# Curvature effect on spin polarization in a three-terminal geometry in
presence of Rashba spin-orbit interaction

Santanu K. Maiti<EMAIL_ADDRESS>Physics and Applied Mathematics
Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata-700
108, India

###### Abstract

The robust effect of curvature on spin polarization is reported in a three-
terminal bridge system where the bridging material is subjected to Rashba
spin-orbit interaction. The results are examined considering two different
geometric configurations, ring- and linear-like, of the material which is
coupled to one input and two output leads. Our results exhibit absolute zero
spin polarization for the linear sample, while finite polarization is obtained
in output leads for the ring-like sample.

###### pacs:

73.23.-b, 72.25.-b, 85.35.Ds, 71.70.Ej

##  I Introduction

The study of spin dependent transport in low-dimensional systems has been
largely dominated in the last few decades due to the rapid advancement in
nano-scale science and technology th1 ; th2 ; th3 ; th4 ; th5 ; th6 ; th7 ;
th8 ; san1 ; san2 ; san3 ; san4 ; th9 ; th10 ; san10 ; ex1 ; ex2 ; ex3 .
Controlling electron’s spin degree of freedom is extremely important for the
development of quantum information processing as well as quantum computation
wolf . The spin-orbit (SO) interaction which couples the electron’s spin to
the charge degree of freedom provides a much deeper insight for generating
spin current and also its manipulation intrinsic1 ; intrinsic2 ; intrinsic3 ;
intrinsic4 ; intrinsic5 ; intrinsic6  rather than the usual methodologies wang
; xie . Earlier, people were mainly using wang ; xie  ferromagnetic leads or
external magnetic field to get spin filtering action though these are not very
suitable especially for low-dimensional systems, since in one case a large
resistivity mismatch is observed while in the other case the main difficulty
appears for confining a huge magnetic field into a narrow region, like a nano-
ring.

Depending on the sources, spin-orbit interaction is classified in two
different categories: one is called extrinsic type which appears mainly due to
magnetic impurities, while the other is defined as intrinsic type that appears
as a result of lacking of inversion symmetry. In this category generally two
kinds of spin-orbit interactions are taken into account. They are called as
Rashba and Dresselhaus SO interactions rashba ; dressel ; winkler . The first
one is associated with the inversion asymmetry of the structure and its
strength can be regulated by means of external gate potential, and the second
one is related to the bulk inversion asymmetry whose coupling strength depends
on the material.

Considering the coupling of spin degree of freedom to the momentum of an
electron, spin polarized currents in output terminals of a multi-terminal
conductor can be achieved from a purely unpolarized electron beam injected to
the input terminal mlt1 ; mlt2 ; mlt3 ; mlt4 ; mlt5 . The existing literature
suggest that a lot of theoretical progress has already been done to explore
spin selective transmission through different model geometries. For example, a
planar T-shaped conductor mlt1  with a ring resonator exhibits polarized spin
currents in outgoing leads in presence of Rashba SO interaction. In other work
Peeters et al. have shown how a ring-like geometry can be utilized as an
electron spin beam splitter exploring the possible quantum interference effect
in presence of SO coupling mlt3 . At the same time Nikolic and his group mlt4
; mlt5  put forward several key ideas in this particular field.

In spite of the considerable volume of work available in this particular area,
a practically unexplored issue is how does the curvature of a material which
is clamped within input and output leads influence spin polarization.

Figure 1: (Color online). Schematic diagram of a three-terminal bridge setup,
where the arrows correspond to the movement of electrons through one input and
two output terminals. Two different geometrical shapes, ring- and linear-like,
of a particular sample subjected to Rashba SO interaction are taken into
account to explore the curvature effect on spin polarization in multi-terminal
system.

To the best of our knowledge, this part is unaddressed so far. In the present
work we essentially focus towards this direction. We investigate the curvature
effect on spin polarization by considering two simple geometries: a simple
linear conductor and a ring-like geometry which is formed by bending the
chain.

The results are quite interesting. Using a tight-binding (TB) framework and
based on Green’s function formalism we show that for a ring shaped conductor
spin polarized currents are obtained in output leads of a multi-terminal
geometry from a completely unpolarized beam of electrons, while absolute zero
spin polarization is obtained for the linear conductor.

The rest of the paper is organized as follows. Section II illustrates two
different models together with a brief theoretical description to obtain spin
polarization in two output leads. Section III contains numerical results and
discussion, and finally, in Section IV we summarize our essential results.

##  II Model and theoretical framework

In this section we describe two different systems of our study and present a
general theory for calculating spin polarization coefficient $P^{z}$ in two
output leads based on Green’s function formalism.

###  II.1 Model and Hamiltonian

The three-terminal bridge setup is schematically shown in Fig. 1, where we
take two different configurations of the same material. In one configuration
we choose a finite one-dimensional ($1$D) chain, which is then bent to form a
$1$D ring to generate another configuration. In both these two cases the
material, subjected to Rashba spin-orbit interaction, is connected with one
input (lead-1) and two output leads (lead-2 and lead-3).

A tight-binding framework is given under the nearest-neighbor hopping
approximation to describe the bridging material (ring/chain) and the side-
attached leads. The TB Hamiltonian of the entire system reads as,

$H=H_{\mbox{\tiny sample}}+H_{\mbox{\tiny leads}}+H_{\mbox{\tiny tun}}$ (1)

where three different terms in the right side correspond to the Hamiltonians
of three different regions of the bridge system those are elaborately
explained below.

The first term, $H_{\mbox{\tiny sample}}$, describes the Hamiltonian of the
conductor placed between the incoming and outgoing leads. Depending on its
geometry (ring-like or chain-like) the Hamiltonian looks different. For a
$N$-site linear conductor subjected to Rashba SO interaction the TB
Hamiltonian liu ; san9  gets the form:

$\displaystyle H_{\mbox{\tiny sample}}$ $\displaystyle=$
$\displaystyle\sum_{n}\mbox{\boldmath$c$}_{n}^{{\dagger}}\mbox{\boldmath$\epsilon$}\mbox{\boldmath$c$}_{n}+\sum_{n}\left(\mbox{\boldmath$c$}_{n}^{{\dagger}}\mbox{\boldmath$t$}\mbox{\boldmath$c$}_{n+1}+h.c.\right)$
(2) $\displaystyle+\sum_{n}\left(i\,t_{\mbox{\small
so}}\mbox{\boldmath$c$}_{n}^{{\dagger}}[\vec{\sigma}\times\hat{d}_{n,n+1}]_{z}\mbox{\boldmath$c$}_{n+1}+h.c.\right)$

where,
$\mbox{\boldmath$c$}^{\dagger}_{n}=\left(\begin{array}[]{cc}c_{n\uparrow}^{\dagger}&c_{n\downarrow}^{\dagger}\end{array}\right);$
$\mbox{\boldmath$c$}_{n}=\left(\begin{array}[]{c}c_{n\uparrow}\\\
c_{n\downarrow}\end{array}\right);$
$\mbox{\boldmath$\epsilon$}=\left(\begin{array}[]{cc}\epsilon&0\\\
0&\epsilon\end{array}\right)$ and
$\mbox{\boldmath$t$}=t\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right).$

In the above expression, $\mbox{\boldmath$c$}_{n\sigma}^{\dagger}$ and
$\mbox{\boldmath$c$}_{n\sigma}$ are the creation and annihilation operators,
respectively, for an electron with spin $\sigma(\uparrow,\downarrow)$ at the
$n$-th atomic site of the sample. $\epsilon$ is the on-site energy and $t$
measures the isotropic nearest-neighbor hopping integral. The parameter
$t_{\mbox{\small so}}$ describes the Rashba SO coupling strength and the term
$\vec{\sigma}$ gives the spin angular momentum of the electron. The unit
vector $\hat{d}_{n,n+1}$ describes the direction of the movement of an
electron between the sites $n$ and $n+1$.

When this $N$-site linear conductor is bent to form a ring the Hamiltonian
becomes san5 ; san6 ; san7 ; san8 ,

$\displaystyle H_{\mbox{\tiny sample}}$ $\displaystyle=$
$\displaystyle\sum_{i}\mbox{\boldmath$c$}_{n}^{{\dagger}}\mbox{\boldmath$\epsilon$}\mbox{\boldmath$c$}_{n}+\sum_{n}\left(\mbox{\boldmath$c$}_{n}^{{\dagger}}\mbox{\boldmath$t$}\mbox{\boldmath$c$}_{n+1}+h.c.\right)$
(3)
$\displaystyle+\sum_{n}\left(\mbox{\boldmath$c$}_{n}^{{\dagger}}\mbox{\boldmath$t$}_{n,n+1}\mbox{\boldmath$c$}_{n+1}+h.c.\right)$

where,
$\mbox{\boldmath$t$}_{n,n+1}=it_{\mbox{\small
so}}\left\\{\mbox{\boldmath$\sigma_{x}$}\cos\left(\frac{\varphi_{n}+\varphi_{n+1}}{2}\right)+\mbox{\boldmath$\sigma_{y}$}\sin\left(\frac{\varphi_{n}+\varphi_{n+1}}{2}\right)\right\\}$
with $\varphi_{n}=2\pi(n-1)/N$. All the other symbols used in Eq. 3 carry
their usual meanings.

In our theoretical framework, three metallic leads are considered to be
identical, semi-infinite and free from any kind of impurities and spin-orbit
interaction. We can express them as,

$H_{\mbox{\tiny leads}}=\sum_{\alpha}H_{\alpha}$ (4)

where $\alpha=1,2,3$ for the three leads. In the absence of any SO coupling
$H_{\alpha}$ takes the form:

$H_{\alpha}=\sum\limits_{i}\mbox{\boldmath$c$}_{i}^{\dagger}\mbox{\boldmath$\epsilon$}_{l}^{\alpha}\mbox{\boldmath$c$}_{i}+\sum_{i}\left(\mbox{\boldmath$c$}_{i}^{\dagger}\mbox{\boldmath$t$}_{l}^{\alpha}\mbox{\boldmath$c$}_{i+1}+h.c.\right)$
(5)

with
$\mbox{\boldmath$\epsilon$}_{l}^{\alpha}=\epsilon_{l}^{\alpha}\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)$ and
$\mbox{\boldmath$t$}_{l}^{\alpha}=t_{l}^{\alpha}\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)$.

where $\epsilon_{l}^{\alpha}$ and $t_{l}^{\alpha}$ are the site energy and
nearest-neighbor hopping integral, respectively, in the $\alpha$-th lead.
Other factors carry their usual meanings as stated earlier. Out of these three
leads, lead-1 is treated as the input terminal, while the other two are
considered as the output terminals and all of them are coupled to the
conductor through the hopping integral $t_{c}$. Here we assume that the lead-1
is always attached to site $1$ and the other two leads are coupled to the
sites $p$ and $q$, those are variables, of the conductor. Following the same
footing as above, we can write the TB Hamiltonian to describe the conductor-
to-lead coupling as,

$H_{\mbox{\tiny tun}}=\sum_{\alpha}H_{{\mbox{\tiny tun}},\alpha}.$ (6)

Here,

$H_{{\mbox{\tiny
tun}},\alpha}=\left[\mbox{\boldmath$c$}^{\dagger}_{i}\mbox{\boldmath$t$}_{c}\mbox{\boldmath$c$}_{n}+\mbox{\boldmath$c$}^{\dagger}_{n}\mbox{\boldmath$t$}_{c}\mbox{\boldmath$c$}_{i}\right]$
(7)

with $\mbox{\boldmath$t$}_{c}=t_{c}\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right).$
The site $i$ corresponds to the boundary site of the lead, and it is coupled
to the $n$-th site of the conductor, which is variable.

###  II.2 Evaluation of polarization coefficient $P^{z}$ in terms of
transmission probabilities

The spin polarization coefficient in the output leads is defined as san2 ;
pola1 ; pola2 ; pola3 ,

$P^{z}=\frac{T_{\uparrow\uparrow}+T_{\downarrow\uparrow}-T_{\uparrow\downarrow}-T_{\downarrow\downarrow}}{T_{\uparrow\uparrow}+T_{\downarrow\uparrow}+T_{\uparrow\downarrow}+T_{\downarrow\downarrow}}$
(8)

where, $T_{\sigma\sigma^{\prime}}$ gives the transmission probability of an
injecting electron with spin $\sigma$ which gets transmitted through the drain
with spin $\sigma^{\prime}$. When $\sigma=\sigma^{\prime}$ we get pure spin
transmission, while for the other case spin flip transmission is obtained.
Equation 8 is the general expression of spin polarization coefficient between
any two leads $i$ and $j$, and, for our three-terminal system we call the
polarization coefficients in two outgoing leads as $P_{1}^{z}$ and
$P_{2}^{z}$. In the present approach we select the quantization direction
along the $Z$ axis for simplification.

To calculate transmission coefficient $T_{\sigma\sigma^{\prime}}$ we use
Green’s function formalism datta1 ; datta2 . In this framework the two-
terminal transmission probability between the leads $i$ and $j$ is defined as
$T_{\sigma\sigma^{\prime}}=\mbox{Tr}\left[\Gamma_{i}^{\sigma}G_{c}^{r}\Gamma_{j}^{\sigma^{\prime}}G_{c}^{a}\right]$.
Here $G_{c}^{r}$ and $G_{c}^{a}$ are the retarded and advanced Green’s
functions, respectively, of the sample considering the effects of the
electrodes. $G_{c}^{r}=\left(E-H_{\mbox{\tiny
sample}}-\sum\limits_{\sigma}\Sigma_{1}^{\sigma}-\sum\limits_{\sigma}\Sigma_{2}^{\sigma}-\sum\limits_{\sigma}\Sigma_{3}^{\sigma}\right)^{-1}$,
where $E$ is the energy of an injecting electron, and $\Sigma_{i}^{\sigma}$’s
($i=1,2,3$) are the self-energies due to coupling of the conductor to the
leads and $\Gamma_{i}^{\sigma}$’s are their imaginary parts. In Refs. datta1 ;
datta2  the detailed calculations of self-energy matrices are available.

