# Sliding Dumbbell Method to search for the CME in heavy ion collisions

Madan M. Aggarwal, Anjali Attri, Sonia Parmar, Anjali Sharma and Jagbir Singh
Department of Physics, Panjab University, Chandigarh, India 160014

###### Abstract

The deconfined state of matter created in the hot and dense medium in ultra-
relativistic heavy-ion collisions causes the back-to-back charge separation
along the direction of magnetic field produced due to the energetic spectator
protons, a phenomenon known as the Chiral Magnetic Effect (CME). A new
technique, the Sliding Dumbbell Method (SDM), is developed to search for the
back-to-back charge separation on an event-by-event basis. The SDM is applied
to AMPT generated Au+Au collision events with string melting configuration at
$\sqrt{s}_{\mathrm{NN}}$ = 200 GeV. The varying percentages i.e., from
$\sim$0.5% to $\sim$8%, of the CME like signals are externally injected in
events by flipping charges of pairs of the particles perpendicular to the
reaction plane. A significant enhancement of the CME sensitive 3-particle
$\gamma$ correlator is reported for the events sub sample exhibiting the
highest back-to-back charge separation for a given collision centrality. For
the first time, we report the linear dependence of $\sqrt{|\gamma|}$
correlator for the same sign charge pairs on the positive charge asymmetry
($\langle A{{}^{+}}\rangle$) across the dumbbell for the CME enriched sub
samples. The CME fraction in the observed $\Delta\gamma$ (difference between
opposite and same sign $\gamma$ correlator) is also presented for different
collision centralities with different percentages of externally injected CME
like signal.

QGP, chiral magnetic effect, charge separation, sliding dumbbell method, 2-
and 3-particle correlators

##  I Introduction

The parity is conserved in strong interactions at vanishing temperature and
isospin density Vaf84 . However, it has been suggested that the parity may be
violated locally in microscopic domains in quantum chromodynamics (QCD) at
finite temperature. In the vicinity of the Quark Gluon Plasma (QGP), formed in
the hot and dense medium in ultra-relativistic heavy-ion collision, metastable
domains of gluon fields with non-zero topological charge may form Lee73 ;
Lee74 ; Kha98 . Interactions of these domains with chiral quarks cause an
imbalance between left- and right-handed quarks. The chiral imbalance, in the
presence of strong magnetic field (generated by the energetic spectator
protons in non-central heavy-ion collisions), causes the charge separation
along the axis of magnetic field, a phenomenon known as the Chiral Magnetic
Effect (CME) Kha06 ; Kha07 ; Kha08 ; Fuk08 . Charged hadrons resulting from
the quarks hadronization can lead to an experimentally detectable charge
separation perpendicular to the reaction plane. The CME is currently under
intense study both theoretically and experimentally in relativistic heavy-ion
collisions at the Relativistic Heavy Ion Collider (RHIC) at BNL and at the
Large Hadron Collider (LHC) at CERN Bzd13 ; Kha16 ; Koc17 ; Sch19 ; Zha19 .

The CME like charge separation can be investigated via the first P-odd sine
term in a Fourier decomposition of the azimuthal distribution of charged
particles Vol04 ; Vol90 :

$\frac{dN^{a}}{d\phi}\propto
1+2v_{1,a}cos(\Delta\phi^{*})+2v_{2,a}cos(2\Delta\phi^{*})...+2a_{1,a}sin(\Delta\phi^{*})+...$
(1)

where $\Delta\phi^{*}$ = $\phi-\Psi_{RP}$, $\phi$ is the azimuthal angle of
the particle and $\Psi_{RP}$ is the reaction plane angle; $v_{1}$ and $v_{2}$
are coefficients referred to as directed and elliptic flows, respectively;
$a_{1}$ represents the parity violating effect, and $a$ indicates type of
particle. These coefficients depend on the transverse momenta and rapidities
of particles. The coefficient, $a_{1}$, vanishes when averaged over many
events, because if spontaneous parity violation occurs, sign of $a_{1}$ will
vary from event to event, leading to $\langle a_{1}\rangle$ = 0. However, one
can observe the effect via the CME sensitive $\gamma$ correlator defined as
Vol04 :

$\begin{split}\gamma&=\langle cos(\phi_{a}+\phi_{b}-2\Psi_{RP})\rangle\\\
&=\langle cos(\Delta\phi_{a}^{*})cos(\Delta\phi_{b}^{*})\rangle-\langle
sin(\Delta\phi_{a}^{*})sin(\Delta\phi_{b}^{*})\rangle\\\
&=v_{1,a}v_{1,b}-a_{1,a}a_{1,b}\end{split}$ (2)

where $\phi_{a}$ and $\phi_{b}$ represent the azimuthal angles of the
particles “a” and “b”, respectively. Here, average is taken over particles in
an event and over the events in a given sample. The $\gamma$ correlator
represents the difference between the in-plane, $\langle
cos(\Delta\phi_{a}^{*})cos(\Delta\phi_{b}^{*})\rangle$, and the out-of-plane,
$\langle sin(\Delta\phi_{a}^{*})sin(\Delta\phi_{b}^{*})\rangle$, correlations.
The out-of-plane term is sensitive to the CME like charge separation (i.e.,
$a_{1,a}a_{1,b}$) whereas the in-plane cosine term serves as a reference since
both are equally sensitive to backgrounds unrelated to the reaction plane
Bor02 . It is possible to measure, using the Eq.( 2), the parity violation
effects (i.e., $a_{1,a}a_{1,b}$) provided directed flow $v_{1}$ = 0. In order
to compute these terms, we also calculated the 2-particle $\delta$ correlator
as:

$\begin{split}\delta&=\langle cos(\phi_{a}-\phi_{b})\rangle\\\ &=\langle
cos(\Delta\phi_{a}^{*})cos(\Delta\phi_{b}^{*})\rangle+\langle
sin(\Delta\phi_{a}^{*})sin(\Delta\phi_{b}^{*})\rangle\end{split}$ (3)

In heavy-ion collisions, the reaction plane angle is not known, so one usually
estimates the second order event plane ($\Psi_{2}$) from particles’ azimuthal
angles. Therefore, the $\gamma$ correlator is determined using $\Psi_{2}$
instead of $\Psi_{RP}$ and is corrected by applying the event plane
resolution. However, Voloshin Vol04  proposed the following equation if the
event plane is determinated using only one particle,

$\langle
cos(\phi_{a}+\phi_{b}-2\phi_{c})\rangle=(v_{1,a}v_{1,b}-a_{1,a}a_{1,b})v_{2,c}$
(4)

here particle “c” is used to determine the event plane and $v_{2,c}$ is its
elliptic flow. The STAR collaboration Abe09 ; Abe10  at the RHIC demonstrated
that, within errors, it is approximately equal to the $\gamma$ correlator
defined in the Eq.( 3) as:

$\gamma=\langle cos(\phi_{a}+\phi_{b}-2\Psi_{RP})\rangle\approx\frac{\langle
cos(\phi_{a}+\phi_{b}-2\phi_{c})\rangle}{v_{2,c}}$ (5)

The $\gamma$ correlators are determined for same sign ($\gamma_{SS}$) and
opposite sign ($\gamma_{OS}$) charge pairs averaged over particles in an event
followed by averaging over events. The difference between opposite and same
sign charge pairs is also estimated as: $\Delta\gamma$ = $\gamma_{OS}$ \-
$\gamma_{SS}$. In case of the CME, one expects
$|\gamma_{OS}|\approx|\gamma_{SS}|$. A positive $\Delta\gamma$ is referred to
as the charge separation signal due to the CME.

The STAR Abe09 ; Abe10 ; Ada13 ; Ada14  and the ALICE Abe13  reported charge
separation signal via multi-particle correlations. The CMS Sir18  has put an
upper limit on the CME contribution to $\Delta\gamma$ of 7% in Pb+Pb
collisions at $\sqrt{s}_{\mathrm{NN}}$ = 2.76 TeV at 95% confidence level. The
ALICE Ach18  also reported an upper limit of 26-33% at 95% confidence level on
the contribution of the CME signal in Pb+Pb collisions at
$\sqrt{s}_{\mathrm{NN}}$ = 2.76 TeV using the Event Shape Engineering (ESE)
technique. Recently, in another measurement of charge separation, the ALICE
Ach20  has put an upper limit, on the fraction of the CME signal ($f_{CME}$),
of 15-18% (20-24%) at 95% (99.7%) confidence level for the 0-40% collision
centralities using the data driven background. Recently, the STAR
collaboration reported some indication of finite CME signal in Au + Au
collisions at $\sqrt{s}_{\mathrm{NN}}$ = 200 GeV using the participating and
spectator planes Spec . However, the STAR collaboration did not observe the
expected enhancement due to the CME signal in the Ru+Ru collisions over the
Zr+Zr collisions in non-central collisions at 200 GeV Iso .

