# Radial Oscillations and Dynamical Instability Analysis for Linear-Quadratic
GUP-modified White Dwarfs

John Paul R. Bernaldez<EMAIL_ADDRESS>Adrian G. Abac
<EMAIL_ADDRESS>Roland Emerito S. Otadoy<EMAIL_ADDRESS>Theoretical and Computational Sciences and Engineering Group, Department of
Physics, University of San Carlos, Cebu City, Philippines 6000  Faculty of
Physics and Astronomy Friedrich-Schiller-Universität Jena, 07743 Jena, Germany
Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am
Mühlenberg 1, D-14476 Potsdam, Germany

###### Abstract

A modification to the Heisenberg uncertainty principle is called the
generalized uncertainty principle (GUP), which emerged due to the introduction
of a minimum measurable length, common among phenomenological approaches to
quantum gravity. One approach to GUP is called linear-quadratic GUP (LQGUP)
which satisfies both the minimum measurable length and the maximum measurable
momentum, resulting to an infinitesimal phase space volume proportional to the
first-order momentum $(1-\alpha p)^{-4}d^{3}xd^{3}p$, where $\alpha$ is the
still-unestablished GUP parameter. In this study, we explore the mass-radius
relations of white dwarfs whose equation of state has been modified by LQGUP,
and provide them with radial perturbations to investigate the dynamical
instability arising from the oscillations. We find from the mass-radius
relations that the main effect of LQGUP is to worsen the gravitational
collapse by decreasing the mass of the relatively massive white dwarfs
(including their limiting mass, while increasing their limiting radius). This
effect gets more prominent with larger values of $\alpha$. We then compare the
results with available observational data. To further investigate the impact
of the GUP parameter, a dynamical instability analysis of the white dwarf was
conducted, and we find that instability sets in for all values of $\alpha$.
With increasing $\alpha$, we also find that the central density at which
instability occurs decreases, resulting to a lower maximum mass. This is in
contrast to quadratic GUP, where instability only sets in below a critical
value of the quadratic GUP parameter.

###### keywords:

white dwarfs; GUP; mass-radius relation; equation of state.

##  1 Introduction

A modification to the fundamental commutation relation, which is predicted by
several quantum gravity phenomenology approaches leads to the generalization
of the Heisenberg’s uncertainty principle (HUP). This modification to the HUP
is called the generalized uncertainty principle (GUP), which emerged due to
the introduction of a minimum measurable length [1, 2, 3, 4, 5, 6, 7, 8]
Several variants to GUP have been developed over the years, including the
first of which was the so-called quadratic GUP (QGUP), which is consistent
with string theory phenomenology [1, 2, 3] and black hole physics [5], and
which leads to a phase space volume measure proportional to the square of
momenta, i. e. $\left(1+\beta p^{2}\right)^{-3}d^{3}xd^{3}p$ [9]. More
recently, a specific form to the generalized uncertainty principle, which
satisfies not only black hole physics but also string theory and doubly
special relativity (DSR), was developed and is called the “linear-quadratic”
GUP (LQGUP) approach [10, 11, 12, 13], which satisfies both the minimum
measurable length and the maximum measurable momentum conditions. An invariant
infinitesimal phase space volume proportional to first-order momentum, i.e.
$\left(1-\alpha p\right)^{-4}d^{3}xd^{3}p$, can be derived from the LQGUP,
where, in this case, the approach is known as the “linear” GUP approach
(indicated by the phase space volume correction only up to order $p$) [10, 12,
13, 14]. Here, $\alpha$ is the quantum gravity (or LQGUP) parameter, which has
no established values, but with upper bounds ranging from
$\alpha_{0}<{10}^{12}-{10}^{24}$ where $\alpha_{0}=\alpha\hbar/l_{p}$ is the
dimensionless LQGUP parameter and $l_{p}$ is the Planck length. From this
point forward in the study, however, for the sake of consistency, we refer to
the GUP used as linear-quadratic GUP (LQGUP).

LQGUP has been studied and applied to numerous systems, such as the internal
structure of compact objects like white dwarfs [13, 15] and neutron stars [14,
16], where LQGUP modifies the equation of state (EoS), and correspondingly,
the mass-radius relations. Several studies which involve investigating the
effects of LQGUP on white dwarfs show that LQGUP resists collapse, by
increasing or completely removing [17] their mass limit (Chandrasekhar
mass[18, 19, 20]), while other studies show that LQGUP worsens the collapse by
reducing the maximum mass limit and increasing the minimum radius limit [13,
15]. It should be noted that there are currently no established values for the
GUP parameter $\alpha_{0}$. From Ref. [21], the upper bound of $\alpha_{0}$ is
suggested to be in the order $10^{17}$ to be compatible with the electroweak
theory. On the other hand, Ref. [22] suggests an upper bound for the GUP
parameter of $\alpha_{0}<10^{24}$, in order to address the concerns with
regards Landau energy shifts for particles that have a mass $m$ and a charge
$e$ within a constant magnetic field and cyclotron frequency. The effects of
linear-quadratic GUP on the Lamb shift also suggest for a GUP parameter upper
bound of $\alpha_{0}<10^{12}$ [8, 10, 11]. From Ref. [23], the gravitational
wave event GW150914 suggests an upper bound of $\alpha_{0}<1.8\times 10^{20}$.
From Ref. [24, 25], the obtained bounds for the GUP parameter $\alpha_{0}$
from gravitational interaction is found to be in the range
$\alpha_{0}<10^{10}\text{ and }\alpha_{0}<10^{35}$. In addition, from Ref. [1,
25, 26, 27], an $\alpha_{0}$ value of the order unity is believed based on
predictions from string theory. As pointed out in Ref. [28] with their review
on the constraints of the quadratic GUP parameter, it was discussed that the
best bound for the dimensionless quadratic GUP parameter is
$\beta_{0}<10^{33}$, which is from scanning tunneling microscope studies.
Assuming the relationship $\beta_{0}\sim\alpha_{0}^{2}$ implies an upper bound
$\alpha_{0}<10^{17}$. However, as demonstrated by the other studies [13, 29],
GUP effects only become significantly noticeable for $\alpha_{0}>10^{17}$, and
values of $\alpha_{0}$ less than the bound will be dominated by numerical
uncertainty when doing a full solution of the EoS and the white dwarf’s
structure equations, so that the values of $\alpha_{0}$ are chosen to better
show the effects of LQGUP. In the context of general relativity, which matters
for the most massive white dwarfs [30], and where GUP effects are known to be
most prominent [13, 31, 32], the dynamical instability of a white dwarf or a
compact object can be investigated by studying its behavior under radial
oscillations [18]. This was done before in the context of the QGUP done by
Ref. [33], where, above some critical value of the dimensionless QGUP
parameter $\beta_{0}$, QGUP resists gravitational collapse by increasing the
Chandrasekhar mass limit [33, 34, 35, 36, 37]. The GUP-modified EoS prevents
further gravitational collapse and limits the formation of compact
astrophysical objects to that of the white dwarfs, which is in contrast with
astrophysical observations, since heavier compact objects such as neutron
stars exist. However, they find that dynamical instability can still set in
for $\beta_{0}\leq 5.38\times 10^{39}$ [38]. Since it was shown by Ref. [13,
14, 15], that LQGUP seems to worsen gravitational collapse, it is reasonable
to explore dynamical instability on linear-quadratic-GUP-modified white
dwarfs, and to see whether this effect is present for a large range of values
of the LQGUP parameter.

We then present the main problem and objectives of this work — investigating
dynamical instability for LQGUP-modified white dwarfs. The study is structured
as follows. We take the framework of general relativity (GR) and calculate the
stellar structure of the LQGUP-modified white dwarfs, where, in Section 2 we
will obtain the exact forms of the LQGUP-modified number density, energy
density, and pressure of ideal Fermi gases at zero-temperature, and from
there, a modified Chandrasekhar equation of state (EoS) will be formulated. We
choose an ideal, (degenerate) electron gas since we are mainly interested in
the phenomenological effects on LQGUP on white dwarfs. This assumption is
sufficient for massive white dwarfs since the effects of the GUP modification
manifests in the high-mass regime and the effects of finite temperature only
occur at low densities and pressures [13, 39]. From the equation of state, in
Section 3 we obtain the analytical form of the LQGUP-modified structure
equations for zero-temperature white dwarfs, and their solutions are found by
using the Tolman-Oppenheimer-Volkoff (TOV) structure equations in general
relativity (GR). From the structure equations, we obtain the modified mass-
radius relation of the white dwarfs. We then carry out in Section 4 the
dynamical instability analysis on the LQGUP-modified white dwarfs so that the
maximal stable configuration of the system is identified. We compare the
results obtained to the work of Ref. [38], where QGUP-modified white dwarfs
are found to have both unstable configurations and super-stable configurations
(where instability does not set in, for various values of the GUP parameter),
and white dwarfs can have arbitrarily large masses. Finally, we present in
Section 5 the conclusions and recommendations for future work.

##  2 Linear quadratic GUP-modified White Dwarfs

In this section, we review the Linear Generalized Uncertainty Principle
(LQGUP), as well as the statistical mechanics of Fermi gases modified by
LQGUP.

###  2.1 Generalized Uncertainty Principle

The generalized uncertainty principle (GUP) is a modification to the
Heisenberg’s uncertainty principle, which emerged due to the existence of a
minimum measurable length [1, 6]. The fundamental commutation relation that
satisfies both black hole physics and string theory is [4, 9, 31]:

$[x,p]=i\hbar(1+\beta p^{2}),$ (1)

which then results to the quadratic GUP given by:

$\Delta x\Delta p\geq\frac{\hbar}{2}[1+\beta(\Delta p)^{2}+\langle
p\rangle^{2}].$ (2)

Another approach to the general uncertainty principle, which not only
satisfies black hole physics but also fits well with string theory, and doubly
special relativity (DSR), was developed and is called the “linear-quadratic”
GUP (LQGUP) approach [10, 11, 12], where the commutation relation is given by:

$\begin{split}\left[x_{i},p_{j}\right]&=i\left[\delta_{ij}-\alpha\left(p\delta_{ij}+\frac{p_{i}p_{j}}{p}\right)+\alpha^{2}\left(p^{2}\delta_{ij}+3p_{i}p_{j}\right)\right]\\\
[x_{i},x_{j}]&=[p_{i},p_{j}]=0\end{split}$ (3)

and the generalized uncertainty principle in one dimension is given by:

$\Delta x\Delta p\geq\frac{\hbar}{2}[1-2\alpha\langle
p\rangle+4\alpha^{2}\langle p\rangle^{2}],$ (4)

where
$\alpha=\frac{\alpha_{0}}{\left(M_{p}c\right)}=\alpha_{0}\left(\frac{l_{p}}{\hbar}\right)$
is the LQGUP parameter [10, 11, 13, 40], $M_{p}$ is the Planck mass, and
$l_{p}$ is the Planck length.