##  III Numerical results and discussion

Based on the above theoretical framework we now analyze our numerical results.
Throughout the analysis we fix the electronic temperature of the system to
absolute zero, and for simplification, we put $c=e=h=1$. Other common
parameters are as follows: $\epsilon=\epsilon_{l}^{\alpha}=0$ and
$t=t_{l}^{\alpha}=t_{c}=1$. The Rashba SO coupling strength $t_{\mbox{\small
so}}$ and all the energy scales are measured in unit of the hopping integral
$t$.

Before focusing to the central point i.e., the curvature effect on spin
polarization in a multi-terminal (more than one output lead) system in
presence of Rashba SO interaction, we want to have a short glimpse on the
system where a conductor subjected to SO interaction is coupled to a single
input and a single output lead i.e., a two-terminal system. Following our
extensive numerical calculations we can conclude that irrespective of the
curvature of the bridging material, only SO interaction cannot induce spin
polarization in output lead of a two-terminal system. We verify it considering
different geometrical shapes of the conductor, e.g., circle, square, triangle,
polygon, linear, etc. In few recent works mlt1 ; moo  it has also been shown
that only SO interaction

Figure 2: (Color online). Energy dependence of spin polarization coefficients
$P_{1}^{z}$ and $P_{2}^{z}$ in two output leads, in (a) and (b) respectively,
for a three-terminal mesoscopic ring when the output leads are attached
symmetrically with respect to the input lead. These coefficients ($P_{1}^{z}$
and $P_{2}^{z}$) are superposed with each other in (c) for comparison. The
parameters are: $N=52$, $t_{\mbox{\small so}}=0.5$, $p=22$ and $q=32$.

is incapable of producing spin polarization. The reason is that, in presence
of SO coupling the time-reversal symmetry is still preserved, and therefore,
it doesn’t break the Kramer’s degeneracy between the $|k\uparrow\rangle$ and
$|-k\downarrow\rangle$ states which results vanishing spin current in the
output lead of a two-terminal system. The degeneracy gets removed when the
system is subjected to any kind of magnetic impurity or external magnetic
field. Under this situation a two-terminal system with SO coupling exhibits
polarized spin currents san2 . This phenomenon has already been established in
the literature, but the essential issue of our present analysis – the
interplay between the curvature of the material and the multi-leads has not
been addressed earlier.

To explore it, in Fig. 2 we present the results for a three-terminal
mesoscopic ring considering $N=52$ and $t_{\mbox{\small so}}=0.5$. Here, the
two outgoing leads are attached

Figure 3: (Color online). Energy dependence of spin polarization coefficients
$P_{1}^{z}$ and $P_{2}^{z}$ in two output leads, in (a) and (b) respectively,
for a three-terminal mesoscopic ring when the output leads are attached
asymmetrically with respect to the source lead. These coefficients
($P_{1}^{z}$ and $P_{2}^{z}$) are superposed with each other in (c) for
comparison. The parameters are: $N=52$, $t_{\mbox{\small so}}=0.5$, $p=27$ and
$q=39$.

symmetrically ($p=22$ and $q=32$) with respect to the incoming lead, as shown
schematically in Fig. 1(a). The upper panel of Fig. 2 corresponds to the
energy dependence of spin polarization coefficient for the one output lead,
while for the other output terminal it is shown in the middle panel of Fig. 2,
and, finally they are placed together in the lower panel of this figure to
compare the polarization coefficients properly. From these spectra it is
observed that finite spin polarizations, associated with the energy
eigenvalues of the ring subjected to only SO interaction, are obtained in both
the two outgoing leads though the system is free from any kind of magnetic
impurities. Most interestingly, we also see that the coefficients $P_{1}^{z}$
and $P_{2}^{z}$ are exactly identical in magnitude and opposite in sign for
each value of the incident electron energy $E$. This phenomenon can be
explained as follows. The spin polarization

Figure 4: (Color online). Energy dependence of spin polarization coefficients
$P_{1}^{z}$ and $P_{2}^{z}$ in a three-terminal mesoscopic chain considering
$N=52$ and $t_{\mbox{\small so}}=0.5$ for two different chain-to-lead
interface geometries, where (a) $p=26$, $q=52$ and (b) $p=20$, $q=40$.

coefficient $P^{z}$ describes the normalized difference among the up and down
spin charge currents propagating through the outgoing leads, since in our
present scheme we assume the quantized direction along the Z direction. In a
multi-lead geometry i.e., when more than one outgoing lead is coupled to the
system the Kramer’s degeneracy between the $|k\uparrow\rangle$ and
$|-k\downarrow\rangle$ gets removed and depending on the allowed paths of the
moving electrons spin dependent scattering takes place. The spin dependent
force associated with the SO coupling is responsible for this scattering. Now,
in the ring-like geometry electrons can go through two different paths, and
accordingly, the up and down spin electrons get deflected in two opposite
directions by the spin dependent force during the movement of electrons
through the ring geometry in presence of SO interaction which results spin
selective transmission through the outgoing leads. This is the key aspect of
observing mesoscopic spin Hall effect san9  and the accumulation of opposite
spin electrons on the opposite edges of a finite width conductor. Since the
two outgoing leads are coupled symmetrically to the ring geometry with respect
to the incoming lead, $P_{1}^{z}$ and $P_{2}^{z}$ get equal magnitude. Their
sign reversals are also understood from Eq. 8.

By breaking the ring-lead interface geometry one can achieve spin
polarizations with unequal magnitudes. The results are shown in Fig. 3 when
the output leads are attached asymmetrically with respect to the source lead.
Here we fix $p=27$ and $q=39$, and all the other parameters are kept identical
as set in Fig. 2. It is clearly seen that, unlike the symmetric configuration,
the magnitudes of $P^{z}$ in two output leads are no longer equal for the
asymmetrically connected ring-lead bridge setup. This is exclusively due to
the quantum interference effect among the electronic waves propagating through
different arms of the ring geometry. Apart from getting unequal magnitudes of
polarization coefficients in two output leads, all the other properties remain
exactly invariant as described earlier in the case of symmetric configuration.

From these results we can emphasize that, for the three-terminal ring geometry
we get spin polarization due to SO interaction only, since the sample is free
from any kind of magnetic impurity or external magnetic field, but a two-
terminal ring geometry cannot provide polarize spin current under this
situation. Here, it is important to note that although the Kramer’s degeneracy
between the $|k\uparrow\rangle$ and $|-k\downarrow\rangle$ gets removed by
coupling the conductor with a third lead or more, but it doesn’t ensure to get
non-vanishing spin polarization in output leads which can be clearly
understood from the following discussion.

The scenario becomes highly significant when the ring-like sample is
transformed into the linear-like one. The results are shown in Fig. 4
considering two different chain-to-lead interface geometries. In one
configuration (Fig. 4(a)) the outgoing leads are coupled to the sites $26$ and
$52$ of the linear conductor, while these coupling sites are $20$ and $40$ for
the other configuration (Fig. 4(b)). The total number of atomic sites $N$ and
the Rashba SO coupling are kept unchanged as taken in Fig. 2. From the spectra
presented in Fig. 4, we interestingly see that both $P_{1}^{z}$ and
$P_{2}^{z}$ drop exactly to zero for the entire energy band spectrum, and,
these results are independent of the chain-to-lead interface geometry as well
as the strength of the SO coupling, which we confirm through our considerable
numerical work. This is really appealing in the sense that a same material
subjected to SO coupling exhibits finite spin polarization for one geometrical
shape, while absolute zero spin polarization is obtained in the other
geometrical configuration for a multi-terminal bridge setup. This is solely
due to the effect of curvature. For the linear chain the up and down spin
electrons can propagate only in a particular direction. Either it can be along
X or Y direction depending on the choice of the co-ordinate system. Under this
situation, the spin dependent force which essentially scatter opposite spin
electrons becomes zero, and therefore, no spin polarization is available even
though the Kramer’s degeneracy is broken for such a multi-terminal bridge
setup. The disappearance of such spin dependent force in a linear sample can
be justified from the following analysis. For a linear chain the Rashba
dependent Hamiltonian in the continuum model representation gets the form xu :
$(t_{\mbox{\small so}}/\hbar)\sigma_{y}p_{x}$, assuming the movement of
electrons along the X direction. Considering this Hamiltonian if we calculate
$\ddot{x}$, $x$ being the position operator, then the output becomes exactly
zero which immediately suggests vanishing spin dependent force since the force
is directly proportional to $\ddot{x}$. To get a finite spin

Figure 5: (Color online). Energy dependence of spin polarization coefficients
$P_{1}^{z}$ and $P_{2}^{z}$ in a three-terminal mesoscopic ring for two
distinct ring-to-lead interface geometries considering $N=200$ and
$t_{\mbox{\small so}}=0.5$. The results for the symmetric configuration are
presented in the first column where we set $p=82$ and $q=120$, while in the
second column they are shown for the asymmetric configuration setting $p=101$
and $q=151$.

dependent force both the components $p_{x}$ and $p_{y}$ are needed in the
Rashba term which is not possible in the case of a linear chain. Therefore,
spin dependent scattering is no longer available, even in the presence of
multi-leads. Now, an important point which should be noted here that when a
linear conductor is properly bent to form a regular ring shaped geometry, the
SO coupling strength may be affected due to its curvature, but that doesn’t at
all change the present physical scenario, and therefore, we consider the
identical coupling strength for both these two geometrical configurations for
the sake of simplification in our model calculations.

The results presented so far to explore the curvature effect on spin
polarization in a three terminal geometry are computed for a conductor with
only $52$ sites. Keeping in mind a possible experimental realization one may
think how such a small sized conductor can be used to design a conductor-lead
bridge setup. To establish this fact now we present the spin polarization
coefficients $P_{1}^{z}$ and $P_{2}^{z}$ taking a $200$-site conductor for its
two different shapes as considered earlier i.e., ring-like and the linear-like
one. For the ring-like geometry, the results are presented in Fig. 5, whereas
for the other case they are shown in Fig. 6, and for both these two cases the
results are computed for two distinct electrode-to-conductor configurations
(symmetric and asymmetric) to justify the robustness of our investigation.
From the spectra given in Fig. 5 it is observed that, like Fig. 2, here also
the polarization coefficients $P_{1}^{z}$ and $P_{2}^{z}$ become exactly
identical in magnitude and opposite in sign (1st column of Fig. 5) when the
ring is attached symmetrically to

Figure 6: (Color online). Energy dependence of spin polarization coefficients
$P_{1}^{z}$ and $P_{2}^{z}$ in a three-terminal mesoscopic chain considering
$N=200$ and $t_{\mbox{\small so}}=0.5$ for two different chain-to-lead
interface geometries, where (a) $p=100$, $q=200$ and (b) $p=50$, $q=150$.

the outgoing leads. These coefficients $P_{1}^{z}$ and $P_{2}^{z}$ are no
longer identical in magnitude for the asymmetric ring-lead configuration (2nd
column of Fig. 5), as expected. This is exactly what we get in a $52$-site
ring as shown in Fig. 3. For the linear-shaped conductor with $N=200$ a
vanishing spin polarization is obtained (Fig. 6) in its output leads for the
entire energy band region irrespective of the chain-to-lead interface geometry
and it is exactly similar in nature which we get earlier in the case of a
$52$-site chain (Fig. 4). From these results we can emphasize that apart from
getting more peaks and dips in $P^{z}$-$E$ spectrum for the ring-shaped
conductor with increasing $N$, all the basic physical properties i.e., non-
vanishing spin polarization coefficients with equal and/or unequal magnitudes
associated with the ring-lead interface geometry and absolute zero spin
polarization in the linear-like conductor irrespective of its coupling
configuration to side-attached leads remain exactly unchanged. For a very
large sized ring several peaks and dips appear in $P^{z}$-$E$ spectrum from
which it may seem that the spectrum is quasi-continuous, but looking it
carefully one can always find distinct peaks since $N$ is finite. Thus, our
essential goal of getting finite and vanishing spin polarizations in a multi-
terminal geometry constructed with the same material with different curvatures
is established.

###### Acknowledgements.

The author is thankful to Prof. S. Sil and M. Dey for useful discussions.

##  IV Conclusion

To summarize, in the present communication we establish the curvature effect
on Rashba spin-orbit interaction induced spin polarization in a three-terminal
bridge setup within a tight-binding framework based on Green’s function
formalism. The results are analyzed considering two different shaped
geometries of the same material. In one configuration we select the bridging
material in a linear-like, which is then bent to form a ring-like geometry.
Quite interestingly, we find that finite polarization is obtained in two
output leads for ring shaped geometry, while absolute zero spin polarization
is noticed when the sample becomes linear. This phenomenon also holds true
even for any other higher-terminal bridge setup and independent of the lead-
conductor interface geometries.

In addition to this ring-like geometry one might expect finite spin
polarization in output leads for other geometrical configurations, except the
linear one, which essentially leads to the robust effect of curvature on spin
polarization in a multi-terminal bridge system.

All the results described in this communication are worked out at absolute
zero temperature, though the finite temperature extension of this analysis is
extremely trivial. The thing is that at finite temperature no new phenomenon
will appear and all the physical pictures presented here remain unaltered even
at finite (low) temperature since the broadening of energy levels of the
conductor due to its coupling with the side-attached leads is too large
compared to the thermal broadening datta1 ; datta2 ; san11 ; san12 ; san13 .

In the present work, we ignore the effect of on-site electron-electron (e-e)
interaction. We can incorporate this effect in our formalism in different
ways. One possible route is the mean field approximation ee1 ; ee2 ; ee3 ; ee4
; ee5 . But, for this particular study e-e interaction doesn’t provide any
such new insight since it cannot scatter up and down spin electrons in
opposite edges of the sample, as spin-orbit interaction does. Only some
modifications in magnitudes of $P_{1}^{z}$ and $P_{2}^{z}$ can be expected.

Before we end, it should be noted that to explore the effect of curvature on
spin polarization in a three-terminal bridge setup we compute our numerical
results considering some typical values of the parameters describing the
systems. But, all these physical properties i.e., vanishing spin polarization
in output leads of a linear-like conductor and finite spin polarization for a
ring-like geometry remain absolute unchanged for any other set of parameter
values. These features certainly demand an experiment in this line.

