# Lifespan estimates for semilinear wave equations with space dependent
damping and potential

Ning-An Lai  Institute of Nonlinear Analysis and Department of Mathematics
Lishui University
Lishui 323000, P. R. China<EMAIL_ADDRESS>,  Mengyun Liu∗ Department of
Mathematics
Zhejiang Sci-Tech University
Hangzhou 310018, P. R. China<EMAIL_ADDRESS>,  Ziheng Tu  School of
Data Science, Zhejiang University of Finance and Economics, 310018 Hangzhou,
P. R. China<EMAIL_ADDRESS>and  Chengbo Wang  School of Mathematical
Sciences
Zhejiang University
Hangzhou 310027, P. R. China<EMAIL_ADDRESS>
###### Abstract.

In this work, we investigate the influence of general damping and potential
terms on the blow-up and lifespan estimates for energy solutions to power-type
semilinear wave equations. The space-dependent damping and potential functions
are assumed to be critical or short range, spherically symmetric perturbation.
The blow up results and the upper bound of lifespan estimates are obtained by
the so-called test function method. The key ingredient is to construct special
positive solutions to the linear dual problem with the desired asymptotic
behavior, which is reduced, in turn, to constructing solutions to certain
elliptic “eigenvalue” problems.

###### Key words and phrases:

semilinear wave equations, damping, potential, lifespan estimate, test
function

###### 2010 Mathematics Subject Classification:

35L05, 35L71, 35B33, 35B44, 35B40, 35B30

* Corresponding author

##  1\. Introduction

The purpose of this paper is to investigate the influence of general damping
and potential terms on the blow-up and lifespan estimates for energy solutions
to power-type semilinear wave equations. The space-dependent damping and
potential functions are assumed to be critical or short range, spherically
symmetric perturbation.

More precisely, let $n\geq 2$, $p>1$, $D,V\in
C(\mathbb{R}^{n}\backslash\\{0\\})$, we consider the following Cauchy problem
of semilinear wave equations, with small data

(1.1) $\left\\{\begin{aligned} &u_{tt}-\Delta
u+D(x)u_{t}+V(x)u=|u|^{p},(t,x)\in(0,T)\times\mathbb{R}^{n},\\\
&u(x,0)=\varepsilon f(x),\quad u_{t}(x,0)=\varepsilon
g(x)\end{aligned}\right.$

Here $f,g\in C^{\infty}_{c}(\mathbb{R}^{n})$, and the small parameter
$\varepsilon>0$ measures the size of the data. As usual, to show blow up, we
assume both $f$ and $g$ are nontrivial, nonnegative and supported in
$B_{R}:=\\{x\in\mathbb{R}^{n}:r\leq R\\}$ for some $R>0$, where $|x|=r$. In
view of scaling, we see that $D$ and $V$ are critical or short range, if
$D=\mathcal{O}(|x|^{-1})$, $V=\mathcal{O}(|x|^{-2})$, near spatial infinity.

There have been many evidences that the critical power, $p_{c}$, for $p$ so
that the problem admits global solutions, seems to be related with two kinds
of the dimensional shift due to the critical damping and potential
coefficients near the spatial infinity. Here we call $p_{c}$ to be a critical
power, if there exists $\delta>0$ such that there are certain class of data
$(f,g)$ so that we have blow up for any $\varepsilon>0$ and
$p\in(p_{c}-\delta,p_{c})$, while there are for any
$p\in(p_{c},p_{c}+\delta)$, we have small data global existence for
$\varepsilon\in(0,\varepsilon_{0})$.

Heuristically, in the sample case $D=d_{\infty}/r$ and $V=0$, the perturbed
linear wave operator for radial solutions is of the following form

$u_{tt}-\Delta
u+D(x)u_{t}=(\partial_{t}^{2}-\partial_{r}^{2}-\frac{n+d_{\infty}-1}{r}\partial_{r})u+\frac{d_{\infty}}{r}(\partial_{t}+\partial_{r})u\
.$

In view of the dispersive nature of the solutions for wave equations,
$(\partial_{t}+\partial_{r})u$ tend to be negligible (good derivative) and
thus it behaves like $n+d_{\infty}$ dimensional wave equations, which suggests
the role of $n+d_{\infty}$. On the other hand, when we consider the elliptic
operator $-\Delta+V$ with $V=v_{\infty}(1+r)^{-2}$, the asymptotic behavior of
the radial solutions seems to be determined by the operator
$\partial_{r}^{2}+\frac{n-1}{r}\partial_{r}-v_{\infty}r^{-2}$, which is a
linear ODE operator of the Euler type and has eigenvalues

(1.2)
$\rho(v_{\infty}):=\sqrt{\big{(}\frac{n-2}{2}\big{)}^{2}+v_{\infty}}-\frac{n-2}{2},-(n-2)-\rho(v_{\infty}).$

This suggests the role of $\rho(v_{\infty})$. In conclusion, the heuristic
analysis strongly suggests that, under some reasonable assumptions on $D$ and
$V$, we have a critical power given by

(1.3) $p_{c}=\max(p_{S}(n+d_{\infty}),p_{G}(n+\rho(v_{\infty})))\ ,$

where, for $m\in\mathbb{R}$,

(1.4) $\ p_{G}(m)=\left\\{\begin{array}[]{ll}1+\frac{2}{m-1}&m>1,\\\
\infty&m\leq 1,\end{array}\right.d_{\infty}=\lim_{r\to\infty}rD(r),\
v_{\infty}=\lim_{r\to\infty}r^{2}V(r),\ $

and $p_{S}(m)$ is related to the Strauss exponent [35], which is defined to be

(1.5)
$p_{S}(m)=\left\\{\begin{array}[]{ll}\frac{m+1+\sqrt{m^{2}+10m-7}}{2(m-1)}&m>1\\\
\infty&m\leq 1\ .\end{array}\right.$

Here, $p_{G}$ is related to the Glassey exponent $p_{G}(n)$ for

$u_{tt}-\Delta u=|u_{t}|^{p}\ ,x\in\mathbb{R}^{n}\ ,$

or the Fujita exponent $p_{F}(n)=p_{G}(n+1)$ for heat or damped wave equations

$u_{tt}+u_{t}-\Delta u=|u|^{p},u_{t}-\Delta u=|u|^{p}\ ,$

see, e.g., [24, 20, 3].

Despite of some partial results, particularly on the blow up part, the problem
of determining the critical power (as well as giving the sharp lifespan
estimates) for the problem (1.1) is still largely open in general.

In this paper, we would like to show that, there exists a large class of the
damping and potential functions of critical/long range, this conjecture is
true, at least in the blow up part. At the same time, we are able to give
upper bounds for the lifespan, which are expected to be sharp for the range
$p\in(p_{c}-\delta,p_{c})$.

Before proceeding, we give the definition of energy solutions.

######  Definition 1.1.

We say that $u$ is an energy solution of (1.1) on $[0,T]$ if

$u\in C([0,T],H^{1}(\mathbb{R}^{n}))\cap
C^{1}([0,T],L^{2}(\mathbb{R}^{n}))\cap L^{p}([0,T]\times\mathbb{R}^{n})$

satisfies $\,\mathop{\\!\mathrm{supp}}u(t,\cdot)\subset B_{t+R}$ and

(1.6)
$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{n}}|u|^{p}\Psi(t,x)dxdt-\left.\int_{\mathbb{R}^{n}}(u_{t}(t,x)+D(x)u(t,x))\Psi(t,x)dx\right|_{t=0}^{T}$
$\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{R}^{n}}u_{t}(t,x)\Psi_{t}(t,x)dxdt+\int_{0}^{T}\int_{\mathbb{R}^{n}}\nabla
u(t,x)\cdot\nabla\Psi(t,x)dxdt$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{R}^{n}}D(x)u(t,x)\Psi_{t}(t,x)dxdt+\int_{0}^{T}\int_{\mathbb{R}^{n}}V(x)u(t,x)\Psi(t,x)dxdt$

for any $\Psi(t,x)\in(C^{0}_{t}H_{loc}^{1}\cap
C^{1}_{t}L_{loc}^{2})([0,T]\times\mathbb{R}^{n})$. When $n=2$, we additionally
suppose $\Psi,V\Psi,D\Psi_{t}\in
L^{1/(1-\delta_{0})}_{loc}([0,T]\times\mathbb{R}^{n})$, and $D(x)\Psi(0,x)\in
L^{1/(1-\delta_{0})}_{loc}$, for some $\delta_{0}>0$, which ensures the
integrals are well-defined. The supremum of all such time of existence, $T$,
is called to be the lifespan to the problem (1.1), denoted by
$T_{\varepsilon}$.

Before presenting our main results, let us first give a brief review of the
history, in a broader context.

(I) Scattering damping $D=\mathcal{O}((1+|x|)^{-\beta})$, $V=0$

When there are no damping and potential, this problem is related to the so-
called Strauss conjecture, for which the critical power is given by
$p_{S}(n)$, which is the positive root of the quadratic equation

(1.7) $\gamma(p,n):=2+(n+1)p-(n-1)p^{2}=0\ ,$

when $n>1$. See [16, 8, 40, 22, 5, 19] for global results and [16, 7, 33, 34,
39, 41] for blow up results (including the critical case $p=p_{S}(n)$).

When there is no potential term, this problem has been widely investigated
with the typical damping $D=\mu(1+|x|)^{-\beta}$

(1.8) $\left\\{\begin{aligned} &u_{tt}-\Delta
u+\mu(1+|x|)^{-\beta}u_{t}=|u|^{p},(t,x)\in[0,T)\times\mathbb{R}^{n},\\\
&u(x,0)=f(x),\quad u_{t}(x,0)=g(x).\end{aligned}\right.$

The asymptotic behavior of the solution to the corresponding linear damped
equations has been comprehensively studied, in view of the works of [12, 14,
27, 30, 31, 32, 36, 37], we have the following results

$\beta\in(-\infty,1)$ | effective |  | solution behaves like
---
that of heat equation
$\beta=1$ |  | scaling invariant
---
weak damping
| the asymptotic behavior
---
depends on $\mu$
$\beta\in(1,\infty)$ | scattering |  | solution behaves like that
---
of wave equation without damping

Turning to the nonlinear problem (1.8), the critical power depends on the
value of $\beta$ and $\mu$. For $\beta\in[0,1)$, Ikehata, Todorova and
Yordanov [15] showed that the critical power of (1.8) is the shifted Fujita
exponent $p_{c}(n)=p_{G}(n-\beta+1)=1+\frac{2}{n-\beta}$. Nishihara [28]
studied the same damping case but with absorbed semilinear term $|u|^{p-1}u$
and proved the diffusion phenomena. For the blow-up solution, Ikeda-Sobajima
[10] gave the sharp upper bound of lifespan for the effective case $\beta<1$
via the test function method, which was developed from Mitidieri-Pokhozhaev
[26]. Recently, Nishihara, Sobajima and Wakasugi [29] verified that the
critical power is still $1+\frac{2}{n-\beta}$ when $\beta<0$. For the critical
case $\beta=1$ with $\mu\geq n$, Li [21] obtained the blow-up result when
$p\leq p_{G}(n)$.

Turning to the scattering case $\beta>1$, as we have discussed, it is natural
to expect that the critical power is exactly the same as that of the Strauss
conjecture, i.e., $p_{c}=p_{S}(n)$, see also the introduction in page $2$ in
Ikehata-Todorova-Yordanov [13] and conjecture (iii) in page 4 in Nishihara-
Sobajima-Wakasugi [29]. This conjecture has been verified at least for the
blow-up part when $\beta>2$ and $n\geq 3$ in Lai-Tu [18], based on a key
observation that the test function $e^{-t}\phi_{1}(x)$ satisfies the dual of
the corresponding linear equation, where
$\phi_{1}(x)=\int_{\mathbb{S}^{n-1}}e^{x\cdot\omega}d\omega$ is the one from
Yordanov-Zhang [38]. On the other hand, when $n=3,4$, Metcalfe-Wang [25]
obtained the global existence when $p>p_{S}(n)$ and $\beta>1$ with
sufficiently small $|\mu|$.

