}|<\delta$, there holds

$\|\mathcal{L}(g,\lambda_{1})-\mathcal{L}(g,\lambda_{2})\|_{L^{p}(Q)}<\epsilon.$

Proof: We start by proving (i). To prove the continuity of
$\mathcal{L}(\cdot,\lambda)$, let us consider $g_{1}$, $g_{2}\in L^{p}(Q)$ and
$u_{1}=\mathcal{L}(g_{1},\lambda)$, $u_{2}=\mathcal{L}(g_{2},\lambda)$ the
corresponding solutions. By (33) and (34) we derive

$\left\\{\begin{array}[]{ll}\displaystyle(u_{1}-u_{2})_{t}-\Delta(u_{1}-u_{2})=-\lambda
l(\phi_{1}-\phi_{2})_{t}&\mbox{ in }Q,\\\
\displaystyle\partial(u_{1}-u_{2})/\partial\nu=0&\mbox{ on }\sum,\\\
\displaystyle(u_{1}-u_{2})(x,0)=0&\mbox{ for all
}x\in\Omega.\end{array}\right.$

We can then apply the classical $L^{q}$-theory (see Ladyzhenskaya, [6, pp.
341]), and obtain the estimate

$\displaystyle\|u_{1}-u_{2}\|_{W^{2,1}_{p}(Q)}\leq
C\|(\phi_{1}-\phi_{2})_{t}\|_{L^{p}(Q)}.$ (36)

Estimate (13) in Proposition 4.1 applied to (35) choosing
$\phi_{01}=\phi_{02}=\phi_{0}$ insures that

$\displaystyle\|\phi_{1}-\phi_{2}\|_{W^{2,1}_{p}(Q)}\leq
C\|g_{1}-g_{2}\|_{L^{p}(Q)}.$ (37)

From (36) and (37) yield a new constant $C>0$ such that

$\displaystyle\|u_{1}-u_{2}\|_{L^{p}(Q)}\leq C\|g_{1}-g_{2}\|_{L^{p}(Q)},$
(38)

which implies the continuity of $\mathcal{L}(\cdot,\lambda)$.

Furthermore, $\mathcal{L}(\cdot,\lambda)$ is a compact operator. In fact, from
(38) we have that $\mathcal{L}(\cdot,\lambda)$, regarded as a mapping from the
space $L^{p}(Q)$ into the space $W^{2,1}_{p}(Q)$, is continuous. From the
compactness of the embedding $W^{2,1}_{p}(Q)\hookrightarrow L^{p}(Q)$ (see
Lions [8, pp. 21, Remark 2.3]), we conclude that
$\mathcal{L}(\cdot,\lambda):L^{p}(Q)\rightarrow L^{p}(Q)$ is continuous and
compact for each fixed $\lambda\in[0,1]$.

Next we prove (ii). For this, fix a bounded set $A\subset L^{p}(Q)$; consider
$\lambda_{1}$, $\lambda_{2}\in[0,1]$ and $g\in A$, and set
$u_{1}~{}=~{}\mathcal{L}(g,\lambda_{1})$, $u_{2}=\mathcal{L}(g,\lambda_{2})$.
Then we have

$\left\\{\begin{array}[]{ll}\displaystyle(u_{1}-u_{2})_{t}-\Delta(u_{1}-u_{2})=-l(\lambda_{1}{\phi_{1}}_{t}-\lambda_{2}{\phi_{2}}_{t})+(\lambda_{1}-\lambda_{2})f&\mbox{
in }Q,\\\ \displaystyle\partial(u_{1}-u_{2})/\partial\nu=0&\mbox{ on }\sum,\\\
\displaystyle(u_{1}-u_{2})(x,0)=(\lambda_{1}-\lambda_{2})u_{0}&\mbox{ for all
}x\in\Omega.\end{array}\right.$ (39)

By an argument similar to that used to derive (37), we get

$\displaystyle\|\phi_{1}-\phi_{2}\|_{W^{2,1}_{p}(Q)}\leq
C|\lambda_{1}-\lambda_{2}|\|g\|_{L^{p}(Q)}\leq C(A)|\lambda_{1}-\lambda_{2}|.$
(40)

By applying the classical $L^{q}$-theory (see Ladyzhenskaya, [6, pp. 341]),
for the first equation of (39) leads to the inequality