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# On the Fujita exponent for a Hardy-Hénon equation with a spatial-temporal
forcing term

Mohamed Majdoub  Department of Mathematics, College of Science, Imam
Abdulrahman Bin Faisal University, P. O. Box 1982, Dammam, Saudi Arabia.
Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal
University, P.O. Box 1982, 31441, Dammam, Saudi Arabia<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The purpose of this work is to analyze the wellposedness and the blow-up of
solutions of the higher-order parabolic semilinear equation

$u_{t}+(-\Delta)^{d}u=|x|^{\alpha}|u|^{p}+\zeta(t){\mathbf{w}}(x)\
\quad\mbox{for }(x,t)\in\mathbb{R}^{N}\times(0,\infty),$

where $d\in(0,1)\cup\mathbb{N}$, $p>1$, $-\alpha\in(0,\min(2d,N))$ or
$\alpha\geqslant 0$ and $\zeta$ as well as ${\mathbf{w}}$ are suitable given
functions. Given $p\geqslant\frac{N-2d\sigma+\alpha}{N-2d\sigma-2d}$ and
setting $p_{c}=\frac{N(p-1)}{2d+\alpha}$,
$\ell=\frac{Np_{c}}{N+2(\sigma+1)dp_{c}}$, we prove that for any data
$u_{0}\in L^{p_{c},\infty}(\mathbb{R}^{N})$ and $\textbf{w}\in
L^{\ell,\infty}(\mathbb{R}^{N})$ with small norms there exists a unique
global-in-time solution under the hypotheses $\zeta(t)=t^{\sigma}$,
$\sigma\in(-1,0)$ and $N>2d$ in the space
$C_{b}([0,\infty);L^{p_{c},\infty}(\mathbb{R}^{N}))$. As a by-product, small
Lebesgue data global existence follows and in particular, unconditional
uniqueness holds in $C_{b}([0,\infty);L^{p_{c}}(\mathbb{R}^{N}))$ provided
$p\in(\frac{N+\alpha}{N-2d},\infty)$. If either $m\in(-\infty,0]$ and
$p\in(1,\frac{N-2dm+\alpha}{N-2dm-2d})$ or $m>0$ and $p>1$ where
$\zeta(t)=O(t^{m})$, $t\rightarrow\infty$ ($m\in\mathbb{R}$), then all
solutions blow up under the additional condition
$\int_{\mathbb{R}^{N}}\textbf{w}(x)\,dx>0$. As a consequence, we deduce that
the corresponding Fujita critical exponent is a function of $\sigma$ and reads
$p_{F}(\sigma)=\frac{N-2d\sigma+\alpha}{N-2d\sigma-2d}$ if $-1<\sigma<0$ and
infinity otherwise.

###### Key words and phrases:

Polyharmonic heat equation, fractional Laplacian, blow-up, local existence,
global solutions, generalized Hardy-Hénon equation.

###### 2020 Mathematics Subject Classification:

35K05, 35K15, 35C15, 35B30, 35B99.

##  1\. Introduction and main results

In this paper we are interested in the study of the critical behaviour of
global in time solutions of the initial value problem

$\begin{cases}u_{t}+(-\Delta)^{d}u=|x|^{\alpha}|u|^{p}+\zeta(t){\mathbf{w}}(x)\
\quad\mbox{for }(x,t)\in\mathbb{R}^{N}\times(0,\infty),\\\
u(x,0)=u_{0}(x),\quad x\in\mathbb{R}^{N},\end{cases}$ (1.1)

where $N\geqslant 1$, $\alpha\in\mathbb{R}$, $p>1$ and $\zeta,{\mathbf{w}}$
are given functions. The operator $(-\Delta)^{d}$ for $d\geqslant 1$ integer
is the $d$-fold composition of the Laplacian with itself while for
$d\in(0,1)$, $(-\Delta)^{d}$ should be interpreted as the fractional Laplacian
defined by
$(-\Delta)^{d}\,u:=\mathcal{F}^{-1}\left(|\xi|^{2d}\mathcal{F}(u)\right)$
where $\mathcal{F}$ and $\mathcal{F}^{-1}$ are the Fourier transform and
inverse Fourier transform, respectively [13]. We will consider the situation
$\alpha\geqslant 0$ in which case $x\in\mathbb{R}^{N}$ and
$0<-\alpha<\min(2d,N)$ with $x\in\mathbb{R}^{N}\setminus\\{0\\}$.

Reaction-diffusion equations with forcing term arises in many physical
phenomena and biological species theories, such as the concentration of
diffusion of some fluid, the density of some biological species, and heat
conduction phenomena, see [8, 6, 21, 17, 18] and references therein.

Our main goal in this paper is to find the critical exponent which separates
the existence and nonexistence of global solutions of (1.1). It goes back to
the work of Fujita [4] that there is an exponent $p_{F}$ (critical Fujita
exponent) for which the behavior of globally defined solutions of the
reaction-diffusion model

$u_{t}-\Delta u=|u|^{p}\ \hskip 2.84544pt\mbox{in}\hskip
2.84544pt\mathbb{R}^{N}\times(0,\infty)$

are classified according to whether one is in the subcritical case
($1<p<p_{F}$), critical ($p=p_{F}$) or supercritical ($p_{F}<p<\infty)$. A
very well-known argument allowing one to predict the correct value of $p_{F}$,
whenever it is finite, for several semilinear parabolic equations consists of
the study of the (linear) stability of the zero solution, see for instance
[12]. A more direct and simple approach pertaining to problems posed in the
whole space and intrinsically connected to the scaling of the underlying
equation has been proposed by the authors in [3]. More specifically, a scaling
analysis on (1.1) without the forcing term, that is,

$u_{t}+(-\Delta)^{d}u=|x|^{\alpha}|u|^{p}\ \hskip 2.84544pt\mbox{in}\hskip
2.84544pt\mathbb{R}^{N}\times(0,\infty),\hskip 5.69046ptu(0)=u_{0}$ (1.2)

shows that the only Lebesgue space which leaves invariant the norm of the
appropriately rescaled initial data is $L^{p_{c}}(\mathbb{R}^{N})$,
$p_{c}=\frac{N(p-1)}{2d+\alpha}$. Thus one may anticipate the critical Fujita
exponent for (1.2) to be given by $1+\frac{2d+\alpha}{N}$ which corresponds to
the value of $p$ for which $p_{c}=1$. The recent work [28] asserts in turn
that this observation is actually correct. Indeed, the global well-posedness
theory for (1.2) among other questions has been investigated in the latter
reference – in particular, small data (in weak-Lebesgue space
$L^{p_{c},\infty}(\mathbb{R}^{N})$ global existence was obtained in
$C_{b}([0,\infty);L^{p_{c},\infty}(\mathbb{R}^{N}))$,
$p_{c}=\frac{N(p-1)}{2d+\alpha}$ provided we require
$\frac{N+\alpha}{N-2d}<p<\infty$ and $0<-\alpha<2d<N$ while under the
conditions $u_{0}\in L^{1}(\mathbb{R}^{N})$ with
$\displaystyle\int_{\mathbb{R}^{N}}u_{0}(x)dx>0$ and
$1<p<1+\frac{2d+\alpha}{N}$, non-existence of weak solutions was established,
see [28]. Note that the previous well-posedness statement does not cover the
range $(1+\frac{2d+\alpha}{N},\frac{N+\alpha}{N-2d})$, a restriction which
essentially emanates from the choice of the functional setting together with
the hypotheses needed for the applicability of certain key estimates heavily
utilized in the analysis. This gap, however, could be closed by using the two-
norms technique as illustrated in [28]. Moving on, it is easy to see that the
previous procedure fails to predict the critical exponent $p^{\star}$ for
(1.1) due to the presence of a forcing term. Nevertheless, as we shall see,
granted the above knowledge on (1.2), the behavior of the inhomogeneous term
plays a crucial role in determining $p^{\star}$. Throughout this manuscript,
unless otherwise specified we will assume that $\zeta:(0,\infty)\to(0,\infty)$
is a continuous function satisfying either

$\zeta(t)\,\underset{t\to
0}{\sim}\,t^{\sigma},\quad\zeta(t)\underset{t\to\infty}{\sim}\,t^{m},\;\;\;(\sigma>-1,\,m\in\mathbb{R}),$
(1.3)

or

$\zeta(t)=t^{\sigma},\;\;\sigma>-1.$ (1.4)

The study of the reaction-diffusion equation involving a source term has been
considered by a number of authors. When the forcing term is purely space-
dependent (i.e. $\zeta\equiv 1$) with $d=1$ and $\alpha=0$, it is known that
the Fujita critical exponent reads

$p_{F}=\left\\{\begin{array}[]{ll}\infty\hskip 5.69046pt\mbox{if}\hskip
5.69046ptN=1,2\\\ \frac{N}{N-2}\hskip 5.69046pt\mbox{if}\hskip
5.69046ptN>2\end{array}\right.$

in the sense that for $p$ below $p_{F}$ and
$\displaystyle\int_{\mathbb{R}^{N}}\textbf{w}(x)dx>0$, Problem (1.1) does not
admit any global solution – a conclusion which persists in the critical case
$p=p_{F}$ under the additional assumptions on w whilst for $p>p_{F}$ global-
in-time solutions exist provided w and $u_{0}$ have strong decay property at
infinity. These statements can be found in [2] where various extensions and
results related to other geometries were obtained, see also [30] for the
geometrical setting given by a manifold. When the source term contains a
polynomial in time just as considered in (1.4) and for $d=1$, $\alpha=0$ it
was proved in [10] that the critical exponent is a function of $\sigma$ which
for $N\geqslant 3$ has a discontinuity at zero,

$p_{F}(\sigma)=\left\\{\begin{array}[]{ll}\frac{N-2\sigma}{N-2\sigma-2}\hskip
5.69046pt\mbox{if}\hskip 5.69046pt-1<\sigma<0\\\ \infty\hskip
5.69046pt\mbox{if}\hskip 5.69046pt\sigma>0.\end{array}\right.$

The situation $d=1$ and $-2<\alpha<0$ has been investigated in [15]. It was
proved that for $N\geqslant 3$, solutions blow up in each of the following
cases: $m\leqslant 0$, $p<\frac{N-2m+\alpha}{N-2m-2}$,
$\int_{\mathbb{R}^{N}}\textbf{w}dx>0$ and $p>1$. Furthermore, these results do
not depend on the behavior of $\zeta$ at small times. However, the study of
global-in-time solutions has not been taken into account and we wish to cover
that in this work.

Our results generalize earlier existing works and reveal new aspects of the
solution. Prescribing initial data $u_{0}$ and w small in Marcinkiewicz space
gives rise to a global solution which lives in a small ball in the solution
space and may as well be singular. As a byproduct, global existence holds in
Lebesgue analogues of the previous functional setting whenever $u_{0}$ and w
are chosen small in the corresponding topologies. In particular, there are
solutions which are initially singular and if additionally data are assumed to
be radially symmetric (resp. radially monotone), then the solution enjoys the
same properties. Moreover, when the principal operator in (1.1) is the
nonlocal fractional Laplacian ($d\in(0,1)$) or the Laplace operator $\Delta$,
($d=1$) then the solution is positive along with $u_{0}$ and w. In the
process, we use an approach which in nature differs from those which have been
employed in the aforementioned papers – for instance, the use of Kato’s two-
norms technique which disallow unconditional uniqueness criterion and excludes
a priori the existence of solutions which at time $t=0$ are singular – our
global wellposedness statement claims the converse. We essentially rely on
smoothing effect type bounds in weak-Lebesgue spaces [28] for the singular
polyharmonic heat semigroup and its integral kernel combined with Yamazaki’s
type estimates [27] which have a wide range of application – they allow us
among others to derive unconditional uniqueness of globally defined solutions
for prescribed data in $L^{p_{c}}(\mathbb{R}^{N})$. Remarkably, we allow
$u_{0}$ and $\mathbf{w}$ to be singular functions on $\mathbb{R}^{N}$.
Concerning blow-up results, our Theorem 1.10 reinforces the fact that blow up
whenever it occurs does not depend on the behavior of the forcing term at
small times. To prove this theorem, we proceed via the test function method
introduced by the authors in [19].

We adopt the following notion of solution.

######  Definition 1.1.

We say that $u$ is a global weak solution of (1.1) if it satisfies the
conditions

$u_{0}\in L^{1}_{loc}(\mathbb{R}^{N}),\quad|x|^{\alpha}|u|^{p}\in
L^{1}_{loc}(\mathbb{R}^{N}\times(0,\infty))$

and

$\displaystyle\int_{\mathbb{R}^{N}\times(0,\infty)}u(-\partial_{t}\psi+(-\Delta)^{d}\psi)dxdt=\int_{\mathbb{R}^{N}}u_{0}(x)\psi(x,0)dx+\int_{\mathbb{R}^{N}\times(0,\infty)}|x|^{\alpha}|u|^{p}\psi
dxdt+$ $\displaystyle\qquad{}\hskip
85.35826pt\int_{\mathbb{R}^{N}\times(0,\infty)}\zeta(t)\textbf{w}\psi\,dxdt$

for all $\psi\in C^{\infty}_{0}(\mathbb{R}^{N}\times[0,\infty))$.

As is a standard practice, (1.1) is equivalent in an appropriate framework to
the Duhamel formulation

$u(x,t)=e^{-t(-\Delta)^{d}}u_{0}+\int_{0}^{t}e^{-(t-s)(-\Delta)^{d}}\left(|\cdot|^{\alpha}|u(s)|^{p}\right)\,ds+\int_{0}^{t}\,\zeta(s)\,e^{-(t-s)(-\Delta)^{d}}{\mathbf{w}}\,ds,$
(1.5)

where $e^{-t(-\Delta)^{d}}$ is the linear semi-group generated by
$(-\Delta)^{d}$. A solution to the integral equation (1.5) is often called
mild solution of (1.1). In our setting, one can see that a mild solution is a
weak solution. See [24, 25, 22, 14] for a discussion on the equivalence
between the differential and integral formulation for the nonlinear heat
equation.