###  2.2 Statistical Mechanics of Fermionic Gases with LQGUP Modification

Statistical mechanics plays an important role to understanding the structure
of stars and compact objects since white dwarfs are generally composed of
degenerate electron gases. With this, there is a need to derive the
thermodynamic properties of the degenerate Fermi gas at zero temperature
obeying Pauli’s exclusion principle. We start with the grand canonical
ensemble [13, 29, 33, 41] formalism in statistical mechanics in order to
derive the relevant thermodynamic quantities, which are the number density
$n$, energy density $e$, and pressure $P$. The partition function can be
written as:

$\ln Z=\sum_{j}\ln[1+e^{\frac{(\mu-E_{j})}{k_{B}T}}].$ (5)

For large volumes, as modified by linear-quadratic GUP, we then have:

$\int\frac{d^{3}xd^{3}p}{(2\pi\hbar)^{3}}\rightarrow\int\frac{d^{3}xd^{3}p}{(2\pi\hbar)^{3}(1-\alpha
p)^{4}}.$ (6)

Thus, the grand canonical potential can then be rewritten as:

$\Phi=-gk_{B}T\int\frac{d^{3}xd^{3}p}{(2\pi\hbar)^{3}(1-\alpha
p)^{4}}\ln[1+e^{\frac{(\mu-E)}{(k_{B}T)}}],$ (7)

where $g$ refers to the multiplicity of states due to the spin of the
particle, and $(2\pi\hbar)^{3}$ refers to the volume occupied by and energy
state in phase-space. From this, we then derive the expressions for the number
density $n$, pressure $P$, electron energy density $\epsilon_{e}$, and total
energy density $\epsilon$. The expression for number density, pressure, and
electron energy density for degenerate ideal Fermi gases are given by [13]:

$n=\frac{8\pi}{(2\pi\hbar)^{3}}\int_{0}^{p_{F}}\frac{p^{2}dp}{(1-\alpha
p)^{4}}$ (8)
$\epsilon_{e}=\frac{8\pi}{(2\pi\hbar)^{3}}\int_{0}^{p_{F}}E\frac{p^{2}dp}{(1-\alpha
p)^{4}}$ (9)
$P=\frac{8\pi}{(2\pi\hbar)^{3}}\int_{0}^{p_{F}}(E_{F}-E)\frac{p^{2}dp}{(1-\alpha
p)^{4}}=E_{F}n-\epsilon_{e},$ (10)

where $p_{F}$ is the Fermi momentum, which corresponds to the maximum possible
momentum, corresponding to the Fermi energy $E_{F}$ in the zero-temperature
case. To optimize the numerical computation, we define a set of dimensionless
quantities:

$\begin{split}\xi\equiv\frac{p_{F}}{m_{e}c}\\\
\widetilde{p}\equiv\frac{p}{m_{e}c}\\\ \sigma\equiv m_{e}c\alpha\end{split}$
(11)

Note that our chosen values of $\alpha\in[10^{17},10^{19},10^{20},7\times
10^{20}]$ (for comparison with previous studies [13, 14, 31, 33, 38] and to
make the GUP effects more noticeable) correspond to $\sigma\in[4.2\times
10^{-6},4.2\times 10^{-4},4.2\times 10^{-3},2.9\times 10^{-2}]$. Therefore,
from this redefinition of the variables, we see that the LQGUP parameter
becomes $\sigma<1.0$ with the dimensionless momentum $\tilde{p}$ compensating
for it by becoming relatively large (by a factor $m_{e}c$), allowing for more
convenient and efficient numerical calculations. Such rescaling of the
variables has also been done in Ref. [13, 14, 31, 33, 38]. As discussed
earlier, LQGUP effects only become more apparent with $\alpha_{0}>10^{17}$,
since any value below that results to deviations that are too small to
distinguish from numerical uncertainty. Therefore, we investigate the implied
effects of LQGUP by using the aforementioned values of $\alpha_{0}$, with the
emphasis that such values, though not necessarily within the bound
$\alpha_{0}<10^{17}$, are still within the other suggested bounds, and from
which we can more noticeably see the behavioral effects of GUP. Thus,
equations (8)-(10) can be rewritten as:

$n=\frac{\pi
m_{e}^{3}c^{3}}{(2\pi\hbar)^{3}}\int_{0}^{\xi}8\frac{\tilde{p}^{2}d\tilde{p}}{(1-\sigma\tilde{p})^{4}}=\frac{K}{m_{e}c^{2}}\tilde{N}$
(12) $\epsilon_{e}=\frac{\pi
m_{e}^{4}c^{5}}{(2\pi\hbar)^{3}}\int_{0}^{\xi}8\tilde{E}\frac{\tilde{p}^{2}d\tilde{p}}{(1-\sigma\tilde{p})^{4}}=K\tilde{\epsilon}_{e}$
(13) $P=\frac{\pi
m_{e}^{4}c^{5}}{(2\pi\hbar)^{3}}\int_{0}^{\xi}8(\tilde{E}_{F}-\tilde{E})\frac{\tilde{p}^{2}d\tilde{p}}{(1-\sigma\tilde{p})^{4}}=K\tilde{P},$
(14)

where $\widetilde{E}=\sqrt{1+{\widetilde{p}}^{2}}$,
${\widetilde{E}}_{F}=\sqrt{1+\xi^{2}}$, and $K=\frac{\pi
m_{e}^{4}c^{5}}{(2\pi\hbar)^{3}}$ (note that $K$ has units of pressure).
$\tilde{N}$, $\tilde{\epsilon}_{e}$, and $\tilde{P}$ refers to the
dimensionless number density, dimensionless electron energy density, and
dimensionless pressure respectively. Evaluating equations (12) and (14) yields
the following dimensionless quantities:

$\tilde{N}(\xi)=\frac{8\xi^{3}}{3(1-\xi\sigma)^{3}}$ (15)
$\begin{split}\widetilde{P}\left(\xi\right)=&\frac{1}{3\sigma^{4}}4\Bigg{\\{}-2\sqrt{1+\xi^{2}}\sigma-\frac{2\sqrt{1+\xi^{2}}\sigma}{\left(-1+\xi\sigma\right)^{3}}-\frac{6\sqrt{1+\xi^{2}}\sigma}{\left(-1+\xi\sigma\right)^{2}}-\frac{6\sqrt{1+\xi^{2}}\sigma}{-1+\xi\sigma}\\\
&+\frac{\sigma\left(6+11\sigma^{2}+2\sigma^{4}\right)}{\left(1+\sigma^{2}\right)^{2}}\\\
&+\frac{\left(\sqrt{1+\xi^{2}}\sigma\left(6+11\sigma^{2}+2\sigma^{4}-3\xi\sigma\left(5+9\sigma^{2}+2\sigma^{4}\right)\right)\right)}{\left(\left(-1+\xi\sigma\right)^{3}\left(1+\sigma^{2}\right)^{2}\right)}\\\
&+\frac{\left(\sqrt{1+\xi^{2}}\sigma\left(\xi^{2}\sigma^{2}\left(11+20\sigma^{2}+6\sigma^{4}\right)\right)\right)}{\left(\left(-1+\xi\sigma\right)^{3}\left(1+\sigma^{2}\right)^{2}\right)}\\\
&-6\sinh^{-1}\left[\xi\right]-\frac{3\left(2+5\sigma^{2}+4\sigma^{4}\right)\ln\left[1-\xi\sigma\right]}{\left(1+\sigma^{2}\right)^{\frac{5}{2}}}\\\
&-\frac{3\left(2+5\sigma^{2}+4\sigma^{4}\right)\ln\left[\sigma+\sqrt{1+\sigma^{2}}\right]}{\left(1+\sigma^{2}\right)^{\frac{5}{2}}}\\\
&+\frac{3\left(2+5\sigma^{2}+4\sigma^{4}\right)\ln\left[\xi+\sigma+\sqrt{1+\xi^{2}}\sqrt{1+\sigma^{2}}\right]}{\left(1+\sigma^{2}\right)^{\frac{5}{2}}}\Bigg{\\}}.\end{split}$
(16)

The total energy density $\epsilon$ is then obtained from the equation:

$\epsilon=\rho c^{2}+\epsilon_{e}$ (17)

where the mass density $\rho$ is equal to:

$\rho(\xi)=m_{u}\mu_{e}n=\frac{K}{qc^{2}}\tilde{N}(\xi)$ (18)

where $q=\frac{m_{e}}{(\mu_{e}m_{u})}$, $m_{u}=1.6605\times 10^{-24}\text{g}$,
which is the atomic mass, and $\mu_{e}=A/Z=2$, where $A$ is the mass number
and $Z$ is the atomic number (assuming a He white dwarf configuration). The
total energy density is:

$\epsilon=\frac{K}{q}\tilde{N}(\xi)+K\tilde{\epsilon}_{e}=\frac{K}{q}\tilde{\varepsilon},$
(19)

where $\widetilde{\varepsilon}$ is the dimensionless total energy density and
is given by:

$\tilde{\varepsilon}=\tilde{N}(\xi)+q\tilde{\epsilon}_{e}$ (20)

Figure 1: We show the equation of state of linear-quadratic GUP-modified white
dwarfs for $\alpha_{0}=0$, $10^{17}$, $10^{19}$, $10^{20}$, $7\times 10^{20}$.

It should be noted that the LQGUP-modified equations (12), (13), and (14) are
singular at $\tilde{p}=\frac{1}{\sigma}$. This implies that there is a
constraint with regards the momentum values such that
$0\leq\tilde{p}<\frac{1}{\sigma}$. This maximum $\tilde{p}$ is of the same
order of magnitude as the maximum momentum shown in [13, 42]. Since $\xi$ is
the Fermi momentum for a white dwarf, the LQGUP structure equation for the
white dwarf can be safely calculated in the said range, which is sufficient in
obtaining the mass-radius relation of white dwarfs. Moreover, it has recently
been discussed in Ref. [16], the first law of thermodynamics may be violated
when GUP is incorporated in the equation of state, and is only preserved when
considering anisotropic pressure. From the first law of thermodynamics, the
following should be satisfied: $\mu=(\epsilon+P)/\rho$, where
$\mu=d\epsilon/d\rho$ is the total chemical potential. With the GUP correction
of the order $(1-\sigma\tilde{p})^{-4}$, we obtain $\sigma<<1$ even for the
largest GUP parameter considered, which is $7\times 10^{20}$ ($\sigma\sim
0.029$). This means that the GUP corrections considered have negligible impact
in the numerical computations since the first law of thermodynamics can only
be significantly violated for $\sigma>>1$, which is not the case in this
study.

Equations (16) and (20) will be used to generate the plot for the equation of
state for the LQGUP-modified white dwarfs. The equation of state for the zero-
temperature ideal electron gases for various values of the LQGUP parameter
$\alpha_{0}$ is shown in Figure 1. From the plots of the equations of states,
it can be seen that the plot for $\alpha_{0}=10^{17}$ and $\alpha_{0}=0$ are
practically the same, which means that the effect of linear quadratic GUP for
a GUP parameter of $10^{17}$ is negligible. It can also be seen from the plot
that there is a slight increase in the slope for $\alpha_{0}=10^{19}$, which
can be seen clearly for larger values of the pressure $\tilde{P}$. A more
notable increase of the slope is visible for both $\alpha_{0}=10^{20}$ and
$\alpha_{0}=7\times 10^{20}$ for large values of $\tilde{P}$, which means that
the effect of linear-quadratic GUP is significant for $\alpha_{0}=10^{19}$ and
$\alpha_{0}=10^{20}$, and is much greater for $\alpha_{0}=7\times 10^{20}$.
From the plots, we can say that the EOS softens with the presence of LQGUP.

##  3 White Dwarf Structure

In this section, we use the results from Chapter 2 to modify the TOV equations
and obtain the mass-radius relations of white dwarfs with LQGUP.

###  3.1 Tolman-Oppenheimer-Volkoff (TOV) Equations

The structure of white dwarfs can be approximated by the Newtonian equations
of stellar structure. However, for the most massive white dwarfs, which are
affected by GUP, general relativity (GR) has to be taken into account [30].
From the Einstein field equations, and using the metric
$ds^{2}=e^{\nu}c^{2}dt^{2}-e^{\mu}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$, one can obtain the structure of GR known as the TOV equations
[43, 44]. It has been suggested that a modification to the TOV equations may
arise from the modification of the effective metric due to GUP. However, this
modification to the TOV equations is outside the scope of the study since we
are only interested on the phenomenological effects of LQGUP and mass-radius
relations from the standard TOV equations, which is consistent to the
approaches of Ref. [13, 14, 33, 38]. The TOV equations are given by [13, 38,
43, 44]:

$\frac{dP}{dr}=-\frac{G}{c^{2}r}(\epsilon+P)\frac{m+\frac{4\pi
r^{3}}{c^{2}}}{r-\frac{2GM}{c^{2}}},\quad P(r=0)=P_{c},\quad
P(r=R_{\star})=0,$ (21) $\frac{dm}{dr}=\frac{4\pi}{c^{2}}\epsilon r^{2},\quad
m(r=0)=0,\quad m(r=R_{\star})=M_{\star},$ (22)

where (21) is the statement of hydrostatic equilibrium, while (22) is the mass
continuity equation together with their boundary conditions. $R_{\star}$ and
the $M_{\star}$ are the star radius and star mass, respectively. From the
boundary conditions, we can see that the pressure is greatest $P_{c}$ at the
center of the star and gradually decreases to zero at $r=R_{\star}$. On the
other hand, the mass at the center of the star is zero, and as we move away
from the center of the star, the star accumulates mass until it reaches a mass
equal to $M_{\star}$ at $r=R_{\star}$.