Our first main result verifies the blow up part of the conjecture for the
scattering damping, which, together with [25], shows that the critical power
is $p_{S}(n)$, at least for small scattering damping function, when $n=3,4$.
Moreover, we improve the lifespan estimates in [18] for $p\leq n/(n-1)$.

######  Theorem 1.2.

Let $n\geq 2$. Consider the Cauchy problem (1.1) with $V(x)=0$. Suppose
$D(x)\in C(\mathbb{R}^{n})\cap C^{\delta}(B_{\delta})$ for some $\delta>0$ and
$0\leq D(x)\leq\mu(1+|x|)^{-\beta}$ with $\beta>1$ and $\mu\geq 0$. Then for
any $1<p\leq p_{S}(n)$, any energy solutions for nontrivial, nonnegative,
compactly supported data will blow up in finite time. In addition, there exist
positive constants $C,\varepsilon_{0}$ such that the lifespan
$T_{\varepsilon}$ satisfies

(1.9)
$T_{\varepsilon}\leq\left\\{\begin{array}[]{ll}C\varepsilon^{-\frac{2p(p-1)}{\gamma_{0}(p)}}&\mbox{for}\
1<p\leq\frac{n}{n-1},\\\
C\varepsilon^{-\frac{2p(p-1)}{\gamma(p,n)}}&\mbox{for}\ \frac{n}{n-1}\leq
p<p_{S}(n),\\\ \exp\left(C\varepsilon^{-p(p-1)}\right)&\mbox{for}\
p=p_{S}(n)\\\ \end{array}\right.$

for any $0<\varepsilon\leq\varepsilon_{0}$, where

(1.10) $\gamma_{0}(p)=-(n-1)p(p-1)+2n(p-1)+2.$

In addition, the results for $p<p_{S}(n)$ apply also for general (short range)
damping function ($D(x)\in L^{n}\cap L^{\infty}(\mathbb{R}^{n})$ without the
sign condition). Here and in what follows, $C$ denotes a positive constant
independent of $\varepsilon$ and may change from line to line.

(II) Critical damping $D=\mathcal{O}((1+|x|)^{-1})$ with short range potential

Concerning the potential term $V$, when it is of short range and $D=0$, as we
have discussed, it is expected that it will not affect the critical power
$p_{S}$. Actually, when $D=0$ and $0\leq V(x)\leq\frac{\mu}{1+|x|^{\beta}}$
with $\beta>2$ and $\mu\geq 0$, Yordanov-Zhang [38] proved blow up result for
$1<p<p_{S}(n)$ when $n\geq 3$.

Our second result addresses the problem with possibly critical damping
$D=\mathcal{O}((1+|x|)^{-1})$, together with short range potential $V$. Here,
the potential $V$ is said to be of short range, if we have $rV(r)\in
L^{1}([1,\infty),dr)$.

######  Theorem 1.3.

Let $n\geq 2$. Suppose that the coefficients $V(x),~{}D(x)$ satisfy:
1. $V(x),\ D(x)\ \in C(\mathbb{R}^{n})\cap C^{\delta}(B_{\delta})$ for some $\delta>0$;
2. $V(r)\geq 0$ ($V$ is nontrivial for $n=2$), $D(r)+V(r)\geq-1$;
3. $rV(r)\in L^{1}([1,\infty),dr)$;
4. $rD(r)=d_{\infty}+rD_{\infty}(r)$, $D_{\infty}(r)\in L^{1}([1,\infty),dr)$, for some $d_{\infty}\in\mathbb{R}$, $rD\in L^{\infty}$.
Then for any $1<p<\max(p_{G}(n),p_{S}(n+d_{\infty}))$, there are no any global
energy solutions $u$ for (1.1) with $f=0$, nontrivial, nonnegative and
compactly supported $g$. Moreover, there exist constants $C,\varepsilon_{0}>0$
such that $T_{\varepsilon}$ is bounded from above by

(1.11)
$\left\\{\begin{array}[]{ll}C\varepsilon^{-(p-1)}&1<p<\frac{n}{n-1},d_{\infty}>(n-1)(\frac{2}{p}-1),\\\
C\varepsilon^{-(p-1)}\left(\ln\varepsilon^{-1}\right)^{(p-1)\max(4-n,1)}&p=\frac{n}{n-1},d_{\infty}>(n-1)(\frac{2}{p}-1),\\\
C\left(\varepsilon^{-1}(\ln\varepsilon^{-1})^{\max(3-n,0)}\right)^{\frac{p-1}{(n+1)-(n-1)p}}&\frac{n}{n-1}<p<p_{G}(n),d_{\infty}>n-1-\frac{2}{p},\\\
C\varepsilon^{-\frac{2p(p-1)}{\gamma_{1}(p)}}\left(\ln\varepsilon^{-1}\right)^{-\frac{2(p-1)}{\gamma_{1}(p)}\max(3-n,0)}&1<p<\frac{n}{n-1},d_{\infty}\leq(n-1)(\frac{2}{p}-1),\\\
C\varepsilon^{-\frac{2p(p-1)}{\gamma(p,n+{d_{\infty}})}}(\ln\varepsilon^{-1})^{\frac{2(p-1)}{\gamma(p,n+{d_{\infty}})}}&p=\frac{n}{n-1},d_{\infty}\leq(n-1)(\frac{2}{p}-1)\\\
C\varepsilon^{-\frac{2p(p-1)}{\gamma(p,n+d_{\infty})}}&\frac{n}{n-1}<p<p_{S}(n+d_{\infty}),d_{\infty}\leq
n-1-\frac{2}{p}\end{array}\right.$

for $0<\varepsilon\leq\varepsilon_{0}$, where

(1.12) $\gamma_{1}(p)=-(n+d_{\infty}-1)p(p-1)+2n(p-1)+2.$

See Figure 1 for the region of $(d_{\infty},p)$ where we have blow up results.
Here $L_{j}$ ($j=1,2,\cdots,6$) denote the $j$-th lifespan, with $L_{S}=L_{6}$
and $L_{G}=L_{3}$.

Figure 1. Theorem 1.3: critical powers and lifespan estimates

As mentioned above, when $D=0$, Yordanov-Zhang [38] has shown blow-up result
for $1<p<p_{S}(n)$ and $n\geq 3$. The restriction for $n\geq 3$ comes from the
proof of the existence and asymptotic behavior of test functions. By using ODE
and elliptic theory, we could include the case $n=2$, in the case of the
radial nontrivial potential.

Moreover, when $d_{\infty}\leq 1-n$, we have $p_{S}(n+d_{\infty})=\infty$ and
so there are no global solutions for any $p\in(1,\infty)$. A specific example
could be $D=\frac{d_{\infty}}{d_{\infty}+r}$ and $V=V_{\infty}(1+r)^{-2}$.

######  Remark 1.4.

Through scaling, we see that the technical condition $1+D(r)+V(r)\geq 0$ could
be replaced by the slightly general condition
$\lambda_{0}^{2}+\lambda_{0}D(r)+V(r)\geq 0$ for some $\lambda_{0}>0$.

(III) Critical damping and potential $D=\mathcal{O}(|x|^{-1})$,
$V=\mathcal{O}(|x|^{-2})$ near spatial infinity

Finally, when both the damping and potential terms exhibit certain critical
nature, it turns out that we have the blow-up phenomenon under the shifted
Strauss exponent and a shifted Glassey exponent, shifted by $\rho(v_{\infty})$
or $d_{\infty}$.

In [4], Georgiev, Kubo and Wakasa showed that the critical power for the
radial solutions is the shifted Strauss exponent $p=p_{S}(3+2)$, for a special
case in $\mathbb{R}^{3}$, with damping and potential coefficients satisfying
the relation

$V(r)=-D^{\prime}(r)/2+D^{2}(r)/4,$

where $D(r)$ is a positive decreasing function in $C([0,\infty))\cap
C^{1}(0,\infty)$ satisfying $D(r)=2/r$ for $r\geq r_{0}>0$.

In the case of the problem with sample scale-invariant “critical” damping and
potential,

$D(x)=\frac{d_{\infty}}{|x|},V(x)=\frac{v_{\infty}}{|x|^{2}}\ ,$

with $0\leq d_{\infty}<n-1+2\rho(v_{\infty})$, $v_{\infty}>-(n-2)^{2}/4$,
thanks to the specific structure of the damping and potential terms, Dai, Kubo
and Sobajima [2] were able to construct explicit test functions by using
hypergeometric functions and obtain the upper bound of the lifespan for

$\frac{n+\rho(v_{\infty})}{n+\rho(v_{\infty})-1}<p\leq p_{c}\ .$

See also Ikeda-Sobajima [9] for the prior blow-up results for
$\frac{n}{n-1}<p\leq p_{S}(n+d_{\infty})$, when $v_{\infty}=0$, $n\geq 3$ and
$0\leq d_{\infty}<(n-1)^{2}/(n+1)$.

As we can see, the above results heavily depends on the specific structure of
the damping and potential terms. Our next theorem addresses on the problem
with a general class of damping and potential terms, which exhibit certain
critical nature.

######  Theorem 1.5.

Consider (1.1) with $n\geq 2$. Assume that the coefficients $V(x),~{}D(x)$
satisfy:

1. $V(r),D(r)\in C(\mathbb{R}_{+})$, $V(r)\geq 0$, $D+V\geq-1$;

2. For the part near spatial infinity, $r>1$,

$\displaystyle r^{2}V(r)=v_{\infty}+rV_{\infty}(r)$, $V_{\infty}(r)\in
L^{1}_{r>1}$, $v_{\infty}\geq 0$ (with $v_{\infty}>0$ for $n=2$);

$\displaystyle rD(r)=d_{\infty}+rD_{\infty}(r),\ D_{\infty}(r)\in
L^{1}_{r>1},\ rD_{\infty}\in L^{\infty}_{r>1}$ with $d_{\infty}\in\mathbb{R}$;

3. For $r\in(0,1]$, $D=\mathcal{O}(r^{\theta-2})$ for some $\theta\in[0,2]$, and

$\displaystyle r^{2}V(r)=v_{0}+rV_{0}(r),\ V_{0}(r)\in L^{1}_{loc}$ for $n\geq
3$ or $v_{0}>0$;

$\displaystyle r^{2}D(r)=d_{0}+rD_{0}(r),\ D_{0}(r)\in L_{loc}^{1}$ for $n\geq
3$ or $v_{0}+d_{0}>0$.
For $n=2$ and the endpoint case, instead of the assumption 3, we assume the
analytic conditions for some $\delta>0$:

3’. $\displaystyle r^{2}V(r)=\sum_{j\geq 1}b_{j}r^{j}$ for $r\in(0,\delta)$ if
$v_{0}=0$;

$\displaystyle r^{2}(V+D)=\sum_{j\geq 1}c_{j}r^{j}$ for $r\in(0,\delta)$ if
$v_{0}+d_{0}=0$.