$\begin{array}[]{l}\|u_{1}-u_{2}\|_{W^{2,1}_{p}(Q)}\\\ \hskip
14.22636pt\displaystyle\leq
C\left(\|\lambda_{1}{\phi_{1}}_{t}-\lambda_{2}{\phi_{2}}_{t}\|_{L^{p}(Q)}+|\lambda_{1}-\lambda_{2}|\|f\|_{L^{p}(Q)}+|\lambda_{1}-\lambda_{2}|\|u_{0}\|_{W^{2-2/p}_{p}(\Omega)}\right)\\\
\hskip 14.22636pt\displaystyle\leq
C|\lambda_{1}-\lambda_{2}|\left(\|{\phi_{1}}_{t}\|_{L^{p}(Q)}+\|u_{0}\|_{W^{2-2/p}_{p}(\Omega)}+\|f\|_{L^{p}(Q)}\right)+C\lambda_{2}\|{\phi_{1}}_{t}-{\phi_{2}}_{t}\|_{L^{p}(Q)}.\end{array}$

Now, estimate (12) ensures that $\|{\phi_{1}}_{t}\|_{L^{p}(Q)}$ is bounded
because $A$ is bounded; therefore, from (40), we get

$\displaystyle\|u_{1}-u_{2}\|_{L^{p}(Q)}\leq
C\|u_{1}-u_{2}\|_{W^{2,1}_{p}(Q)}\leq C(A)|\lambda_{1}-\lambda_{2}|.$

It follows from (40) and the last inequality the proof of (ii) with
$\delta=\epsilon/(2C(A))$, for each fixed $\epsilon>0$.

In the next result, we prove estimates for any possible fixed points of
$\mathcal{L}(\cdot,\lambda)$.

######  Lemma 5.2.

Under assumptions $(H_{0})-(H_{4})$, there exists a number $\rho>0$,
independent of $\lambda\in[0,1]$, with the following property

$\mathcal{L}(u,\lambda)=u\textup{, for any }\lambda\in[0,1],\ \Rightarrow\
\|u\|_{L^{p}(Q)}<\rho.$ (41)

Proof: Let us consider $u$ a possible fixed point of
$\mathcal{L}(\cdot,\lambda)$. Then $u$ satisfies $(\ref{P.6})$, where $\phi$
is the unique solution of the nonlinear parabolic boundary value problem (see
Proposition 4.1)

$\left\\{\begin{array}[]{ll}\displaystyle\phi_{t}-\Delta\phi=F(x,t,\phi)+u&\mbox{
in }Q,\\\ \displaystyle\partial\phi/\partial\nu=0&\mbox{ on }\sum,\\\
\displaystyle\phi(x,0)={\phi}_{0}(x)&\mbox{ for all
}x\in\Omega.\end{array}\right.$

By Proposition 3.1, $u\in L^{p}(Q)$. Then, by Proposition 4.1, there is a
unique $\phi\in W^{2,1}_{p}(Q)$ solving the above problem. In addition,
estimate (12) is true with $u$ replacing $g$. From (12), we get

$\displaystyle\|\phi_{t}\|_{L^{p}(Q)}\leq\|\phi\|_{W^{2,1}_{p}(Q)}\leq
C\left(1+\|\phi_{0}\|_{W^{2-2/p}_{p}(\Omega)}+\|u\|_{L^{p}(Q)}\right),$ (42)

where $C$ is a constant which depends only $|\Omega|,T,p,r,a,c_{0},d_{0}$.

By multiplying the first equation in (34) by $|u|^{p-2}u$, integrating over
$Q_{t}$, with $t\in(0,T]$, and using Green’s formula and Young’s inequality,
we obtain

$\begin{array}[]{c}\displaystyle\frac{1}{p}\int_{\Omega}|u|^{p}(t)dx+(p-1)\int_{Q_{t}}|\nabla
u|^{2}|u|^{pr-2}dxds\\\
\displaystyle\leq\frac{1}{p}\|u_{0}\|^{p}_{L^{p}(\Omega)}+\frac{1}{p}\int_{Q_{t}}|\phi_{t}|^{p}dxds+\frac{2(p-1)}{p}\int_{Q_{t}}|u|^{p}dxds+\frac{1}{p}\int_{Q_{t}}|f|^{p}dxds,\end{array}$

for all $t\in(0,T]$. By combining (42) (with $Q_{t}$ in place of $Q$) and the
last inequality, it turns out that