We can then re-express the TOV equations (21) and (22) in terms of the
dimensionless mass given by $v\equiv\frac{m}{m_{0}}$, dimensionless radius
$\eta\equiv\frac{r}{r_{0}}$, where $m_{0}\equiv\frac{q^{2}c^{4}}{G\sqrt{4\pi
G\ K}}$ and $r_{0}\equiv\frac{qc^{2}}{\sqrt{4\pi G\ K}}$. With this, we start
with

$\frac{d\widetilde{P}}{d\eta}=\frac{d\widetilde{P}}{d\xi}\frac{d\xi}{d\eta}\rightarrow\frac{d\xi}{d\eta}\
=\frac{1}{\frac{d\widetilde{P}}{d\xi}}\frac{d\widetilde{P}}{d\eta}\ ,$ (23)

where $\frac{d\widetilde{P}}{d\xi}$ and $\frac{d\widetilde{P}}{d\eta}$ are:

$\frac{d\widetilde{P}}{d\xi}=\frac{\xi}{\sqrt{1+\xi^{2}}}\widetilde{N}\left(\xi\right)$
(24)

and

$\frac{d\widetilde{P}}{d\eta}=\frac{-\left(\widetilde{\varepsilon}+q\widetilde{P}\right)}{\eta}\frac{v+q\widetilde{P}\eta^{3}}{\eta-2qv}$
(25)

respectively. We can then obtain the dimensionless hydrostatic equilibrium
equation for $\frac{d\xi}{d\eta}$

$\frac{d\xi}{d\eta}=-\frac{\sqrt{1+\xi^{2}}}{\xi}\frac{1+q\sqrt{1+\xi^{2}}}{\eta}\frac{v+q\tilde{P}\eta^{3}}{\eta-2qv},\quad\xi(\eta=0)=\xi_{c},\quad\xi(\eta=\eta_{R})=0.$
(26)

Equation (22) can then be rewritten in terms of dimensionless quantities and
is given by:

$\frac{m_{0}dv}{r_{0}d\eta}=\frac{4\pi}{c^{2}}\frac{K}{q}\widetilde{\varepsilon}\left(\eta
r_{0}\right)^{2},$ (27)

which can be simplified into the dimensionless mass-continuity equation:

$\frac{dv}{d\eta}=\frac{4\pi}{c^{2}}\frac{K}{q\
m_{0}}\widetilde{\varepsilon}{r_{0}}^{3}\eta^{2}=\frac{4\pi}{c^{2}}\frac{K}{q\
m_{0}}\widetilde{\varepsilon}\eta^{2}\frac{\left(\frac{qc^{2}}{\sqrt{4\pi G\
K}}\right)^{3}}{\frac{q^{2}c^{4}}{G\sqrt{4\pi G\ K}}},$ (28)

and we finally obtain:

$\frac{dv}{d\eta}=\tilde{\varepsilon}\eta^{2},\quad v(\eta=0)=0,\quad
v(\eta=\eta_{R})=v_{R}$ (29)

where $v_{R}$ and $\eta_{R}$ are the dimensionless final star mass
($v_{R}=M_{\star}/m_{0}$) and final star radius ($\eta_{R}=R_{\star}/r_{0}$),
respectively. Notice that in this case, we have redefined our TOV equations in
terms of the Fermi momentum $\xi$ instead of the usual pressure $p$, for
convenience in dealing with the dynamical instability analysis later on. This
is different from approach of Ref. [13] (where the pressure TOV equation was
not reexpressed in terms of the Fermi momentum), but should nevertheless yield
the same results.

###  3.2 Mass-Radius Relations

Figure 2: The figure shows the dimensionless Fermi momentum $\xi$ as a
function of radius $\eta\,r_{0}$ (in km) for GUP parameter values
$\alpha_{0}=0$, $10^{17}$, $10^{19}$, $10^{20}$, $7\times 10^{20}$, given an
initial central momentum $\xi_{c}=1.0$. The final star radius $\eta_{R}$ is
obtained when $\xi=0$.

Figure 2 shows the Fermi momentum versus $\eta$ plot for both the LQGUP-
modified white dwarfs and the white dwarf with no linear-quadratic GUP
corrections. The figure shows that the Fermi momentum of the star decreases
from the center of the white dwarf and vanishes at the radius of the white
dwarf. We also see that the GUP $\alpha_{0}=10^{17}$ appears to have a
negligible effect on the white dwarf. On the other hand, the white dwarfs with
$\alpha_{0}=$ $10^{19}$, $10^{20}$, and $7\times 10^{20}$ have variations to
their plots with respect to the ideal case, where they have smaller final star
radius than the white dwarfs with no LQGUP effects. This trend is more evident
when looking at the GUP parameter of $7\times 10^{20}$, where the plot is
shown to have a smaller value of $\eta_{R}$.

Figure 3: The accumulated star mass $vm_{0}$ (in $M_{\odot}$) as a function of
accumulated star radius $\eta\,r_{0}$ (in km) for GUP parameter values
$\alpha_{0}=0$, $10^{17}$, $10^{19}$, $10^{20}$, $7\times 10^{20}$. , with
inital $\xi_{c}=1.0$. Note that in this case, the star accumulates mass until
it reaches its final star mass $v_{R}$ at $\eta=\eta_{R}$. The value of
$v_{R}$ gets smaller with increasing $\alpha_{0}$.

Figure 3 shows the $v$ versus $\eta$ plot for varying values of the GUP
parameter $\alpha_{0}$. The plot with $\alpha_{0}=10^{17}$ shows negligible
effects as it shows a similar trend to the plot for $\alpha_{0}=0$. On the
other hand, $\alpha_{0}=10^{19}$, and $\alpha_{0}=10^{20}$ show significant
LQGUP effects on the white dwarfs, where LQGUP appears to decrease the star
mass of the white dwarf, which is notable at larger values of $\eta$. The plot
also shows that for $\alpha_{0}=7\times 10^{20}$, $v_{R}$ is much smaller
compared to those with smaller $\alpha_{0}$.

Figure 4: Final star radius $\eta_{R}\,r_{0}$ (in km) as a function of the
central Fermi momentum $\xi_{c}$ for GUP parameter values $\alpha_{0}=0$,
$10^{17}$, $10^{19}$, $10^{20}$, $7\times 10^{20}$.

We then solve the TOV equations for a chosen range of values of the central
Fermi momentum $\xi_{c}$, taking caution that $\xi_{c}<1/\sigma$, and plot the
final radius $\eta_{R}$ and star mass $v_{R}$ as a function of $\xi_{c}$. The
star radius $\eta_{R}$ versus central Fermi momentum plot $\xi_{c}$ (shown in
Figure 4) indicates that various values of $\xi_{c}$ correspond to different
star radius values. Larger values of $\xi_{c}$ corresponds to a smaller star
radius, while smaller values of $\xi_{c}$ corresponds to a more prominent star
radius. This checks out with the fact that a larger $\xi_{c}$ indicates a more
compact object, thus, a smaller radius. It should be noted that for a GUP
parameter of $7\times 10^{20}$, $\eta_{R}$ decreases faster at low values of
$\xi_{c}$ compared with smaller $\alpha_{0}$. However, $\eta_{R}$ reaches a
minimum at some $\xi_{c}$ before increasing for larger values of $\xi_{c}$,
while for the other $\alpha_{0}$, $\eta_{R}$ just decreases in value in this
range of central Fermi momentum $\xi_{c}\in[0,25]$. Note that the range of the
central Fermi momenta that we used is well within the constraints we placed
for the Fermi momentum. This is sufficient to describe the trend of the mass
radius relation, as well as the dynamical instability analysis for LQGUP-
modified white dwarfs.

Figure 5: Final star mass $v_{R}\,m_{0}$ (in $M_{\odot}$) as a function of
dimensionless central Fermi momentum $\xi_{c}$ for GUP parameter values
$\alpha_{0}=0$, $10^{17}$, $10^{19}$, $10^{20}$, $7\times 10^{20}$.

Figure 5 corresponds to the star mass versus the central Fermi momentum plot,
which shows the noticeable effects of GUP on the star mass. The plot shows
that the effect of GUP is to decrease the maximumam star mass, that is the
Chandrasekhar limit. The GUP parameter $\alpha_{0}=10^{17}$ exhibits
negligible effects to the maximum star mass, while $\alpha_{0}=10^{19}$ shows
a noticeable decrease and a drastic decrease of the maximum star mass for
$\alpha_{0}=10^{20}$. For $\alpha_{0}=7\times 10^{20}$, $v_{R}$ drastically
decreases smaller values of $\xi_{c}$, however, for larger values of Fermi
momentum $\xi_{c}$, $v_{R}$ is shown to increase.

Figure 6: The plot shows the final star mass $v_{R}\,m_{0}$ (in $M_{\odot}$)
as a function of final star radius $\eta_{R}\,r_{0}$ (in km) for GUP parameter
values $\alpha_{0}=0$, $10^{17}$, $10^{19}$, $10^{20}$, $7\times 10^{20}$
together with observational data taken from Ref. [45].

The mass-radius relations can be seen in Figure 6, (together with
observational data taken from Ref. [45]) which contains the parametric plots
between star mass vs star radius. The plots show that as the star radius
decreases, the star mass increases, which means that the white dwarf is
heavier when it’s dimensionally smaller, demonstrating its compact nature.
Examining the plot from the largest star radius decreasing towards zero (right
to left), we see that the star mass eventually reaches a maximum value, before
decreasing towards zero radius. The plots also show that for larger values of
the GUP parameter $\alpha_{0}$, the maximum mass is smaller, while
$\alpha_{0}=10^{17}$ shows that the effects of the GUP parameter are
negligible. There is also a larger limiting radius, which is the radius by
which the star reaches its maximum mass, for the larger GUP parameters. In
addition, for a GUP parameter of $7\times 10^{20}$, as the final radius of the
white dwarf becomes smaller, the mass increases but eventually decreases after
reaching the maximum mass of the white dwarf. At the lower values of the final
radius, the plot starts to spiral, where both the final mass and final radius
increases. It should be noted that direct comparisons of the observation data
and our results to determine the most compatible GUP parameter is not
straightforward to establish and one has to take caution in doing so, mainly
because the regime where GUP is expected to dominate have very little
observational data. As another note of caution, we also state that the GUP
parameters used are larger than some of the bounds discussed earlier
(particularly $\alpha_{0}<10^{17}$), and do not necessarily mean that such GUP
parameters exist, since the main point of this study is to see how the changes
from the ideal case would look like given an increasing (non-zero) value of
the LQGUP parameter; in other words, the effects may be exaggerated than what
one may expect in nature. From Figure 6, it can be seen that most of the
observation data can be found from the low to intermediate mass stars and have
very few data points for high mass stars, where the GUP effects is very
noticeable. In addition, the assumption of ideal electron gas for the equation
of state of the white dwarfs makes it difficult to constrain the possible
values of the GUP parameter. Nevertheless, we perform a chi-square analysis in
the vein of Ref. [46] to see which GUP parameter looks “most compatible” with
current observational data. From Ref. [46], we can use chi-square analysis to
investigate whether the observation data used in this study favors the LQGUP
model. The chi-square between the observational data and theoretical data is
given by:
$\Delta\chi_{i}^{2}=(M-M_{i})^{2}/\sigma^{2}_{M,i}+(R-R_{i})^{2}/\sigma^{2}_{R,i}$,
where $M_{i}$ and $R_{i}$ are the observation mass and radius data, $M$ and
$R$ are from our LQGUP-modified mass-radius relations, while $\sigma_{M,i}$
and $\sigma_{r,i}$ are the mass and radius standard deviation of the
observational data, respectively. From this, we extract the minimum
$\Delta\chi^{2}_{i}$ value for each GUP parameter value, and calculate the
$\chi^{2}$ value by $\chi^{2}=\Sigma_{i}\Delta X_{i}^{2}$. We find the
following chi-square values $\chi^{2}=[432.610,432.606,432.1,428.7,558.0]$ for
$\alpha_{0}=[0,10^{17},10^{19},10^{20},7\times 10^{20}]$. The $\chi^{2}$
values decrease from $\alpha_{0}=0$ to $10^{20}$ before jumping to a large
value at $7\times 10^{20}$. More sophisticated equations of state, such as the
inclusion of other compositions beside the ideal electron gas, as well as
improved independent mass and radius measurements (where neither relies on an
existing mass-radius relation to calculate the radius from the mass or vice-
versa) is needed to further constrain the GUP parameters, as discussed on Ref.
[47]. The increasing sophistication of the EoS to make it more realistic, not
to mention the incorporation of GUP (both in the EoS and in other cases, the
metric itself of the underlying theory of gravity), are non-trivial, and is
outside the scope of this study.