Suppose that

$p_{0}:=\frac{n+\rho(v_{0})}{n+\rho(v_{0}+\min(0,d_{0}))+\theta-2}<p_{c}=\max(p_{S}(n+d_{\infty}),p_{G}(n+\rho(v_{\infty})),$

then for any $p_{0}<p<p_{c}$, any energy solutions $u$ for (1.1), with $f=0$,
nontrivial, nonnegative and compactly supported $g$, will blow up in finite
time. Moreover, there exist constants $C,\varepsilon_{0}>0$ such that
$T_{\varepsilon}$ has to satisfy

(1.13)
$T_{\varepsilon}\leq\left\\{\begin{array}[]{ll}C\varepsilon^{-(p-1)}&\mbox{if}\
p\in(p_{2},p_{3})\neq\emptyset\\\
C\varepsilon^{-(p-1)}\left(\ln\varepsilon^{-1}\right)^{p-1}&\mbox{if}\
p=p_{3}>p_{2}\\\
C\varepsilon^{-\frac{p-1}{n+\rho(v_{\infty})+1-(n+\rho(v_{\infty})-1)p}}&\mbox{if}\
p\in(\max(p_{2},p_{3}),p_{G}(n+\rho(v_{\infty})))\neq\emptyset\\\
C\varepsilon^{-\frac{2p(p-1)}{\gamma_{2}(p)}}&\mbox{if}\
p\in(p_{0},\min(p_{3},p_{5}))\neq\emptyset\\\
C\varepsilon^{-\frac{2p(p-1)}{\gamma(p,n+{d_{\infty}})}}(\ln\varepsilon^{-1})^{\frac{2(p-1)}{\gamma(p,n+{d_{\infty}})}}&\mbox{if}\
p=p_{3}\in(p_{0},p_{S}(n+{d_{\infty}}))\neq\emptyset\\\
C\varepsilon^{-\frac{2p(p-1)}{\gamma(p,n+{d_{\infty}})}}&\mbox{if}\
p\in(\max(p_{0},p_{3}),p_{S}(n+{d_{\infty}}))\neq\emptyset\\\
C\varepsilon^{-\frac{2(p-1)}{(n+d_{\infty}+1)-(n+d_{\infty}-1)p}}&\mbox{if}\
p\in(p_{4},p_{0}]\cap(p_{4},p_{G}(n+d_{\infty}))\neq\emptyset\end{array}\right.$

for $0<\varepsilon\leq\varepsilon_{0}$. Here $p_{i}$ are defined as follows:

$p_{0}=\max(p_{1},p_{2}),\
p_{1}:=\frac{n+\rho(v_{0})}{n+\rho(v_{0}+d_{0})+\theta-2},\
p_{2}:=\frac{n+\rho(v_{0})}{n+\rho(v_{0})+\theta-2}$
$p_{3}:=\frac{n+\rho(v_{\infty})}{n+\rho(v_{\infty})-1}\ ,\
p_{4}:=\frac{n+\rho(v_{0}+d_{0})}{n+\rho(v_{0}+d_{0})+\theta-2}.$
$p_{5}=\left\\{\begin{array}[]{ll}\frac{3n+d_{\infty}+2\rho(v_{\infty})-1+\sqrt{(3n+d_{\infty}+2\rho(v_{\infty})-1)^{2}-8(n+d_{\infty}-1)(n+\rho(v_{\infty})-1)}}{2(n+d_{\infty}-1)}&n+d_{\infty}>1\\\
\infty&n+d_{\infty}\leq 1\ ,\end{array}\right.$

and

(1.14)
$\gamma_{2}(p):=2(n+\rho(v_{\infty})-1)(p-p_{3})+\gamma(p,n+d_{\infty})\ ,$

where $\gamma$ is given in (1.7) and $p_{5}$ is the positive root of
$\gamma_{2}(p)=0$ when $n+d_{\infty}>1$.

######  Remark 1.6.

We observe that, in our statement, besides the expected upper bound of the
blow up range, we have certain lower bound, which depends on the local
behavior of $V(x)$ and $D(x)$ near the origin, as well as $v_{\infty}$ for
$p_{3}$. Heuristically, we expect that the lower bounds for the blow-up range
are just technical conditions. In other words, we conjecture that we still
have blow up results for any $p\in(1,p_{c})$.

######  Remark 1.7.

When $n=3$, $v_{0}=d_{0}=0$, $v_{\infty}=d_{\infty}=2$, and $\theta=2$, we
have $p_{1}=p_{2}=1$, $p_{3}=4/3$, and thus recover the blow-up result in [4]
for $1<p<p_{S}(5)$. When $d_{0}=0$, $d_{\infty}=A$, $v_{0}=v_{\infty}=B$ and
$\theta=1$, our result recovers and generalizes the subcritical results in
[2]. The blow-up and lifespan estimate for the “critical” power are lost in
our result, which seems to be much more delicate to obtain, in our general
setting for the damping and potential functions. One of the unexpected
features is that we could obtain blow up results for highly singular damping
term near the origin. For example, for any
$d_{0},v_{0},d_{\infty},v_{\infty}>0$ such that $p_{0}<p_{c}$, the problem
with

$D=\frac{d_{0}}{r^{2}}+\frac{d_{\infty}}{r},\
V=\frac{v_{0}}{r^{2}}+\frac{2(v_{\infty}-v_{0})\arctan r}{\pi r^{2}}\ ,$

could not admit global solutions in general, for $p\in(p_{0},p_{c})$, despite
the strong damping effect near the origin.

Concerning the comparison of the exponents $p_{i}$, $p_{S}$ and $p_{G}$, they
depend on the the damping, potential and dimension. For example, we observe
from (1.14) and (1.7) for $\gamma_{2}$ and $\gamma(p,n)$, whose positive roots
give $p_{5}$ and $p_{S}$, as well as the obvious relation
$p_{3}<p_{G}(n+\rho(v_{\infty}))$, that we always have

$\min(p_{3},p_{5})<p_{c}\ .$

As $n+\rho(d_{\infty})-1>0$, we have $\gamma_{2}(p)<\gamma(p,n+d_{\infty})$
for $p<p_{3}$, which shows that the current lifespan estimate for
$p_{0}<p<\min(p_{3},p_{5})$ is weaker than the standard one. Also, it is clear
that $p_{1}<p_{2}$ iff $d_{0}>0,\theta<2$, which is also equivalent to
$p_{4}<p_{2}$.

In relation with Remark 1.6, we would like to show blow up results for any
$p\in(1,p_{c})$. It is clear that when $\theta=2$ (which ensures that
$d_{0}=0$), we have $p_{0}=1$ and so is the blow up results for
$p\in(1,p_{c})$. Moreover, by examining the proof of Theorem 1.5, in
particular the estimates (4.8) and (4.3), we find that the technical condition
$p_{3}$ for the lifespan estimates could also be avoided if the damping term
vanishes $D=0$ near spatial infinity. That is, when $D(r)\in
C_{c}([0,\infty))$ (so that $d_{0}=d_{\infty}=0$ and $\theta=2$), we have

$F_{0}=\int_{T/2}^{T}\int_{B_{t+R}}T^{-2p^{\prime}}\phi_{0}dxdt{\lesssim}T^{-2p^{\prime}+\rho(v_{\infty})+n+1}\
,$

and so is the following

######  Corollary 1.8.

Under the same assumptions as in Theorem 1.5, with an additional assumption
that $D(|x|)\in C_{c}([0,\infty))$. Then we have blow up results for any
$1<p<p_{c}=\max(p_{G}(n+\rho(v_{\infty})),p_{S}(n))$. In addition, we have,
for some constants $C,\varepsilon_{0}>0$,

(1.15)
$T_{\varepsilon}\leq\left\\{\begin{array}[]{ll}L_{S}=C\varepsilon^{-\frac{2p(p-1)}{\gamma(p,n)}}&\mbox{if}\
p\in(1,p_{S}(n))\\\
L_{G}=C\varepsilon^{-\frac{p-1}{n+\rho(v_{\infty})+1-(n+\rho(v_{\infty})-1)p}}&\mbox{if}\
p\in(1,p_{G}(n+\rho(v_{\infty})))\end{array}\right.$

for any $0<\varepsilon\leq\varepsilon_{0}$.

See Figure 2 for the region of $(v_{\infty},p)$ where we have blow up results
for $n=2$. As we assume $v_{\infty}\geq 0$, we always have
$p_{S}(n)>p_{G}(n+\rho(v_{\infty}))$ and the second upper bound $L_{G}$ is
effective (i.e., $L_{G}\leq L_{S}$ for $\varepsilon\ll 1$ if and only if
$(2\rho(v_{\infty})+n-1)p\leq 2$, which is nonempty only if $n\leq 2$.

$2/p=2\rho(v_{\infty})+n-1$$T_{\varepsilon}\leq
L_{G}$$p$$v_{\infty}$$1/4$$\frac{2}{n-1}$$p_{G}(n)$$p_{S}(n)$$p_{S}(n)$$p=p_{G}(n+\rho(v_{\infty}))$$T_{\varepsilon}\leq
L_{S}$$T_{\varepsilon}\leq L_{S}$$(0,1)$ Figure 2. Corollary 1.8: lifespan
estimates for $n=2$

For the strategy of proof, we basically follow the test function method. The
key ingredient is to construct special positive standing wave solutions, of
the form $w=e^{-\lambda t}\phi_{\lambda}(x)$, to the linear dual problem,

$(\partial_{t}^{2}-\Delta-D\partial_{t}+V)w=0\ ,$

with the desired asymptotic behavior. In turn, it is reduced to constructing
solutions to certain elliptic “eigenvalue” problems:

(1.16) $(\lambda^{2}-\Delta+D\lambda+V)\phi_{\lambda}=0\ .$

Concerning the subcritical blow up results, $p<p_{c}$, it suffices to
construct solutions for (1.16) with $\lambda=0$ and some $\lambda_{0}>0$.
While for the critical case, we will also need to find solutions with uniform
estimates with respect to all $\lambda\in(0,\lambda_{0})$.

#### Outline

Our paper is organized as follows. In Section 2, we present the existence
results of special solutions for the elliptic “eigenvalue” problems (1.16),
with certain asymptotic behavior, by applying elliptic and ODE theory. These
solutions play a key role in constructing the test functions and the proof of
blow up results. The proof of the existence results are given in Section 3.
Equipped with the eigenfunctions, the test function method is implemented in
Section 4, to give the proof of Theorem 1.5. Then, in Section 5, we give the
proof of Theorem 1.3, when $n\geq 2$ and the potential is of short range. In
essence, with the help of Lemma 2.2 and 2.4, Theorem 1.3 could be viewed as a
corollary of Theorem 1.5 when $n\geq 3$. At last, in Section 6, we present the
required test function for the critical case, as well as the upper bound and
lower bound estimates. Equipped with the test function, a relatively routine
argument (see, e.g., [18] or [17]) will yield Theorem 1.2.

#### Notations.

We close this section by listing the notation. Let $\langle
x\rangle=\sqrt{1+|x|^{2}}$ for $x\in\mathbb{R}^{n}$. We will also use
$A{\lesssim}B$ to stand for $A\leq CB$ where the constant $C$ may change from
line to line.

##  2\. Solutions to elliptic equations

In this section, we present the existence results of special solutions for
some elliptic “eigenvalue” problems, with certain asymptotic behavior, which
would be used to construct test functions to derive the expected lifespan
estimates. We will consider two types of elliptic equations.

###  2.1. Eigenfunction for $V-\Delta$

At first, we consider the “zero-eigenfunction” for the positive elliptic
operator $V-\Delta$.

######  Lemma 2.1.

Suppose $0\leq V\in C(\mathbb{R}^{n}\backslash\\{0\\})$ and

$r^{2}V(r)=v_{\infty}+rV_{\infty}(r)=v_{0}+rV_{0}(r)\ ,V_{\infty}(r)\in
L^{1}_{r>1}\ ,V_{0}(r)\in L^{1}_{r<1}\ ,$

Then if $n\geq 3$ or $v_{0},v_{\infty}>0$, there exists a solution
$\phi_{0}\in H_{loc}^{1}(\mathbb{R}^{n})\cap W_{loc}^{2,1+}(\mathbb{R}^{n})$
of

(2.1) $\Delta\phi_{0}=V\phi_{0},\ x\in\mathbb{R}^{n},\ \ n\geq 2,\ $

satisfying

(2.2) $\begin{split}\phi_{0}\simeq\begin{cases}r^{\rho(v_{0})},\ r\leq 1,\\\
r^{\rho(v_{\infty})},\ r\geq 1\ .\end{cases}\end{split}$

In addition, for $n=2$ with $v_{0}=0$, if we assume $r^{2}V$ is analytic in
$(0,\delta)$ for some $\delta>0$, i.e.,

$r^{2}V(r)=\sum^{\infty}_{j=1}b_{j}r^{j},\ 0<r<\delta\ ,$

then the same result holds.