##  II Sliding Dumbbell Method

In this letter, we present a new method, the Sliding Dumbbell Method (SDM),
designed to search minutely for the back-to-back charge separation on an
event-by-event basis in heavy-ion collisions. It is noted that the environment
in each heavy-ion collision may not be conducive to give rise to the Chiral
Magnetic Effect. In view of this, it is extremely crucial to search for the
CME signal in each and every event to isolate potential events exhibiting the
back-to-back charge separation rather than looking for the signal averaging
over events in a given collision centrality. In the present analysis, an
attempt is made to find the sub sample enriched with the CME like signal in a
given collision centrality employing the SDM. The SDM is tested by externally
injected CME like signal in each AMPT generated event which has no CME signal
initially. Also, we report, for the first time, the linear dependence of
$\sqrt{|\gamma|}$ correlator for the same sign charge pairs on the positive
charge asymmetry ($\langle A{{}^{+}}\rangle$) across the dumbbell for the CME
enriched sub samples.

We generated 16.3 M Au+Au events at $\sqrt{s}_{\mathrm{NN}}$ = 200 GeV using
AMPT Monte Carlo event generator with string melting. The reaction plane angle
($\Psi_{RP}$) was set to zero while generating AMPT events and these events
are divided into different collision centralities (i.e., 0-5% (most central
events), 5-10%, 10-20%, ……, 90-100% (peripheral events)) depending on the
total number of participating nucleons. In each event, experimental track cuts
Abe09 ; Abe10  were imposed i.e., tracks having $0.15<p_{t}<2.0$ GeV/c and
$-1<\eta<1$ with full azimuthal coverage were selected for the analysis. We
measure $\gamma$ correlator in the rapidity region symmetric with respect to
the mid-rapidity as $v_{1}$ = 0 for this region and we can measure the parity
violation effects (i.e., $a_{1,a}a_{1,b}$) using the Eq.( 4).

In order to inject the CME like signal, in each event, randomly selected two
negative particles in the azimuth $45^{\circ}$-$135^{\circ}$ were flipped to
two positive particles whereas in the azimuth $225^{\circ}$-$315^{\circ}$, two
positive particles were flipped to two negative particles. Thus, a back-to-
back charge separation is introduced perpendicular to the reaction plane (here
$\Psi_{RP}$=0) as expected in case of the CME. The sample thus formed is
termed as CME. It is to be noted that though the CME signal injected in each
event is fixed but the percentage of particles flipped varies depending on the
number of particles in an event. Therefore, the injected CME signal varies
from $\sim$8% (for 60-70% collision centrality) to $\sim$0.5%(for 10-20%
centrality). The sample prepared by injecting the CME like signal and the
default AMPT with no signal injection, are analysed in a similar way to test
feasibility of the SDM.

Figure 1: Pictorial depiction of transverse (azimuthal) plane with hits of
positive (+) and negative (-) particles in an event. The red coloured dumbbell
of size of $90^{\circ}$ with “a” and “b” sides is also displayed. The black
arrows indicate the anti clockwise sliding of the dumbbell and the dashed
dumbbell slid by $\delta\phi$ = $1^{\circ}$ is also shown.

Fig. 1 displays hits of positive and negative charge particles in an event
represented by + and -, respectively, on the transverse (azimuthal) plane
i.e., plane perpendicular to the beam direction. $N^{+}$ ($N^{-}$) represents
the number of positive (negative) charge particles in an event. The dumbbell
is also shown using a solid red color covering the azimuthal angles
$0^{\circ}$-$90^{\circ}$ on one side and $180^{\circ}$-$270^{\circ}$ on the
other side denoted as “a” and “b” sides, respectively. Below we define some
terms associated with the dumbbell:

  * •

The sum of the positive and the negative charge fractions across the dumbbell
is defined as:

$Db_{+-}=\frac{n_{+}^{a}}{(n_{+}^{a}+n_{-}^{a})}+\frac{n_{-}^{b}}{(n_{+}^{b}+n_{-}^{b})}$
(6)

where $n_{\pm}^{a}$ and $n_{\pm}^{b}$ are the number of positive (negative)
charge particles, in $90^{\circ}$ azimuth, on “a” and “b” sides of the
dumbbell, respectively. The $Db_{+-}$=2 corresponds to 100% back-to-back
charge separation whereas $Db_{+-}$=1 means no back-to-back charge separation.

  * •

The charge excess asymmetry across the dumbbell is defined as:

$Db_{+-}^{asy}=\frac{(N_{+}^{ex}-N_{-}^{ex})}{(N_{+}^{ex}+N_{-}^{ex})}$ (7)

where $N_{+}^{ex}$ ($=n_{+}^{a}-n_{-}^{a}$) is positive particles’ excess on
the “a” side of the dumbbell and $N_{-}^{ex}$ ($=n_{-}^{b}-n_{+}^{b}$) is
negative particles’ excess on the “b” side of the dumbbell. The
$Db_{+-}^{asy}$=0 corresponds to 100% back-to-back charge separation whearas
$Db_{+-}^{asy}$=1 means one sided either positive or negative particles’
excess.

  * •

The positive charge asymmetry across the dumbbell in an event is given as:

$A^{+}=\frac{|n_{+}^{a}-n_{+}^{b}|}{N^{+}}$ (8)

The A+ is related to the coefficient, $a_{1}$ Kha05 , describing the parity
violating effect in the Eq. (1). In the Sliding Dumbbell Method, a whole
transverse plane (Fig. 1) is scanned minutely to search for the maximum value
of the $Db_{+-}$ by sliding the dumbbell in steps of $\delta\phi=1^{\circ}$ as
shown in the figure by black arrows while calculating $Db_{+-}$ and
$Db_{+-}^{asy}$ for each sliding setup. In other words, we obtain 360 values
of $Db_{+-}$ and $Db_{+-}^{asy}$ using Eqs.(6) and (7), respectively, for
different azimuthal coverages on “a” and “b” sides (i.e.,
$0^{\circ}$-$90^{\circ}$, $180^{\circ}$-$270^{\circ}$;
$1^{\circ}$-$91^{\circ}$, $181^{\circ}$-$271^{\circ}$;………;
$359^{\circ}$-$89^{\circ}$, $179^{\circ}$-$269^{\circ}$). Out of these 360
values of $Db_{+-}$, we find maximum value of $Db_{+-}$ i.e., $Db_{+-}^{max}$,
imposing the condition that $|Db_{+-}^{asy}|<0.25$ to select the events
similar to the CME like events rather than one sided either positive or
negative particles’ excess across the dumbbell. The above procedure is
repeated for each event to get $Db_{+-}^{max}$. The back-to-back fractional
charge separation across the dumbbell, $f_{DbCS}$, can be written as:

$f_{DbCS}=Db_{+-}^{max}-1$ (9)

Hereinafter, $f_{DbCS}$ is referred to as back-to-back charge separation. For
each collision centrality, we obtain $f_{DbCS}$ distribution. In order to get
the CME like enriched sample, we sliced $f_{DbCS}$ distribution for each
centrality into 10 percentile bins (i.e., 0-10%, 10-20%, ……, 90-100%). The
0-10% class corresponds to the most potential CME like candidates with higher
back-to-back charge separation (i.e., higher $f_{DbCS}$ values) whereas
90-100% class corresponds to lower back-to-back charge separation (i.e., lower
$f_{DbCS}$ values). We study the CME sensitive $\gamma$ correlator for each
sliced $f_{DbCS}$ bin in each centrality to investigate its dependence on
charge separation across the dumbbell.

##  III Background Estimation

The total background contribution leading to the back-to-back charge
separation resulting the CME like correlations is estimated as below:

  * •

The charges of particles in each event shuffled randomly to destroy the charge
dependent correlations amongst charged particles but keeping $\theta$ and
$\phi$ of each particle unchanged in an event so that the flow does not get
affected. We determine the $\gamma$ correlator for such sample and denote it
as $\gamma_{ChS}$.

  * •

The shuffling of charges of particles in an event keeping the flow in, kills
not only the CME like correlations but also correlations amongst produced
particles in an event. In order to retrieve correlations amongst particles, we
restore the original charges in each event which were shuffled. It may be
mentioned that there is no correlations between the $f_{DbCS}$ due to charge
shuffled and $f_{DbCS}$ of real data. So, the correlated background, one gets
after restoring the charges, is from real events spreading over almost all
$f_{DbCS}$ of real data but corresponding to the $f_{DbCS}$ bin of charge
shuffled sample for a particular collision centrality. Again $\gamma$
determined for such sample is represented as $\gamma_{Corr}$.