In parallel to the questions discussed above, we also study local existence of
solutions. Our first result in this direction deals with the case of initial
data in the space of bounded continuous functions in $\mathbb{R}^{N}$.

######  Theorem 1.2.

Suppose we have (1.3), ${\mathbf{w}}\in BC(\mathbb{R}^{N})$,
$0<-\alpha<\min(2d,N)$ and $N\geqslant 2$. Given $u_{0}\in
BC(\mathbb{R}^{N})$, there exists a time $T:=T(u_{0})>0$ and a unique mild
solution $u\in C([0,T];BC(\mathbb{R}^{N}))$ of Eq. (1.1) such that
$u(0)=u_{0}.$

Our next goal is to investigate the local well-posedness in Lebesgue spaces.
The first result of this flavor was obtained in [1] for (1.1) with $d=1$ and
without a forcing term.

######  Theorem 1.3.

Let $p_{c}=\frac{N(p-1)}{2d+\alpha}$. Assume $0<-\alpha<\min(2d,N)$ and
$q>\max\big{\\{}p_{c},\frac{Np}{N+\alpha}\big{\\}}$ with $q>1$. Suppose that
(1.3) holds, $u_{0},{\mathbf{w}}\in L^{q}(\mathbb{R}^{N})$ and $N\geqslant
2$.Then, there exists $T:=T(u_{0})>0$ and a unique function $u\in
C([0,T];L^{q}(\mathbb{R}^{N}))$ mild solution of Eq. (1.1). Let $T_{\max}$ be
the maximal time of existence of this solution. If $T_{\max}<\infty$, then one
has

$\lim_{t\rightarrow T_{\max}}\|u(t)\|_{L^{q}(\mathbb{R}^{N})}=\infty.$

######  Remark 1.4.

Observe that although the solution obtained in Theorem 1.2 has the regularity
of the initial data, it is not true in general that this solution enjoys a
better regularity, see [31] for this evidence in some special situation. A
similar result can be stated for data in $C_{0}(\mathbb{R}^{N})$, the set of
continuous function vanishing at infinity in which case, the solution
naturally belongs to $C_{0}(\mathbb{R}^{N})$. In theorem 1.3, the fact that
one can compare $p_{c}$ and $\frac{Np}{N+\alpha}$ leads to the question
whether or not local solutions can be constructed if the condition imposed on
$q$ fails to hold true. This involves three different cases namely
$p_{c}>\frac{Np}{N+\alpha}$, $p_{c}=\frac{Np}{N+\alpha}$ and
$p_{c}<\frac{Np}{N+\alpha}$ which are worth investigating. We do not pursue in
this direction here. We point out, however, that local wellposedness in the
critical case $p_{c}>\frac{Np}{N+\alpha}$ appears as a byproduct of Theorem
1.6 below.

Next, motivated by the study of local-in-time solutions which are smooth
$C^{\infty}$ instantly, we consider $\alpha>0$ and introduce the space
$C_{\Lambda}(\mathbb{R}^{N})$ which collects continuous functions defined on
$\mathbb{R}^{N}$ satisfying the condition $\|\Lambda
u\|_{L^{\infty}(\mathbb{R}^{N})}<\infty$ where
$\Lambda(x)=(1+|x|)^{\frac{\alpha}{p-1}}$ and $p>1$. We endow this space with
the norm given by

$\|u\|_{\Lambda}=\|\Lambda u\|_{L^{\infty}(\mathbb{R}^{N})}.$

The following result establishes local existence for initial data in
$C_{\Lambda}(\mathbb{R}^{N})$.

######  Theorem 1.5.

Let $\alpha>0$ and $\zeta$ satisfying (1.4). Then, Eq. (1.5) is locally
wellposed. More precisely, given $u_{0},{\mathbf{w}}\in
C_{\Lambda}(\mathbb{R}^{N})$, there exist $T\in(0,\infty)$ and a unique
solution $u$ of (1.5) in $C([0,T];C_{\Lambda}(\mathbb{R}^{N}))$. This solution
in turn is classical and can be extended to a maximal interval $[0,T_{\max})$
where $T_{\max}\leqslant\infty$ and $\displaystyle\lim_{t\rightarrow
T_{\max}}\|u(t)\|_{\Lambda}=\infty$ whenever $T_{\max}<\infty$.

Switching to the analysis of the global theory, our main results in this
direction are the followings.

######  Theorem 1.6 (Global existence).

Let $0<-\alpha<2d<N$. Granted (1.4) with $-1<\sigma<0$, assume that
$p\geqslant\frac{N-2d\sigma+\alpha}{N-2d\sigma-2d}$ and set
$\ell=\frac{Np_{c}}{N+2(\sigma+1)dp_{c}}$. Then for any $u_{0}\in
L^{p_{c},\infty}(\mathbb{R}^{N})$ and ${\mathbf{w}}\in
L^{\ell,\infty}(\mathbb{R}^{N})$ with the property that
$\|u_{0}\|_{L^{p_{c},\infty}}+\|{\mathbf{w}}\|_{L^{\ell,\infty}}$ is
sufficiently small, Eq. (1.5) admits a global-in-time solution $u$ which
converges to $u_{0}$ as $t\rightarrow 0^{+}$ in
$\mathcal{S}^{\prime}(\mathbb{R}^{N})$. Moreover, this solution obeys the
following properties.

  1. ($\textbf{a}_{1})$

$(\textbf{Radial symmetry})$. If $u_{0}$ and ${\mathbf{w}}$ are radial
functions, then the solution $u$ is radial in the variable $x$.

  2. ($\textbf{a}_{2})$

$(\textbf{Radial monotonicity})$. Assume $u_{0}$ and ${\mathbf{w}}$ are
radially nonincreasing. Then $u(t)$ is radially nonincreasing in $x$ for all
$t\in[0,\infty)$.

  3. ($\textbf{a}_{3})$

$(\textbf{Positiveness})$. Assume $d\in(0,1]$ and let $u_{0}$ and $\mathbf{w}$
be positive functions. Then the solution $u$ is also positive.

######  Remark 1.7.

The above theorem, in particular shows that the functions
$u_{0}(x)=|x|^{-\frac{2d+\alpha}{p-1}}$ and
$\mathbf{w}(x)=|x|^{-\frac{N}{\ell}}$ as well as their translated analogues
$|x-x_{0}|^{-\frac{2d+\alpha}{p-1}}$ and $|x-x_{0}|^{-\frac{N}{\ell}}$ for
each $x_{0}\in\mathbb{R}^{N}$ are admissible choices as they belong to
$L^{p_{c},\infty}(\mathbb{R}^{N})$ and $L^{\ell,\infty}(\mathbb{R}^{N})$,
respectively. A direct consequence of Theorem 1.6 is that for $\mathbf{w}\in
L^{\ell}(\mathbb{R}^{N})$, there exist solutions which initially belong to the
critical Lebesgue space $L^{p_{c}}(\mathbb{R}^{N})\subset
L^{p_{c},\infty}(\mathbb{R}^{N})$.

######  Theorem 1.8 (Unconditional uniqueness).

Assume that $-\alpha\in(0,2d)$ and let $2d<N$. Further assume that
$p>\frac{N+\alpha}{N-2d}$. If $u$ and $v$ are two solutions of (1.5) in
$C([0,\infty);L^{p_{c}}(\mathbb{R}^{N}))$ obtained under smallness of
$\|u_{0}\|_{L^{p_{c}}}+\|\mathbf{w}\|_{L^{\ell}},$ then $u=v$ a.e. in
$\mathbb{R}^{N}\times(0,\infty)$.

######  Remark 1.9.

The result of Theorem 1.8 is true for the inhomogeneous nonlinearity
$|x|^{\alpha}\,|u|^{p-1}u$. Note that the same result was obtained in [23] for
$d=1$ and $\zeta\,{\mathbf{w}}=0$ and in [28] for $d\in(0,1)\cup\mathbb{N}$
and $\zeta\,{\mathbf{w}}=0$. See [23, Theorem 1.1 (ii), p. 4] for $d=1$. The
idea of the proof in [28] is the same as in [23]. Here, we adapt the arguments
used in [23] in our setting. See also [23, Proposition 3.7, p. 14] and [23,
Lemma 4.2, p. 16].

######  Theorem 1.10 (Blow-up).

Suppose that $d\geqslant 1$ is an integer, $0<-\alpha<2d<N$, and that
${\mathbf{w}}$ belongs to $C_{0}(\mathbb{R}^{N})\cap L^{1}(\mathbb{R}^{N})$,
obeys $\displaystyle\int_{\mathbb{R}^{N}}{\mathbf{w}}(x)dx>0$. Let $\zeta$
defined as in (1.3).

  * (i)

If $m\leqslant 0$ and $1<p<\frac{N-2dm+\alpha}{N-2md-2d}$, then for any
initial data $u_{0}\in C_{0}(\mathbb{R}^{N})$, then Pb. (1.1) has no global
solution in the sense of definition 1.1.

  * (ii)

In case $m>0$, the same conclusion holds whenever $p>1$, i.e. Pb. (1.1) has no
global weak solution.

######  Remark 1.11.

The statements of Theorem 1.10 and Theorem 1.6 both reveals that

$p_{F}(\sigma)=\left\\{\begin{array}[]{ll}\frac{N-2d\sigma+\alpha}{N-2\sigma
d-2d}&\text{if}\hskip 2.84544pt-1<\sigma<0\\\ \infty&\text{if}\hskip
2.84544pt\sigma>0\end{array}\right.$

separates the nonexistence/existence regime of global-in-time solutions of
(1.1) if $\zeta$ satisfies (1.4).

######  Remark 1.12.

The blow-up result does not cover the fractional case $d\in(0,1)$. The ideas
of the proof when $d\in\mathbb{N}$ do not directly apply to the latter case
because of the nature of the fractional Laplacian. See (3.9) below. New ideas
are therefore required.

The rest of this paper is organized as follows. In the next section, we recall
some basic facts and useful tools. Section 3 is devoted to the proofs of our
main results.

##  2\. Background materials

In this section we collect auxiliary results which will later find their
usefulness and applicability in the proofs of our main results. Recall the
space $BC(\mathbb{R}^{N})$ which collects bounded and continuous functions in
$\mathbb{R}^{N}$ and $C_{0}(\mathbb{R}^{N})$ denoting the space of continuous
functions in $\mathbb{R}^{N}$ vanishing at infinity.

We will need some basic properties of Lorentz spaces. For
$f:\mathbb{R}^{N}\rightarrow\mathbb{R}$ a measurable function on
$\mathbb{R}^{N}$, denote by
$f^{\ast}(\lambda)=\inf\\{\tau>0:|\\{|f|>\tau\\}|\leqslant\lambda\\}$ its
decreasing function and let
$f^{\ast\ast}(\tau)=\frac{1}{\tau}\int_{0}^{\tau}f^{\ast}(s)ds$. The Lorentz
space $L^{p,q}(\mathbb{R}^{N})$, (weak-Lebesgue or Marcinkiewicz space in case
$q=\infty$) collects all measurable functions $f$ such that$\|f\|_{L^{p,q}}$
is finite,

$\|f\|_{L^{p,q}}=\left\\{\begin{array}[]{ll}\bigg{(}\dfrac{q}{p}\displaystyle\int^{\infty}_{0}[t^{1/p}f^{\ast\ast}(t)]^{q}\dfrac{dt}{t}\bigg{)}^{1/q},&\mbox{if}\hskip
5.69046ptp\in(1,\infty),\hskip 5.69046ptq\in[1,\infty)\\\
\displaystyle\sup_{t>0}t^{1/p}f^{\ast\ast}(t),&\mbox{if}\hskip
5.69046ptp\in(1,\infty],\hskip 5.69046ptq=\infty.\end{array}\right.$

These spaces increase with the second exponent and contain Lebesgue spaces as
subspace. In fact, $L^{p,1}(\mathbb{R}^{N})\subset
L^{p,q_{1}}(\mathbb{R}^{N})\subset L^{p}(\mathbb{R}^{N})\subset
L^{p,q_{2}}(\mathbb{R}^{N})$, $1<p<\infty$, $1\leqslant q_{1}\leqslant
p\leqslant q_{2}\leqslant\infty$ with continuous injection. Moreover,
$L^{p,q}(\mathbb{R}^{N})$ may alternatively be realized as the real
interpolation space between the Lebesgue $L^{1}(\mathbb{R}^{N})$ and
$L^{\infty}(\mathbb{R}^{N})$,

$L^{p,q}(\mathbb{R}^{N})=[L^{1}(\mathbb{R}^{N}),L^{\infty}(\mathbb{R}^{N})]_{1-1/p,q},\hskip
5.69046ptp\in(1,\infty),q\in[1,\infty]$

and more generally, for $1<p_{1}<p_{2}<\infty$,
$\dfrac{1}{p}=\dfrac{1-\theta}{p_{1}}+\dfrac{\theta}{p_{2}}$, $\theta\in(0,1)$
and $1\leqslant q,q_{1},q_{2}\leqslant\infty$ we have the interpolation
identity

$[L^{p_{1},q_{1}}(\mathbb{R}^{N}),L^{p_{2},q_{2}}(\mathbb{R}^{N})]_{\theta,q}=L^{p,q}(\mathbb{R}^{N}).$

Consider the linear polyharmonic heat equation

$\partial_{t}u+(-\Delta)^{d}u=0\hskip 5.69046pt\text{in}\hskip
5.69046pt\mathbb{R}^{N}\times(0,\infty).$ (2.1)

It is well known that the operator $(-\Delta)^{d}$, $d\geqslant 1$ is a
generator of a semigroup $e^{-t(-\Delta)^{d}}$ whose kernel $E_{d}$ is smooth,
radial and satisfies the scaling property

$E_{d}(x,t)=t^{-\frac{N}{2d}}E_{d}(t^{-\frac{1}{2d}}x,1).$ (2.2)

Hence, a solution of (2.1) subject to initial data $u(0)=u_{0}$ may be
formally realized via convolution by

$u(x,t)=e^{-t(-\Delta)^{d}}u_{0}(x)=\left(E_{d}(\cdot,t)\ast u_{0}\right)(x)$

whenever this representation makes sense (e.g. when $u_{0}\in
BC(\mathbb{R}^{N})$ or $u_{0}\in\mathscr{S}^{\prime}(\mathbb{R}^{N})$ is a
Schwartz distribution). Note that when $d\in(0,1)$, the operator
$(-\Delta)^{d}$ defined via Fourier transform by
$\widehat{(-\Delta)^{d}\varphi}(\xi)=|\xi|^{2d}\widehat{\varphi}(\xi)$
($\varphi\in\mathcal{S}(\mathbb{R}^{N})$) also generates a semigroup whose
kernel as an algebraic decay at infinity, see [9]. We recall the following
$L^{r}-L^{q}$ estimate proved in [16] by using the majorizing kernel
established in [5]. See [16, Proposition 6.1, p. 521].