When $V$ is Hölder continuous near the origin, the regularity of the solutions
could be improved.

######  Lemma 2.2.

Suppose $0\leq V\in C(\mathbb{R}^{n})$, $V=V(r)=\mathcal{O}(\langle
r\rangle^{-\beta})$ for some $\beta>2$, and $V\in C^{\delta}(B_{\delta})$ for
some $\delta>0$. In addition, we assume $V$ is nontrivial when $n=2$. Then
there exists a $C^{2}$ solution of (2.1) satisfying

(2.3) $\begin{split}\phi_{0}\simeq\begin{cases}\ln(r+2),\ n=2,\\\ 1,\ \ n\geq
3.\ \end{cases}\end{split}$

Moreover, when $n=2$ and $V\equiv 0$, it is clear that $\phi_{0}=1$ is a
solution (2.1).

When $n\geq 3$, $V$ is locally Hölder continuous (not necessarily radial) and
$0\leq V\leq\frac{C}{1+|x|^{2+\delta}}$ with $C,\delta>0$, Lemma 2.2 was known
from Yordanov-Zhang [38].

###  2.2. “Eigenfunction” for $V+D-\Delta$ with negative “eigenvalue”

In this subsection, we consider the “eigenfunction” for $V+D-\Delta$ with
negative “eigenvalue”:

(2.4)
$\Delta\phi_{\lambda_{0}}=(\lambda_{0}^{2}+\lambda_{0}D+V)\phi_{\lambda_{0}},\
\lambda_{0}>0,\ \ x\in\mathbb{R}^{n}.\ $

######  Lemma 2.3.

Let $V=V(r),D=D(r)\in C(0,\infty)$. Suppose $V\geq 0$,
$D\geq-\lambda_{0}-\frac{V}{\lambda_{0}}$ for some $\lambda_{0}>0$, and

$D(r)=\frac{d_{0}}{r^{2}}+\frac{1}{r}D_{0}(r),\
V(r)=\frac{v_{0}}{r^{2}}+\frac{1}{r}V_{0}(r),\ r\leq 1,\ $

with $D_{0}(r)\in L^{1}_{r<1}$, $V_{0}(r)\in L^{1}_{r<1}$. In addition, we
assume for some $d_{\infty}\in\mathbb{R}$,

$V(r)\in L^{1}_{r>1},\ D(r)=\frac{d_{\infty}}{r}+D_{\infty}(r),\
D_{\infty}(r)\in L^{1}_{r>1}.\ $

Then if $n\geq 3$ or $v_{0}+\lambda_{0}d_{0}>0$, there exists a
$H_{loc}^{1}(\mathbb{R}^{n})\cap W_{loc}^{2,1+}(\mathbb{R}^{n})$ solution of
(2.4) satisfying

(2.5)
$\begin{split}\phi_{\lambda_{0}}\sim\begin{cases}r^{\rho(v_{0}+\lambda_{0}d_{0})},\
r\leq 1,\\\ r^{-\frac{n-1-d_{\infty}}{2}}e^{\lambda_{0}r},\ r\geq
1.\end{cases}\end{split}$

In addition, we have the same result for $n=2$ with
$v_{0}+\lambda_{0}d_{0}=0$, when $r^{2}(V+\lambda_{0}D)$ is analytic near $0$:

$r^{2}(V+\lambda_{0}D)=\sum^{\infty}_{j=1}c_{j}r^{j}\ ,0<r<\delta\ .$

######  Lemma 2.4.

Let $V=V(r),D=D(r)\in C(\mathbb{R}^{n})\cap C^{\delta}(B_{\delta})$ for some
$\delta>0$. Suppose $V\geq 0$, $D\geq-\lambda_{0}-\frac{V}{\lambda_{0}}$, and
also that for some $d_{\infty}\in\mathbb{R}$, $R>1$ and $r\geq R$, we have

$V(r)\in L^{1}([R,\infty))\ ,\ D(r)=\frac{d_{\infty}}{r}+D_{\infty}(r),\
D_{\infty}(r)\in L^{1}([R,\infty))\ .$

Then there exists a $C^{2}$ solution of (2.4) satisfying

(2.6) $\phi_{\lambda_{0}}\simeq\langle
r\rangle^{-\frac{n-1-d_{\infty}}{2}}e^{\lambda_{0}r}\ .$

###  2.3. Eigenvalue problem with parameters

To handle the critical problem, we will need to construct a class of
(unbounded) positive solutions for the eigenvalue problem, with certain
uniform estimates for small parameters.

######  Lemma 2.5.

Let $n\geq 2$, $\beta>1$, $\mu\geq 0$. Suppose $D(x)\in C(\mathbb{R}^{n})\cap
C^{\delta}(B_{\delta})$ for some $\delta>0$ and $0\leq
D(x)\leq\frac{\mu}{(1+|x|)^{\beta}}$. Then there exists $c_{1}\in(0,1)$ such
that for any $0<\lambda\leq 1$, there is a $C^{2}$ solution of

(2.7) $\Delta\phi_{\lambda}-\lambda
D(x)\phi_{\lambda}=\lambda^{2}\phi_{\lambda}$

satisfying

(2.8)
$c_{1}\langle\lambda|x|\rangle^{-\frac{n-1}{2}}e^{\lambda|x|}<\phi_{\lambda}(x)<c_{1}^{-1}\langle\lambda|x|\rangle^{-\frac{n-1}{2}}e^{\lambda|x|}\
.$

##  3\. Proof of Lemmas 2.1-2.5

In this section, we present the proof of Lemmas 2.1-2.5. by applying elliptic
and ODE theory.

###  3.1. Proof of Lemma 2.1

We will find a radial solution $\phi_{0}(x)=\phi_{0}(|x|)=\phi_{0}(r)$.

At first, for the region $0<r\leq 1$, we observe that the equation in $r$ is
of the Euler type:

$\Delta\phi_{0}=(\partial^{2}_{r}+\frac{n-1}{r}\partial_{r})\phi_{0}=V\phi_{0}\
,$

for which it is natural to introduce a new variable $t$ with $r=e^{-t}$. Then
$f(t)=\phi_{0}(e^{-t})$ satisfies

$\partial^{2}_{t}f-(n-2)\partial_{t}f=(v_{0}+e^{-t}V_{0}(e^{-t}))f\ ,t\geq 0\
.$

Let $Y(t)=(Y_{1}(t),Y_{2}(t))^{T}$ with $Y_{1}=f$, $Y_{2}=\partial_{t}f$, then
we have

$Y^{\prime}=(A+B(t))Y,A=\left(\begin{array}[]{cc}0&1\\\
v_{0}&n-2\end{array}\right),B(t)=\left(\begin{array}[]{cc}0&0\\\
e^{-t}V_{0}(e^{-t})&0\end{array}\right)\ .$

As $V_{0}(r)\in L^{1}_{r<1}$, that is, $B\in L^{1}[0,\infty)$, we could apply
the Levinson theorem (see, e.g., [1, Chapter 3, Theorem 8.1]) to the system.
Then there exists $t_{0}\in[0,\infty)$ so that we have two independent
solutions, which have the asymptotic form as $t\to\infty$

$Y_{+}(t)=\left(\begin{array}[]{c}1+o(1)\\\
\lambda_{1}+o(1)\end{array}\right)e^{\lambda_{1}t}\ ,\
Y_{-}(t)=\left(\begin{array}[]{c}1+o(1)\\\
\lambda_{2}+o(1)\end{array}\right)e^{\lambda_{2}t}\ ,$

where $\lambda_{1}=\sqrt{(\frac{n-2}{2})^{2}+v_{0}}+\frac{n-2}{2}$,
$\lambda_{2}=-\sqrt{(\frac{n-2}{2})^{2}+v_{0}}+\frac{n-2}{2}=-\rho(v_{0})$.

We choose $Y(t)=Y_{-}(t)$ so that

$\phi_{0}(r)=f(-\ln r)=(1+o(1))r^{\rho(v_{0})}$

as $r\to 0$. It is easy to check that $\phi_{0}\in H^{1}(B_{\delta})\cap
W^{2,1+}(B_{\delta})$, and

$\partial_{r}\phi_{0}=-r^{-1}\partial_{t}f|_{t=-\ln
r}=(\rho(v_{0})+o(1))r^{\rho(v_{0})-1}$

for $r\in(0,1]$. Based on the assumption $n\geq 3$ or $v_{0}>0$, we have
$\rho(v_{0})>0$ and so there exists $\delta_{0}>0$ such that

(3.1) $\partial_{r}\phi_{0}\simeq r^{\rho(v_{0})-1}\ ,r\in(0,\delta_{0})\ .$

In addition, for the case $n=2$ and $v_{0}=0$ when $r^{2}V$ is analytic in
$(0,\delta)$, by applying the Frobenius method, there is an analytic solution

$\phi_{0}=r^{\lambda_{1}}\sum^{+\infty}_{j=0}a_{j}r^{j}\sim 1,\ r<\delta\ ,$

with $a_{0}=1$, where $\lambda_{1}=\rho(v_{0})=0$ is the root of
$\lambda^{2}=0$. If $V\equiv 0$ near $0$, then $\phi_{0}=1$. Otherwise, there
exists $k>0$ such that $b_{j}=0$ for all $j<k$ and $b_{k}>0$, in which we have
$a_{j}=b_{j}/j^{2}$ for any $j\in[1,k]$. Thus once again we have, for some
$\delta_{0}>0$,

$\partial_{r}\phi_{0}\simeq(\frac{b_{k}}{k}+o(1))r^{k-1}\ ,r\in(0,\delta_{0})\
.$

Similarly, for $r>1$, let $r=e^{t}$, then $F(t)=\phi_{0}(e^{t})$ satisfies

$\partial^{2}_{t}F+(n-2)\partial_{t}F=(v_{\infty}+e^{t}V_{\infty}(e^{t}))F\
,t\geq 0\ .$

Applying the Levinson theorem again, we know that there exists $c_{1},c_{2}$
such that

$\phi_{0}(e^{t})=c_{1}(1+o(1))e^{\rho(v_{\infty})t}+c_{2}(1+o(1))e^{(-\rho(v_{\infty})-(n-2))t},$
$\partial_{t}\phi_{0}(e^{t})=c_{1}(\rho(v_{\infty})+o(1))e^{\rho(v_{\infty})t}+c_{2}(2-n-\rho(v_{\infty})+o(1))e^{(-\rho(v_{\infty})-(n-2))t}$

as $t\to\infty$. Notice that $\rho(v_{\infty})>0$, due to the assumption
$n\geq 3$ or $v_{\infty}>0$.