The background contribution ($\gamma_{bkg}$) to $\gamma$ is estimated as:

$\gamma_{bkg}=\gamma_{ChS}+\gamma_{Corr}$ (10)

The Q-cumulant technique Bil11  is used for computing 2- and 3- particle
correlators. Also, the $v_{2}$ is estimated from 2- and 4- particle
Q-cumulants. The $v_{2}$ is taken as the average of $v_{2}${2} and $v_{2}${4}
to calculate the 3-particle $\gamma$ correlator Abe13 . The correlations for
the positive-positive and the negative-negative charge pairs are found to
agree with each other within the statistical uncertainties and are combined
into one set of points, denoted as the same sign charge pairs’ correlations.

##  IV Results and Discussion

The centrality dependences of the 2-particle $\delta$ correlator for different
charge combinations for AMPT and CME samples are shown in Fig.2 (Left). As the
correlation values are quite small for the default AMPT, so y-axes with
magnified scales are given on the right hand sides.

Figure 2: Centrality dependence of same and opposite sign charge pairs for
AMPT and CME events of Left: 2-particle $\delta$ correlator, Middle:
3-particle $\gamma$ correlator and Right: $\Delta\gamma$. For the default AMPT
sample y-axes are given on the right hand sides.

It is observed that the opposite sign (OS) charge pairs’ correlations are
stronger and negative whereas the same sign (SS) charge pairs’ correlations
are positive. For both same and opposite signs charge pairs’ correlations, the
magnitude of correlations increases with decrease in the centrality. The
centrality dependence of the 3-particle $\gamma$ correlator for OS and SS
charge pairs is displayed in Fig.2 (Middle) for default AMPT (with no signal
injection) and CME samples. Strong correlations are seen for the SS charge
pairs whereas weaker correlations are observed for the OS charge pairs. The
strength of correlations decreases with increasing collision centrality as the
percentage of the externally injected signal decreases with increasing
collision centrality. Further, it is worth noting that for the SS charge
pairs, $\delta$ is positive and $\gamma$ is negative whereas for the OS charge
pairs exactly reverse is observed i.e., $\delta$ is negative and $\gamma$ is
positive as expected in case of the CME. Also, magnitude of these $\delta$ and
$\gamma$ are approximately equal. Fig.2 (Right) exhibits $\Delta\gamma$
variation with collision centrality for different AMPT samples. For the AMPT
events with no signal injection, $\Delta\gamma$ is small whereas for CME
sample, $\Delta\gamma$ has maximum value for 60-70% collision centrality and
it decreases with increasing collision centrality i.e., decreasing externally
injected CME signal.

Figure 3: Comparison of $f_{DbCS}$ distributions for 50-60% (Left) and 30-40%
(Right) centralities as indicated in the figure for AMPT, charge shuffled
denoted as ChS and CME ($\sim$4% for 50-60% centrality and $\sim$1% for 30-40%
centrality) samples.

Now using the Sliding Dumbbell Method, we obtained the fractional dumbbell
charge separation ($f_{DbCS}$) across the dumbbell (Eq.9) for each event. The
$f_{DbCS}$ distributions for different AMPT samples i.e., default AMPT, charge
shuffled (denoted as ChS) and CME are compared in Fig.3 for 50-60% ($\sim$4%
CME signal) and 30-40% ($\sim$1% CME signal) collision centralities. The
$f_{DbCS}$ distributions shift towards higher $f_{DbCS}$ values for the AMPT
data set with higher percentage of externally injected CME like signal for a
given centrality. The $f_{DbCS}$ distributions move towards lower values of
$f_{DbCS}$ with increasing collision centrality. It is noticed that $f_{DbCS}$
distributions with externally injected CME like signal extend to higher
$f_{DbCS}$ values than those of default AMPT and charge shuffled samples.
Further, these $f_{DbCS}$ distributions are sliced into 10 percentile bins
with 0-10% bin containing top $f_{DbCS}$ values extending upto 1 and 90-100%
with lower $f_{DbCS}$ values approaching zero.

Figure 4: Left: The dependence of $\gamma$ correlator on the charge separation
($f_{DbCS}$) for OS (top Left) and SS (bottom Left) charge pairs for the
30-40% collision centrality for CME ($\sim$1% externally injected CME like
signal), default AMPT, charge shuffled particles denoted as ChS and respective
correlated backgrounds represented as $Corr_{CME}$ and $Corr_{AMPT}$. Right:
Similar plots for $\delta$.

Fig.4 (Left) for the 30-40% collision centrality displays the dependence of
3-particle $\gamma$ correlator on $f_{DbCS}$ for $\sim$1% injected CME signal,
default AMPT, charge shuffled (ChS) and correlated background (Corr) for
different charge combinations. The magnitude of $\gamma$ enhances many times
for both SS and OS charge pairs for the top $f_{DbCS}$ bins as compared to
that of the average value for a given collision centrality. It is seen that
the opposite sign charge pairs are weakly correlated having positive $\gamma$
whereas the same sign charge pairs are strongly correlated having negative
$\gamma$ for top $f_{DbCS}$ bins. Correlations for both SS and OS charge pairs
decrease with decreasing $f_{DbCS}$ values and become approximately zero for
lower $f_{DbCS}$ bins. For instance, $\gamma_{SS}$ is negative whereas
$\gamma_{OS}$ is positive for the top 0-40% $f_{DbCS}$ bins, indicating the
back-to-back charge separation in these events whereas for the rest 40-100%
$f_{DbCS}$ values, signs of OS correlation change, resulting the normal trend.
Similar trends are seen for other centralities for lesser (more) number of top
$f_{DbCS}$ bins for higher (lower) collision centralities. The $\delta$
correlator plots for various $f_{DbCS}$ bins for the same collision centrality
are displayed in Fig.4 (Right) for opposite and same signs charge pairs. Here,
again it is noticed that the CME data points for $\delta$ are enhanced many
times for the top $f_{DbCS}$ bins than that of overall value (Fig.2 (Left))
and enhancement in the correlation values are of the same order as in the case
of $\gamma$. Similar trends are observed for other collision centralities.

Figure 5: Decomposition of the correlators into in (red symbols)- and out
(blue symbols)-of-plane correlations for the opposite (top Right) and the same
(bottom Right) sign charge pairs.

We have also computed in- and out-of-planes correlations for SS and OS charge
pairs for different $f_{DbCS}$ bins to confirm whether or not the observed SS
correlations are out-of-plane as expected in case of the CME. Fig.5 displays
the dependence of in- and out-of-planes correlations on the charge separation
($f_{DbCS}$) for different charge combinations for the 30-40% collision
centrality. The SS correlations for the top 0-40% $f_{DbCS}$ are out-of-plane
whereas in-plane weak correlations are observed as expected for the CME like
correlations. The out-of-plane correlations are stronger than those of in-
plane correlations for OS charge pairs for the top 0-40% $f_{DbCS}$ which is
due to charge separation. These correlations decrease with decreasing
$f_{DbCS}$ values. Similar trends are also observed for other collision
centralities resulting increase/decrease in correlation magnitude with
decreasing/increasing collision centrality.

Figure 6: The dependence of $\sqrt{|\gamma_{SS}|}$ on positive charge
asymmetry ($\langle A^{+}\rangle$) across the dumbbell for the top 10% charge
separation, $f_{DbCS}$, from each centrality for the CME sample.

It is observed that for the CME sample $\sqrt{|\gamma_{SS}|}$ for the top 10%
$f_{DbCS}$ events from each collision centrality varies linearly with positive
charge asymmetry ($\langle A^{+}\rangle$) across the dumbbell, displayed in
Fig.6 as expected (Eq. 2) since ($\langle A^{+}\rangle$) is proportional to
the parity violating parameter $a_{1}$ (i.e., A+ = $\pi a_{1}/4$) Kha05 . The
data points of $\sqrt{|\gamma_{SS}|}$ versus $\langle A^{+}\rangle$ are fitted
with a straight line, $\sqrt{|\gamma_{SS}|}=p_{0}+p_{1}\langle A^{+}\rangle$.
For the CME, the fitted values $p_{0}$ and $p_{1}$ are -0.077 $\pm$ 0.00702
and 1.04 $\pm$ 0.0502, respectively, and the total $\chi^{2}$ value is 1.22.