$\alpha_{0}$ |  $v_{R}$ ($M\odot$) |  $\eta_{R}$(km)
---|---|---
0 | 1.411 | 1077.129
$10^{17}$ | 1.411 | 1077.129
$10^{19}$ | 1.393 | 1269.338
$10^{20}$ | 1.289 | 2033.731
$7\times 10^{20}$ | 0.984 | 3409.005
Table 1: Values for the maximum value of star mass $v_{R}$ with the
corresponding star radius $\eta_{R}$ for different values of the GUP
parameter.

The table shows the value of maximum mass and minimum radius of the white
dwarf for every $\alpha_{0}$. We see that for $\alpha_{0}=10^{17}$ are similar
with $\alpha_{0}=0$, which means that the effect of LQGUP is negligible. In
addition, we can see that the maximum value of $v_{R}$ decreases with
increasing value of the GUP parameter. On the other hand, the corresponding
$\eta_{R}$ increases when $\alpha_{0}$ is larger. This means that for larger
values of the GUP parameter $\alpha_{0}$, LQGUP-modified white dwarfs have a
lower mass limit. The behaviours observed in the structure equations and mass
radius relations are similar to those obtained from Ref. [13], albeit using a
different approach, where we wrote the TOV equations in terms of the Fermi
momentum $\xi$. We will see in the next section that white dwarfs beyond the
maximum mass limit are unstable configurations.

##  4 Dynamical Instability Analysis

In Subsection 4.1, we review some general results in analyzing the dynamical
instability of white dwarfs, following the procedures laid out by Ref. [18,
19, 20, 31, 33, 38, 48]. In Subsection 4.2, we present our own derived
expressions for the dynamical instability analysis containing LQGUP
modifications.

###  4.1 Radial Oscillations and Dynamical Instability of White Dwarfs

For the dynamical stability analysis, we start with the metric interior of the
star, which can be expressed as [20, 31, 38]:

$ds^{2}=e^{\nu+\delta\nu}c^{2}dt^{2}-e^{\mu+\delta\mu}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (30)

From the equation, $\nu\left(r\right)$ and $\mu\left(r\right)$ here refers to
the equilibrium metric potentials, while $\delta\nu\left(r,t\right)$ and
$\delta\mu\left(r,t\right)$ refers to the perturbations due to small radial
Lagrangian displacements $\zeta\left(r,t\right)$ given by
$\zeta\left(r,t\right)=r^{-2}e^{\frac{\nu}{2}\psi\left(r\right)e^{i\omega t}}$
which induces perturbations $\delta P\left(r,t\right)$ and
$\delta\epsilon\left(r,t\right)$. The radial oscillation equation can then be
obtained in the Sturm-Liouville form and is given by[38, 48]:

$\frac{d}{dr}\Big{(}U\frac{d\psi}{dr}\Big{)}+(V+\frac{\omega^{2}}{c^{2}}W)\phi=0,$
(31)

With the boundary conditions $\psi=0$ at $r=0$, and $\delta
P=-e^{\frac{\nu}{2}}\left(\frac{\gamma
P}{r^{2}}\right)\frac{d\psi}{dr}=0\,\,\text{at}\,\,r=R$, the terms in equation
(3.2) are:

$U(r)=e^{\frac{(\mu+3\nu)}{2}}\frac{\gamma P}{r^{2}},$ (32)
$V(r)=-4\frac{e^{\frac{\mu+3\nu}{2}}}{r^{3}}\frac{dP}{dr}-\frac{8\pi
G}{c^{4}}\frac{e^{3\frac{\mu+\nu}{2}}}{r^{2}}P\left(P+\epsilon\right)+\frac{e^{\frac{\mu+3\nu}{2}}}{r^{2}}\frac{1}{P+\epsilon}\left(\frac{dP}{dr}\right)^{2},$
(33)
$W\left(r\right)=\frac{e^{\frac{3\mu+\nu}{2}}}{r^{2}}\left(P+\epsilon\right),$
(34)

where $\gamma$ is the adiabatic index equal to:

$\gamma=\frac{\epsilon+P}{P}\left(\frac{dP}{d\epsilon}\right).$ (35)

Distributing $\psi$ to equation (31) and then integrating, where we let
$\psi^{\prime}=\frac{d\psi}{dr}$, we obtain:

$J\left[\psi\right]=\int_{0}^{R}{U\psi^{\prime
2}-V\psi^{2}-\frac{\omega^{2}}{c^{2}}W\psi^{2}}dr.$ (36)

Minimizing equation (40) with respect to $\psi$, we can then obtain the Sturm-
Liouville equation in equation (31), which means we have a variational basis
for determining the lowest characteristic eigenfrequency and is expressed as:

$\frac{\omega_{0}^{2}}{c^{2}}=\min_{\psi\left(r\right)}\frac{\int_{0}^{R}{(U}\psi^{\prime
2}-V\psi^{2})dr}{\int_{0}^{R}W\psi^{2}dr}$ (37)

From equation (43), we will then obtain a set of $\omega_{0}^{2}$ values
corresponding to a range of central Fermi momentum $\xi$. A star with an
$\omega_{0}^{2}$ that is greater than zero means that the star is stable,
while a star that has a negative $\omega_{0}^{2}$ value means that dynamical
instability is present for that certain central Fermi momentum value. The
trial function $\psi$ of the fundamental mode can be approximated in the form
$\psi(r)=c_{0}r^{3}$ [20, 38, 49] The equation for the central mass density
can be found in equation (18). With this, the mass density at which the white
dwarf shows gravitational collapse will be the critical density $\rho_{c}$*
and can be identified with the zero eigenfrequency solution of the equation.

The interior Schwarzschild metric potentials, which satisfy Einstein’s field
equations, are given by [38, 43, 44]

$e^{-\mu(r)}=1-\frac{2GM}{c^{2}r},$ (38)

and

$e^{\nu}=\left(1-\frac{2GM}{c^{2}R}\right)\left(\exp{\left[-2\int_{0}^{P\left(r\right)}\frac{dP}{\varepsilon+P}\right]}\right).$
(39)

###  4.2 Dynamical Instability Analysis for LQGUP-modified White Dwarfs

We then rewrite the expressions in Section 4.1, this time taking into account
the dimensionless quantities we defined in Section 2 together with LQGUP
modification. The dimensionless adiabatic index $\tilde{\gamma}$ can be
obtained using the definition of $\frac{dP}{d\tilde{\varepsilon}}$ given by:

$\frac{dP}{d\tilde{\varepsilon}}=\left[\frac{\xi^{2}\left(1-\xi\sigma\right)}{3\sqrt{1-\xi^{2}}\
\left(1+q\sqrt{1+\xi^{2}}\right)}\right],$ (40)

where $\tilde{\gamma}$ will be:

$\tilde{\gamma}=\frac{\frac{K}{q}\widetilde{\varepsilon}+K\widetilde{P}}{K\widetilde{P}}\left(\frac{d\tilde{P}}{d\tilde{\varepsilon}}\right)=\frac{\widetilde{\varepsilon}+q\widetilde{P}}{\widetilde{P}}\left[\frac{\xi^{2}\left(1-\xi\sigma\right)}{3\sqrt{1-\xi^{2}}\
\left(1+q\sqrt{1+\xi^{2}}\right)}\right],$ (41)

and can be simplified to:

$\tilde{\gamma}=\tilde{N}\xi^{2}\frac{\left(1-\sigma\
\xi\right)}{3\tilde{P}\sqrt{1+\xi^{2}}}$ (42)

Equation (37) can then be rewritten in terms of dimensionless quantities given
by [38]:

$\omega_{0}^{2}=\left(\frac{qc^{2}}{r_{0}^{2}}\right)\frac{\tilde{U}+\tilde{V}}{\tilde{W}},$
(43)

where the equations for $\tilde{U}$, $\tilde{V}$, and $\tilde{W}$, are given
by:

$\tilde{U}=\int_{0}^{\eta_{R}}{e^{\left(\mu+3\nu\right)/2}\frac{\gamma\widetilde{P}}{\eta^{2}}}\psi^{\prime
2}d\eta,$ (44)
$\tilde{V}=\int_{0}^{\eta_{R}}{\frac{e^{\left(\mu+3\nu\right)/2}}{\eta^{2}}\left[\frac{4}{\eta}\frac{d\widetilde{P}}{d\eta}+2qe^{\mu}\widetilde{P}\left(\widetilde{\varepsilon}+q\widetilde{P}\right)-\frac{q}{\widetilde{\varepsilon}+q\widetilde{P}}\left(\frac{d\widetilde{P}}{d\eta}\right)^{2}\right]\psi^{2}}d\eta,$
(45)
$\tilde{W}=\int_{0}^{\eta_{R}}{e^{\left(3\mu+\nu\right)/2}\frac{\widetilde{\varepsilon}+q\widetilde{P}}{\eta^{2}}}\psi^{2}d\eta,$
(46)

respectively. The interior Schwarzschild metric potential from Equations (38)
and (39) can also be re-expressed in terms of dimensional quantities given by:

$e^{-\mu}=1-\frac{2Gvm_{0}}{c^{2}\eta
r_{0}}=1-\frac{2Gv\frac{\left(qc^{2}\right)^{2}}{\left(G^{3/2}\sqrt{4\pi
K}\right)}}{c^{2}\eta\frac{qc^{2}}{\sqrt{4\pi G\ K}}}=1-2q\frac{v}{\eta}$ (47)

and

$e^{\nu}=\left(1-2q\frac{v_{R}}{\eta_{R}}\right)\left(exp{\left[-2\int_{0}^{P\left(r\right)}\frac{qKd\widetilde{P}}{K(\widetilde{\varepsilon}+q\widetilde{P})}\right]}\right),$
(48)

which can be simplified into

$e^{\nu\left(\eta\right)}=\left(1-2q\frac{v_{R}}{\eta_{R}}\right)\left[\frac{\left(1+q\right)^{2}}{\left(1+q\sqrt{1+\xi^{2}}\right)^{2}}\right].$
(49)

Figure 7: The figure above shows the eigenfrequency of the fundamental mode
$\omega_{0}^{2}$ versus central mass density $\rho_{c}$ for for GUP parameter
values $\alpha_{0}=0$, $10^{17}$, $10^{19}$, $10^{20}$, $7\times 10^{20}$.

Plotting the numerical integrations from the equations above gives us the
eigenfrequency $\omega_{0}^{2}$ versus central density $\rho_{c}$ plot for
varying values of the GUP parameter $\alpha_{0}$ shown in Figure 7. From the
plot, we can see that white dwarfs with central densities
$\rho_{c}\lesssim{10}^{7}\hskip 2.84544pt\text{g}\hskip
2.84544pt\text{cm}^{-3}$ have overlapping plots, which means that the effect
of the GUP parameter does not have a notable impact on the white dwarfs in the
indicated range of central densities. This also means that all the values of
the GUP parameter in the range of $0\leq\alpha_{0}\leq{10}^{17}$ will yield
similar results, which are overlapping plots with the ideal case
($\alpha_{0}=0$) and have approximately the same value of the central density
by which gravitational collapse begins. This is also apparent in the mass
radius relation where low mass white dwarfs are not significantly affected by
GUP. The plots of the different GUP parameters start to deviate in the higher
density regime, where a larger GUP parameter such as for
$\alpha_{0}={10}^{20}$, $\alpha_{0}={10}^{19}$, and $\alpha_{0}={7\times
10}^{20}$ deviates at a lower $\rho_{c}$ than the other values of
$\alpha_{0}$. The plot for $\alpha_{0}=7\times 10^{20}$ deviates from the
ideal case at an even smaller $\rho_{c}$, reaching the maximum value of
$\omega_{0}^{2}\approx 1$ and is shown to decrease immediately from there at
smaller values of the central density. On the other hand, a GUP parameter
equal to ${10}^{17}$ has negligible effects on the white dwarfs. A larger GUP
parameter shows that the white dwarf has a smaller central density by which
its eigenfrequency reaches a maximum value. This is true for GUP parameters
${10}^{19}$, $10^{20}$, and ${7\times 10}^{20}$, while the plot of
$\alpha_{0}={10}^{17}$ has a large $\omega_{0}^{2}$ value, and corresponding
$\rho_{c}$* value. However, it should be noted that all values of the GUP
parameter yield similar behaviors, which show a descending value for
eigenfrequency in higher central densities. This means that there exists a
zero eigenfrequency solution at critical central densities $\rho_{c}$* for
different values of the LQGUP parameter (even at very large $\alpha_{0}$),
beyond which, at $\omega_{0}^{2}\leq 0$ the white dwarfs will undergo
gravitational collapse. This is very much in contrast with the results from
Ref. [38], which found a range of values of the quadratic GUP parameter for
which the white dwarf can still undergo instability, and beyond which, the
white dwarf is super-stable and can have any arbitrary mass, which is very
much in contrast with astrophysical observations. This is also means that
LQGUP removes any need for quadratic GUP to impose additional constraints in
the super-stable case just so that the white dwarf can retain its maximum
limiting mass even in a super-stable configuration. Additionally, we can infer
from the mass-radius relations or the dynamical instability analysis that even
with LQGUP in GR, the condition [50] $(dM/d\rho_{c})<0$ is preserved, unlike
the one with QGUP but also in the context of GR. It may then be interesting to
see how LQGUP could affect this condition for beyond-GR theories, but this is
outside the scope of the study and we reserve delving deeper into this
question for future extensions.