By the fundamental well-posed theory of linear ordinary differential equation,
we know that $\phi_{0}(r)\in C^{2}(0,\infty)$. We claim that
$\partial_{r}\phi_{0}(r)\geq 0$ for all $r>0$. Actually, we have seen from
(3.1) that $\phi_{0}>0$ and $\partial_{r}\phi_{0}\geq 0$ for
$r\in(0,\delta_{0})$. Suppose, by contradiction, there exists a
$r_{2}>\delta_{0}$ such that $\phi^{\prime}_{0}(r_{2})<0$ with
$\phi_{0}(r)\geq 0$ for any $r\in(0,r_{2}]$. Then there is a $r_{1}<r_{2}$
such that $\partial_{r}\phi_{0}(r_{1})=0$. Recall that

$\partial_{r}(r^{n-1}\partial_{r}\phi_{0})=r^{n-1}V\phi_{0}\ .$

By integrating it form $r_{1}$ to $r_{2}$, we get

$r^{n-1}_{2}\phi^{\prime}_{0}(r_{2})=\int^{r_{2}}_{r_{1}}\tau^{n-1}V\phi_{0}(\tau)d\tau\geq
0\ ,$

which is a contradiction. Hence we get $c_{1}>0$ and

$\phi_{0}\sim r^{\rho(v_{\infty})},\partial_{r}\phi_{0}\sim
r^{\rho(v_{\infty})-1}\ r\gg 1\ .$

###  3.2. Proof of Lemma 2.2

Suppose $V$ is Hölder continuous in $\overline{B_{\delta}}$ for some
$\delta>0$. Then we consider the Dirichlet problem

(3.2) $\begin{cases}\Delta\phi_{0}=V\phi_{0},x\in B_{\delta}\\\
\phi_{0}|_{\partial B_{\delta}}=1\end{cases}$

By Gilbarg-Trudinger [6, Theorem 6.14], there exists a unique
$C^{2}(\overline{B_{\delta}})$ solution, which must be radial. For $r>0$, with
$\phi_{0}(x)=\phi_{0}(r)$, (3.2) becomes an ordinary differential equation

(3.3)
$\begin{cases}\phi^{\prime\prime}_{0}+\frac{n-1}{r}\phi^{\prime}_{0}-V\phi_{0}=0\\\
\phi_{0}(\delta)=1,\phi^{\prime}_{0}(\delta)=C\end{cases}$

Then by the theory of ordinary differential equation, there is a unique
solution $\phi_{0}(r)\in C^{2}(0,\infty)$, which agrees with $\phi_{0}$ in
$B_{\delta}$. Thus we get a solution $\phi_{0}(x)\in C^{2}(\mathbb{R}^{n})$
and we could apply strong maximum principle to get $\partial_{r}\phi_{0}\geq
0$ for all $r>0$ and $\phi_{0}(0)>0$.

Recall that

(3.4) $\partial_{r}(r^{n-1}\partial_{r}\phi_{0})=r^{n-1}V\phi_{0}\ .$

By integrating it from $0$ to $r$, we get

(3.5) $r^{n-1}\partial_{r}\phi_{0}=\int^{r}_{0}\tau^{n-1}V\phi_{0}d\tau\
{\lesssim}\ \phi_{0}\int^{r}_{0}(1+\tau)^{n-1-\beta}d\tau\ .$

Thus, as $\beta>2$, if $n\geq 3$ and $r\geq 1$, we have

$\partial_{r}\phi_{0}\ {\lesssim}\
\phi_{0}r^{1-n}\int^{r}_{0}(1+\tau)^{n-1-\beta}d\tau{\lesssim}\phi_{0}r^{-1-\delta}\
,$

for some $\delta>0$. By Gronwall’s inequality, we obtain
$\phi_{0}{\lesssim}\phi_{0}(1)$, for all $r\geq 1$, which yields

$\phi_{0}\simeq 1\ .$

For $n=2$, since $V$ is nontrivial, then there exists some $R>0$ such that
$V\neq 0$ when $R\leq r\leq 2R$. By (3.4), we have

$r\partial_{r}\phi_{0}=R\partial_{r}\phi_{0}(R)+\int^{r}_{R}\tau
V\phi_{0}d\tau\ \geq\ \phi_{0}(R)R\int^{2R}_{R}V(\tau)d\tau\geq C\ ,\ r\geq
2R\ ,$

hence we get

$\phi_{0}(r)\geq\phi_{0}(R)+\int^{r}_{R}\frac{C}{\tau}d\tau\geq
C\ln\frac{r}{R}+\phi_{0}(R)\ {\gtrsim}\ \ln(2+r),\ r\geq 2R\ .$

On the other hand, by (3.5), we have

$r\partial_{r}\phi_{0}=\int^{r}_{0}\tau V\phi_{0}d\tau\ {\lesssim}\
\phi_{0}(r)\int^{\infty}_{0}(1+\tau)^{1-\beta}d\tau\ {\lesssim}\ \phi_{0}(r)\
,$

which gives us

(3.6) $\phi_{0}\ {\lesssim}\ 1+r\ ,\forall\ r\geq 0\ .$

By inserting (3.6) into (3.5), we have for any $\delta_{1}\in(0,\beta/2-1)$
and $r\geq 1$

(3.7)
$\partial_{r}\phi_{0}{\lesssim}\left\\{\begin{array}[]{ll}r^{-1},&\beta>3,\\\
r^{2-\beta+\delta_{1}},&\ 2<\beta\leq 3\ .\end{array}\right.$

Then it easy to obtain the desired upper bound $\ln(2+r)$ for $\beta>3$, while
for $\beta\leq 3$, we get $\phi_{0}(r){\lesssim}(1+r)^{3-\beta+\delta_{1}}$
for any $r\geq 0$ which is better than (3.6).

Thus, to obtain the expected upper bound, we do the iteration, by inserting
the improved upper bound
$\phi_{0}(r){\lesssim}(1+r)^{3-\beta+\delta_{1}}=(1+r)^{k}$ into (3.5) to get,
for $r>1$,

$\partial_{r}\phi_{0}{\lesssim}\begin{cases}r^{-1},\ k+2<\beta\\\
r^{k+1-\beta+\delta_{2}},\
k+2\geq\beta,\end{cases}\phi_{0}{\lesssim}\begin{cases}\ln(2+r),\ k+2<\beta\\\
(1+r)^{k+2-\beta+\delta_{2}},\ k+2\geq\beta\ ,\end{cases}$

for any $\delta_{2}\in(0,\beta/2-1)$. For $k+2\geq\beta$, we can get the
improved upper bound. By repeating (finitely) steps, we can finally obtain

$\phi_{0}(r){\lesssim}\ln(2+r)\ ,\forall\ r\geq 0\ .$

###  3.3. Proof of Lemma 2.3

To start with, we record a lemma from Liu-Wang [23] which we will use later.

######  Lemma 3.1 (Lemma 3.1 in [23]).

Let $\lambda\in(0,\lambda_{0}]$, $\delta_{0}\in(0,1)$, $\epsilon>0$,
$y_{0}>0$, $K\in(\delta_{0},\delta_{0}^{-1})$,

(3.8)
$\|K^{\prime}\|_{L^{1}([\epsilon\lambda_{0}^{-1},\infty))}\leq\delta_{0}^{-1},\|G\|_{L^{1}([\epsilon\lambda^{-1},\infty))}\leq\delta_{0}^{-1}\lambda,\forall\lambda\in(0,\lambda_{0}]\
.$

Considering

(3.9)
$\begin{cases}y^{\prime\prime}-\lambda^{2}K^{2}(r)y+G(r)y=0,r>\epsilon\lambda^{-1}\\\
y(\epsilon\lambda^{-1})=y_{0},y^{\prime}(\epsilon\lambda^{-1})=y_{1}\in(0,\delta_{0}^{-1}\lambda
y_{0})\end{cases}$

Then for any solution $y$ with $y,y^{\prime}>0$, we have the following uniform
estimates, independent of $\lambda\in(0,\lambda_{0}]$,

(3.10) $y\simeq y_{0}e^{\lambda\int^{r}_{\epsilon/\lambda}K(\tau)d\tau}\ ,\
r\geq\epsilon\lambda^{-1}\ .$

Assume in addition $1-\lambda^{-2}K^{-2}G\in(\delta_{0},\delta_{0}^{-1})$,
then the solution $y$ to (3.9) satisfies $y,y^{\prime}>0$ and we have

(3.11) $y^{\prime}\simeq
y_{1}+y_{0}\lambda(e^{\lambda\int^{r}_{\epsilon/\lambda}K(\tau)d\tau}-1)\ .$

###### Proof of Lemma 2.3.

For $r\leq\delta$, by the similar proof of Lemma 2.1, there is a solution
$\phi_{\lambda_{0}}$ satisfying

$\phi_{\lambda_{0}}\sim r^{\rho(v_{0}+\lambda_{0}d_{0})}\
,\phi_{\lambda_{0}}\in H^{1}(B_{\delta})\cap
W^{2,1+}(B_{\delta}),\partial_{r}\phi_{\lambda_{0}}\geq 0\ .$

When $r\geq\delta$, with $\phi_{\lambda_{0}}(x)=\phi_{\lambda_{0}}(r)$, we
need only to consider the following ordinary differential equation

(3.12)
$\phi^{\prime\prime}_{\lambda_{0}}+\frac{n-1}{r}\phi^{\prime}_{\lambda_{0}}=(\lambda_{0}^{2}+\lambda_{0}D+V)\phi_{\lambda_{0}}\
,\phi_{\lambda_{0}}(\delta)=C_{1}>0,\
\partial_{r}\phi_{\lambda_{0}}(\delta)=C_{2}\geq 0\ .$

Then, as in the proof of Lemma 2.1, there is a unique solution
$\phi_{\lambda_{0}}(r)\in C^{2}(0,\infty)$ and
$\partial_{r}\phi_{\lambda_{0}}(r)\geq 0$ for all $r>0$.

For $r\geq R$, we shall consider (3.12) with $\phi_{\lambda_{0}}(R)>0$ and
$\phi^{\prime}_{\lambda_{0}}(R)\geq 0$. Let
$y=r^{\frac{n-1}{2}}\phi_{\lambda_{0}}$, then the new function $y$ satisfies

(3.13)
$\begin{cases}y^{\prime\prime}-(\lambda_{0}^{2}+\frac{d_{\infty}}{r}\lambda_{0})y-(\frac{(n-1)(n-3)}{4r^{2}}+V+\lambda_{0}D_{\infty})y=0\\\
y(R)=R^{\frac{n-1}{2}}\phi_{\lambda_{0}}(R),y^{\prime}(R)\geq\frac{n-1}{2}R^{\frac{n-3}{2}}\phi_{\lambda_{0}}(R)>0\end{cases}$

Thus by Lemma 3.1, we have

$y\simeq
e^{\int^{r}_{R}(\sqrt{\lambda_{0}^{2}+\lambda_{0}\frac{d_{\infty}}{\tau}})d\tau}\simeq
r^{\frac{d_{\infty}}{2}}e^{\lambda_{0}r},r\geq R\ ,$

which yields

$\phi_{\lambda_{0}}\simeq r^{-\frac{n-1-d_{\infty}}{2}}e^{\lambda_{0}r},r\geq
R\ .$

∎

###  3.4. Proof of Lemma 2.4

Suppose $\lambda_{0}^{2}+V+\lambda_{0}D$ is Hölder continuous in
$\overline{B_{\delta}}$ for some $\delta>0$. Then we consider the Dirichlet
problem

(3.14)
$\begin{cases}\Delta\phi_{\lambda_{0}}=(\lambda_{0}^{2}+\lambda_{0}D+V)\phi_{\lambda_{0}},x\in
B_{\delta}\\\ \phi_{\lambda_{0}}|_{\partial B_{\delta}}=1.\end{cases}$

By Theorem 6.14 in Gilbarg-Trudinger [6], there exists a unique
$C^{2}(\overline{B_{\delta}})$ solution. Hence we could apply maximum
principle to get $0<\phi_{\lambda_{0}}\leq 1$ in $B_{\delta}$ and
$\partial_{r}\phi_{\lambda_{0}}\geq 0$ for all $0<r\leq\delta$. When
$r\geq\delta$, with $\phi_{\lambda_{0}}(x)=\phi_{\lambda_{0}}(r)$, (3.14)
becomes an ordinary differential equation

(3.15)
$\begin{cases}\phi^{\prime\prime}_{\lambda_{0}}+\frac{n-1}{r}\phi^{\prime}_{\lambda_{0}}-(\lambda_{0}^{2}+\lambda_{0}D+V)\phi_{\lambda_{0}}=0\\\
\phi_{\lambda_{0}}(\delta)=1,\phi^{\prime}_{\lambda_{0}}(\delta)=C_{2}\geq
0.\end{cases}$

Then $\phi_{\lambda_{0}}\in C^{2}(\mathbb{R}^{n})$ and we could apply strong
maximum principle again to get $\partial_{r}\phi_{\lambda_{0}}\geq 0$ for all
$r>0$.