Figure 7: Top Left : The dependence of $\Delta\gamma$ on $f_{DbCS}$for the
30-40% collision centrality for AMPT, CME ( $\sim$1% CME) and charge shuffled
(ChS) alongwith corresponding correlated backgrounds for the CME
($Corr_{CME}$) and AMPT ($Corr_{AMPT}$) as indicated in the figure. Inset
shows the magnified $\Delta\gamma$ for 50-100% $f_{DbCS}$. Top Right : Similar
to Top plot but for 50-60% ($\sim$ 4% CME) centrality. Bottom : The dependence
of $\Delta\gamma_{norm}$ on $f_{DbCS}$ for the 30-40% and 50-60% collision
centralities.

Fig.7 (Top Left) and Fig.7 (Top Right) present the $\Delta\gamma$ versus
charge separation ($f_{DbCS}$) plots for 30-40% ($\sim$1% CME) and 50-60%
($\sim$4% CME) collision centralities, respectively. The values of
$\Delta\gamma$ for correlated background (Corr) and charge shuffled (ChS) are
also displayed in figures. It is observed that charge shuffled and default
AMPT data points agree within statistical uncertainties for various $f_{DbCS}$
bins whereas the data points corresponding to the CME sample have larger
correlation values. It is also seen that the $\Delta\gamma$ for the CME have
larger values than those of combined charged shuffled and correlated
backgrounds for the top $f_{DbCS}$ bins. The variation of $\Delta\gamma$
normalized to the average value ($\Delta\gamma_{avg}$) for a given collision
centrality (i.e., $\Delta\gamma_{norm}=\Delta\gamma/\Delta\gamma_{avg}$)
versus charge separation $f_{DbCS}$ is displayed in Fig.7 (bottom) for the
50-60% and 30-40% collision centralities. It is seen that
$\Delta\gamma_{norm}$ is greater than 1 for the 50-60% centrality
(corresponding to $\sim$4% injected CME signal) for the top 40% $f_{DbCS}$
whereas it is greater than 1 for the top 30% $f_{DbCS}$ for the 30-40%
centrality( i.e., injected CME signal $\sim$1%). Further, it is seen that for
the top 10% $f_{DbCS}$ the $\Delta\gamma_{norm}$ is $\sim$8 and it decreases
rapidly with decreasing externally injected CME signal and
$\Delta\gamma_{norm}$ approaches zero for $f_{DbCS}$ 90-100%.

An attempt is made to get the fractional CME ($f_{CME}$) contribution in the
observed $\Delta\gamma$ for the top $f_{DbCS}$ bins wherein the
$\Delta\gamma_{norm}>1$ using the following equation:

$f_{CME}=\frac{(\Delta\gamma-\Delta\gamma_{bkg})}{\Delta\gamma}$ (11)

$f_{DbCS}$ | CME | AMPT
---|---|---
50-60% Centrality | 30-40% Centrality | 50-60% Centrality | 30-40% Centrality
($\sim$4% CME) | ($\sim$1% CME) | (No CME) | (No CME)
0-10% | 0.508$\pm$0.079 | 0.555$\pm$0.094 | 0.01$\pm$0.124 | 0.045$\pm$0.081
10-20% | 0.533$\pm$0.146 | 0.615$\pm$0.175 | 0.07$\pm$0.238 | -
20-30% | 0.354$\pm$0.155 | 0.061$\pm$0.304 | - | -
30-40% | 0.010$\pm$0.246 | - | - | -

Table 1: The $f_{CME}$ values for different $f_{DbCS}$ bins with
$\Delta\gamma_{norm}>1$ for 50-60% and 30-40% centralities for AMPT and CME
samples.

Table 1 lists the fraction of CME contributions for the top $f_{DbCS}$ bins
for 50-60% centrality ($\sim$4% injected CME signal) and 30-40% collision
centrality ($\sim$1% injected CME signal) wherein $\Delta\gamma_{norm}>1$ for
default AMPT and CME samples. It is observed that fraction is significant in
the top $f_{DbCS}$ bins and the number of $f_{DbCS}$ bins exhibiting such
trend decreases with the decreasing percentage of externally injected CME like
signal. However, extracted $f_{CME}$ is approximately zero within statistical
errors for the default AMPT with no externally injected CME like signal. In
view of these, it is worth noting that to confirm the existence of the CME in
heavy ion collisions one needs to look in to the top $f_{DbCS}$ bins having
the highest back-to-back charge separation for a given collision centrality.

In summary, we have analyzed AMPT Monte Carlo generated Au+Au events at
$\sqrt{s}_{\mathrm{NN}}=200$ GeV along with the AMPT sample obtained by
externally injected CME signal, using the sliding dumbbell method. For the
first time, it became possible to corner the events responsible for the CME
like signal corresponding the top 10% $f_{DbCS}$ bins wherein the CME
sensitive 3-particle $\gamma$ correlator enhances many times as it is evident
from the Fig.4 leading to enhanced value of $\Delta\gamma$ ($\sim$8 times) for
the top $f_{DbCS}$ bin (Fig.7).

It is found that $\sqrt{|\gamma_{SS}|}$ varies linearly with the positive
charge asymmetry across the dumbbell ($\langle A^{+}\rangle$). Therefore, for
the first time, it became possible to access the parity violating parameter,
$a_{1}$, through $A^{+}$. The fraction of CME extracted in the top $f_{DbCS}$
bin is significant ($>$0.5), however the extracted CME fraction is zero within
the statistical uncertainty for the default AMPT sample. The $f_{CME}$
decreases with increasing collision centrality (decreasing the percentage of
injecting CME like signal) for particular AMPT sample. In view of the above,
it is possible with the SDM to detect even a smallest CME signal ($\sim$0.5%)
generated due to flipping the charges of pair of particles in an event. The
SDM can be applied to Au+Au, isobaric (Zr+Zr, Ru+Ru) collisions at the RHIC
and Pb+Pb collisions at the LHC energies to search minutely for the CME
signal.

The financial assistance from the Department of Science $\&$ Technology and
University Grants Commission, Government of India, are gratefully
acknowledged.

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# Blow-up of classical solutions of quasilinear wave equations in one space
dimension

Yuki Haruyama 111 Master course, Mathematical Institute, Tohoku University,
Aoba, Sendai 980-8578, Japan. email<EMAIL_ADDRESS>and
Hiroyuki Takamura 222Mathematical Institute, Tohoku University, Aoba, Sendai
980-8578, Japan. e-mail<EMAIL_ADDRESS>
(  $\begin{array}[]{ll}\mbox{\footnotesize{\bf Keywords:}}&\mbox{\footnotesize
quasilinear wave equation, one dimension, blow-up, classical solution,
lifespan}\\\ \mbox{\footnotesize{\bf MSC2020:}}&\mbox{\footnotesize primary
35L72, secondary 35B44}\\\ \end{array}$ )

###### Abstract

This paper studies the upper bound of the lifespan of classical solutions of
the initial value problems for one dimensional wave equations with quasilinear
terms of space-, or time-derivatives of the unknown function. The results are
same as those of the semilinear case. But it is quite meaningful to consider
this kind of problems for the purpose to cover the optimality of the general
theory for nonlinear wave equations by many model equations as far as
possible.

##  1 Introduction

In this paper, we first consider the following initial value problems of a
scaler unknown function $v=v(x,t)$ of spatial variables;

$\left\\{\begin{array}[]{ll}v_{tt}-v_{xx}=A|v_{x}|^{p-2}v_{x}v_{xx}&\mbox{in}\
\in{\bf R}\times(0,T),\\\ v(x,0)={\varepsilon}f(x),\
v_{t}(x,0)={\varepsilon}g(x),&x\in{\bf R},\end{array}\right.$ (1.1)

where $p>1,A,T>0$. We assume that $f$ and $g$ are given smooth functions of
compact support and a parameter ${\varepsilon}>0$ is “small enough”. We are
interested in the lifespan $T({\varepsilon})$, the maximal existence time, of
classical solutions of (1.1).