$\alpha_{0}$ |  $\rho_{c}$* |  $v_{R}$ ($M\odot$) |  $\eta_{R}$(km)
---|---|---|---
0 | 1.036 $\times 10^{11}$ | 1.415 | 1025.493
$10^{17}$ | 1.036 $\times 10^{11}$ | 1.415 | 1033.486
$10^{19}$ | 1.009 $\times 10^{11}$ | 1.396 | 1243.393
$10^{20}$ | 9.332 $\times 10^{10}$ | 1.291 | 2031.892
$7\times 10^{20}$ | 8.433 $\times 10^{10}$ | 0.985 | 3407.039
Table 2: Values for the critical mass density $\rho_{c}$*, star mass $v_{R}$,
star radius $\eta_{R}$ for varying values of the GUP parameter $\alpha_{0}$ at
the onset of dynamical instability.

Table 2 shows the values of the critical mass density $\rho_{c}$*, star mass,
star radius for various values of $\alpha_{0}$. The table shows that the
critical mass density values, star radius, and star mass are similar for
$\alpha_{0}=0$ and $\alpha_{0}=10^{17}$. However, for GUP parameter
$\alpha_{0}=10^{19}$, $\alpha_{0}=10^{20}$, and $\alpha_{0}=7\times 10^{20}$,
the critical mass density and final mass are shown to decrease with increasing
values of the GUP parameter. This means that larger values of the GUP
parameter reduces the final mass before it undergoes gravitational collapses.
On the other hand, the larger the GUP parameter, the larger the final radius
of the white dwarf before it collapses. Since the final mass of the white
dwarf decreases with increasing values of the GUP parameter, the Chandrasekhar
mass limit of $1.42\text{M}_{\odot}$ remains unaltered (for $\alpha_{0}\leq
10^{17}$). In addition, the maximum value of the critical central density from
the table is much smaller than the nuclear matter density of around
$10^{14}\hskip 2.84544pt\text{g}\hskip 2.84544pt\text{cm}^{-3}$, thus the
assumption of a free fermionic equation of state holds true to the stable
regime of the white dwarfs. Comparing the table above to Table 1, we can see
that the corresponding $v_{R}$ and $\eta_{R}$ values of the peaks from the
mass-radius relation plot in Figure 6 are practically similar to Table 2 for
the same values of $\alpha_{0}$. This means that from the mass-radis relation
plots, we can identify that instability occurs for larger $\alpha_{0}$ values
at larger star radius and smaller star mass. With this, we know that dynamical
instability happens after reaching the maximum mass of the star, when
examining the plots from the largest star radius decreasing towards zero. When
the radius of the star is smaller than the corresponding radius of the star
after reaching its maximum mass, dynamical instability occurs.

Figure 8: Eigenfrequency of the fundamental mode $\omega_{0}^{2}$ versus
central mass density $\rho_{c}$ for different values of $\alpha_{0}$

Figure 8 shows the complete plot of $\omega_{0}^{2}$ versus central mass
density $\rho_{c}$ for $\alpha_{0}=7\times 10^{20}$, which shows all the
values of $\omega_{0}^{2}$ below zero. From this plot, we can see that
$\omega_{0}^{2}$ decreases from zero, until it reaches the minimum value of
$\omega_{0}^{2}$, where it increases again from there. This behavior confirms
the spiraling that occurs in the mass-radius plot, which are still deemed
unstable.

Figure 9: Plot for adiabatic index $\gamma$ as a function of compactness $C$
for different values of $\alpha_{0}$

In addition, we solve for the adiabatic index for different values of
$\alpha_{0}$, and plot in 9 adiabatic index $\gamma$ as a function of the
compactness $C$, where $C=\frac{GM}{Rc^{2}}$, plotted up to the compactness of
the maximal stable mass in the mass-radius relations. We observe very similar
trends for the adiabatic index for different (relatively) smaller values of
$\alpha_{0}$, whereas at $\alpha_{0}=7\times 10^{20}$, there is significant
deviation of $\gamma$ from $\alpha_{0}=0$ case at relatively large
compactness. This also helps confirm our observation in Table 2 where the star
gets less compact with relatively larger values of $\alpha_{0}$. In
investigating the effects of LQGUP on the adiabatic index of the system, we
first note that we have imposed a constraint to our Fermi momenta $\xi$, i.e.
$0\leq\xi<1/\sigma$ (where $\sigma=m_{e}\alpha_{0}/M_{p}<1$) to prevent the
onset of singularity, and as already noted above, this is already sufficient
in obtaining the mass-radius relations of the white dwarfs (as well as
dynamical instability analysis) from a desired range of $\xi_{c}$. In the
limiting case then, we take the upper bound $\xi\rightarrow 1/\sigma$, instead
of the usual $\xi\rightarrow\infty$. This yields $\lim\gamma|_{\xi\rightarrow
1/\sigma}=2/3$, implying that this limiting value is independent of the LQGUP
parameter $\sigma$. Note that for $\sigma\rightarrow 0$ (no GUP),
$\lim\gamma|_{\xi\rightarrow\infty}=4/3$. This result is also different from
the result of Ref. [38], where they obtained a result of
$\lim\gamma|_{\xi\rightarrow\infty}=(\pi/16)\beta^{3}$, which is dependent on
the value of the quadratic GUP parameter $\beta$.

##  5 Conclusions and Recommendations

The mass-radius relations plots from Ref. [13] are successfully simulated in
this study, albeit using a different approach. In general, the LQGUP decreases
the mass of the white dwarf for a given radius, and this is particularly most
manifested in the most massive white dwarfs. The results of the mass-radius
relations and dynamical instability analysis show that dynamical instability
sets in at different values of $\eta_{R}$ and $v_{R}$ for various values of
the GUP parameter $\alpha_{0}$. The mass-radius relations plot shows us that
after reaching a certain maximum mass, the white dwarfs will become unstable
and will continuously decrease its mass and radius due to instability. On the
other hand, based on the results from dynamical instability analysis, we find
that the dynamical instability sets in for all values of the GUP parameters
$\alpha_{0}$, albeit at different mass limits. Therefore, unlike in the case
of QGUP [38], there seems to be no bound to the LQGUP parameter for which
dynamical instability does not set in; in other words the condition [50]
$(dM/d\rho_{c})<0$ is satisfied for the wide range of LQGUP parameters used
here. This verifies the results obtained from the mass-radius relations plots.
In addition, we find that the Chandrasekhar mass limit is preserved for LQGUP-
modified white dwarfs as seen from the onset of gravitational collapse, where
as we increase the value of $\alpha_{0}$, the mass limit will tend to
decrease. The results show that when the eigenfrequency of the fundamental
mode $\omega_{0}^{2}$ is plotted against the central density $\rho_{c}$ for
various values of the GUP parameter $\alpha_{0}$, the eigenfrequencies
$\omega_{0}^{2}$ continuously decrease up to the critical central density
$\rho_{c}$* indicating gravitational collapse.

It can be observed that the eigenfrequency for a GUP parameter
$\alpha_{0}={10}^{17}$ coincides with the plot for the ideal case, which has
no GUP correction. With this, for GUP parameters within the range of
$\alpha_{0}=0$ and $\alpha_{0}={10}^{17}$, the critical central density
$\rho_{c}*$ for the onset of gravitational collapse will be approximately the
same. For values of the GUP parameter in the range
$0\leq\alpha_{0}\leq{10}^{17}$, their corresponding mass-radius relation plot
shows maximal points corresponding to the Chandrasekhar mass limit at around
$1.42\text{M}_{\odot}$. On the other hand, larger values of $\alpha_{0}$ is
shown to have a lower maximum value when compared to $\alpha_{0}$ values less
than $10^{17}$, which implies that the mass limit by which instability sets in
is lower. It is given that upon reaching a mass beyond the Chandrasekhar mass
limit, the star would collapse and will form a more compact object. With this,
the LQGUP-modification supports gravitational collapse by decreasing the
Chandrasekhar mass limit of the white dwarf. This is opposite to the results
obtained when using QGUP [38], where at some values of the GUP parameter
$\beta_{0}$, the white dwarf will remain stable at masses beyond the
Chandrasekhar mass limit. We also note that in both the LQGUP case studied
here and QGUP case, the dynamical instability treatment is within the context
of general relativity (GR). Therefore, incorporating GUP itself onto GR via an
effective metric or to other modified gravity theories would be an interesting
extension to this study to see whether or not $(dM/d\rho_{c})<0$ still holds
in any of these cases [46, 51, 52].

The nature of the GUP parameter and phase space volume approach used affects
the resulting equation of state and mass radius relations of white dwarfs.
Thus, Ref. [51] notes that it is important to work with the full expression of
the GUP correction from the LQGUP model instead of using the series
truncation, which is one of the limitations for this study. With this, future
studies can work on the effect the GUP parameter to the white dwarf structures
using higher-order GUP approaches as suggested from Ref. [53, 54]. In
addition, modifications to the TOV equations due to the effect of GUP to
effective metric is expected. Thus, analyzing the effects of the modifications
to the TOV equation for GUP-modified white dwarfs is recommended for future
studies. From Ref. [16], it should be noted that there is a possibility for
GUP modifications to the equation of state could violate the first law of
thermodynamics for large values of the GUP parameter. One solution for this is
to assume anisotropic pressures for the system, which wasn’t implemented for
this study and should be taken account for future work. Furthermore, the study
considers an ideal electron gas, which is one of the limitations of the study.
Hence, considering refinements and modifications on the equation of state
including many-body interactions, envelopes, finite temperatures, and magnetic
fields is needed for a more realistic description of the white dwarf’s
interior [47, 55], and with the addition of GUP, this will pose a challenge
for next studies, not to mention the inclusion also of GUP onto theories of
gravity from which the structures of the white dwarfs will be solved. Aside
from that, one can also consider denser compact objects such as neutron stars
and black holes which may increase the chance of constraining the GUP
parameters, since these objects warp spacetime more significantly than white
dwarfs [14].

Interestingly, we also find that the qualitative effect of LQGUP onto the
mass-radius relations is similar to that of dark matter, where it has been
shown in Ref. [32, 56, 57] that the mass of the neutron star is decreased with
the inclusion of dark matter. It would be worthwhile to investigate how dark
matter will affect such system that is considered in this study, and how to
break apart the potential degeneracies that can occur if the effects of both
dark matter and LQGUP were to be considered.

The works of Ref. [37] suggested that the quadratic GUP parameter $\beta$ can
be negative, where the negative GUP parameter can prevent the white dwarf from
having arbitrary large masses. This negative value has also been suggested in
the uncertainty relations in the crystal lattice [58], as well as the
Magueijo-Smolin formulation [59] of doubly special relativity (DSR) [60]. On
the other hand, a negative GUP parameter $\alpha_{0}$ have been considered by
Ref. [46], but in the context of entropic gravity from Ref. [61], which is not
considered in this study. Additionally, dynamical instability analysis can be
used for other astronomical systems such as neutron stars. Dynamical
instability analysis can be used to analyze the radial oscillations of neutron
stars and compare its behavior to other compact objects when modified by
LQGUP. Lastly, future studies can explore using other numerical methods for
solving the equations above to improve the accuracy and efficiency of the
calculations. With this, using larger values of $\alpha_{0}$ would be
possible, which will be helpful in identifying the upper bound of the LQGUP
parameter to be used for white dwarfs.