Furthermore, for $r\geq R$, we consider (3.15) from $r=R$, then we have
$\phi_{\lambda_{0}}(R)>0,\phi^{\prime}_{\lambda_{0}}(R)\geq 0$. By the same
argument as in the proof of Lemma 2.3, we get

$\phi_{\lambda_{0}}\simeq r^{-\frac{n-1-d_{\infty}}{2}}e^{\lambda_{0}r},r\geq
R\ ,$

which completes the proof.

###  3.5. Proof of Lemma 2.5

We first show that (2.7) admits a $C^{2}(\mathbb{R}^{n})$ solution. For
$r\leq\lambda^{-1}$, we consider the Dirichlet problem within
$B_{\lambda^{-1}}$

(3.16) $\begin{cases}\Delta\phi_{\lambda}-\lambda
D(x)\phi_{\lambda}=\lambda^{2}\phi_{\lambda},x\in B_{\lambda^{-1}}\\\
\phi_{\lambda}|_{\partial B_{\lambda^{-1}}}=1\ .\end{cases}$

By Theorem 6.14 in Gilbarg-Trudinger [6], there exists a unique (and hence
radial) $C^{2}(B_{\lambda^{-1}})$ solution. For $r>0$, the equation is reduced
(2.7) to a second order ordinary differential equation

$\phi^{\prime\prime}_{\lambda}+\frac{n-1}{r}\phi^{\prime}_{\lambda}-\lambda
D(r)\phi_{\lambda}=\lambda^{2}\phi_{\lambda}\ ,$

which ensures that $\phi_{\lambda}\in C^{2}(\mathbb{R}^{n})$. To obtain (2.8),
we need to divide $\mathbb{R}^{n}$ into two parts: $B_{1/\lambda}$ and
$\mathbb{R}^{n}\backslash B_{1/\lambda}$.

(I) Inside the ball $B_{1/\lambda}$

Considering the Dirichlet problem (3.16) within $B_{1/\lambda}$, it is easy to
see that $0<\phi_{\lambda}\leq 1$ in $B_{1/\lambda}$. In fact, if there exists
$x_{0}\in B_{1/\lambda}$ such that $\phi_{\lambda}(x_{0})\leq 0$, then by
strong maximum principle, we get $\phi_{\lambda}$ is constant within
$B_{1/\lambda}$, which is a contradiction. By Hopf’s lemma, we have
$\partial_{r}\phi_{\lambda}>0$ for $0<r\leq 1/\lambda$.

To get the uniform lower bound of $\phi_{\lambda}$, we define rescaled
function $f_{\lambda}(x)=\phi_{\lambda}(x/\lambda)$, which satisfies

$\displaystyle\begin{cases}\Delta
f_{\lambda}-\frac{1}{\lambda}D(\frac{r}{\lambda})f_{\lambda}=f_{\lambda},x\in
B_{1}\\\ f_{\lambda}|_{\partial B_{1}}=1\ .\end{cases}$

Since $f_{\lambda}$ is radial increasing, we know that

$\liminf_{0<\lambda\leq 1}\inf_{x\in B_{1}}f_{\lambda}=\liminf_{0<\lambda\leq
1}f_{\lambda}(0):=C\geq 0\ .$

To complete the proof, we need only to prove $C>0$. By definition, there
exists a sequence $\lambda_{j}\to 0$ such that $f_{\lambda_{j}}(0)\to C$ as
$j\to\infty$.

(i) Derivative estimates of $f_{\lambda}$.

As $f_{\lambda}$ is radial, we have

$\Delta f_{\lambda}=r^{1-n}\partial_{r}(r^{n-1}\partial_{r}f_{\lambda})\ ,$

and so

(3.17)
$\partial_{r}(r^{n-1}\partial_{r}f_{\lambda})=\Big{(}\frac{1}{\lambda}D(\frac{r}{\lambda})+1\Big{)}r^{n-1}f_{\lambda}.$

Recall that $f_{\lambda}\in(0,1]$ and $D(r)\leq\frac{\mu}{(1+|x|)^{\beta}}$,
by integrating it from $0$ to $r$, we get

$\displaystyle r^{n-1}\partial_{r}f_{\lambda}=$
$\displaystyle\int_{0}^{r}\Big{(}\frac{1}{\lambda}D(\frac{\tau}{\lambda})+1\Big{)}\tau^{n-1}f_{\lambda}d\tau$
$\displaystyle\leq$
$\displaystyle\int_{0}^{r}\Big{(}\frac{\mu\lambda^{\beta-1}}{(\lambda+\tau)^{\beta}}+1\Big{)}\tau^{n-1}f_{\lambda}d\tau$
$\displaystyle\leq$
$\displaystyle\int_{0}^{r}\Big{(}\frac{\mu}{\tau}+1\Big{)}\tau^{n-1}d\tau$
$\displaystyle\leq$ $\displaystyle\frac{\mu}{n-1}r^{n-1}+\frac{r^{n}}{n}\ ,$

that is,

(3.18) $\partial_{r}f_{\lambda}\leq\frac{\mu}{n-1}+\frac{r}{n}\ .$

(ii) Convergence of $f_{\lambda}$.

Since $\|f_{\lambda}\|_{H^{1}(B_{1})}$ are uniformly bounded, there exists a
subsequence of $\lambda_{j}$ (for simplicity we still denote the subsequence
as $\lambda_{j}$) such that $f_{\lambda_{j}}$ converges weakly to some $f$ in
$H^{1}(B_{1})$ as $j$ goes to infinity. Moreover, by the Arzela-Ascoli
theorem, $f_{\lambda_{j}}$ converges uniformly to $f$ in $C(\bar{B}_{1})$ as
$j$ goes to infinity, thus $f=1$ on $\partial B_{1}$ and $f(0)=C$.

In view of the equations satisfied by $f_{\lambda_{j}}$, we see that, for any
$\phi\in C^{\infty}_{c}(B_{1})$, we have

(3.19) $\int_{B_{1}}\nabla
f_{\lambda_{j}}\nabla\phi+f_{\lambda_{j}}\phi+\frac{1}{\lambda_{j}}D\big{(}\frac{r}{\lambda_{j}}\big{)}f_{\lambda_{j}}\phi
dx=0\ .$

Let
$g_{\lambda_{j}}=\frac{1}{\lambda_{j}}D(\frac{r}{\lambda_{j}})f_{\lambda_{j}}\phi$,
then it is easy to see

$|g_{\lambda_{j}}|\leq|\phi|\frac{\mu\lambda_{j}^{\beta-1}}{(\lambda_{j}+|x|)^{\beta}}\to
0,\ \forall x\in B_{1}\backslash\\{0\\}\ ,$

as $\lambda_{j}\to 0$. Notice that

$|g_{\lambda_{j}}|\leq\ \frac{\mu|\phi|}{\lambda_{j}+|x|}\ \leq\
\frac{\mu\|\phi\|_{L^{\infty}}}{|x|}\in L^{1}(B_{1})\ ,$

by dominated convergence theorem, we have

$\lim_{\lambda_{j}\to 0}\int_{B_{1}}g_{\lambda_{j}}dx=0\ .$

Thus let $\lambda_{j}\to 0$ in (3.19), we get

(3.20) $\int_{B_{1}}(\nabla f\cdot\nabla\phi+f\phi)dx=0\ ,$

for any $\phi\in C^{\infty}_{c}(B_{1})$. This tells us that $f\in
H^{1}(B_{1})$ is a weak solution to the Poisson equation

$\displaystyle\begin{cases}\Delta f=f,x\in B_{1},\\\ f|_{\partial B_{1}}=1\
,f(0)=C\ .\end{cases}$

By regularity and strong maximum principle, we know that $f\in
C^{\infty}(B_{1})$ and $f(0)=C>0$, which completes the proof of the claim
$C>0$.

(II) Outside the ball $B_{1/\lambda}$

Inspired by Lemma 3.1, we try to reduce (2.7) to a second order ordinary
differential equation by finding a radially symmetric solution when $r\geq
1/\lambda$. Before proceeding, we need to estimate the derivative of
$\phi_{\lambda}$. Recall that
$f_{\lambda}(r)=\phi_{\lambda}(\frac{r}{\lambda})$, then by (3.18), we have

$(\partial_{r}\phi_{\lambda})(\lambda^{-1})=\lambda\partial_{r}f_{\lambda}(1)\leq\left(\frac{\mu}{n-1}+\frac{1}{n}\right)\lambda=C_{1}\lambda\
.$

We consider the second order ordinary differential equation

(3.21)
$\begin{cases}\phi^{\prime\prime}_{\lambda}+\frac{n-1}{r}\phi^{\prime}_{\lambda}-\lambda
D(r)\phi_{\lambda}=\lambda^{2}\phi_{\lambda}\\\
\phi_{\lambda}(\frac{1}{\lambda})=1,\phi^{\prime}_{\lambda}(\frac{1}{\lambda})\in(0,C_{1}\lambda]\
.\end{cases}$

Let $\phi_{\lambda}(r)=r^{-\frac{n-1}{2}}y(r)$, then $y$ satisfies

(3.22) $y^{\prime\prime}-\lambda^{2}y-\Big{(}\frac{(n-1)(n-3)}{4r^{2}}+\lambda
D(r)\Big{)}y=0$

with initial data

$y(\lambda^{-1})=\lambda^{-\frac{n-1}{2}},y^{\prime}(\lambda^{-1})=\frac{n-1}{2}\lambda^{-\frac{n-3}{2}}+\lambda^{-\frac{n-1}{2}}\phi_{\lambda}^{\prime}(\lambda^{-1})\in(0,C_{2}\lambda
y(\lambda^{-1})]\ ,$

where $C_{2}=\frac{n-1}{2}+C_{1}$. Thus by Lemma 3.1 with
$K=1,\epsilon=1,\lambda_{0}=1$, we have

$y\simeq\lambda^{-\frac{n-1}{2}}e^{\lambda r},r\lambda\geq 1\ ,$

which yields

$\phi_{\lambda}(x)\simeq(\lambda|x|)^{-\frac{n-1}{2}}e^{\lambda|x|},\lambda|x|\geq
1\ .$

Combining (I), (II), we conclude that there exist uniform $c_{1}\in(0,1)$ and
solutions $\phi_{\lambda}$ of (2.7) with $\lambda\in(0,1]$ satisfying the
uniform estimates (2.8).

##  4\. Proof of Theorem 1.5

In this section, we prove Theorem 1.5.

###  4.1. Test function method

Equipped with the test functions, we could construct two kinds of radial
solutions to the linear dual problem

$(\partial_{t}^{2}-\Delta-D\partial_{t}+V)\Phi=0\ ,$

that is,

$\Phi_{0}(t,x)=\phi_{0}(x),\ \Phi_{\lambda}(t,x)=e^{-\lambda
t}\phi_{\lambda}(x)\ .$

In addition to these solutions, we will also introduce a smooth cut-off
function. Let $\eta(t)\in C^{\infty}([0,\infty))$ such that

$\eta(t)=1,t\leq\frac{1}{2}\ ,\ \ \eta(t)=0,t\geq 1\ .$

Then for $T\in(2,T_{\varepsilon})$, we set $\eta_{T}(t)=\eta(t/T)$.

Let $\Psi_{\lambda}=\eta_{T}^{2p^{\prime}}(t)\Phi_{\lambda}(t,x)$, where
$\Phi_{\lambda}\in C^{2}([0,T]\times(\mathbb{R}^{n}\backslash\\{0\\}))\cap
C^{0}_{t}H_{loc}^{1}\cap C^{1}_{t}L_{loc}^{2})([0,T]\times\mathbb{R}^{n})$.
Then by the definition of energy solution (1.6), we have

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{n}}|u|^{p}\Psi_{\lambda}dxdt-\left.\int_{\mathbb{R}^{n}}(u_{t}(t,x)+D(x)u(t,x))\Psi_{\lambda}(t,x)dx\right|_{t=0}^{T}$
$\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{R}^{n}}u_{t}(t,x)\partial_{t}\Psi_{\lambda}(t,x)dxdt+\int_{0}^{T}\int_{\mathbb{R}^{n}}\nabla
u(t,x)\cdot\nabla\Psi_{\lambda}(t,x)dxdt$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{R}^{n}}D(x)u(t,x)\partial_{t}\Psi_{\lambda}(t,x)dxdt+\int_{0}^{T}\int_{\mathbb{R}^{n}}V(x)u(t,x)\Psi_{\lambda}(t,x)dxdt$
$\displaystyle=$
$\displaystyle-\int_{\mathbb{R}^{n}}u(t,x)\partial_{t}\Psi_{\lambda}(t,x)dx|_{t=0}^{T}+\int_{0}^{T}\int_{B_{t+R}}u(\partial_{t}^{2}-\Delta-D\partial_{t}+V)\Psi_{\lambda}dxdt\
,$

where all of the integration by parts and integrals could be justified by the
properties of $\Phi_{\lambda}$ and the support assumption
$\,\mathop{\\!\mathrm{supp}}u(t,\cdot)\subset B_{t+R}$.