Our motivation to investigate (1.1) comes from Sasaki, Takamatsu and Takamura
[13] in which the equation

$v_{tt}-v_{xx}=|v_{x}|^{p}$ (1.2)

is considered and its result is that the lifespan $T({\varepsilon})$ has to
satisfy

$C_{1}{\varepsilon}^{-(p-1)}\leq T({\varepsilon})\leq
C_{2}{\varepsilon}^{-(p-1)},$ (1.3)

where $C_{1}$ and $C_{2}$ are positive constants independent of
${\varepsilon}$. We know that (1.3) is also available for the equation

$v_{tt}-v_{xx}=|v_{t}|^{p}.$ (1.4)

Such a result is due to Zhou [15] for the upper bound and Kitamura, Morisawa
and Takamura [4] for the lower bound. We note that (1.4) is important to show
the optimality of the general theory for nonlinear wave equations by Li, Yu
and Zhou [6, 7]. Recently, the general theory is improved by “combined
effect”. See Takamatsu [14] for this direction as well as Morisawa, Sasaki and
Takamura [8, 9], Kido, Takamatsu, Sasaki and Takamura [3] for its optimality.
The advantage to investigate (1.2) beyond (1.4) is not only to cover the
optimality of the theory by model equations as many as possible, but also to
extend the analysis on the “blow-up boundary”. See Sasaki [12] for details.
The key fact is that a point-wise positiveness of the solution is available
for (1.4), but (1.2) is out of the case. In this way, the semilinear model
equations are well-studied.

On contrary, as far as the authors know, the quasilinear model equations in
one space dimension has been studied only for the equation

$v_{tt}-c^{2}(v_{x})v_{xx}=0$ (1.5)

by John [1], or John [2] on page 7, where a given smooth function $c$
satisfies

$c(v_{x})>0\quad\mbox{and}\quad c^{\prime}(v_{x})\neq 0.$ (1.6)

In this situation, the method of the characteristic curves gives us the blow-
up of the solution and the lifespan estimate of

$T({\varepsilon})\leq C{\varepsilon}^{-1}.$ (1.7)

We note that the setting $c(v_{x})=\sqrt{1+A|v_{x}|^{p-2}v_{x}}$ in our
problem (1.1) cannot be adopted to the assumption (1.6) except for $p=2$.
Related to this exceptional case, the equation of a different type,

$v_{tt}-\Delta v=2v_{t}v_{tt},$

in three space dimension is studied by John [2] on page 27, and it is easy to
reduce its result and proof to one dimensional case. We will see it at the
last section of this paper by studying generalized equations like the one in
(1.1),

$v_{tt}-v_{xx}=A|v_{t}|^{p-2}v_{t}v_{tt}.$ (1.8)

Our main purpose in this paper is to extend John’s result on (1.5) to (1.1),
and to obtain the same upper bound in (1.3). The optimality of such a result
is already guaranteed by the general theory when $p$ is any even integer. We
note that our method is different from [1, 2] after a reduction to a wave
equation with positive terms as we employ the argument in Sasaki, Takamatsu
and Takamura [13] which is based on the weighted functional method by Rammaha
[10, 11]. We also note that the analysis including the comparison argument on
the reduced ordinary differential inequality in [10, 11] is replaced with a
simple iteration argument. Such a hybrid method was first introduced by Lai
and Takamura [5] to avoid any restriction on the lower bound of the exponent
in power-type nonlinear terms in showing a blow-up result for semiliner damped
wave equations.

##  2 Theorems

All the results expected to be shown are listed in this section. In any case,
our assumption on the initial data is $(f,g)\in C_{0}^{2}({\bf R})\times
C_{0}^{1}({\bf R})$ satisfying

$\mbox{supp}\ (f,g)\subset\\{x\in{\bf R}:|x|\leq\sigma\\}\ (\sigma\geq 1).$
(2.1)

Our main result is the following which will be proved in the next section.

###### Theorem 2.1

Assume (2.1) and that $f$ and $g$ satisfy

$f(x),g(x)\geq 0,\ \mbox{and}\ f(x)\not\equiv 0,$ (2.2)

or

$f(x),g(x)\geq 0,\ \mbox{and}\ g(x)\not\equiv 0.$ (2.3)

Then, there exists a positive constant
${\varepsilon}_{0}={\varepsilon}_{0}(f,g,p,A,\sigma)>0$ such that any
classical solution of (1.1) in the time interval $[0,T]$ cannot exist as far
as $T$ satisfies

$T>C{\varepsilon}^{-(p-1)},$ (2.4)

where $0<{\varepsilon}\leq{\varepsilon}_{0}$, and $C$ is a positive constant
independent of ${\varepsilon}$.

As stated in Introduction, we have a similar result on the equation of time-
derivatives whose proof will be given at the last section.

###### Theorem 2.2

Even if the equation in (1.1) is replaced with (1.8), the conclusion of
Theorem 2.1 is still valid.

###### Remark 2.1

The estimate (2.4) implies that

$T({\varepsilon})\leq C{\varepsilon}^{-(p-1)}\quad\mbox{for}\
0<{\varepsilon}\leq{\varepsilon}_{0}.$

##  3 Proof of Theorem 2.1

First of all, we note that we may assume $A=p$ without loss of generality by
scaling of the unknown function. We shall employ the contradiction argument.
Let $v$ be a classical solution of (1.1) in the time interval $[0,T)$. It is
well-known that

$\mbox{supp}\ v(x,t)\subset\\{(x,t)\in{\bf R}\times[0,T):|x|\leq t+\sigma\\}.$
(3.1)

See Appendix in John [2] for example. Hence

$u(x,t):=\int_{-\infty}^{x}v(y,t)dy$ (3.2)

is well-defined in ${\bf R}\times[0,T)$ and

$\mbox{supp}\ u(x,t)\subset\\{(x,t)\in{\bf R}\times[0,T):x\geq-t-\sigma\\}.$
(3.3)

Then, it follows from the equation in (1.1) that

$\frac{\partial}{\partial x}(u_{tt}-u_{xx}-|u_{xx}|^{p})=0\quad\mbox{in}\ {\bf
R}\times(0,T).$

Integrating this identity in $x$ and making use of (3.3), we obtain that

$\left\\{\begin{array}[]{ll}u_{tt}-u_{xx}=|u_{xx}|^{p}&\mbox{in}\ {\bf
R}\times(0,T),\\\ \displaystyle u(x,0)={\varepsilon}\int_{-\sigma}^{x}f(y)dy,\
u_{t}(x,0)={\varepsilon}\int_{-\sigma}^{x}g(y)dy,&x\in{\bf
R}.\end{array}\right.$

The function $u$ is a classical solution of this problem, so that $u$ has to
satisfy an integral equation,

$u=u_{0}+L(|u_{xx}|^{p})\quad\mbox{in}\ {\bf R}\times[0,T),$ (3.4)

where we set

$\begin{array}[]{rl}u_{0}(x,t)&\displaystyle:=\frac{1}{2}\left\\{u(x+t,0)+u(x-t,0)+\int_{x-t}^{x+t}u_{t}(y,0)dy\right\\},\\\
L(w)(x,t)&\displaystyle:=\frac{1}{2}\int_{0}^{t}ds\int_{x-t+s}^{x+t-s}w(y,s)dy.\end{array}$
(3.5)

Now, following Rammaha [10, 11], we shall introduce a new function $H=H(t)$
defined by

$H(t):=\int_{0}^{t}(t-s)ds\int_{-s-\sigma}^{-s-\sigma_{0}}u(x,s)dx,$ (3.6)

where $\sigma_{0}$ is a constant satisfying $0<\sigma_{0}<\sigma$ which will
be fixed appropriately later. Then, it follows that

$H^{\prime}(t)=\int_{0}^{t}ds\int_{-s-\sigma}^{-s-\sigma_{0}}u(x,s)dx,\quad
H^{\prime\prime}(t)=\int_{-t-\sigma}^{-t-\sigma_{0}}u(x,s)dx.$

Hence (3.4) yields that

$\begin{array}[]{ll}H^{\prime\prime}(t)=&\displaystyle\frac{1}{2}\int_{-t-\sigma}^{-t-\sigma_{0}}\left\\{u(x+t,0)+u(x-t,0)+\int_{x-t}^{x+t}u_{t}(y,0)dy\right\\}dx\\\
&\displaystyle+\frac{1}{2}F(t),\end{array}$ (3.7)

where $F=F(t)$ is defined by

$F(t):=\int_{-t-\sigma}^{-t-\sigma_{0}}dx\int_{0}^{t}ds\int_{x-t+s}^{x+t-s}|u_{xx}(y,s)|^{p}dy.$
(3.8)

First, we shall prove the theorem under the assumption (2.2). Then, we have
that

$H^{\prime\prime}(t)\geq\frac{1}{2}\int_{-t-\sigma}^{-t-\sigma_{0}}u(x+t,0)dx=\frac{1}{2}\int_{-\sigma}^{-\sigma_{0}}u(x,0)dx=\frac{{\varepsilon}}{2}\int_{-\sigma}^{-\sigma_{0}}dx\int_{-\sigma}^{x}f(y)dy$

which implies that

$H^{\prime\prime}(t)\geq\frac{{\varepsilon}}{2}\int_{-(\sigma+\sigma_{0})/2}^{-\sigma_{0}}dx\int_{-\sigma}^{-(\sigma+\sigma_{0})/2}f(y)dy=C_{f}{\varepsilon}\quad\mbox{for}\
t\in[0,T),$

where $\sigma_{0}$ is fixed as

$C_{f}:=\frac{\sigma-\sigma_{0}}{4}\int_{-\sigma}^{-(\sigma+\sigma_{0})/2}f(y)dy>0.$
(3.9)

We note that we may assume $f\not\equiv 0$ in
$(-\sigma,-(\sigma+\sigma_{0})/2)$ without loss of generality by parallel
translation in $x$. Therefore it follows from $H(0)=H^{\prime}(0)=0$ that

$H(t)\geq Bt^{2}\quad\mbox{for}\ t\in[0,T),$ (3.10)

where $B:=C_{f}{\varepsilon}/2$. This will be the first step of the iteration
procedure.