## Acknowledgments

The authors thank the Department of Science and Technology, Philippines and
Ironwood Corporation for providing financial support for this project.

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utf8,latin1 adieresis=ä, germandbls=ß,

# Non-simultaneous blow-up for a system with local and non-local diffusion

Leandro M. Del Pezzo  Leandro M. Del Pezzo CONICET and UTDT Departamento de
Matemáticas y Estadística Universidad Torcuato Di Tella Av. Figueroa Alcorta
7350 (C1428BCW) Buenos Aires, ARGENTINA<EMAIL_ADDRESS>http://cms.dm.uba.ar/Members/ldpezzo/ and  Raúl Ferreira  Raúl Ferreira
Departamento de Matemática Aplicada, Fac. de C.C. Químicas, U. Complutense de
Madrid, 28040, Madrid, Spain<EMAIL_ADDRESS>
###### Abstract.

We study the possibility of non-simultaneous blow-up for positive solutions of
a coupled system of two semilinear equations, $u_{t}=J*u-u+u^{\alpha}v^{p}$,
$v_{t}=\Delta v^{+}u^{q}v^{\beta}$, $p,q,\alpha,\beta>0$ with homogeneous
Dirichlet boundary conditions and positive initial data. We also give the
blow-up rates for non-simultaneous blow-up.

Keywords. Local operator, non-local operator, system, blow-up, non-
simultaneous blow-up

##  1\. Introduction

Let $\Omega$ be a smooth boundary domain of $\mathbb{R}^{N},$ $p,q>0$ and
$\alpha,\beta\geq 0.$ Our main interest lies in the blow-up phenomena for the
following local-nonlocal reaction diffusion system

(1) $\begin{cases}u_{t}=\mathcal{L}u+u^{\alpha}v^{p}\qquad&x\in\Omega,t>0,\\\
v_{t}=\Delta v+u^{q}v^{\beta}&x\in\Omega,t>0,\end{cases}$

under Dirichlet “boundary” conditions

(2) $\begin{cases}u(x,t)=0&x\in\mathbb{R}^{n}\setminus\Omega,t>0,\\\
v(x,t)=0&x\in\partial\Omega,t>0,\end{cases}$

and initial data

(3) $\begin{cases}u(x,0)=u_{0}(x)&x\in\Omega,\\\
v(x,0)=v_{0}(x)&x\in\Omega.\end{cases}$

Here $\mathcal{L}u=J*u-u$ is a non-local diffusion operator for which the
convolution kernel $J$ satisfies the following assumptions which will be
assumed through out all this article

$J\in C_{0}(\mathbb{R}^{N},\mathbb{R})\text{ is a nonnegative function with
}J(0)>0\text{ and }\int_{\mathbb{R}^{N}}J(x)\,dx=1.$

Regarding the initial data, we assume that $u_{0},v_{0}\in
C(\overline{\Omega})$ are non-negative functions.

This type of system appears in some biological models, more precisely in the
last year, the study of dispersal strategies and the comparison between local
and nonlocal diffusive behaviours have attracted a great attention both in
terms of experiments and from the purely mathematical point of view. It is
worth mentioning that, the scientific community has not arrive to agreement
about which non-local models better describe the dispersion of a real
biological population, for this reason many non-local terms have been taken
into account in order to comprise long-range effects. But the present
literature on the subject of non-local dispersal mostly considers convolution
operators. See for instance [7, 10, 11, 4, 6, 12] and the references therein
for more details.

Problem (1)-(3) provide a simple example of a reaction-diffusion system of
Fujita type with two different difusión, local in one component and no-local
in the other. When the two diffusion are given by the same diffusion

(4) $\left\\{\begin{array}[]{ll}u_{t}=\Delta u+u^{\alpha}v^{p}\\\ v_{t}=\Delta
v+u^{q}v^{\beta}\\\
\end{array}\right.\qquad\left\\{\begin{array}[]{ll}u_{t}=\mathcal{L}u+u^{\alpha}v^{p},\\\
v_{t}=\mathcal{L}v+u^{q}v^{\beta},\\\ \end{array}\right.$

under homogeneous boundary conditions, it is well known that there exists
blow-up solutions if and only if

$\alpha>1,\quad\beta>1,\quad\text{and}\quad pq>(1-\alpha)(1-\beta),$

see for instance [9, 5, 15, 2] for the local case and [13] for the nonlocal
case. At time $T$ we have,

$\limsup_{t\to T}\
\max\\{\|u(\cdot,t)\|_{\infty},\|v(\cdot,t)\|_{\infty}\\}=\infty.$

Note that the solution blows up if only one of the components blows up, it is
what we call non-simultaneous blow-up. In [14, 13] the authors consider the
systems (4) and they prove that if only $v$ blows up then $\beta>1+p$. The
fact that $\beta>1$ ensures the existence of initial data such that $v$ blows
up and the condition $p<\beta-1$ implies that the coupling of the system is
weak enough to get non-simultaneous blow-up.

As previous paper in the study of blow-up with problems that mix local and
non-local diffusions, we refer to [3] where the the system

$\left\\{\begin{array}[]{l}u_{t}=\mathcal{L}u+v^{p}\\\ v_{t}=\Delta
v+u^{q}\end{array}\right.$

has been analysed for mixed boundary condition (Dirichlet for the non-local
component and Neumann for the local one). However, this system is weakly
coupled ($\alpha=0$ and $\beta=0$), then non-simultaneous blow-up cannot
occurs.

Existence of a solution can easily be achieved, but uniqueness is subtle due
to the fact that the parameters $\alpha,\beta,p,q$ can be less than 1. The
difficulty comes from the non-Lipschitz character of the reaction, see for
instance [8]. Nevertheless, in this case we can prove existence of a maximal
solution just taking the limit

$(u,v)=\lim_{\varepsilon\to 0}(u_{\varepsilon},v_{\varepsilon})$

where $(u_{\varepsilon},v_{\varepsilon})$ is the solution to (1) with initial
data
$(u_{\varepsilon}(x,0),v_{\varepsilon}(x,0))=(u_{0}(x)+\varepsilon,v_{0}(x)+\varepsilon)$,
boundary data $u=\varepsilon$ outside $\Omega$ and
$v_{\varepsilon}=\varepsilon$ at the boundary of $\Omega$ and the non-
Lipschitz power replace by

$f_{\varepsilon}(w,\gamma)=\left\\{\begin{array}[]{ll}w^{\gamma}&w\geq\varepsilon,\\\
\varepsilon^{\gamma}&w\leq\varepsilon.\end{array}\right.$

Moreover, a comparison principle among maximal solutions can easily be
obtained, see for instance [3, Lemma 24]. Notice that if the reaction terms
are Lipschitz we have uniqueness. From now on, we only consider maximal
solutions.

### Main results

The goal of this paper is to study the non-simultaneous blow-up when we mix
local and non-local diffusions. Our first goal in the study of system (1) is
to obtain, in term of the parameters, the existence of blow-up solutions.

######  Theorem 1.1.

Let $(u,v)$ be a solution of (1)-(3). Then,

  1. (i)

If $\alpha<1$, $\beta<1$ and $pq\leq(1-\alpha)(1-\beta)$ all solutions are
global.

  2. (ii)

Other wise there exist both, blow-up and global solutions.

Now, we look for the possibility that just one component of the system (1)
blows up. To see this we restrict ourselves to solutions increasing in time.

######  Theorem 1.2.

Let $(u,v)$ be a solution of (1)-(3) with

$\begin{array}[]{l}\mathcal{L}u_{0}+u_{0}^{\alpha}v_{0}^{p}>0,\\\ \Delta
v_{0}+u_{0}^{q}v_{0}^{\beta}>0.\end{array}$

  1. (i)

If $u$ blows up while $v$ remains bounded, then

$\max_{x\in\overline{\Omega}}u(x,t)\sim(T-t)^{\frac{-1}{\alpha-1}}.$

  2. (ii)

If $v$ blows up while $u$ remains bounded, then $\beta>1+p$ and

$\max_{x\in\overline{\Omega}}v(x,t)\sim(T-t)^{\frac{-1}{\beta-1}}.$

To conclude this introduction, we would like to point out that working with
mixed local and non-local diffusion it turned out to be a not easy
mathematical problem since the known tools were not adapted in a trivial way.

### The rest of this manuscript is organized as follows

In Section 2, we prove that there exist global solutions (the first item of
Theorem 1.1). Then, in Section 3, we prove that there is a blow-up solution
(the second item of Theorem 1.1). Lastly, in Section 4, we study the
possibility that one component of the system (1) blows up at time $T<\infty$
while the other remains bounded (proof of Theorem 1.2).

##  2\. Global existence

In this section, we prove the first item of Theorem 1.1. To this end, we
borrow ideas of [13, Lemma 2.1].

######  Lemma 2.1.

If $p,q>0$ then there exist global solutions. Moreover

  * •

If $\max\\{\alpha,\beta\\}<1$ and $\kappa=pq-(1-\alpha)(1-\beta)<0$ then every
solution is bounded.

  * •

If $\max\\{\alpha,\beta\\}<1$ and $\kappa=0$ then every solution is global.

###### Proof.

Let $w,z$ be the positive bounded solutions of

$\begin{cases}-\mathcal{L}(u)=1&x\in\Omega,\\\
u=0&x\in\mathbb{R}^{N}\setminus\Omega,\end{cases}$

and

$\begin{cases}-\Delta u=1&x\in\Omega,\\\ u=0&x\in\partial\Omega,\end{cases}$

respectively. We next set

$\bar{u}=Aw\quad\text{and}\quad\bar{v}=Bz$

where $A$ and $B$ are positive constants that will be selected so that the
pair $(\bar{u},\bar{v})$ will be a supersolution of (1). Specifically
$(\bar{u},\bar{v})$ is a supersolution of (1) if

(5) $1\geq
A^{\alpha-1}B^{p}\|w\|_{\infty}^{\alpha}\|z\|_{\infty}^{p}\quad\text{and}\quad
1\geq A^{q}B^{\beta-1}\|w\|_{\infty}^{q}\|z\|_{\infty}^{\beta}.$

Taking

$1=A^{\alpha-1}B^{p}\|w\|_{\infty}^{\alpha}\|z\|_{\infty}^{p},$

that is

$B=\dfrac{A^{\nicefrac{{(1-\alpha)}}{{p}}}}{\|w\|_{\infty}^{\nicefrac{{\alpha}}{{p}}}\|z\|_{\infty}},$

the second inequality of (5) reads

$A^{\nicefrac{{\kappa}}{{p}}}\leq\|w\|_{\infty}^{-\nicefrac{{(pq+\alpha(1-\beta))}}{{p}}}\|z\|_{\infty}^{-1}$

where $\kappa=pq-(1-\alpha)(1-\beta)\neq 0.$ Therefore

  * •

If $\kappa>0$ then the inequalities (5) hold for $A$ small.

  * •

If $\kappa<0$ then the inequalities (5) hold for $A$ large.

Thus, in both cases, we obtain global solution by comparison.

Moreover if $\kappa<0$ and $\max\\{\alpha,\beta\\}<1,$ the inequalities (5)
hold for $A$ and $B$ large. For this reason, in this case, every solution is
bounded.

We now study the case $k=0$ and $\min\\{\alpha,\beta\\}\geq 1.$ In this case,
we take

$B=\dfrac{1}{\|w\|_{\infty}^{\nicefrac{{\alpha}}{{p}}}\|z\|_{\infty}}$

and the inequalities (5) read

$1\geq
A^{\alpha-1}\quad\text{and}\quad\|w\|_{\infty}^{-\nicefrac{{(pq+\alpha(1-\beta))}}{{p}}}\|z\|_{\infty}^{-1}\geq
A^{q}.$

Therefore the the inequalities (5) hold for $A$ small. Again, we obtain global
solution by comparison.