Noticing that

$\Psi_{\lambda}(T)=\partial_{t}\Psi_{\lambda}(T)=0,\partial_{t}\Psi_{\lambda}=-\lambda\Psi_{\lambda}+\partial_{t}(\eta_{T}^{2p^{\prime}})\Phi_{\lambda}\
,$
$\displaystyle\partial_{t}\eta_{T}^{2p^{\prime}}=\frac{2p^{\prime}}{T}\eta_{T}^{2p^{\prime}-1}\eta^{\prime}(\frac{t}{T})=\mathcal{O}(\frac{\eta_{T}^{2p^{\prime}-1}}{T}\chi_{[\frac{T}{2},T]}(t)),$
$\displaystyle\partial_{t}^{2}\eta_{T}^{2p^{\prime}}=\frac{2p^{\prime}(2p^{\prime}-1)}{T^{2}}\eta_{T}^{2p^{\prime}-2}|\eta^{\prime}|^{2}+\frac{2p^{\prime}}{T^{2}}\eta_{T}^{2p^{\prime}-1}\eta^{\prime\prime}=\mathcal{O}(\frac{{\eta_{T}^{2(p^{\prime}-1)}}}{T^{2}}\chi_{[\frac{T}{2},T]}(t))\
.$

The integral identity could be reorganized into the following form:

(4.1)
$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{n}}|u|^{p}\Psi_{\lambda}dxdt+\varepsilon\int_{\mathbb{R}^{n}}(g(x)+(\lambda+D(x))f(x))\Phi_{\lambda}(x)dx$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{B_{t+R}}u(\partial_{t}^{2}-\Delta-D\partial_{t}+V)\Psi_{\lambda}dxdt$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{B_{t+R}}u(\partial_{t}^{2}(\eta_{T}^{2p^{\prime}})+2\partial_{t}(\eta_{T}^{2p^{\prime}})\partial_{t}-D\partial_{t}(\eta_{T}^{2p^{\prime}}))\Phi_{\lambda}dxdt$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{B_{t+R}}u(\partial_{t}^{2}(\eta_{T}^{2p^{\prime}})-2\lambda\partial_{t}(\eta_{T}^{2p^{\prime}})-D\partial_{t}(\eta_{T}^{2p^{\prime}}))\Phi_{\lambda}dxdt$
$\displaystyle\leq$ $\displaystyle
C\int_{T/2}^{T}\int_{B_{t+R}}|u|\eta_{T}^{2(p^{\prime}-1)}(T^{-2}+(2\lambda+|D|)T^{-1})\Phi_{\lambda}dxdt\
.$

Basically, the test function method is to construct specific test function, so
that we could try to use the integral inequality to control the right hand
side by the left hand side, which gives the lifespan estimates.

Before proceeding, let us present the following technical Lemma.

######  Lemma 4.1.

Let $\beta>0$, $\alpha,\ \gamma\in\mathbb{R}$ and $R>0$, there exists a
constant $C$, independent of $t>2$, so that

(4.2)
$\displaystyle\int_{0}^{t+R}(1+r)^{\alpha}\ln^{\gamma}(1+r)e^{-\beta(t-r)}dr\leq
C(t+R)^{\alpha}\ln^{\gamma}(t+R).$

Proof. We split the proof into two parts. First it is easy to see

$\displaystyle\int_{\frac{t+R}{2}}^{t+R}(1+r)^{\alpha}\ln^{\gamma}(1+r)e^{-\beta(t-r)}dr$
$\displaystyle\leq$ $\displaystyle Ce^{-\beta
t}(t+R)^{\alpha}\ln^{\gamma}(t+R)\int_{\frac{t+R}{2}}^{t+R}e^{\beta r}dr$
$\displaystyle\leq$ $\displaystyle C(t+R)^{\alpha}\ln^{\gamma}(t+R).$

For the remaining case, we have

$\displaystyle\int_{0}^{\frac{t+R}{2}}(1+r)^{\alpha}\ln^{\gamma}(1+r)e^{-\beta(t-r)}dr$
$\displaystyle\leq$ $\displaystyle Ce^{-\frac{\beta
t}{2}}\int_{0}^{\frac{t+R}{2}}(1+r)^{|\alpha|+1}dr$ $\displaystyle\leq$
$\displaystyle Ce^{-\frac{\beta t}{2}}(t+R)^{|\alpha|+2}$ $\displaystyle\leq$
$\displaystyle C(t+R)^{\alpha}\ln^{\gamma}(t+R)\ ,$

which completes the proof.

###  4.2. First choice of the test function $\Phi_{0}(t,x)$ with $\lambda=0$

Let $\lambda=0$, we have $\phi_{0}$ ensured by Lemma 2.1. With help of
$\Phi_{0}=\phi_{0}$, the inequality (4.1) reads as follows

$\displaystyle
C_{1}(f,g)\varepsilon+\int_{0}^{T}\int_{\mathbb{R}^{n}}|u|^{p}\eta_{T}^{2p^{\prime}}\phi_{0}dxdt$
$\displaystyle\leq$ $\displaystyle
C\int_{0}^{T}\int_{\mathbb{R}^{n}}|u|\eta_{T}^{2(p^{\prime}-1)}(T^{-2}+|D|T^{-1})\phi_{0}dxdt$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{0}^{T}\int_{\mathbb{R}^{n}}|u|^{p}\eta_{T}^{2p^{\prime}}\phi_{0}dxdt+C\int_{T/2}^{T}\int_{B_{t+R}}(T^{-2}+|D|T^{-1})^{p^{\prime}}\phi_{0}dxdt\
,$

where $C_{1}(f,g)=\int_{\mathbb{R}^{n}}(g(x)+D(x)f(x))dx$ and we have used the
Hölder and Young’s inequality in last inequality. In conclusion,

(4.3)
$C_{1}(f,g)\varepsilon+\int_{0}^{T}\int_{\mathbb{R}^{n}}|u|^{p}\eta_{T}^{2p^{\prime}}\phi_{0}dxdt\leq
C\int_{T/2}^{T}\int_{B_{t+R}}(T^{-2}+|D|T^{-1})^{p^{\prime}}\phi_{0}dxdt=F_{0}\
,$

Concerning the right hand side of (4.3), by our assumption
$D=\mathcal{O}(r^{\theta-2})$ locally, $D=\mathcal{O}(r^{-1})$ near spatial
infinity, and Lemma 2.1, we see that $F_{0}$ could be controlled by

$\displaystyle
T^{-p^{\prime}}\int_{T/2}^{T}\int_{0}^{1}(T^{-1}+r^{\theta-2})^{p^{\prime}}r^{\rho(v_{0})+n-1}drdt$
$\displaystyle+T^{-p^{\prime}}\int_{T/2}^{T}\int_{1}^{t+R}(T^{-1}+r^{-1})^{p^{\prime}}r^{\rho(v_{\infty})+n-1}drdt$
$\displaystyle{\lesssim}$ $\displaystyle
T^{-2p^{\prime}}\left(T+T^{\rho(v_{\infty})+n+1}\right)+T^{1-p^{\prime}}+T^{1-p^{\prime}}\int_{1}^{T+R}r^{-p^{\prime}+\rho(v_{\infty})+n-1}dr$
$\displaystyle{\lesssim}$
$\displaystyle\left\\{\begin{array}[]{ll}\max\\{T^{1-2p^{\prime}},T^{-2p^{\prime}+\rho(v_{\infty})+n+1},T^{1-p^{\prime}}\\}&p^{\prime}\neq\rho(v_{\infty})+n\\\
T^{1-p^{\prime}}\ln T&p^{\prime}=\rho(v_{\infty})+n\end{array}\right.$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}T^{-2p^{\prime}+\rho(v_{\infty})+n+1}&p^{\prime}<\rho(v_{\infty})+n\\\
T^{1-p^{\prime}}\ln T&p^{\prime}=\rho(v_{\infty})+n\\\
T^{1-p^{\prime}}&p^{\prime}>\rho(v_{\infty})+n\ ,\end{array}\right.$

where, to ensure the integrability of first term of second bracket, we need to
assume

(4.6) $(\theta-2)p^{\prime}+\rho(v_{0})+n>0\Leftrightarrow
p>p_{2}:=\frac{n+\rho(v_{0})}{n+\rho(v_{0})+\theta-2}\ .$

Also, we observe that

(4.7) $p^{\prime}<\rho(v_{\infty})+n\Leftrightarrow
p>p_{3}:=\frac{n+\rho(v_{\infty})}{n+\rho(v_{\infty})-1}\ .$

In conclusion, provided that $p>p_{2}$, we have

(4.8)
$F_{0}{\lesssim}\left\\{\begin{array}[]{ll}T^{-2p^{\prime}+\rho(v_{\infty})+n+1}&p>\max(p_{2},p_{3})\\\
T^{1-p^{\prime}}\ln T&p=p_{3}>p_{2}\\\
T^{1-p^{\prime}}&p_{2}<p<p_{3},\mathrm{if}\ p_{2}<p_{3}\ .\end{array}\right.$

Recalling (4.3), if we have data such that $C_{1}(f,g)>0$, which is always
true for $f=0$ and nontrivial, nonnegative and compactly supported $g$, we
have $\varepsilon{\lesssim}F_{0}$. Then we obtain the blow up results for
$p_{2}<p<p_{G}$, whenever $(p_{2},p_{G}(n+\rho(v_{\infty})))\neq\emptyset$,
and

$-2p^{\prime}+\rho(v_{\infty})+n+1<0\Leftrightarrow
p<p_{G}(n+\rho(v_{\infty}))\ .$

At the same time, we could extract the first three lifespan estimates in
(1.13).

###  4.3. Second choice of the test function $\Phi_{1}(t,x)$ with $\lambda=1$

Let $\lambda=1$, we have $\phi_{1}$ ensured by Lemma 2.3 with $\lambda_{0}=1$.
With help of $\Phi_{1}=e^{-t}\phi_{1}$,
$\Psi_{1}=\eta_{T}^{2p^{\prime}}(t)\Phi_{1}$, as for (4.3), the inequality
(4.1) gives us

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{n}}|u|^{p}\Psi_{1}dxdt+C_{2}(f,g)\varepsilon$
$\displaystyle\leq$ $\displaystyle
C\int_{T/2}^{T}\int_{B_{t+R}}|u|\eta_{T}^{2(p^{\prime}-1)}\frac{2+|D|}{T}\Phi_{1}dxdt$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{T/2}^{T}\int_{\mathbb{R}^{n}}|u|^{p}\eta_{T}^{2p^{\prime}}\Phi_{1}dxdt+C\int_{T/2}^{T}\int_{B_{t+R}}\left(\frac{2+|D|}{T}\right)^{p^{\prime}}\Phi_{1}dxdt\
,$

and so

(4.10) $C_{2}(f,g)\varepsilon\
{\lesssim}\int_{T/2}^{T}\int_{B_{t+R}}\left(\frac{2+|D|}{T}\right)^{p^{\prime}}\Phi_{1}dxdt\
,$

where $C_{2}(f,g):=\int_{\mathbb{R}^{n}}\phi_{1}(Df+f+g)dx$.