Next, we shall derive an iteration frame from (3.7). Rewrite $F$ in (3.8) as

$F(t)=\int_{0}^{t}ds\int_{-t-\sigma}^{-t-\sigma_{0}}dx\int_{x-t+s}^{x+t-s}|u_{xx}(y,s)|^{p}dy$

and assume that

$t\geq\sigma_{1}:=\frac{\sigma-\sigma_{0}}{2}.$ (3.11)

Then, switching the order of $x$-$y$ integral, we have that

$\begin{array}[]{l}\displaystyle\int_{-t-\sigma}^{-t-\sigma_{0}}dx\int_{x-t+s}^{x+t-s}|u_{xx}(y,s)|^{p}dy\\\
\displaystyle=\left(\int_{-2t+s-\sigma}^{-2t+s-\sigma_{0}}dy\int_{-t-\sigma}^{y+t-s}dx+\int_{-2t+s-\sigma_{0}}^{-s-\sigma}dy\int_{-t-\sigma}^{-t-\sigma_{0}}dx\right)|u_{xx}(y,s)|^{p}\\\
\displaystyle\quad+\int_{-s-\sigma}^{-s-\sigma_{0}}dy\int_{y-t+s}^{-t-\sigma_{0}}|u_{xx}(y,s)|^{p}dx\\\
\displaystyle\geq\int_{-s-\sigma}^{-s-\sigma_{0}}(-s-\sigma_{0}-y)|u_{xx}(y,s)|^{p}dy\end{array}$

for $0\leq s\leq t-\sigma_{1}$. Similarly, we also have that

$\begin{array}[]{l}\displaystyle\int_{-t-\sigma}^{-t-\sigma_{0}}dx\int_{x-t+s}^{x+t-s}|u_{xx}(y,s)|^{p}dy\\\
\displaystyle=\int_{-2t+s-\sigma}^{-s-\sigma}dy\int_{-t-\sigma}^{y+t-s}|u_{xx}(y,s)|^{p}dx\\\
\displaystyle\quad+\left(\int_{-s-\sigma}^{-2t+s-\sigma_{0}}dy\int_{y-t+s}^{y+t-s}dx+\int_{-2t+s-\sigma_{0}}^{-s-\sigma_{0}}dy\int_{y-t+s}^{-t-\sigma_{0}}dx\right)|u_{xx}(y,s)|^{p}\\\
\displaystyle\geq\int_{-s-\sigma}^{-2t+s-\sigma_{0}}2(t-s)|u_{xx}(y,s)|^{p}dy+\int_{-2t+s-\sigma_{0}}^{-s-\sigma_{0}}(-s-\sigma_{0}-y)|u_{xx}(y,s)|^{p}dy\end{array}$

for $t-\sigma_{1}\leq s\leq t$. Summing up the estimates above and making use
of

$1=\frac{-s-\sigma_{0}-y}{-s-\sigma_{0}-y}\geq\frac{-s-\sigma_{0}-y}{\sigma-\sigma_{0}}\geq\frac{-s-\sigma_{0}-y}{2t}\quad\mbox{for}\
y\geq-s-\sigma$

in supp $u(y,s)$ by (3.3), we obtain that

$\begin{array}[]{ll}F(t)\geq&\displaystyle\int_{0}^{t-\sigma_{1}}ds\int_{-s-\sigma}^{-s-\sigma_{0}}(-s-\sigma_{0}-y)|u_{xx}(y,s)|^{p}dy\\\
&\displaystyle+\int_{t-\sigma_{1}}^{t}\frac{t-s}{t}ds\int_{-s-\sigma}^{-2t+s-\sigma_{0}}(-s-\sigma_{0}-y)|u_{xx}(y,s)|^{p}dy\\\
&\displaystyle+\int_{t-\sigma_{1}}^{t}ds\int_{-2t+s-\sigma_{0}}^{-s-\sigma_{0}}(-s-\sigma_{0}-y)|u_{xx}(y,s)|^{p}dy\end{array}$

which yields that

$F(t)\geq\frac{1}{t}\int_{0}^{t}(t-s)ds\int_{-s-\sigma}^{-s-\sigma_{0}}(-s-\sigma_{0}-y)|u_{xx}(y,s)|^{p}dy\quad\mbox{for}\
t\geq\sigma_{1}.$

Therefore, neglecting the first integral in (3.7), we obtain that

$H^{\prime\prime}(t)\geq\frac{1}{2t}\int_{0}^{t}(t-s)ds\int_{-s-\sigma}^{-s-\sigma_{0}}(-s-\sigma_{0}-y)|u_{xx}(y,s)|^{p}dy\quad\mbox{for}\
t\geq\sigma_{1}.$ (3.12)

Now we shall estimate the right-hand side of (3.12) by $H$. It follows from
the integration by parts and (3.3) that

$\begin{array}[]{ll}H(t)&\displaystyle=-\int_{0}^{t}(t-s)ds\int_{-s-\sigma}^{-s-\sigma_{0}}\partial_{y}(-s-\sigma_{0}-y)u(y,s)dy\\\
&\displaystyle=\int_{0}^{t}(t-s)ds\int_{-s-\sigma}^{-s-\sigma_{0}}(-s-\sigma_{0}-y)u_{x}(y,s)dy\\\
&\displaystyle=-\int_{0}^{t}(t-s)ds\int_{-s-\sigma}^{-s-\sigma_{0}}\partial_{y}\left\\{\frac{1}{2}(-s-\sigma_{0}-y)^{2}\right\\}u_{x}(y,s)dy\\\
&\displaystyle=\int_{0}^{t}(t-s)ds\int_{-s-\sigma}^{-s-\sigma_{0}}\frac{1}{2}(-s-\sigma_{0}-y)^{2}u_{xx}(y,s)dy,\end{array}$

so that the we have

$|H(t)|\leq\frac{1}{2}\left(\int_{0}^{t}(t-s)ds\int_{-s-\sigma}^{-s-\sigma_{0}}(-s-\sigma_{0}-y)|u_{xx}(y,s)|^{p}dy\right)^{1/p}I(t)^{1-1/p},$

where

$\begin{array}[]{rl}I(t):=&\displaystyle\int_{0}^{t}(t-s)ds\int_{-s-\sigma}^{-s-\sigma_{0}}(-s-\sigma_{0}-y)^{(2p-1)/(p-1)}dy\\\
=&\displaystyle\frac{p-1}{2(3p-2)}(\sigma-\sigma_{0})^{(3p-2)/(p-1)}t^{2}\\\
=&\displaystyle\frac{p-1}{3p-2}2^{(2p-1)/(p-1)}\sigma_{1}^{(3p-2)/(p-1)}t^{2}.\end{array}$

Therefore we obtain that

$H^{\prime\prime}(t)\geq Dt^{1-2p}H(t)^{p}\quad\mbox{for}\ t\geq\sigma_{1},$
(3.13)

where

$D:=2^{-p}\left(\frac{p-1}{3p-2}\right)^{1-p}\sigma_{1}^{-(3p-2)}>0.$

From now on, we employ a different method to investigate (3.10) and (3.13)
from Rammaha [10, 11], which is a combination of ordinary differential
inequality and an iteration argument firstly introduced in Lai and Takamura
[5]. Namely, assume that

$H(t)\geq C_{j}(t-\sigma_{1})^{a_{j}}t^{(1-2p)b_{j}}\quad\mbox{for}\
t\geq\sigma_{1},$ (3.14)

where $C_{j}>0$, $a_{j}\geq 0$, $b_{j}\geq 0$ $(j\in{\bf N})$. Our first step
(3.10) should be the case of $j=1$ as $C_{1}=B,\ a_{1}=2,\ b_{1}=0$. It
follows from (3.13) and $H^{\prime}(0)=H^{\prime\prime}(0)=0$ that