Finally, we study the case $\kappa=0,$ and $\max\\{\alpha,\beta\\}<1.$ We
propose as a supersolution of (1) the pair

$\bar{u}(x,t)=Ae^{Bt}\quad\text{and}\quad\bar{v}(x,t)=Ce^{Dt}$

where $A,B,C,D$ are positive constants such that

$Be^{((1-\alpha)B-pD)t}\geq A^{\alpha-1}C^{p}\quad\text{ and }\quad
De^{((1-\beta)B-qD)t}\geq A^{q}C^{1-\beta}.$

If we take $D=B\tfrac{(1-\alpha)}{p},$ the last inequalities reads

$B\geq A^{\alpha-1}C^{p}\quad\text{ and }\quad\frac{(1-\alpha)}{p}B\geq
A^{q}C^{1-\beta}.$

since $\kappa=0.$ Therefore the previous inequalities hold for any $A>0$ and
$C>0$ taking $B$ large enough. Then, in this case, we also obtain global
solution by comparison. Moreover all the solution of (1) are global. ∎

##  3\. blow-up solutions

Here we prove that there is a blow-up solution (the second item of Theorem
1.1).

######  Lemma 3.1.

Let $\lambda_{1},\mu_{1}$ be the first Direichlet eigenvalues of $\mathcal{L}$
and $\Delta$ respectively, and $(u,v)$ be a solution of (1)–(3). Then there is
a positive constant $C$ such that

$u(x,t)\geq Ce^{-\lambda_{1}t},\quad v(x,t)\geq Ce^{-\mu_{1}t}\psi(x)\quad
x\in\overline{\Omega},t>0$

where $\psi$ is a positive eigenfunction of $\Delta$ associated to $\mu_{1}.$

###### Proof.

Let $\phi$ and $\psi$ be positive eigenfunctions of $\mathcal{L}$ and $\Delta$
associate to $\lambda_{1}$ and $\mu_{1},$ respectively. It is well known that
$\phi(x)\geq\delta>0$ in $\overline{\Omega},$ see for instance [1]. Then it is
easy to see that there is a positive constant $k$ so small that
$(k\phi,k\psi)$ is a subsolution of (1)–(3). Hence, by comparison,

$u(x,t)\geq ke^{-\lambda_{1}t}\phi(x)\geq k\delta e^{-\lambda_{1}t},\quad
v(x,t)\geq ke^{-\mu_{1}t}\psi(x)\quad x\in\overline{\Omega},t>0.$

This completes the proof. ∎

######  Lemma 3.2.

There is a blow-up solution either $\max\\{\alpha,\beta\\}>1$ or else
$\max\\{\alpha,\beta\\}<1$ and $\kappa=pq-(1-\alpha)(1-\beta)>0.$

###### Proof.

We split the proof in three cases.

Case 1: $\alpha>1.$ Let $(u,v)$ be a solution of (1)–(3). Then, by Lemma 3.1,
there is a positive constant $C$ such that

$u_{t}(x,t)\geq Cu^{\alpha}(x,t)e^{-\mu_{1}pt}\psi_{1}(x)-u(x,t)\quad
x\in\Omega,t>0,$

where $\mu_{1}$ is the first Direichlet eigenvalue of $\Delta$ and $\psi$ is a
positive eigenfunction associated to $\mu_{1}.$ Then

$e^{t}u_{t}(x,t)+e^{t}u(x,t)\geq
C(e^{t}u(x,t))^{\alpha}e^{(1-\mu_{1}p-\alpha)t}\psi_{1}^{p}(x)\quad
x\in\Omega,t>0.$

Taking $w(x,t)=e^{t}u(x,t),$ we get

$\dfrac{w_{t}(x,t)}{w^{\alpha}(x,t)}\geq
Ce^{(1-\mu_{1}p-\alpha)t}\psi_{1}^{p}(x)\quad x\in\Omega,t>0,$

and integrating the last inequality, we deduce

$\dfrac{1}{1-\alpha}\left(w^{1-\alpha}(x,t)-w^{1-\alpha}(x,0)\right)\geq\dfrac{C}{1-\mu_{1}p-\alpha}e^{(1-\mu_{1}p-\alpha)t}\psi_{1}^{p}(x)\quad
x\in\Omega,t>0.$

Now, using $\alpha>1$ and $w(x,t)=e^{t}u(x,t),$ we have that

$u(x,t)\geq
e^{-t}\left[u_{0}^{1-\alpha}(x)-\dfrac{C(\alpha-1)}{\alpha+\mu_{1}p-1}\left(1-e^{(1-\mu_{1}p-\alpha)t}\right)\psi_{1}^{p}(x)\right]^{-\frac{1}{\alpha-1}}\,x\in\Omega,t>0.$

Therefore, if there is $x_{0}\in\Omega$ such that

$u_{0}(x_{0})>\left(\dfrac{C(\alpha-1)}{\alpha+\mu_{1}p-1}\psi_{1}^{p}(x_{0})\right)^{-\frac{1}{\alpha-1}},$

then $u$ blows up at finite time.

Case 2: $\beta>1.$ As before, given $(u,v)$ a solution of (1)–(3), by Lemma
3.1, there is a positive constant $C$ such that

$v_{t}(x,t)\geq\Delta v(x,t)+Cv^{\beta}(x,t)e^{-\lambda_{1}qt}\quad
x\in\Omega,t>0,$

where $\lambda_{1}$ is the first Direichlet eigenvalue of $\mathcal{L}$. Now,
we apply Kapplan’s method to prove blow-up. To do that we multiply by
$\psi_{1}$ and integrate in $\Omega$ to obtain that

$I(t)=\int_{\Omega}u(x,t)\psi_{1}(x)dx$

satisfies

$I^{\prime}\geq-\mu_{1}I+Ce^{-\lambda_{1}qt}I^{\beta}.$

Solving this inequality

$I^{1-\beta}(t)\leq
e^{\mu_{1}t}\left(I^{1-\beta}(0)-\frac{\beta-1}{\lambda_{1}q+\mu_{1}(\beta-1)}\left(1-e^{-(\lambda_{1}q+\mu_{1}(\beta-1))t}\right)\right)$

Therefore, as $\beta>1$ we have that for $I(0)$ large enough the function
$I(t)$ blows up in finite time. Which implies that $v$ also blows up.

Case 3: $\max\\{\alpha,\beta\\}<1$ and $\kappa>0.$ In this case, the existence
of blow-up solution are obtained by comparison. So, for a fix point
$x_{0}\in\Omega$, we can consider the system (1) in a ball
$B_{L}=B(x_{0},L)\subset\subset\Omega$. For the component $u$ we define

$\underline{u}=\varphi(t)\chi_{R}(x),$

where $\chi_{R}$ is the characteristic function in $B_{R}$ with $R<L$ and the
function $\varphi$ is given by

$\varphi(t)=(T-t)^{-2/q}e^{\frac{1-\beta}{q}(T-t)^{-1}}.$

Let us observe that the function $\psi(t)=e^{(T-t)^{-1}}$ solves the equation

$\psi^{\prime}=\varphi^{q}\psi^{\beta}.$

On the other hand, for $v$ component we consider $\underline{v}=w$ the
solution of

(6) $\left\\{\begin{array}[]{ll}w_{t}=\Delta
w+\varphi^{q}\chi_{R}w^{\beta}&x\in B_{L},t\in(0,T),\\\ w=0&x\in\partial
B_{L},t>0.\end{array}\right.$

Therefore, $(\underline{u},\underline{v})$ is a subsolution of our problem if
in the ball $B_{R}$ the following inequality holds

$\varphi^{\prime}\leq\varphi\left(\int_{B_{R}}J(x-y)dy-1\right)+\varphi^{\alpha}w^{p}.$

We claim that there exists $\varepsilon>0$ such that the solution of (6)
satisfies

(7) $w(x,t)\geq\varepsilon e^{(1-\varepsilon)^{2}(T-t)^{-1}}\qquad
x\in\overline{B_{R}},t\in(0,T).$

Assuming this claim we have that $(\underline{u},\underline{v})$ is a
subsolution of our problem if in the ball $B_{R}$ the following inequality
holds

$\varphi^{\prime}\leq\varphi\left(\int_{B_{R}}J(x-y)dy-1\right)+\varphi^{\alpha}\varepsilon^{p}e^{(1-\varepsilon)^{2}p(T-t)^{-1}},$

that is, if

$\begin{array}[]{rl}\frac{2}{q}(T-t)^{-1}+\frac{1-\beta}{q}(T-t)^{-2}\leq&\displaystyle\int_{B_{R}}J(x-y)dy-1\\\
&+(T-t)^{-\frac{2(1-\alpha)}{q}}e^{\left((1-\varepsilon)^{2}p-\frac{(1-\alpha)(1-\beta)}{q}\right)(T-t)^{-1}}.\end{array}$

Since $\kappa>0,$ we can take $\varepsilon$ small enough such that
$(1-\varepsilon)^{2}p-\frac{(1-\alpha)(1-\beta)}{q}>0$. Then, the previous
inequality trivially holds taking $T$ small enough.

In order to prove the claim, we construct two blow-up subsolutions. First, let
$\phi_{R}(x)$ be the first eigenfunction of the laplacian in $B_{R}$ with
homogeneous Dirichlet boundary condition normalize to
$\|\phi_{R}\|_{\infty}=1$, then it is easy to see that the function

$\underline{w}=A\psi(t)\phi_{R}(x).$

extended by zero, it is a subsolution of (6) provided that $A$ is small enough
to have

$A^{1-\beta}\left(1+\lambda_{1}(T-t)^{2}\right)\leq 1.$

Even more, using this lower bound we get that $w$ is a supersolution of the
problem

$\left\\{\begin{array}[]{ll}z_{t}=\Delta z&x\in B_{L}\setminus
B_{R-\delta},t\in(0,T),\\\ z=Ce^{(T-t)^{-1}}&x\in\partial
B_{R-\delta},t\in(0,T),\\\ z=0&x\in\partial
B_{L},t\in(0,T).\end{array}\right.$

But, for this problem it is easy to prove that the function

$\underline{z}(x,t)=C\left\\{\begin{array}[]{ll}e^{(1-k(|x|-R+\delta))^{2}(T-t)^{-1}}-1&|x|\in(R-\delta,R-\delta+\frac{1}{k}),t\in(0,T),\\\
0&|x|\in(R-\delta+\frac{1}{k},L),t\in(0,T),\end{array}\right.$

is a subsolution provided that

$C\leq A\phi_{R}(R-\delta)\quad\mbox{and}\quad
k\geq\max\left\\{\frac{1}{4}\,,\frac{d-1}{R-\delta}\,,\frac{1}{L-R+\delta}\right\\}.$

Moreover, $\underline{z}(R,t)>0$ if $k\delta<1$. Notice that for $\delta$
small enough, we can choose $k$ independent of $\delta$ and by Hopf’s Lemma
$\phi_{R}(R-\delta)\sim\delta$. Then estimate (7) holds taking
$\varepsilon=k\delta$. ∎

##  4\. Non simultaneous blow-up

In this section we study the possibility that one component of the system (1)
blows up at time $T<\infty$ while the other remains bounded. Thus, in this
section we assume that the solution $(u,v)$ blows up in finite time $T$.
Notice that by Lemma 3.2 there exist blow-up solutions whenever one of the
following conditions holds

$\alpha>1,\quad\beta>1,\quad pq>(1-\alpha)(1-\beta).$

If only the third condition is satisfied, it is easy to see that both
components blow up at the same time. Indeed, if we assume that $v$ is bounded,
we get that $u$ is a subsolution of the problem

$\left\\{\begin{array}[]{ll}z_{t}=\mathcal{L}z+Kz^{\alpha}&x\in\Omega,\
t>0,\\\ z(x,t)=0&x\in\mathbb{R}^{N}\setminus\Omega,\ t>0,\\\
z(x,0)=u_{0}(x)&x\in\Omega.\end{array}\right.$

It is easy to see that for $\alpha\leq 1$ and $A$ large enough
$\overline{z}=Ae^{t}$ is a global super-solution. Therefore, by comparison $u$
is global. Thus, to get non-simultaneous blow-up we need
$\max\\{\alpha,\beta\\}>1$.