$\begin{array}[]{cc}\displaystyle\frac{1}{p}\int_{\Omega}|u|^{p}(t)dx+(p-1)\int_{Q_{t}}|\nabla
u|^{2}|u|^{pr-2}dxds\\\
\displaystyle\leq\frac{1}{p}\|u_{0}\|^{p}_{L^{p}(\Omega)}+\frac{C}{p}\left(1+\|\phi_{0}\|^{p}_{W^{2-2/p}_{p}(\Omega)}\right)\\\
\displaystyle+\left[\frac{C}{p}+\frac{2(p-1)}{p}\right]\int_{Q_{t}}|u|^{p}dxds+\frac{1}{p}\int_{Q_{t}}|f|^{p}dxds,&\forall\
t\in(0,T].\end{array}$

By using Gronwall’s lemma in last inequality, we obtain a positive constant
$C=C(|\Omega|,T,p,r,a,c_{0},d_{0})$ such that

$\displaystyle\|u\|_{L^{p}(Q)}\leq
C(1+\|u_{0}\|_{L^{p}(\Omega)}+\|\phi_{0}\|_{W^{2-2/p}_{p}(\Omega)}+\|f\|_{L^{p}(Q)}).$
(43)

By taking for instance
$\rho=C(1+\|u_{0}\|_{L^{p}(\Omega)}+\|\phi_{0}\|_{W^{2-2/p}_{p}(\Omega)}+\|f\|_{L^{p}(Q)})+1$,
we then have property (41).

###  5.2. Proof of Theorem 3.1

From Lemma (5.2), we know the existence of a number $\rho>0$ which satisfies
the property stated in (41); moreover, any fixed point $\bar{u}$ corresponding
to $\lambda=0$, that is, $\bar{u}$ satisfying
$\mathcal{L}(\bar{u},0)=\bar{u}$, is exactly the solution of the standard
linear heat equation

$\left\\{\begin{array}[]{ll}\displaystyle\bar{u}_{t}-\Delta\bar{u}=0&\mbox{ in
}Q,\\\ \displaystyle\partial\bar{u}/\partial\nu=0&\mbox{ on }\sum,\\\
\displaystyle\bar{u}(x,0)=0&\mbox{ for all }x\in\Omega,\end{array}\right.$

which has a unique solution.

It then follows from Leray-Schauder’s fixed point theorem (see Friedman [4,
pp. 189, Theorem 3]) that problem

$\mathcal{L}(u,1)=u,$

has a solution $u$.

This implies that equation (34) has a solution $u\in W^{2,1}_{p}(Q)$, with a
corresponding $\phi\in W^{2,1}_{p}(Q)$ which is the unique solution of (35)
with $g=u$. That is, there is a solution $(u,\phi)\in W^{2,1}_{p}(Q)\times
W^{2,1}_{p}(Q)$ of (1), which, by estimate (12) in Proposition 4.1, satisfies

$\|\phi\|_{W^{2,1}_{p}(Q)}\leq
C(|\Omega|,T,p,r,a,c_{0},d_{0})\left(1+\|\phi_{0}\|_{W^{2-2/p}_{p}(\Omega)}+\|u\|_{L^{p}(Q)}\right).$
(44)

By using Proposition 3.1 with first equation in (1), combined with the last
inequality, we get

$\begin{array}[]{l}\displaystyle\|u\|_{W^{2,1}_{p}(Q)}\leq
C\left(\|u_{0}\|_{W^{2-2/p}_{p}(\Omega)}+\|\phi_{t}\|_{L^{p}(Q)}+\|f\|_{L^{p}(Q)}\right)\\\
\displaystyle\phantom{\|u\|_{W^{2,1}_{p}(Q)}}\leq
C\left(1+\|\phi_{0}\|_{W^{2-2/p}_{p}(\Omega)}+\|u_{0}\|_{W^{2-2/p}_{p}(\Omega)}+\|u\|_{L^{p}(Q)}+\|f\|_{L^{p}(Q)}\right).\end{array}$

From this result and (43), we obtain that there is a positive constant
$C=C(|\Omega|,T,p,r,a,c_{0},d_{0})$ such that