######  Proposition 2.1.

There exists a positive constant ${\mathcal{H}}_{d}$ such that for all
$1\leqslant p\leqslant q\leqslant\infty$, we have

$\|e^{-t(-\Delta)^{d}}\varphi\|_{L^{q}}\leqslant{\mathcal{H}}_{d}\,t^{-\frac{N}{2d}(\frac{1}{p}-\frac{1}{q})}\|\varphi\|_{L^{p}}$
(2.3)

for all $t>0$ and $\varphi\in L^{p}(\mathbb{R}^{N})$.

Interpolating estimate (2.3) yields the following smoothing effect in Lorentz
spaces

$\|e^{-t(-\Delta)^{d}}\varphi\|_{L^{q,r}}\leqslant{\mathcal{H}}_{d}\,t^{-\frac{N}{2d}(\frac{1}{p}-\frac{1}{q})}\|\varphi\|_{L^{p,r}}$
(2.4)

for all $t>0$, $\varphi\in L^{p,r}(\mathbb{R}^{N})$ and for $1\leqslant
r\leqslant\infty$, $1<p\leqslant q<\infty$. For $N\geqslant 2$ and
$\alpha\in(0,N)$, define the operator

${\mathbf{S}}_{d,\alpha}(t)=e^{-t(-\Delta)^{d}}|\cdot|^{-\alpha}.$

######  Proposition 2.2 ([28]).

Let $N\geqslant 2$, $\alpha\in(0,N)$, $1<p_{1},p_{2}\leqslant\infty$ and
$q\in[1,\infty]$ such that

$p_{1}>\frac{N}{N-\alpha},\hskip
14.22636ptp_{2}\geqslant\frac{N\,p_{1}}{N+\alpha\,p_{1}}.$ (2.5)

Then ${\mathbf{S}}_{d,\alpha}(t)$ has the following mapping properties:

  1. ($\textbf{A}_{1})$

${\mathbf{S}}_{d,\alpha}(t)$ maps continuously $L^{p_{1}}(\mathbb{R}^{n})$
into $C_{0}(\mathbb{R}^{n})$ for all $t>0$.

  2. ($\textbf{A}_{2})$

For any $\gamma\in\mathbb{N}^{N}_{0}$, ${\mathbf{S}}_{d,\alpha}(t)$ maps
continuously $L^{p_{1},\infty}(\mathbb{R}^{N})$ into
$L^{p_{2},q}(\mathbb{R}^{N})$ for all $t>0$. Moreover, there exits a positive
constant $C:=C(N,p_{1},p_{2},d,\alpha,\gamma)$ such that

$\|\partial^{\gamma}{\mathbf{S}}_{d,\alpha}(t)\varphi\|_{L^{p_{2},q}}\leqslant
Ct^{-\frac{N}{2d}(\frac{1}{p_{1}}-\frac{1}{p_{2}})-\frac{\alpha}{2d}-\frac{|\gamma|}{2d}}\|\varphi\|_{L^{p_{1},\infty}}$
(2.6)

for all $t>0$ and $\varphi\in L^{p_{1},\infty}(\mathbb{R}^{N})$.

######  Remark 2.3.

Proposition 2.2 is known for $d=1,\gamma=0$ in [1] for Lebesgue spaces and in
[23, Proposition 3.3, p. 13] for Lorentz spaces. The original ideas of the
proofs are like that of [1, 23]. See also [28].

######  Lemma 2.4 ([28]).

Given $\eta>0$, there exits a positive constant $C:=C(\eta,N)$ such that for
$\beta\in[0,2]$ and $y\in\mathbb{R}^{N}$,

$\int_{\mathbb{R}^{N}}|g(x)|(1+|y-\beta x|)^{-\eta}dx\leqslant
C(1+|y|)^{-\eta},$ (2.7)

where $g(x)=E_{d}(x,1)$ and $E_{d}$ is as in (2.2).

Let $\phi:\mathbb{R}^{N}\times(0,\infty)\rightarrow\mathbb{R}$. We introduce
the nonlinear operator

$\mathscr{M}\phi(x)=\displaystyle\int_{0}^{\infty}{\mathbf{S}}_{d,\alpha}(s)|\phi(x,s)|^{p}ds$

for a suitable $\phi$ such that the integral on the right hand side is
meaningful.

######  Lemma 2.5 ([28]).

Let $p>1$ and $0<\alpha<2d<N$. Given $k,q>1$, assume that

$k>\frac{Np}{N-\alpha}\hskip 5.69046pt\text{and}\hskip
5.69046pt\frac{1}{q}=\frac{p}{k}-\frac{2d-\alpha}{N}.$

There exists a constant $C>0$ such that

$\|\mathscr{M}\phi\|_{L^{q,\infty}}\leqslant
C\sup_{t>0}\|\phi(t)\|^{p}_{L^{k,\infty}},\hskip 5.69046pt\phi\in
L^{\infty}([0,\infty);L^{k,\infty}(\mathbb{R}^{N})).$ (2.8)

In particular we can take $k=\frac{N(p-1)}{2d-\alpha}$ in (2.8) and if
$\phi\in L^{\infty}([0,\infty);L^{p_{c},\infty}\cap
L^{r,\infty}(\mathbb{R}^{N}))$ with $r>1$ and
$1<r^{\prime}<\frac{N}{2d-\alpha}$, $\frac{1}{r}+\frac{1}{r^{\prime}}=1$, then
we have

$\|\mathscr{M}\phi\|_{L^{r,\infty}}\leqslant\,C\sup_{t>0}\|\phi(t)\|_{L^{r,\infty}}\sup_{t>0}\|\phi(t)\|^{p-1}_{L^{p_{c},\infty}}.$
(2.9)

##  3\. Proofs of the main results

Starting with the proofs of theorems pertaining to the local theory, recall
that $\mathbf{S}_{d,\alpha}(t)$ is a $C_{0}$ semigroup on Lebesgue spaces and
on $BC(\mathbb{R}^{N})$ so that we can systematically appeal to the abstract
result in [26]. For this end, we define for $0<-\alpha<N$ the map

$\mathscr{K}_{t}u=\mathbf{S}_{d,-\alpha}(t)|u|^{p}+\mathbf{S}_{d,0}(t)\zeta(\cdot)\mathbf{w},\hskip
5.69046ptt>0.$

###  3.1. Proof of Theorem 1.2

Given $a,b\in\mathbb{R}$, we can find a constant $\eta:=\eta(p)>0$ such that

$\big{|}|a|^{p}-|b|^{p}\big{|}\leqslant\eta|a-b|(|a|^{p-1}+|b|^{p-1}),\hskip
5.69046ptp>1.$ (3.1)

By invoking Proposition 2.2 with $p_{2}=p_{1}=\infty$, it is easy to verify
that $\mathscr{K}_{t}$ is bounded on the space $BC(\mathbb{R}^{N})$ and the
following bound holds, namely $\|\mathbf{S}_{d,0}u_{0}\|_{L^{\infty}}\leqslant
c\|u_{0}\|_{L^{\infty}}$. Next, call $\overline{B}_{L}(0)\subset
BC(\mathbb{R}^{N})$ the ball with center at zero and radius $L>0$, we prove
that $\mathscr{K}_{t}$ is Lipschitz continuous on $\overline{B}_{L}(0)$ for
each $t>0$. Take $u,v\in\overline{B}_{L}(0)$, by (3.1) and making use of
Proposition 2.2 once again we arrive at

$\displaystyle\|\mathscr{K}_{t}u-\mathscr{K}_{t}v\|_{L^{\infty}}\leqslant
c_{L}(t)\|u-v\|_{L^{\infty}}$

with the Lipschitz constant given by
$c_{L}(t)=CL^{p-1}t^{\frac{\alpha}{2d}}\in L^{1}(0,\varepsilon)$ for some
$\varepsilon>0$ and for each $L>0$ since $-\alpha<2d$. Moreover, the kernel
$E_{d}$ has the property

$E_{d}(x,t)=\int_{\mathbb{R}^{N}}E_{d}(x-y,t-s)E_{d}(y,s)dy$

for all $x\in\mathbb{R}^{N}$ and $s,t>0$; which obviously implies
$\mathbf{S}_{d,0}(t)\phi=\mathbf{S}_{d,0}(t-s)[\mathbf{S}_{d,0}(s)\phi]$,
$0<s<t$. Hence,

$\displaystyle\mathbf{S}_{d,0}(s)\mathcal{K}_{t}u$
$\displaystyle=\mathbf{S}_{d,0}(s)\mathbf{S}_{d,0}(t)|\cdot|^{\alpha}|u|^{p}+\mathbf{S}_{d,0}(s)\mathbf{S}_{d,0}(t)\zeta(\cdot)\mathbf{w}$
$\displaystyle=\mathbf{S}_{d,0}(t+s)|\cdot|^{\alpha}|u|^{p}+\mathbf{S}_{d,0}(t+s)\zeta(\cdot)\mathbf{w}$
$\displaystyle=\mathbf{S}_{d,-\alpha}(t+s)|u|^{p}+\mathbf{S}_{d,0}(t+s)\zeta(\cdot)\mathbf{w}$
$\displaystyle=\mathcal{K}_{t+s}u.$

On the other hand, we have
$\mathcal{K}_{t}(0)=\mathbf{S}_{d,0}(t)\zeta(\cdot)\mathbf{w}$, so that for
some $\varepsilon>0$ and by using (1.3), there holds

$\displaystyle\big{\|}\|\mathscr{K}_{t}(0)\|_{L^{\infty}}\big{\|}_{L^{1}(0,\varepsilon)}$
$\displaystyle=\big{\|}\|\mathbf{S}_{d,0}(t)\zeta(\cdot)\mathbf{w}\|_{L^{\infty}}\big{\|}_{L^{1}(0,\varepsilon)}$
$\displaystyle\leqslant
C\|\mathbf{w}\|_{L^{\infty}}\|\zeta\|_{L^{1}(0,\varepsilon)}\leqslant C.$

At this point, a simple application of Theorem 1 in [26] yields the desired
conclusion.

###  3.2. Proof of Theorem 1.3

Here, the reasoning goes in the same spirit as in the above proof. Let
$u_{0},\mathbf{w}\in L^{q}(\mathbb{R}^{N})$. Then granted (1.3), under the
condition $q>\frac{Np}{2d+\alpha}$, we can apply Proposition 2.2 with
$(p_{1},p_{2})=(q/p,q)$ to obtain that $\mathscr{K}_{t}$ is bounded on
$L^{q}(\mathbb{R}^{N})$ for each $t>0$ and there holds
$\|\mathbf{S}_{d,0}u_{0}\|_{L^{q}}\leqslant C\|u_{0}\|_{L^{q}}$ by Proposition
2.1. We eventually want to verify that the following conditions are fulfilled:

  1. (1)

$\mathscr{K}_{t}:L^{q}(\mathbb{R}^{N})\rightarrow L^{q}(\mathbb{R}^{N})$ is a
locally Lipschitz map with constant $c_{L}(t)\in L^{1}(0,\varepsilon)$ for
each $L>0$ and for some $\varepsilon>0$.

  2. (2)

The map $t\mapsto\|\mathscr{K}_{t}(0)\|_{L^{q}}$ belongs to
$L^{1}(0,\varepsilon)$ for some $\varepsilon>0.$

  3. (3)

$\mathbf{S}_{d,0}(s)\mathscr{K}_{t}=\mathscr{K}_{t+s}$ for $s,t>0.$

The third property has already been established in the previous lines. As
regards the second, for an arbitrary $\varepsilon>0$, we have that

$\displaystyle\big{\|}\|\mathscr{K}_{t}(0)\|_{L^{q}}\big{\|}_{L^{1}(0,\varepsilon)}$
$\displaystyle=\big{\|}\|\mathbf{S}_{d,0}(t)\zeta(\cdot)\mathbf{w}\|_{L^{q}}\big{\|}_{L^{1}(0,\varepsilon)}$
$\displaystyle\leqslant
C\|\mathbf{w}\|_{L^{q}}\|\zeta\|_{L^{1}(0,\varepsilon)}\leqslant C.$

Let us now show that the first condition is equally satisfied. Fix $L>0$ and
denote by $\overline{B}_{L}(0)$ the closed ball in $L^{q}(\mathbb{R}^{N})$
centered at the origin and with radius $L$. We compute

$\displaystyle\|\mathscr{K}_{t}u-\mathscr{K}_{t}v\|_{L^{q}}$
$\displaystyle=\|\mathbf{S}_{d,-\alpha}(t)(|u|^{p}-|v|^{p})\|_{L^{q}}$
$\displaystyle\leqslant c_{L}(t)\|u-v\|_{L^{q}}$

with $c_{L}(t)=CL^{p-1}t^{-\frac{N(p-1)}{2qd}+\frac{\alpha}{2d}}$. Note that
in order to get this bound, we use the pointwise estimate (3.1) and invoked
Proposition 2.2 with $(p_{1},p_{2})=(\frac{q}{p},q)$. Since
$q>\frac{N(p-1)}{2d+\alpha}=p_{c}$, it follows that $c_{L}(t)\in
L^{1}(0,\varepsilon)$ for all $L>0$ and for all $\varepsilon>0$. We are now in
position to apply Theorem 1 in [26] which yields the statement of Theorem 1.3.