$H(t)\geq
D\int_{\sigma_{1}}^{t}ds\int_{\sigma_{1}}^{s}\tau^{1-2p}H(\tau)^{p}d\tau\quad\mbox{for}\
t\geq\sigma_{1}.$ (3.15)

This is the iteration frame. Substituting $H$ in the right-hand side of (3.15)
by (3.14), we have that

$\begin{array}[]{ll}H(t)&\displaystyle\geq
DC_{j}^{p}\int_{\sigma_{1}}^{t}ds\int_{\sigma_{1}}^{s}\tau^{(1-2p)(pb_{j}+1)}(\tau-\sigma_{1})^{pa_{j}}d\tau\\\
&\displaystyle\geq
DC_{j}^{p}t^{(1-2p)(pb_{j}+1)}\int_{\sigma_{1}}^{t}ds\int_{\sigma_{1}}^{s}(\tau-\sigma_{1})^{pa_{j}}d\tau\\\
&\displaystyle=\frac{DC_{j}^{p}}{(pa_{j}+1)(pa_{j}+2)}(t-\sigma_{1})^{pa_{j}+2}t^{(1-2p)(pb_{j}+1)}\\\
&\displaystyle\geq\frac{DC_{j}^{p}}{(pa_{j}+2)^{2}}(t-\sigma_{1})^{pa_{j}+2}t^{(1-2p)(pb_{j}+1)}.\end{array}$

Hence (3.14) is true for all $j\in{\bf N}$ if the sequence
$\\{C_{j}\\},\\{a_{j}\\},\\{b_{j}\\}\ (j\in{\bf N})$ satisfy

$\left\\{\begin{array}[]{ll}\displaystyle
C_{j+1}\geq\frac{DC_{j}^{p}}{(pa_{j}+2)^{2}},&C_{1}=B,\\\
a_{j+1}=pa_{j}+2,&a_{1}=2,\\\ b_{j+1}=pb_{j}+1,&b_{1}=0.\end{array}\right.$

Solving $a_{j}$ and $b_{j}$, we have

$a_{j}=\frac{2}{p-1}(p^{j}-1),\quad b_{j}=\frac{1}{p-1}(p^{j-1}-1)$

for $j\in{\bf N}$, so that $C_{j}$ satisfies

$C_{j+1}\geq\frac{DC_{j}^{p}}{a_{j+1}^{2}}\geq\frac{(p-1)^{2}DC_{j}^{p}}{4p^{2(j+1)}}$

which implies that

$\begin{array}[]{ll}\log C_{j+1}&\displaystyle\geq p\log C_{j}-\log
p^{2(j+1)}+\log\frac{D(p-1)^{2}}{4}\\\ &\displaystyle\geq p(p\log C_{j-1}-\log
p^{2j}+\log\frac{D(p-1)^{2}}{4}\\\ &\displaystyle\quad+p\log C_{j}-\log
p^{2(j+1)}+\log\frac{D(p-1)^{2}}{4}\\\ &\displaystyle\geq p^{j}\log
C_{1}-\sum_{i=1}^{j}p^{j-i}\left(\log
p^{2(i+1)}-\log\frac{D(p-1)^{2}}{4}\right).\end{array}$

Therefore $C_{j}$ should be defined by

$C_{j}:=\exp\\{p^{j-1}(\log B-E)\\},$

where

$E:=\sum_{i=1}^{\infty}\frac{1}{p^{i}}\left(\log
p^{2i}+\left|\log\frac{D(p-1)^{2}}{4}\right|\right)<\infty.$

We note that the convergence of the infinite sum can be verified by d’Alembert
criterion.

Turning back to (3.14), we have that

$H(t)\geq\frac{t^{(2p-1)/(p-1)}}{(t-\sigma_{1})^{2/(p-1)}}\exp\\{p^{j-1}J(t)\\}\quad\mbox{for}\
t\geq\sigma_{1},\ j\in{\bf N},$ (3.16)

where $J$ is defined by

$J(t):=\log(t-\sigma_{1})^{(2p)/(p-1)}-\log t^{(2p-1)/(p-1)}+\log B-E.$

By virtue of (3.16), we reach to a contradiction $H(t)=\infty$ to the
assumption that $u$ is a classical solution by taking limit as
$j\rightarrow\infty$ if $t\in[\sigma_{1},T)$ satisfies

$\frac{B(t-\sigma_{1})^{(2p)/(p-1)}}{t^{(2p-1)/(p-1)}\exp E}>1$ (3.17)

which yields $J(t)>0$.

Now, assume further that

$t\geq 2\sigma_{1}$

Then (3.17) follows from

$\frac{Bt^{1/(p-1)}}{2^{(2p)/(p-1)}\exp E}>1$

which is equivalent to

$t>2^{3p-1}\left(\frac{\exp E}{C_{f}}\right)^{p-1}{\varepsilon}^{-(p-1)}.$

Therefore, if ${\varepsilon}_{0}$ is defined to satisfy

$2^{3p-1}\left(\frac{\exp
E}{C_{f}}\right)^{p-1}{\varepsilon}_{0}^{-(p-1)}=2\sigma_{1},$

then the classical solution of $u$ as well as $v$ of (1.1) cannot exists as
far as $T$ satisfies

$T>2^{3p-1}\left(\frac{\exp E}{C_{f}}\right)^{p-1}{\varepsilon}^{-(p-1)},$
(3.18)

where ${\varepsilon}$ is any positive number of
$0<{\varepsilon}\leq{\varepsilon}_{0}$.

Finally, we shall prove the theorem under the assumption (2.3) in stead of
(2.2). In this case, viewing (3.7), we have that

$H^{\prime\prime}(t)\geq\frac{{\varepsilon}}{2}\int_{-t-\sigma}^{-t-\sigma_{0}}dx\int_{x-t}^{x+t}dy\int_{-\sigma}^{y}g(z)dz\quad\mbox{for}\
t\geq 0.$

We note that there is a freedom to fix $\sigma_{0}\in(0,\sigma)$ in this case
as we don’t assume (3.9). Here we may assume that

$\sigma>3\sigma_{0}\quad\mbox{and}\quad\int_{-\sigma}^{-(\sigma+\sigma_{0})/2}g(z)dz>0.$

Then it follows from

$x-t\leq-2t-\sigma_{0}$

that

$H^{\prime\prime}(t)\geq\frac{{\varepsilon}}{2}\int_{-t-(\sigma+\sigma_{0})/2}^{-t-\sigma_{0}}dx\int_{-(\sigma+\sigma_{0})/2}^{x+t}dy\int_{-\sigma}^{y}g(z)dz\quad\mbox{for}\
t\geq\frac{\sigma_{1}}{2},$

where $\sigma_{1}$ is the one in (3.11). Because

$-2t-\sigma_{0}\leq-\frac{\sigma+\sigma_{0}}{2}$

is equivalent to $t\geq\sigma_{1}/2$. We also note that
$\sigma_{0}<\sigma_{1}<\sigma$ holds. Hence we have that

$H^{\prime\prime}(t)\geq\frac{{\varepsilon}}{2}\int_{-t-(\sigma+\sigma_{0})/2}^{-t-\sigma_{0}}dx\int_{-(\sigma+\sigma_{0})/2}^{x+t}dy\int_{-\sigma}^{-(\sigma+\sigma_{0})/2}g(z)dz\quad\mbox{for}\
t\geq\frac{\sigma_{1}}{2}$

which implies

$H^{\prime\prime}(t)\geq C_{g}{\varepsilon}\quad\mbox{for}\
t\geq\frac{\sigma_{1}}{2},$

where

$C_{g}:=\frac{\sigma_{1}^{2}}{4}\int_{-\sigma}^{-(\sigma+\sigma_{0})/2}g(z)dz>0.$

Since $H^{\prime\prime}(t)\geq 0$ for $t\geq 0$ and $H(0)=H^{\prime}(0)=0$, we
have $H(\sigma_{1}/2),H^{\prime}(\sigma_{1}/2)\geq 0$ which yields that

$H(t)\geq\frac{C_{g}}{2}{\varepsilon}\left(t-\frac{\sigma_{1}}{2}\right)^{2}\quad\mbox{for}\
t\geq\frac{\sigma_{1}}{2}.$

Therefore we obtain the first step of the iteration,

$H(t)\geq\frac{C_{g}}{8}{\varepsilon}t^{2}\quad\mbox{for}\ t\geq\sigma_{1}$

instead of (3.10). The conclusion can be established by completely same way as
in the previous case in which $C_{f}$ is replaced with $C_{g}/8$. The proof is
now completed. $\Box$