$\|u\|_{W^{2,1}_{p}(Q)}\leq
C(1+\|\phi_{0}\|_{W^{2-2/p}_{p}(\Omega)}+\|u_{0}\|_{W^{2-2/p}_{p}(\Omega)}+\|f\|_{L^{p}(Q)}).$
(45)

By collecting the results in estimates (43), (44) and (45), we conclude that
there is a positive constant $C=C(|\Omega|,T,p,r,a,c_{0},d_{0})$ such that
arrive the inequality

$\|u\|_{W^{2,1}_{p}(Q)}+\|\phi\|_{W^{2,1}_{p}(Q)}\leq
C(1+\|\phi_{0}\|_{W^{2-2/p}_{p}(\Omega)}+\|u_{0}\|_{W^{2-2/p}_{p}(\Omega)}+\|f\|_{L^{p}(Q)}),$

which proves (4).

Next, we prove the uniqueness of the solution.

For this, take $(\phi_{1},u_{1})$ and $(\phi_{2},u_{2})$ two solutions as in
the statement of Theorem 3.1 corresponding to the same $\phi_{0}$, $u_{0}$ and
$f$. By denoting $u:=u_{1}-u_{2}$ and $\phi:=\phi_{1}-\phi_{2}$, we have that
$u$ and $\phi$ satisfy the following problem:

$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle u_{t}-\Delta
u=-l\phi_{t}&\mbox{ in }Q,\\\
\displaystyle\phi_{t}-\Delta\phi=F(x,t,\phi_{1})-F(x,t,\phi_{2})+u&\mbox{ in
}Q,\\\ \displaystyle\partial u/\partial\nu=\partial\phi/\partial\nu=0&\mbox{
on }\sum,\\\ \displaystyle u(x,0)=\phi(x,0)=0&\mbox{ for all
}x\in\Omega.\end{array}\right.$ (50)

To complete the proof, it is enough to show that $\|u\|_{L^{2}(Q)}=0$ and
$\|\phi\|_{L^{2}(Q)}=0$. For this, we multiply first equation in (50) by
$u+l\phi$, integrate on $Q_{t}$ and use Green’s formula and Young’s inequality
to get

$\displaystyle\frac{1}{2}\int_{\Omega}(u+l\phi)^{2}(t)dx+\int_{Q_{t}}(|\nabla
u|^{2}+l\nabla u.\nabla\phi)dxds=0.$

By multiplying the second equation in (50) by $\phi$, integrating on $Q_{t}$
and using the Green’s formula $(H_{1})$ and Young’s inequality, we conclude
that

$\begin{array}[]{c}\displaystyle\frac{1}{2}\int_{\Omega}\phi^{2}(t)dx+\int_{Q_{t}}|\nabla\phi|^{2}dxds\\\
\displaystyle=\int_{Q_{t}}(F(x,s,\phi_{1})-F(x,s,\phi_{2}))\phi\
dxds+\int_{Q_{t}}u.\phi\ dxds\\\ \displaystyle\leq
C\left(\int_{Q_{t}}\phi^{2}dxds+\int_{Q_{t}}u^{2}dxds\right).\end{array}$

Next, by multiplying this last inequality by an arbitrary positive constant
$A$ and adding the result to the previous inequality, we obtain

$\begin{array}[]{c}\displaystyle\frac{1}{2}\int_{\Omega}((u+l\phi)^{2}+A\phi^{2})(t)dx+\int_{Q_{t}}(|\nabla
u|^{2}+l\nabla u.\nabla\phi+A|\nabla\phi|^{2})dxds\\\ \displaystyle\leq
C\int_{Q_{t}}(u^{2}+\phi^{2})dxds,\end{array}$

By using suitably Hölder’s inequality and taking $A=1+l^{2}$ and then
Gronwall’s lemma, we arrive at

$\displaystyle\int_{\Omega}(u^{2}+\phi^{2})(t)dx+\int_{Q_{t}}(|\nabla
u|^{2}+|\nabla\phi|^{2})dxds\leq 0.$

Therefore, $\|u\|_{L^{2}(Q)}=0$ and $\|\phi\|_{L^{2}(Q)}=0$ and thus,
$\phi_{1}=\phi_{2}$ e $u_{1}=u_{2}$.

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