###  3.3. Proof of Theorem 1.5

We turn to the proof of the local well-posedness result subject to data in
$C_{\Lambda}(\mathbb{R}^{n})$. Recall
$\Lambda(x)=(1+|x|)^{\frac{\alpha}{p-1}}$. In this case, we proceed in a
slightly different fashion with the aim of reaching the hypotheses that will
allow an application of a Banach fixed point theorem.

Let $0<T<1$ be a time to be chosen later and $\delta>0$. Consider the space

$X_{T,\Lambda}=\bigg{\\{}u\in C([0,T];C_{\Lambda}(\mathbb{R}^{N})):\|\Lambda
u(t)\|_{L^{\infty}(\mathbb{R}^{N})}\leqslant\delta,t\in(0,T)\bigg{\\}}$

on which we put the metric
$d(u,v)=\displaystyle\sup_{t\in[0,T]}\|\Lambda(u(t)-v(t))\|_{L^{\infty}(\mathbb{R}^{N})}$.
We will prove in the sequel that the operator $\mathcal{Q}$ defined as

$\mathcal{Q}u=\mathbf{S}_{d,0}(t)u_{0}+\int_{0}^{t}\mathbf{S}_{d,-\alpha}(t-s)|u(s)|^{p}ds+\int_{0}^{t}\mathbf{S}_{d,0}(t-s)\zeta(s)\mathbf{w}ds$
(3.2)

possesses a fixed point in $C([0,T];C_{\Lambda}(\mathbb{R}^{N}))$. We first
show that $\mathcal{Q}$ is a self-mapping from $X_{T,\Lambda}$ onto itself.
Using Lemma 2.4, we have

$\displaystyle|\mathbf{S}_{d,0}(t)u_{0}|$
$\displaystyle=|e^{-t(-\Delta)^{d}}u_{0}|=\bigg{|}\int_{\mathbb{R}^{N}}\,E_{d}(x-y,t)u_{0}(y)\,dy\bigg{|}$
$\displaystyle\leqslant\int_{\mathbb{R}^{N}}\,|E_{d}(x-y,t)||u_{0}(y)|dy$
$\displaystyle\leqslant\int_{\mathbb{R}^{N}}\,t^{\frac{-N}{2d}}|g(t^{-\frac{1}{2d}}(x-y))||u_{0}(y)|dy$
$\displaystyle\leqslant\|u_{0}\|_{C_{\Lambda}(\mathbb{R}^{N})}\,\int_{\mathbb{R}^{N}}\,t^{\frac{-N}{2d}}|g(t^{-\frac{1}{2d}}(x-y))|\Lambda^{-1}(y)\,dy$
$\displaystyle\leqslant\|u_{0}\|_{C_{\Lambda}(\mathbb{R}^{N})}\,\int_{\mathbb{R}^{N}}|g(y)|\left(1+|x-t^{1/2d}\,y|\right)^{-\frac{\alpha}{p-1}}\,dy$
$\displaystyle\leqslant
C(\alpha,p,N)\|u_{0}\|_{C_{\Lambda}(\mathbb{R}^{N})}\,(1+|x|)^{-\frac{\alpha}{p-1}}.$

Similarly, setting
$Bu(t)=\displaystyle\int_{0}^{t}e^{-(t-s)(-\Delta)^{d}}|\cdot|^{\alpha}|u(s)|^{p}\,ds$,
$u\in X_{T,\Lambda}$, an application of Lemma 2.4 once again permits us to
write

$\displaystyle|B(t)u|$
$\displaystyle=\bigg{|}\int_{0}^{t}\int_{\mathbb{R}^{N}}\,(t-s)^{-\frac{N}{2d}}g\big{(}(t-s)^{-\frac{1}{2d}}(x-y)\big{)}|y|^{\alpha}|u(s)|^{p}\,dyds\bigg{|}$
$\displaystyle\leqslant
C\int_{0}^{t}\int_{\mathbb{R}^{N}}\,(t-s)^{-\frac{N}{2d}}|g\big{(}(t-s)^{-\frac{1}{2d}}(x-y)\big{)}||y|^{\alpha}|\Lambda(y)u(y,s)|^{p}\Lambda^{-p}(y)\,dyds$
$\displaystyle\leqslant
C\delta^{p}\int_{0}^{t}\,(t-s)^{-\frac{N}{2d}}\int_{\mathbb{R}^{N}}\,|g\big{(}(t-s)^{-\frac{1}{2d}}(x-y)\big{)}|y|^{\alpha}(1+|y|)^{-\frac{\alpha\,p}{p-1}}\,dyds$
$\displaystyle\leqslant
C\delta^{p}\int_{0}^{t}\int_{\mathbb{R}^{N}}\,|g(z)|(1+|x-(t-s)^{\frac{1}{2d}}z|)^{-\frac{\alpha}{p-1}}\,dzds$
$\displaystyle\leqslant
C\delta^{p}\int_{0}^{T}\,(1+|x|)^{-\frac{\alpha}{p-1}}\,ds$
$\displaystyle\leqslant C\delta^{p}(1+|x|)^{-\frac{\alpha}{p-1}}T.$

Finally, setting
$Du(t)=\displaystyle\int_{0}^{t}\,e^{-(t-s)(-\Delta)^{d}}{\mathbf{w}}\zeta(s)\,ds$,
we invoke Lemma 2.4 to arrive at

$\displaystyle|D(t)u|$
$\displaystyle=\bigg{|}\int_{0}^{t}\int_{\mathbb{R}^{N}}\,(t-s)^{-\frac{N}{2d}}g\big{(}(t-s)^{-\frac{1}{2d}}(x-y)\big{)}|{\mathbf{w}}(y)|\zeta(s)\,dyds\bigg{|}$
$\displaystyle\leqslant\|{\mathbf{w}}\|_{C_{\Lambda}(\mathbb{R}^{N})}\int_{0}^{t}\int_{\mathbb{R}^{N}}\,(t-s)^{-\frac{N}{2d}}\big{|}g\big{(}(t-s)^{-\frac{1}{2d}}(x-y)\big{)}\big{|}|\Lambda(y)|^{-1}\zeta(s)\,dyds$
$\displaystyle\leqslant\|{\mathbf{w}}\|_{C_{\Lambda}(\mathbb{R}^{N})}\int_{0}^{t}\int_{\mathbb{R}^{N}}\,|g(z)|\zeta(s)(1+|x-(t-s)^{\frac{1}{2d}}z|^{-\frac{\alpha}{p-1}}\,dzds$
$\displaystyle\leqslant
C\|{\mathbf{w}}\|_{C_{\Lambda}(\mathbb{R}^{N})}\frac{T^{\sigma+1}}{\sigma+1}\Lambda^{-1}(x)$

so that $\|\Lambda Du(t)\|_{L^{\infty}}\leqslant
C\|{\mathbf{w}}\|_{C_{\Lambda}(\mathbb{R}^{N})}\dfrac{T^{\sigma+1}}{\sigma+1}$.
Gluing together all the above estimates, we find that

$\|\Lambda\mathcal{Q}u(t)\|_{L^{\infty}}\leqslant
C\big{(}\|u_{0}\|_{C_{\Lambda}(\mathbb{R}^{N})}+\delta^{p}T+\frac{T^{\sigma+1}}{\sigma+1}\|{\mathbf{w}}\|_{C_{\Lambda}(\mathbb{R}^{N})}\big{)},\;\;0\leqslant
t\leqslant T,$ (3.3)

where $C$ is a constant depending on $N,\alpha$ and $p$. Let $\delta>0$ such
that $C\|u_{0}\|_{C_{\Lambda}(\mathbb{R}^{N})}<\delta$. Pick $T<1$ with the
property that
$C\big{(}\|u_{0}\|_{C_{\Lambda}(\mathbb{R}^{N})}+\delta^{p}T+\frac{T^{\sigma+1}}{\sigma+1}\|{\mathbf{w}}\|_{C_{\Lambda}(\mathbb{R}^{N})}\big{)}\leqslant\delta$
to draw the conclusion that $\mathcal{Q}$ maps $X_{T,\Lambda}$ into itself.
Next, let $u,v\in X_{T,\Lambda}$ and let’s additionally impose the condition
$C\delta^{p-1}T<1$ where $C$ is that constant appearing in (3.3). Then
$\mathcal{Q}$ is a contraction. To see this write

$\displaystyle|\mathcal{Q}u(t)-\mathcal{Q}v(t)|$
$\displaystyle=\bigg{|}\int_{0}^{t}\int_{\mathbb{R}^{N}}E_{d}(x-y,t-s)|y|^{\alpha}\big{|}|u(s)|^{p}-|v(s)|^{p}\big{|}\,dyds\bigg{|}$
$\displaystyle\leqslant
C\int_{0}^{t}\int_{\mathbb{R}^{N}}|E_{d}(x-y,t-s)||y|^{\alpha}|u-v|(|u|^{p-1}+|v|^{p-1})\,dyds$
$\displaystyle\leqslant
C\bigg{(}\int_{0}^{t}\int_{\mathbb{R}^{N}}|E_{d}(x-y,t-s)||y|^{\alpha}|u-v|\Lambda(y)|\Lambda
u|^{p-1}\Lambda^{-p}(y)dyds+$
$\displaystyle\qquad\int_{0}^{t}\int_{\mathbb{R}^{N}}|E_{d}(x-y,t-s)||y|^{\alpha}|u-v|\Lambda(y)|\Lambda
v|^{p-1}\Lambda^{-p}(y)\,dyds\bigg{)}$ $\displaystyle\leqslant
Cd(u,v)\bigg{(}\|\Lambda
u(t)\|^{p-1}_{L^{\infty}}\int_{0}^{t}\int_{\mathbb{R}^{N}}|g(z)||x-(t-s)^{\frac{1}{2d}}z|^{-1}dzds+$
$\displaystyle\qquad\qquad\qquad\qquad\|\Lambda
v(t)\|^{p-1}_{L^{\infty}}\int_{0}^{t}\int_{\mathbb{R}^{N}}|g(z)||x-(t-s)^{\frac{1}{2d}}z|^{-1}\,dzds\bigg{)}.$

This yields, utilizing Lemma 2.4 the bound

$d\left(\mathcal{Q}u,\mathcal{Q}v\right)\leqslant C\delta^{p-1}Td(u,v).$

Existence of a solution as claimed in Theorem 1.5 is now a simple consequence
of the Banach fixed point theorem. The blow up alternative is verified using
standard arguments. Since $C_{\Lambda}(\mathbb{R}^{N})\subset
C_{b}(\mathbb{R}^{N})$, we have that $u\in
L^{\infty}([0,T];C_{b}(\mathbb{R}^{N}))$ and by parabolic regularity theory,
$u$ satisfies (1.1) in the classical sense. The proof of Theorem 1.5 is now
complete.

Our next focus is on the proofs of global existence and non-existence results.
Recall the definition of the beta function

$\mathcal{B}(a,b)=\int_{0}^{1}s^{a-1}(1-s)^{b-1}ds=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)},\quad
a,b>0.$

where $\Gamma$ is the standard gamma function.

###  3.4. Proof of Theorem 1.6

We will distinguish between two cases assuming first that
$p>\frac{N+\alpha}{N-2d}$ with $-\alpha<2d<N$. The second case
$p\in[p_{F},\frac{N+\alpha}{N-2d}]$ will be handle subsequently. Let $u_{0}\in
L^{p_{c},\infty}(\mathbb{R}^{N})$, $\mathbf{w}\in
L^{\ell,\infty}(\mathbb{R}^{N})$ such that
$\|u_{0}\|_{L^{p_{c},\infty}}+\|\mathbf{w}\|_{L^{\ell,\infty}}<\varepsilon_{0}$
for some $\varepsilon_{0}>0$. We wish to show that the equation

$u=w+\mathcal{F}(u)\hskip 5.69046pt\mbox{in}\hskip
5.69046pt\mathbb{R}^{N}\times(0,\infty);\quad
w=\mathbf{S}_{d,0}(t)u_{0}+\int_{0}^{t}\mathbf{S}_{d,0}(t-s)\zeta(s)\mathbf{w}ds$
(3.4)

has a unique fixed point in a closed ball $\overline{B}_{2\varepsilon}(0)$ of
$C_{b}((0,\infty);L^{p_{c},\infty})$,
$\varepsilon=\varepsilon(\varepsilon_{0})$ where for $t>0$,

$\mathcal{F}(u)(t)=\int_{0}^{t}\mathbf{S}_{d,-\alpha}(t-s)|u(s)|^{p}ds.$

Apply Lemma 2.5 with $k=p_{c}$ (i.e $q=p_{c}$) in (2.8) and
$|\phi(s)|^{p}=\begin{cases}|u(t-s)|^{p}\hskip 5.69046pt\text{if}\hskip
5.69046pts\in(0,t)\\\ 0\hskip 5.69046pt\text{if}\hskip 5.69046pts\geqslant
t\end{cases}$ to obtain the estimate

$\|\mathcal{F}u\|_{L^{\infty}((0,\infty);L^{p_{c},\infty})}\leqslant
C\sup_{t>0}\|u(t)\|^{p}_{L^{p_{c},\infty}}.$ (3.5)

Arguing in a similar fashion, we have that for $u$ and $v$ in
$L^{\infty}((0,\infty);L^{p_{c},\infty})$,

$\|\mathcal{F}(u)-\mathcal{F}(v)\|_{L^{\infty}(\mathbb{R}_{+};L^{p_{c},\infty})}\leqslant
C\|u-v\|_{L^{\infty}(\mathbb{R}_{+};L^{p_{c},\infty})}(\|u\|^{p-1}_{L^{\infty}(\mathbb{R}_{+};L^{p_{c},\infty})}+\|v\|^{p-1}_{L^{\infty}((0,\infty);L^{p_{c},\infty})}).$
(3.6)

Observe that $\ell=\frac{Np_{c}}{N+2(\sigma+1)dp_{c}}>1$ in view of the
condition imposed on $p$ and that