######  Lemma 4.1.

Assume that there exists $\delta_{i}\geq 0$ such that the initial data
satisfies

(8)
$\begin{array}[]{l}\mathcal{L}u_{0}+u_{0}^{\alpha}v_{0}^{p}-\delta_{1}u_{0}^{\alpha}\geq
0,\\\ \Delta v_{0}+u_{0}^{q}v_{0}^{\beta}-\delta_{2}v_{0}^{\beta}\geq
0.\end{array}$

Then

$u_{t}\geq\delta_{1}u^{\alpha}\qquad\text{ and }\qquad
v_{t}\geq\delta_{2}v^{\beta}.$

###### Proof.

Let us define $\mathcal{H}=u_{t}-\delta_{1}u^{\alpha}$ and
$H=v_{t}-\delta_{2}v^{\beta}$ which satisfy

$\left\\{\begin{array}[]{ll}{\mathcal{H}_{t}-\mathcal{L}\mathcal{H}-\alpha
u^{\alpha-1}v^{p}\mathcal{H}-pv^{p-1}u^{\alpha}H\geq
0}&x\in\Omega,t\in(0,T),\\\ H_{t}-\Delta H-\beta
v^{\beta-1}u^{q}H-qu^{q-1}v^{\beta}\mathcal{H}\geq
0&x\in\Omega,t\in(0,T),\end{array}\right.$

with boundary conditions

$\left\\{\begin{array}[]{ll}\mathcal{H}=0&x\in\mathbb{R}^{N}\setminus\Omega,t\in(0,T),\\\
H=0&x\in\partial\Omega,t\in(0,T),\end{array}\right.$

and initial data

$\left\\{\begin{array}[]{ll}\mathcal{H}(x,0)=\mathcal{L}u_{0}+u_{0}^{\alpha}v_{0}^{\beta}-\delta
u_{0}^{\alpha}\geq 0,\\\ H(x,0)=\Delta v_{0}+v_{0}^{\beta}u_{0}^{q}-\delta
v_{0}^{\beta}\geq 0.\end{array}\right.$

Then, by comparison the result follows. ∎

Now, given $\varepsilon>0,$ we consider two regular functions in $\Omega$, $f$
and $g$, such that:

$f(x)>0\quad x\in\Omega_{\varepsilon},\quad f(x)=0\quad
x\in\Omega\setminus\Omega_{\varepsilon},\qquad\mathcal{L}f>0\quad
x\in\Omega\setminus\Omega_{2\varepsilon}$

and $g$ is a positive function in $\Omega$ vanishing at the boundary such that

$g(x)\geq m>0\quad x\in\Omega_{\varepsilon},\qquad\Delta g(x)>0\quad
x\in\Omega\setminus\Omega_{2\varepsilon},$

where
$\Omega_{\varepsilon}=\\{x\in\Omega\,:\,d(x,\partial\Omega)>\varepsilon\\}$.

The idea to prove non-simultaneous blow-up is to consider initial data
$(u_{0},v_{0})=(Af(x),Bg(x))$ to see that if $B$ is bounded and $A$ is large
enough only $u$ blows up, on the contrary if $A$ is bounded and $B$ is large
enough only $v$ blows up.

######  Lemma 4.2.

If $\alpha>1+q$, then there exists initial data such that $u$ blows up while
$v$ remains bounded.

###### Proof.

Let $(u,v)$ be the solution of (1) with initial data $(Af,g)$. In order to
apply the previous lemma, we note that for
$x\in\Omega\setminus\Omega_{2\varepsilon}$ (near the boundary)

$\Delta v_{0}+u_{0}^{q}v_{0}^{\beta}=\Delta g+A^{q}f^{q}g^{\beta}\geq\Delta
g>0$

while for $x\in\Omega_{2\varepsilon}$ (in the interior)

$\Delta v_{0}+u_{0}^{q}v_{0}^{\beta}=\Delta g+A^{q}\gamma^{q}m^{\beta}>0$

for $A$ large enough. On the other hand, for
$x\in\Omega\setminus\Omega_{\varepsilon}$

$\mathcal{L}u_{0}+u_{0}^{\alpha}v_{0}^{p}-\delta_{1}u_{0}^{\alpha}=J*u_{0}\geq\mu>0$

while for $x\in\Omega_{\varepsilon}$

$\begin{array}[]{rl}\mathcal{L}u_{0}+u_{0}^{\alpha}v_{0}^{p}-\delta_{1}u_{0}^{\alpha}=&AJ*f-Af+A^{\alpha}f^{\alpha}(g^{p}-\delta_{1})\\\
\geq&AJ*f-Af+A^{\alpha}f^{\alpha}(m^{p}-\delta_{1})\\\
\geq&{AJ*f-C_{\alpha}(m^{p}-\delta_{1})^{\frac{-1}{\alpha-1}}}>0\end{array}$

for $\delta_{1}<m^{p}$ and $A$ large. Applying now Lemma 4.1, we get

$u_{t}\geq\delta_{1}u^{\alpha},\qquad\text{ and }\qquad v_{t}\geq 0.$

Hence, $u$ blows up a finite time $T_{A}$, which becomes small if $A$ is
large, and

$u\leq\left(\delta_{1}(\alpha-1)(T_{A}-t)\right)^{\frac{-1}{\alpha-1}}.$

Using this inequality in the equation of $v$ we have that $v$ is a subsolution
of

$z_{t}=\Delta z+C(T-t)^{\frac{-q}{\alpha-1}}z^{\beta}.$

Now, we compare with the flat supersolution

$\overline{z}(t)=\left\\{\begin{array}[]{ll}\displaystyle\left(\|z_{0}\|_{\infty}^{1-\beta}+\frac{(\alpha-1)(1-\beta)C}{\alpha-1-q}(T_{A}^{\frac{\alpha-1-q}{\alpha-1}}-(T_{A}-t)^{\frac{\alpha-1-q}{\alpha-1}})\right)^{\frac{1}{1-\beta}}&\beta\neq
1,\\\
\exp{\displaystyle\left(\ln(\|z_{0}\|_{\infty})+\frac{(\alpha-1)(1-\beta)C}{\alpha-1-q}(T_{A}^{\frac{\alpha-1-q}{\alpha-1}}-(T_{A}-t)^{\frac{\alpha-1-q}{\alpha-1}})\right)}&\beta=1,\end{array}\right.$

to obtain that $v$ is bounded if either, $\beta\leq 1$, or, $\beta>1$ and
$T_{A}$ small enough, which implies $A$ large. ∎

######  Lemma 4.3.

If $\beta>1+p$, then there exists initial data such that $v$ blows up while
$u$ remains bounded.

###### Proof.

Let $(u,v)$ be the solution of (1) with initial data $(f,Bg)$ and
$(u_{l},v_{l})$ the solution of the linear problem

$(u_{l})_{t}=\mathcal{L}u_{l},\qquad(v_{l})_{t}=\Delta v_{l}$

with initial datum $(f,g)$. As, it is well known that $u_{l}\geq C(t)$ for
$x\in\overline{\Omega}$ we can apply comparison to get that

$u(x,t)\geq u_{l}(x,t)\geq C(t),\qquad v(x,t)\geq Bv_{l}(x,t).$

Moreover, by continuity, we can take $t_{1}$ small enough to have $\Delta
v(x,t_{1})\geq 0$ and $\mathcal{L}u\geq 0$ in
$\Omega\setminus{\Omega_{2\varepsilon}}$. So there is $\delta_{2}>0$ such that
at time $t_{1}$

$\begin{array}[]{l}\displaystyle\Delta
v+u^{q}v^{\beta}-\delta_{2}v^{\beta}\geq\Delta
v+(C^{q}(t_{1})-\delta_{2})B^{\beta}v_{l}^{\beta},\\\
\displaystyle\mathcal{L}u+u^{\alpha}v^{p}\geq\mathcal{L}u+C(t_{1})Bv_{l}^{p},\end{array}$

and both are non-negative for $\delta_{2}<C(t_{1})$ and $B$ large. Then, by
Lemma 4.1

$v_{t}\geq\delta_{2}v^{\beta},\qquad u_{t}\geq 0.$

Arguing as before, we have that if $B$ is large only the $v$ component blows
up. ∎

######  Remark 4.1.

Observe that by the continuity of blow-up time with respect to the initial
data, there exists $A$ and $B$ such that the solution of (1) with initial data
$(Af,Bg)$ blows up simultaneously provided that $\alpha>1+q$ and $\beta>1+p$.

######  Lemma 4.4.

Assume that (8) holds with $\delta_{2}>0$. If $v$ blows up at time $T<\infty$
and $u$ remains bounded, then $\beta>p+1$ and

$\|v(\cdot,t)\|_{\infty}\sim(T-t)^{\frac{-1}{\beta-1}}.$

###### Proof.

Let us define $x(t)$ as $v(x(t),t)=\max_{x\in\overline{\Omega}}v(\cdot,t)$,
which is a Lipschitz continuous function. Then $\partial v(x(t),t)$ exists
almost everywhere and satisfies that $\partial v(x(t),t)=v_{t}(x(t),t)$. On
the other hand, by Lemma 4.1

$v_{t}(x(t),t)\geq\delta_{2}v^{\beta}(x(t),t).$

Integrating this inequality we get the upper blow up rate

$\|v(\cdot,t)\|_{\infty}\leq\left(\delta_{2}(\beta-1)(T-t)\right)^{\frac{-1}{\beta-1}}.$

To obtain the lower bound, we observe that as $u$ is bounded the component $v$
is a subsolution of the equation $w_{t}=\Delta w+Cw^{\beta}$ and

$w(t)=\left(C(\beta-1)(T-t)\right)^{\frac{-1}{\beta-1}}$

is a solution which blows up at the same time. Assume by contradiction that
there is $t_{0}\in(0,T)$ so that

$\max_{x\in\overline{\Omega}}v(x,t_{0})<w(t_{0})$

Therefore

$\max_{x\in\overline{\Omega}}v(x,t_{0})<w(t_{0}-\varepsilon)$

for some $\varepsilon>0$. Then, by comparison, we have that

$v(x,t)\leq w(t-\varepsilon)\quad\forall(x,t)\in\Omega\times(t_{0},T).$

It follows that $v$ is bounded, a contradiction.

Finally to prove that $\beta>p+1$ we observe that as $v$ is increasing in time
we have that $v(x(t_{0}),t)\geq v(x(t_{0}),t_{0})$. Putting this estimate into
the equation of $u$ we get

(9) $u_{t}(x(t_{0}),t)\geq
u^{\alpha}(x(t_{0}),t)v^{p}(x(t_{0}),t_{0})-u(x(t_{0}),t).$

We can integrate this expression to obtain

$\int_{e^{t_{0}}u(x(t_{0}),t_{0})}^{e^{T}u(x(t_{0}),T)}s^{-\beta}\,ds\geq
v^{p}(x(t_{0}),t_{0})\int_{t_{0}}^{T}e^{(1-\alpha)t}\,dt.$

As $u$ is a continuous bounded function, the left hand side goes to zero as
$t_{0}$ goes to $T$. On the other hand, using the lower blow-up rate we have
that the right hand side satisfies

$v^{p}(x(t_{0}),t_{0})\int_{t_{0}}^{T}e^{(1-\alpha)t}\,dt\geq\widetilde{C}(T-t_{0})^{1-\frac{p}{\beta-1}}.$

Therefore, $\beta>p+1$. ∎

######  Lemma 4.5.

Assume that (8) holds with $\delta_{1}>0$. If $u$ blows up at time $T<\infty$
and $v$ remains bounded, then

$\|u(\cdot,t)\|_{\infty}\sim C(T-t)^{\frac{-1}{\alpha-1}}.$

###### Proof.

We argue as before. The upper bound follows by Lemma 4.1 and the lower bound
from the fact that $v$ is bounded, which implies that $u$ is a sub-solution of
the equation $w_{t}=\mathcal{L}w+Cw^{\alpha}$ and

$w(t)=e^{-at}\left((\alpha-1)C(T-t)\right)^{-\frac{1}{\alpha-1}},\qquad
a=1-\max_{x\in\overline{\Omega}}\int_{\Omega}J(x-y)dy$

is a solution which blows up at the same time. ∎

## Acknowledgements

L.M.D.P. was partially supported by the European Union’s Horizon 2020 research
and innovation program under the Marie Sklodowska-Curie grant agreement No
777822, (Agencia Nacional de Promoción de la Investigación, el Desarrollo
Tecnológico y la Innovación PICT-2018-3183, and PICT-2019-00985 and UBACYT
20020190100367.

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