##  4 Proof of Theorem 2.2

The basic strategy in this section is almost the same as John [1, 2]. We
employ the contradiction argument again and, similarly to the previous
section, we may assume $A=p$. The finite propagation speed of the wave (3.1)
is still valid, but in this case we define a new function $u$ by

$u(x,t):=\int_{0}^{t}v(x,s)ds.$ (4.1)

Then the equation, $v_{tt}-v_{xx}=p|v_{t}|^{p-2}v_{t}v_{tt}$, yields that

$\frac{\partial}{\partial
t}\left(u_{tt}-u_{xx}-|u_{tt}|^{p}\right)=0\quad\mbox{in}\ {\bf
R}\times(0,T).$

Since it follows from (3.1) in which $t$ is replaced with $s\in[0,t]$ that

$\mbox{supp}\ u(x,t)\subset\\{(x,t)\in{\bf R}\times[0,T):|x|\leq t+\sigma\\},$
(4.2)

we have, by integration in time, a new problem;

$\left\\{\begin{array}[]{ll}u_{tt}-u_{xx}=|u_{tt}|^{p}+{\varepsilon}g-{\varepsilon}^{p}g^{p}&\mbox{in}\
{\bf R}\times(0,T),\\\ u(x,0)=0,\ u_{t}(x,0)={\varepsilon}f(x),&x\in{\bf
R}\end{array}\right.$ (4.3)

because of

$u_{tt}(x,0)=v_{t}(x,0)={\varepsilon}g(x),\quad
u_{xx}(x,0)=\int_{0}^{0}v_{xx}(x,s)ds=0.$

Hence, similarly to (3.4), $u$ has to satisfy

$u=u_{0}+L(|u_{tt}|^{p}+{\varepsilon}g-{\varepsilon}^{p}g^{p})\quad\mbox{in}\
{\bf R}\times[0,T),$ (4.4)

where

$u_{0}(x,t)=\frac{{\varepsilon}}{2}\int_{x-t}^{x+t}f(y)dy\geq 0.$

In this section, we assume that

${\varepsilon}^{p-1}\|g\|_{L^{\infty}({\bf R})}^{p-1}\leq\frac{1}{2}$ (4.5)

which yields that

${\varepsilon}g-{\varepsilon}^{p}g^{p}={\varepsilon}g\\{1-{\varepsilon}^{p-1}g^{p-1}\\}\geq\frac{1}{2}{\varepsilon}g\quad\mbox{in}\
{\bf R}.$ (4.6)

First, we shall prove the theorem under the assumption (2.2). From now on, we
set

$t=x+\sigma\quad\mbox{for}\ x\geq\sigma.$ (4.7)

Then, we have that

$u_{0}(x,t)=\frac{{\varepsilon}}{2}\int_{-\sigma}^{\sigma}f(y)dy=:\widetilde{C_{f}}{\varepsilon}>0$

and

$\begin{array}[]{ll}L(|u_{tt}|^{p})(x,t)&\displaystyle=\frac{1}{2}\int_{0}^{t}ds\int_{x-t+s}^{x+t-s}|u_{tt}(y,s)|^{p}dy\\\
&\displaystyle\geq\frac{1}{2}\int_{\sigma}^{x}dy\int_{y-\sigma}^{y+\sigma}|u_{tt}(y,s)|^{p}ds.\end{array}$

We shall estimate the integrand of the right-hand side in the inequality
above. (4.2) yields that

$u(y,y+\sigma)=u(y,y-\sigma)+\int_{y-\sigma}^{y+\sigma}u_{t}(y,s)ds=\int_{y-\sigma}^{y+\sigma}u_{t}(y,s)ds,$

so that we have

$u(y,y+\sigma)=\int_{y-\sigma}^{y+\sigma}(y+\sigma-s)u_{tt}(y,s)ds\quad\mbox{for}\
y\geq\sigma$

by integration by parts. Hence it follows from Hölder’s inequality that

$\begin{array}[]{ll}|u(y,y+\sigma)|&\displaystyle\leq\int_{y-\sigma}^{y+\sigma}(y+\sigma-s)|u_{tt}(y,s)|ds\\\
&\displaystyle\leq\left(\int_{y-\sigma}^{y+\sigma}(y+\sigma-s)^{p/(p-1)}ds\right)^{1-1/p}\left(\int_{y-\sigma}^{y+\sigma}|u_{tt}(y,s)|^{p}ds\right)^{1/p}.\end{array}$

Therefore, setting

$U(x):=u(x,x+\sigma)\quad\mbox{for}\ x\geq\sigma,$

we obtain by (4.4) and (4.6) that

$U(x)\geq\widetilde{C_{f}}{\varepsilon}+\frac{1}{2F}\int_{\sigma}^{x}U(y)^{p}dy\quad\mbox{for}\
x\geq\sigma,$ (4.8)

where a constant $F$ is defined by

$F:=\left(\frac{p-1}{2p-1}(2\sigma)^{(2p-1)/(p-1)}\right)^{p-1}>0.$

If $W$ is a solution of

$W(x)=\widetilde{C_{f}}{\varepsilon}+\frac{1}{2F}\int_{\sigma}^{x}W(y)^{p}dy,$

Gronwall’s inequality yields that

$U(x)\geq W(x)\quad\mbox{for}\ x\geq\sigma.$ (4.9)

Since $W$ has an expricite form,

$W(x)=\left\\{\frac{2F(\widetilde{C_{f}})^{p-1}{\varepsilon}^{p-1}}{2F-(p-1)(\widetilde{C_{f}})^{p-1}{\varepsilon}^{p-1}(x-\sigma)}\right\\}^{1/(p-1)},$

our setting (4.7) says that $t$ has to satisfy

$(2\sigma\leq)\
t<\frac{2F(\widetilde{C_{f}})^{-(p-1)}}{p-1}{\varepsilon}^{-(p-1)}+2\sigma$

whenever $U$ exists. It means that, if ${\varepsilon}_{1}$ is defined to
satisfy

$\frac{2F(\widetilde{C_{f}})^{-(p-1)}}{p-1}{\varepsilon}_{1}^{-(p-1)}=\sigma$

in this case, the classical solution of $u$ as well as $v$ cannot exists as
far as $T$ satisfies

$T>\frac{6F(\widetilde{C_{f}})^{-(p-1)}}{p-1}{\varepsilon}^{-(p-1)}\quad\mbox{for}\
0<{\varepsilon}\leq{\varepsilon}_{1}.$

In view of (4.5), the theorem in this case is established by taking

${\varepsilon}_{0}=\min\\{{\varepsilon}_{1},2^{-1/(p-1)}\|g\|_{L^{\infty}({\bf
R})}^{-1}\\}.$

we note that $g\equiv 0$ is included in this case.

Finally, we shall prove the theorem under the assumption (2.3) instead of
(2.2). It follows from (4.6) that

$L({\varepsilon}g-{\varepsilon}^{p}g^{p})(x,t)\geq\frac{{\varepsilon}}{4}\int_{0}^{t}ds\int_{x-t+s}^{x+t-s}g(y)dy.$

Then, setting (4.7) again, we have that

$\begin{array}[]{ll}L({\varepsilon}g-{\varepsilon}^{p}g^{p})(x,t)&\displaystyle\geq\frac{{\varepsilon}}{4}\int_{-\sigma}^{x}g(y)dy\int_{0}^{y+\sigma}ds\\\
&\displaystyle\quad+\frac{{\varepsilon}}{4}\int_{x}^{x+2\sigma}g(y)dy\int_{0}^{-y+x+2\sigma}ds\\\
&\displaystyle\geq\frac{{\varepsilon}}{4}\int_{-\sigma}^{x}(y+\sigma)g(y)dy.\end{array}$

Hence, neglecting $u^{0}$ in (4.4), we obtain that

$U(x)\geq\widetilde{C_{g}}{\varepsilon}+\frac{1}{2F}\int_{\sigma}^{x}U(y)^{p}dy\quad\mbox{for}\
x\geq\sigma$

instead of (4.8), where

$\widetilde{C_{g}}:=\frac{1}{4}\int_{-\sigma}^{\sigma}(y+\sigma)g(y)dy>0.$

Therefore the same conclusion holds, in which $\widetilde{C_{f}}$ is replaced
with $\widetilde{C_{g}}$. The proof is now completed. $\Box$

## Acknowledgement

The second author is partially supported by the Grant-in-Aid for Scientific
Research (A) (No. 22H00097), Japan Society for the Promotion of Science.

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