$\displaystyle\|w\|_{L^{p_{c},\infty}}\leqslant
C\|u_{0}\|_{L^{p_{c},\infty}}+\bigg{\|}\int_{0}^{t}\mathbf{S}_{d,0}(t-s)\zeta(s)\mathbf{w}ds\bigg{\|}_{L^{p_{c},\infty}}$

where the second term is further estimated via duality as follows

$\displaystyle\bigg{\|}\int_{0}^{t}\mathbf{S}_{d,0}(t-s)\zeta(s)\mathbf{w}ds\bigg{\|}_{L^{p_{c},\infty}}$
$\displaystyle=\sup_{\begin{subarray}{c}\psi\in L^{p_{c}^{\prime},1}\\\
\|\psi\|_{L^{p_{c}^{\prime},1}}=1\end{subarray}}\bigg{|}\langle\int_{0}^{t}\mathbf{S}_{d,0}(t-s)\zeta(s)\mathbf{w}ds,\psi\rangle\bigg{|}$
$\displaystyle\leqslant\sup_{\begin{subarray}{c}\psi\in
L^{p_{c}^{\prime},1}\\\
\|\psi\|_{L^{p_{c}^{\prime},1}}=1\end{subarray}}\bigg{|}\int_{0}^{t}\int_{\mathbb{R}^{N}}\mathbf{S}_{d,0}(t-s)\zeta(s)\mathbf{w}\psi(x)dxds\bigg{|}$
$\displaystyle\leqslant\sup_{\begin{subarray}{c}\psi\in
L^{p_{c}^{\prime},1}\\\
\|\psi\|_{L^{p_{c}^{\prime},1}}=1\end{subarray}}\bigg{(}\int_{0}^{t}\|\mathbf{S}_{d,0}(t-s)\psi\|_{L^{\ell^{\prime},1}}\|\zeta(s)\mathbf{w}\|_{L^{\ell,\infty}}ds\bigg{)}$
$\displaystyle\leqslant C\sup_{\begin{subarray}{c}\psi\in
L^{p_{c}^{\prime},1}\\\
\|\psi\|_{L^{p_{c}^{\prime},1}}=1\end{subarray}}\bigg{(}\int_{0}^{t}(t-s)^{-\frac{N}{2d}(\frac{1}{p_{c}^{\prime}}-\frac{1}{\ell^{\prime}})}\|\psi\|_{L^{p_{c}^{\prime},1}}\|\zeta(s)\mathbf{w}\|_{L^{\ell,\infty}}ds\bigg{)}$
$\displaystyle\leqslant C\sup_{\begin{subarray}{c}\psi\in
L^{p_{c}^{\prime},1}\\\
\|\psi\|_{L^{p_{c}^{\prime},1}}=1\end{subarray}}\|\psi\|_{L^{p_{c}^{\prime},1}}\|\mathbf{w}\|_{L^{\ell,\infty}}\int_{0}^{t}(t-s)^{-\frac{N}{2d}(\frac{1}{\ell}-\frac{1}{p_{c}})}\zeta(s)ds$
$\displaystyle\leqslant C\sup_{\begin{subarray}{c}\psi\in
L^{p_{c}^{\prime},1}\\\
\|\psi\|_{L^{p_{c}^{\prime},1}}=1\end{subarray}}\|\psi\|_{L^{p_{c}^{\prime},1}}\|\mathbf{w}\|_{L^{\ell,\infty}}\int_{0}^{1}\tau^{\sigma}(1-\tau)^{-(\sigma+1)}d\tau$
$\displaystyle\leqslant
C\|\mathbf{w}\|_{L^{\ell,\infty}}\mathcal{B}(\sigma+1,-\sigma)$

where we have systematically used Fubini’s Theorem, smoothing effect in
Lorentz space (2.4), the generalized Hölder’s inequality and in the estimate
before the last we performed the change of variables $\tau=s/t$ together with
the fact that
$\frac{1}{p_{c}^{\prime}}-\frac{1}{\ell^{\prime}}=\frac{2(\sigma+1)d}{N}$,
$-1<\sigma<0$ and $\ell^{\prime}$ being the conjugate exponent of $\ell$.
Hence, we have that

$\|w\|_{L^{p_{c},\infty}}\leqslant
C_{1}(\|u_{0}\|_{L^{p_{c},\infty}}+\|\mathbf{w}\|_{L^{\ell,\infty}})\leqslant
C_{1}\varepsilon_{0}.$

Put $\varepsilon=C_{1}\varepsilon_{0}$ and $R=(2^{p}\,C)^{-\frac{1}{p-1}}$.
Assuming that $\varepsilon<R$, we deduce from (3.5) and (3.6) that Equation
(3.4) has a unique fixed point in $\overline{B}_{2\varepsilon}(0)$ which
indeed is the solution of (1.5) we were looking for. Now assume that
$p_{F}\leqslant p\leqslant\frac{N+\alpha}{N-2d}$. We shall argue differently
in this case since the strategy employed before will clearly fail to work well
here. This being said, pick a number $r>1$ such that the inequality below is
satisfied,

$\max\bigg{\\{}\frac{\alpha
p+2d}{Np(p-1)},\frac{1}{p_{c}}+\frac{2d\sigma}{N}\bigg{\\}}<\frac{1}{r}<\min\bigg{\\{}\frac{1}{p_{c}},\frac{N+\alpha}{Np}\bigg{\\}},\quad
r>p.$ (3.7)

This choice is possible. In fact, in view of the condition $p\geqslant p_{F}$,
one can easily check that all inequalities in (3.7) are satisfied expect
possibly for $\dfrac{1}{p_{c}}+\dfrac{2d\sigma}{N}<\dfrac{N+\alpha}{Np}$ which
we verify as follows. The function $f(a)=2d\sigma
p^{2}+p(2d-2d\sigma-a)+a+\alpha$ defined for
$a\geqslant\rho=2d\sigma+\dfrac{2dp+\alpha}{p-1}$ is decreasing so that in
particular, $f(N)\leqslant f(\rho)<0$ because $p\geqslant
p_{F}=\frac{N-2d\sigma+\alpha}{N-2d(\sigma+1)}$. Moving on, observe that from
(3.7) follows the restrictions $1\leqslant\ell<p_{c}<r$ and if we put

$\mu=\frac{N}{2d}\bigg{(}\frac{1}{p_{c}}-\frac{1}{r}\bigg{)},$

then $0<\mu<\dfrac{1}{p}$. Moreover, we have

$\mu=\frac{N(p-1)}{2rd}-\frac{\alpha}{2d}+p\mu-1=\frac{N}{2d}\bigg{(}\frac{1}{\ell}-\frac{1}{r}\bigg{)}-\sigma-1.$

Introduce the function space $\mathbf{X}$ defined by

$\mathbf{X}=\bigg{\\{}u\in
C_{b}\big{(}(0,\infty);L^{p_{c},\infty}(\mathbb{R}^{N})\big{)}:t^{\mu}u\in
C_{b}\big{(}(0,\infty);L^{r,\infty}(\mathbb{R}^{N})\bigg{\\}}$

which carries out the structure of a complete metric space when equipped with
the distance
$d(u,v)=\displaystyle\sup_{t>0}t^{\mu}\|u(t)-v(t)\|_{L^{r,\infty}}:=\|u-v\|_{\mathbf{X}}$.
We want to show that Eq. (3.4) has a fixed point in $\mathbf{X}$ which is
unique in $\overline{B}_{K}(0)\subset\mathbf{X}$ a closed ball centered at the
origin and with radius $K>0$ sufficiently small. We estimate separately each
of the terms of the right hand side of (3.4). Making use of the smoothing
estimate (2.4) we obtain

$\|\mathbf{S}_{d,0}(t)u_{0}\|_{L^{r,\infty}}\leqslant
Ct^{-\frac{N}{2d}\big{(}\frac{1}{p_{c}}-\frac{1}{r}\big{)}}\|u_{0}\|_{L^{p_{c},\infty}}\leqslant
Ct^{-\mu}\|u_{0}\|_{L^{p_{c},\infty}}.$

Since $r>\frac{Np}{N+\alpha}$, it follows from Proposition 2.2 that

$\displaystyle\bigg{\|}\int_{0}^{t}\mathbf{S}_{d,-\alpha}(t-s)|u(s)|^{p}ds\bigg{\|}_{L^{r,\infty}}$
$\displaystyle\leqslant
C\int_{0}^{t}(t-s)^{-\frac{N}{2d}\big{(}\frac{p}{r}-\frac{1}{r}\big{)}+\frac{\alpha}{2d}}\||u|^{p}\|_{L^{r/p,\infty}}ds$
$\displaystyle\leqslant
C\int_{0}^{t}\,(t-s)^{-\frac{N}{2d}\big{(}\frac{p}{r}-\frac{1}{r}\big{)}+\frac{\alpha}{2d}}s^{-p\mu}(s^{\mu}\|u\|_{L^{r,\infty}})^{p}ds$
$\displaystyle\leqslant
C(\sup_{t>0}t^{\mu}\|u\|_{L^{r,\infty}})^{p}\int_{0}^{t}\,(t-s)^{-\frac{N(p-1)}{2rd}+\frac{\alpha}{2d}}s^{-p\mu}ds$
$\displaystyle\leqslant
Ct^{-\frac{N(p-1)}{2rd}+\frac{\alpha}{2d}-p\mu+1}\|u\|^{p}_{\mathbf{X}}\int_{0}^{1}(1-s)^{-\frac{N(p-1)}{2rd}+\frac{\alpha}{2d}}s^{-p\mu}ds$
$\displaystyle\leqslant
Ct^{-\mu}\|u\|^{p}_{\mathbf{X}}\mathcal{B}\bigg{(}1-p\mu,1-\frac{N(p-1)}{2rd}+\frac{\alpha}{2d}\bigg{)}$
$\displaystyle\leqslant Ct^{-\mu}\|u\|^{p}_{\mathbf{X}}.$

To estimate the last term, we utilize (2.4) and proceed as follows

$\displaystyle\bigg{\|}\int_{0}^{t}\mathbf{S}_{d,0}(t-s)\mathbf{w}\zeta(s)\bigg{\|}_{L^{r,\infty}}ds$
$\displaystyle\leqslant
C\int_{0}^{t}(t-s)^{-\frac{N}{2d}\big{(}\frac{1}{\ell}-\frac{1}{r}\big{)}}\|\mathbf{w}\|_{L^{\ell,\infty}}\zeta(s)ds$
$\displaystyle\leqslant
C\|\mathbf{w}\|_{L^{\ell,\infty}}t^{-\frac{N}{2d}\big{(}\frac{1}{\ell}-\frac{1}{r}\big{)}+\sigma+1}\int_{0}^{1}(1-s)^{-\frac{N}{2d}\big{(}\frac{1}{\ell}-\frac{1}{r}\big{)}}s^{\sigma}ds$
$\displaystyle\leqslant
C\|\mathbf{w}\|_{L^{\ell,\infty}}t^{-\frac{N}{2d}\big{(}\frac{1}{\ell}-\frac{1}{r}\big{)}+\sigma+1}\mathcal{B}\bigg{(}\sigma+1,1-\frac{N}{2d}\big{(}\frac{1}{\ell}-\frac{1}{r}\big{)}\bigg{)}$
$\displaystyle\leqslant C\|\mathbf{w}\|_{L^{\ell,\infty}}t^{-\mu}.$

Summarizing, if $\mathcal{Q}u=w+\mathcal{F}(u),\hskip
2.84544ptw=\mathbf{S}_{d,0}(t)u_{0}+\int_{0}^{t}\mathbf{S}_{d,0}(t-s)\zeta(s)\mathbf{w}$,
then

$\displaystyle\sup_{t>0}t^{\mu}\|\mathcal{Q}u(t)\|_{L^{r,\infty}}\leqslant
C\left(\|u_{0}\|_{L^{p_{c},\infty}}+K^{p}+\|\mathbf{w}\|_{L^{\ell,\infty}}\right)\leqslant
C\varepsilon_{0}.$

Upon taking $\varepsilon_{0}$ and $K>0$ sufficiently small, one can achieve
$C(\|u_{0}\|_{L^{p_{c}}}+\|\mathbf{w}\|_{L^{\ell,\infty}})\leqslant K$ and
thus $\displaystyle\sup_{t>0}t^{\mu}\|\mathcal{Q}(t)\|_{L^{r,\infty}}\leqslant
K$ so that $\mathcal{Q}$ maps $\overline{B}_{K}(0)$ into itself. Arguing as
above, we can show without ambiguity that $\mathcal{Q}$ is a contraction map
on $\overline{B}_{K}(0)$ for appropriately chosen $K$ (small). Applying the
Banach fixed point theorem, we thus obtain the existence of a solution $u$ to
(1.5) in $\mathbf{X}$ which is unique in $\overline{B}_{K}(0)$. Also,
$u(t)\rightarrow u_{0}$, $t\rightarrow 0^{+}$ in
$\mathcal{S}^{\prime}(\mathbb{R}^{N})$. To see this let
$\langle\cdot,\cdot\rangle$ denotes the duality bracket between
$\mathcal{S}^{\prime}(\mathbb{R}^{N})$ and $\mathcal{S}(\mathbb{R}^{N})$. It
suffices to show that $\displaystyle\lim_{t\rightarrow 0^{+}}|\langle
u(t)-u_{0},\varphi\rangle|=0$ for all $\varphi\in\mathcal{S}(\mathbb{R}^{N})$.
The justification of this fact seemingly follows the lines of the proof in
[28]. The details are therefore omitted.
For the remaining bit of the proof, Part ($\textbf{a}_{1})$ is proved as
follows. The solution constructed above via a fixed point argument can be
realized as the limit of the following sequence of approximations

$u_{1}={\mathbf{S}}_{d,0}u_{0},\quad
u_{j+1}=u_{1}+{\mathbf{A}}_{1}(t)u_{j}+W(t,x),\quad j=1,2,3,...$

where

${\mathbf{A}}_{1}(t)v=\int_{0}^{t}{\mathbf{S}}_{d,-\alpha}(t-s)|v(s)|^{p}\,ds,\quad
W(x,t)=\int_{0}^{t}{\mathbf{S}}_{d,0}(t-s){\mathbf{w}}\zeta(s)ds.$

Assume $u_{0},{\mathbf{w}}$ are radial functions. Since the kernel $E_{d}$ is
radial in the $x$-variable, we deduce that $u_{1}$ is radial in
$x\in\mathbb{R}^{N}$ as the convolution of two radial functions. For the same
reason, $W(x,t)$ is radial in $x$ for all $0<t<\infty$. Likewise, note that
${\mathbf{A}}_{1}(t)v$ is radial in the spatial variable provided $v$ is so.
Hence, an induction argument shows that each element of the sequence
$(u_{j})_{j\geqslant 1}$ is radial in $x\in\mathbb{R}^{N}$. On the other hand,
$u_{j}$ converges to $u$ in a weak-$\star$ sense in
$C_{b}([0,\infty);L^{p_{c},\infty}(\mathbb{R}^{N}))$ and up to a subsequence
which we still denote by $(u_{j})_{j}$; $u_{j}\rightarrow u$ almost everywhere
as $j\rightarrow\infty$ for all $t\in(0,\infty)$. The conclusion follows from
the fact that almost everywhere convergence preserves radial symmetry. The
second ($\textbf{a}_{2})$ and last part ($\textbf{a}_{3})$ are established by
essentially mimicking the previous argument bearing in mind that radial
monotonicity and positivity (under the condition $d\in(0,1]$, $E_{d}$ has a
positive kernel) are properties which are preserved under convolution. This
achieves the proof of Theorem 1.6.

###  3.5. Proof of Theorem 1.8

The strategy to prove Theorem 1.8 is exactly the same as that employed in [23,
Theorem 1.1, (ii)] and [28, Theorem 2.4]. Therefore we omit the details and
simply refer the interested reader to [1, 23, 28].

###  3.6. Proof of Theorem 1.10

Set $p_{F}=p_{F}(m)=\frac{N-2md+\alpha}{N-2dm-2d}$, $m\in\mathbb{R}$ and let
$p\in(1,p_{F})$. Assume $\textbf{w}\in C_{0}(\mathbb{R}^{N})\cap
L^{1}(\mathbb{R}^{N})$ such that
$\displaystyle\int_{\mathbb{R}^{N}}\textbf{w}(x)\,dx>0$ and suppose that the
function $\zeta$ is given according to (1.3). By way of contradiction, assume
that Problem (1.1) has a global weak solution in the sense of Definition 1.1.
Consider the cut-off functions $\psi_{k}\in C^{\infty}_{0}([0,\infty))$,
$k=1,2$ with $0\leqslant\psi_{k}\leqslant 1$ and

$\psi_{1}(s)=\begin{cases}1\hskip 3.41418pt\text{if}\hskip
3.41418pt1/2\leqslant s\leqslant 3/4\\\ 0\hskip 3.41418pt\text{if}\hskip
3.41418pts\in[0,1/4]\cup[4/5,\infty)\end{cases},\quad\psi_{2}(s)=\begin{cases}1\hskip
3.41418pt\text{if}\hskip 3.41418pts\in[0,1]\\\ 0\hskip
3.41418pt\text{if}\hskip 3.41418pts\geqslant 2.\end{cases}$

Next, pick $T>0$ large enough and let us introduce the function

$\psi_{T}(x,t)=\psi_{1}\bigg{(}\frac{t}{T}\bigg{)}^{\frac{p}{p-1}}\psi_{2}\bigg{(}\frac{|x|^{2d}}{T}\bigg{)}^{\frac{2dp}{p-1}}.$

Since $u$ is a global weak solution of (1.1), we have that

$\displaystyle-\int_{\mathbb{R}^{N}}\int_{0}^{T}u\partial_{t}\psi_{T}-\int_{\mathbb{R}^{N}}\psi_{T}(x,0)u_{0}dx$
$\displaystyle=\int_{\mathbb{R}^{N}}\int_{0}^{T}u(-\Delta)^{d}\psi_{T}+\int_{\mathbb{R}^{N}}\int_{0}^{T}|x|^{\alpha}|u|^{p}\psi_{T}$
$\displaystyle\hskip
99.58464pt\qquad{}+\int_{\mathbb{R}^{N}}\int_{0}^{T}\textbf{w}(x)\zeta(t)\psi_{T}.$

Remark that $\displaystyle\int_{\mathbb{R}^{N}}\psi_{T}(x,0)u_{0}(x)dx=0$ so
that the above expression implies

$\displaystyle\int_{\mathbb{R}^{N}}\int_{0}^{T}|x|^{\alpha}|u|^{p}\psi_{T}+\int_{\mathbb{R}^{N}}\int_{0}^{T}\textbf{w}(x)\zeta(t)\psi_{T}$
$\displaystyle\leqslant\int_{\mathbb{R}^{N}}\int_{0}^{T}|u(-\Delta)^{d}\psi_{T}|+$
$\displaystyle\hskip
65.44142pt\qquad{}\int_{\mathbb{R}^{N}\times[0,T]}|u||\partial_{t}\psi_{T}|.$
(3.8)

At this point, set
$I_{1}=\displaystyle\int_{\mathbb{R}^{N}}\int_{0}^{T}|u(-\Delta)^{d}\psi_{T}|$
and
$I_{2}=\displaystyle\int_{\mathbb{R}^{N}\times[0,T]}|u||\partial_{t}\psi_{T}|$.
We wish to find suitable bounds for both $I_{1}$ and $I_{2}$. Start by
observing that via an induction argument, one has

$|(-\Delta)^{d}\psi_{T}|\leqslant
CT^{-1}\psi_{2}\bigg{(}\frac{|x|^{2d}}{T}\bigg{)}^{\frac{2d}{p-1}}\psi_{1}\left(\frac{t}{T}\right)^{\frac{p}{p-1}}.$
(3.9)

Hence, by utilizing the $\varepsilon$-Young inequality, we arrive at

$I_{1}\leqslant\frac{1}{2}\int_{\mathbb{R}^{N}}\int_{0}^{T}|x|^{\alpha}|u|^{p}\psi_{T}dxdt+C\int_{\mathbb{R}^{N}}\int_{0}^{T}|(-\Delta)^{d}\psi_{T}|^{\frac{p}{p-1}}|x|^{-\frac{\alpha}{p-1}}\psi_{T}^{-\frac{1}{p-1}}dxdt$

and by invoking (3.9), the second expression (call it $I_{11}$) in the right
hand side of the above estimate can further be estimated as follows

$\displaystyle I_{11}$ $\displaystyle\leqslant
cT^{-\frac{p}{p-1}}\bigg{(}\int_{0}^{T}\psi_{1}\bigg{(}\frac{t}{T}\bigg{)}^{\frac{p^{2}}{(p-1)^{2}}-\frac{p}{(p-1)^{2}}}dt\bigg{)}\bigg{(}\int_{\\{|y|<2^{\frac{1}{2d}}\\}}T^{-\frac{\alpha}{2d(p-1)}+\frac{N}{2d}}|y|^{-\frac{\alpha}{p-1}}dy\bigg{)}$
$\displaystyle\leqslant
cT^{-\frac{p}{p-1}+1}T^{\frac{N}{2d}-\frac{\alpha}{2d(p-1)}}\bigg{(}\int_{0}^{1}\psi_{1}(\tau)^{\frac{p}{p-1}}d\tau\bigg{)}\int_{\\{|y|<2^{\frac{1}{2d}}\\}}|y|^{-\frac{\alpha}{p-1}}dy$
$\displaystyle\leqslant
cT^{-\frac{1}{p-1}-\frac{\alpha}{2d(p-1)}+\frac{N}{2d}}$

where we have made in the first and second lines the change of variable
$x=T^{1/2d}y$ and $T\tau=t$, respectively. This bound implies in particular
that

$\displaystyle I_{1}$
$\displaystyle\leqslant\frac{1}{2}\int_{\mathbb{R}^{n}\times[0,T]}|u|^{p}|x|^{\alpha}\psi_{T}dxdt+cT^{-\frac{1}{p-1}-\frac{\alpha}{2d(p-1)}+\frac{N}{2d}}.$
(3.10)

Similarly, one has

$\displaystyle I_{2}$
$\displaystyle\leqslant\frac{1}{2}\int_{\mathbb{R}^{N}\times[0,T]}|u|^{p}|x|^{\alpha}\psi_{T}+C\int_{\mathbb{R}^{N}\times[0,T]}\bigg{|}\partial_{t}\psi_{1}\bigg{(}\frac{t}{T}\bigg{)}^{\frac{p}{p-1}}\bigg{|}^{\frac{p}{p-1}}\psi_{1}(t/T)^{-\frac{p}{(p-1)^{2}}}\psi_{2}^{\frac{2dp}{p-1}}(|x|^{2d}/T)|x|^{-\frac{\alpha}{p-1}}$
$\displaystyle\leqslant\frac{1}{2}\int_{\mathbb{R}^{N}\times[0,T]}|u|^{p}|x|^{\alpha}\psi_{T}+C\bigg{(}\int_{0}^{T}T^{-\frac{p}{p-1}}|\psi_{1}^{\prime}(t/T)|^{\frac{p}{p-1}}dt\bigg{)}\int_{\mathbb{R}^{N}}|x|^{-\frac{\alpha}{p-1}}\psi_{2}(|x|^{2d}/T)^{\frac{2dp}{p-1}}dx$
$\displaystyle\leqslant\frac{1}{2}\int_{\mathbb{R}^{N}\times[0,T]}|u|^{p}|x|^{\alpha}\psi_{T}+CT^{\frac{-\alpha}{2d(p-1)}+\frac{N}{2d}-\frac{1}{p-1}}\bigg{(}\int_{0}^{1}[\psi^{\prime}_{1}(s)]^{\frac{p}{p-1}}ds\bigg{)}\int_{|y|<2^{\frac{1}{2d}}}|y|^{-\frac{\alpha}{p-1}}\psi_{2}(|y|^{2d})^{\frac{2dp}{p-1}}dy$

so that

$I_{2}\leqslant\frac{1}{2}\int_{\mathbb{R}^{N}\times[0,T]}|u|^{p}|x|^{\alpha}\psi_{T}dxdt+CT^{\frac{-\alpha}{2d(p-1)}+\frac{N}{2d}-\frac{1}{p-1}}.$
(3.11)

On the other hand, since $T$ is chosen large we bound the second term in the
left-hand side of (3.6) from below as follows

$\displaystyle\int_{\mathbb{R}^{N}\times[0,T]}\textbf{w}(x)\zeta(t)\psi_{T}(x,t)dxdt$
$\displaystyle\geqslant\int_{T/2}^{T}\int_{\mathbb{R}^{N}}\textbf{w}(x)\zeta(t)\psi_{T}(x,t)dxdt$
$\displaystyle\geqslant\bigg{(}\int_{T/2}^{T}t^{m}\psi_{1}\bigg{(}\frac{t}{T}\bigg{)}^{\frac{p}{p-1}}dt\bigg{)}\bigg{(}\int_{\mathbb{R}^{N}}\textbf{w}(x)\psi_{2}\bigg{(}\frac{|x|^{2d}}{T}\bigg{)}^{\frac{2dp}{p-1}}dx\bigg{)}$
$\displaystyle\geqslant
T^{m+1}\bigg{(}\int_{1/2}^{1}\psi_{1}(\tau)^{\frac{p}{p-1}}d\tau\bigg{)}\int_{\mathbb{R}^{N}}\textbf{w}(x)\psi_{2}\bigg{(}\frac{|x|^{2d}}{T}\bigg{)}^{\frac{2dp}{p-1}}dx.$

From (3.6) and by combining the above bound with (3.10) and (3.11), we obtain
that

$\int_{\mathbb{R}^{N}}\textbf{w}(x)\psi_{2}\bigg{(}\frac{|x|^{2d}}{T}\bigg{)}^{\frac{2dp}{p-1}}dx\leqslant
CT^{-\frac{\alpha}{2d(p-1)}+\frac{N}{2d}-\frac{p}{p-1}-m}.$

We deduce from the latter that

$\int_{\mathbb{R}^{N}}\textbf{w}(x)dx\leqslant 0$

which yields a contradiction in view of the assumption imposed on w along with
the condition $p<p_{F}$. In fact, the function
$\psi_{2}\bigg{(}\frac{|x|^{2d}}{T}\bigg{)}^{\frac{2dp}{p-1}}$ converges in a
pointwise sense to $1$ as $T$ approaches infinity so that by applying the
Dominated Convergence Theorem, there holds

$\lim_{T\rightarrow\infty}\int_{\mathbb{R}^{N}}\textbf{w}(x)\psi_{2}\bigg{(}\frac{|x|^{2d}}{T}\bigg{)}^{\frac{2dp}{p-1}}dx=\int_{\mathbb{R}^{N}}\textbf{w}(x)dx>0.$

As a consequence, for sufficiently large $T$,
$\displaystyle\int_{\mathbb{R}^{N}}\textbf{w}(x)\psi_{2}\bigg{(}\frac{|x|^{2d}}{T}\bigg{)}^{\frac{2dp}{p-1}}dx\geqslant\gamma\int_{\mathbb{R}^{N}}\textbf{w}(x)dx$
for some $\gamma<1$. This achieves the proof of the first part of Theorem
1.10. When $m>0$, one can easily adapt the preceding approach with a slightly
modified test function. More precisely, for $R>0$, by replacing $\psi_{T}$ by
the function
$\psi_{T,R}(x,t)=\psi_{1}\big{(}\frac{t}{T}\big{)}^{\frac{p}{p-1}}\psi_{2}\big{(}|x|^{2d}R^{-2d}\big{)}^{\frac{2dp}{p-1}}$,
we reach the same conclusion
$\displaystyle\int_{\mathbb{R}^{N}}\textbf{w}(x)dx\leqslant 0$ which leads to
a contradiction. Theorem 1.10 is now completely proved.

##  Acknowledgements

The author would like to express his deep thanks to Gael Diebou Yomgne for
interesting discussions. He also thanks Slim Tayachi for helpful remarks and
suggestions.

Funding. Funding information is not applicable / No funding was received.

Declarations. On behalf of all authors, the corresponding author states that
there is no conflict of interest. No data-sets were generated or analyzed
during the current study.

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