As in the previous section 4.2, we could extract certain lifespan estimate
from (4.10). Actually, by (4.10), Lemma 4.1 and Lemma 2.3 with
$\lambda_{0}=1$, we have

(4.11) $\displaystyle\varepsilon{\lesssim}$ $\displaystyle
T^{-p^{\prime}}\int_{\frac{T}{2}}^{T}\int_{|x|\leq
t+R}(2+|D(x)|^{p^{\prime}})\Phi_{1}dxdt$ $\displaystyle{\lesssim}$
$\displaystyle T^{-p^{\prime}}\Bigg{(}\int_{\frac{T}{2}}^{T}\int_{r\leq
1}(1+r^{(\theta-2)p^{\prime}})e^{-t}r^{\rho(v_{0}+d_{0})+n-1}drdt$
$\displaystyle+\int_{\frac{T}{2}}^{T}\int_{1\leq r\leq
t+R}(1+r^{-p^{\prime}})e^{-t}e^{r}r^{\frac{1-n+d_{\infty}}{2}+n-1}drdt\Bigg{)}$
$\displaystyle{\lesssim}$ $\displaystyle
T^{-p^{\prime}}\left(e^{-\frac{T}{2}}+T^{\frac{n+1+d_{\infty}}{2}}\right){\lesssim}T^{-p^{\prime}+\frac{n+1+d_{\infty}}{2}},$

provided that $C_{2}(f,g)>0$, and

$\rho(v_{0}+d_{0})+n+(\theta-2)p^{\prime}>0,i.e.,p>p_{4}:=\frac{n+\rho(v_{0}+d_{0})}{n+\rho(v_{0}+d_{0})+\theta-2}\
.$

Based on (4.11), when

$-p^{\prime}+\frac{n+1+d_{\infty}}{2}<0\Leftrightarrow p<p_{G}(n+d_{\infty})\
,$

we obtain the last lifespan estimate in (1.13) for any
$p\in(p_{4},p_{G}(n+d_{\infty}))\neq\emptyset$, whenever $f=0$ and nontrivial
$g\geq 0$.

###  4.4. Combination

It turns out that we have not exploited the full strength of $\Psi_{0}$ and
$\Psi_{1}$. Actually, a combination of (4.3) and (4.3) could give us more
information on the lifespan estimates.

To connect (4.3) with that appeared in (4.3), we try to control the middle
term in (4.3) by the left of (4.3), that is,

$\displaystyle\varepsilon T{\lesssim}$
$\displaystyle\int_{T/2}^{T}\int_{B_{t+R}}|u|\eta_{T}^{2(p^{\prime}-1)}(2+|D|)\Phi_{1}dxdt$
$\displaystyle{\lesssim}$
$\displaystyle\left(\int_{0}^{T}\int_{\mathbb{R}^{n}}\eta_{T}^{2p^{\prime}}\phi_{0}|u|^{p}dxdt\right)^{\frac{1}{p}}\left(\int_{\frac{T}{2}}^{T}\int_{|x|\leq
t+R}(2+|D(x)|)^{p^{\prime}}\phi_{0}^{-\frac{p^{\prime}}{p}}\Phi_{1}^{p^{\prime}}dxdt\right)^{\frac{1}{p^{\prime}}}.$

Thus, combining it with (4.3), we derive that

(4.12)
$\varepsilon^{p}T^{p}{\lesssim}F_{0}\left(\int_{\frac{T}{2}}^{T}\int_{|x|\leq
t+R}(2+|D(x)|)^{p^{\prime}}\phi_{0}^{-\frac{p^{\prime}}{p}}\Phi_{1}^{p^{\prime}}dxdt\right)^{p-1}:=F_{0}F_{1}^{p-1}\
.$

Concerning $F_{1}$, by Lemma 2.1 and 2.3, we have

$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle\int_{\frac{T}{2}}^{T}\int_{|x|\leq
t+R}(2+|D|^{p^{\prime}})\phi_{0}^{-\frac{p^{\prime}}{p}}\Psi_{0}^{p^{\prime}}dxdt$
$\displaystyle{\lesssim}$
$\displaystyle\int_{\frac{T}{2}}^{T}e^{-tp^{\prime}}\int_{0}^{1}(2+r^{(\theta-2)p^{\prime}})r^{-\frac{p^{\prime}}{p}\rho(v_{0})+p^{\prime}\rho(v_{0}+d_{0})+n-1}drdt$
$\displaystyle+\int_{\frac{T}{2}}^{T}\int_{1}^{t+R}(2+|D|)^{p^{\prime}}r^{-\frac{p^{\prime}}{p}\rho(v_{\infty})-p^{\prime}\frac{n-1-d_{\infty}}{2}+n-1}e^{(r-t)p^{\prime}}drdt.$

To ensure the integrability near $r=0$, we need to require that

$(\theta-2)p^{\prime}-\frac{p^{\prime}}{p}\rho(v_{0})+p^{\prime}\rho(v_{0}+d_{0})+n-1>-1,$

that is

(4.13) $p>p_{1}:=\frac{n+\rho(v_{0})}{n+\rho(v_{0}+d_{0})+\theta-2}\ .$

While for the second integral, we utilize Lemma 4.1 with $\beta=1$, as well as
$|D|{\lesssim}r^{-1}$ for $r>1$, to conclude

$\displaystyle\int_{1}^{t+R}(2+|D|)^{p^{\prime}}r^{-\frac{p^{\prime}}{p}\rho(v_{\infty})-p^{\prime}\frac{(n-1-d_{\infty})}{2}+n-1}e^{(r-t)\lambda_{0}p^{\prime}}dr$
$\displaystyle{\lesssim}$
$\displaystyle\int_{1}^{t+R}r^{-\frac{p^{\prime}}{p}\rho(v_{\infty})-p^{\prime}\frac{n-1-d_{\infty}}{2}+n-1}e^{(r-t)\lambda_{0}p^{\prime}}dr$
$\displaystyle{\lesssim}$
$\displaystyle(t+R)^{-\frac{p^{\prime}}{p}\rho(v_{\infty})-p^{\prime}\frac{n-1-d_{\infty}}{2}+n-1}.$

Thus we have

(4.14)
$F_{1}{\lesssim}e^{-Tp^{\prime}/2}+T^{-\frac{p^{\prime}}{p}\rho(v_{\infty})-p^{\prime}\frac{n-1-d_{\infty}}{2}+n}{\lesssim}T^{-\frac{\rho(v_{\infty})}{p-1}-p\frac{n-1-d_{\infty}}{2(p-1)}+n}\
,$

if $p>p_{1}$.

In view of (4.8), (4.14) and (4.12), we arrive at, for $p>\max(p_{1},p_{2})$,

$\varepsilon^{p}{\lesssim}T^{-p}F_{0}F_{1}^{p-1}{\lesssim}\left\\{\begin{array}[]{ll}T^{\frac{n+d_{\infty}-1}{2}p-2p^{\prime}+1}&p>p_{3}\\\
T^{\frac{n+d_{\infty}-1}{2}p-2p^{\prime}+1}\ln T&p=p_{3}\\\
T^{-\rho(v_{\infty})+p\frac{n+d_{\infty}-1}{2}-n+1-p^{\prime}}&p<p_{3}\
,\end{array}\right.$

by which we are able to obtain the final set of lifespan estimates.

When $p>\max\\{p_{1},p_{2},p_{3}\\}$, we could obtain an upper bound of the
lifespan, if

$\frac{n+d_{\infty}-1}{2}p-2p^{\prime}+1<0\Leftrightarrow\frac{n+d_{\infty}-1}{2}p(p-1)<p+1,$

that is, $p<p_{S}(n+d_{\infty})$. This gives us the sixth lifespan estimate in
(1.13):

$T_{\varepsilon}\
{\lesssim}\varepsilon^{-\frac{2p(p-1)}{\gamma(p,n+d_{\infty})}}.$

In the critical case, $p=p_{3}\in(\max\\{p_{1},p_{2}\\},p_{S}(n+d_{\infty}))$,
we obtain the estimate with log loss:

$T_{\varepsilon}{\lesssim}\varepsilon^{-\frac{2p(p-1)}{\gamma(p,n+{d_{\infty}})}}(\ln\varepsilon^{-1})^{\frac{2(p-1)}{\gamma(p,n+{d_{\infty}})}}\
.$

For the remaining case of $\max\\{p_{1},p_{2}\\}<p<p_{3}$, we could obtain an
upper bound of the lifespan, provided that
${-\rho(v_{\infty})+p\frac{n+d_{\infty}-1}{2}-n+1-p^{\prime}}<0$, i.e.,

$\frac{n+d_{\infty}-1}{2}p(p-1)+(n+\rho(v_{\infty})-1)(p_{3}-p)<p+1\Leftrightarrow\gamma_{2}>0\Leftrightarrow
p<p_{5}\ ,$

where $\gamma_{2}$ is given in (1.14). Thus we have blow up result and
lifespan estimate, the fourth lifespan estimate in (1.13), if
$\max\\{p_{1},p_{2}\\}<p<\min(p_{3},p_{5})$.

Finally, we remark that the last upper bound of the lifespan in (1.13) was
obtained for $p\in(p_{4},p_{G}(n+d_{\infty}))\neq\emptyset$. However, after
comparison with the corresponding estimates in the first and fourth case, we
see that it gives better upper bound only for
$p\in(p_{4},p_{0}]\cap(p_{4},p_{G}(n+d_{\infty}))$, if we have
$p_{4}<\min(p_{0},p_{G}(n+d_{\infty}))$. This is the reason we state it for
this restricted range in (1.13).

##  5\. Proof of Theorem 1.3

In this section, we prove Theorem 1.3.

According to Lemma 2.2 and 2.4 we have,

$\phi_{0}\simeq\begin{cases}\ln(r+2)\ ,n=2\\\ 1\ ,\ n\geq 3\ ,\end{cases}\
\mbox{and}\ \ \phi_{1}\simeq\langle r\rangle^{-\frac{n-1-d_{\infty}}{2}}e^{r}\
.$

Thus, when $n\geq 3$, the situation considered in Theorem 1.3 could be viewed
as a particular case of $v_{0}=d_{0}=v_{\infty}=0$ and $\theta=2$, in Theorem
1.5. Then $p_{1}=p_{2}=1$, $p_{3}=\frac{n}{n-1}$, and so Theorem 1.3 for
$n\geq 3$ could be obtained as the Corollary of Theorem 1.5. In the following,
it suffices for us to present the proof for $n=2$, for which we still have
(4.3), (4.3), (4.10) and (4.12), with only replacements from $\phi_{0}\sim 1$
to $\phi_{0}\sim\ln(r+2)$.

Concerning the right hand side of (4.3), we have

(5.4) $\displaystyle\varepsilon\ {\lesssim}F_{0}$ $\displaystyle{\lesssim}$
$\displaystyle T^{-p^{\prime}}\int_{T/2}^{T}\int_{0}^{t+R}(T^{-1}+\langle
r\rangle^{-1})^{p^{\prime}}r\ln(r+2)drdt$ $\displaystyle{\lesssim}$
$\displaystyle T^{-2p^{\prime}}T^{3}\ln
T+T^{-p^{\prime}}\int_{T/2}^{T}\int_{0}^{t+R}\langle
r\rangle^{1-p^{\prime}}\ln(r+2)drdt$ $\displaystyle{\lesssim}$
$\displaystyle\left\\{\begin{array}[]{ll}T^{1-p^{\prime}}&p<2\\\
T^{1-p^{\prime}}(\ln T)^{2}&p=2\\\ T^{3-2p^{\prime}}\ln T&p>2\
.\end{array}\right.$

Based on this inequality, we could extract the first three lifespan estimates
in (1.11) for $p\in(1,3)$ and $n=2$, which have certain log loss for the case
$p\geq n/(n-1)=2$.

Concerning $F_{1}$ in (4.12), by Lemma 4.1, we have

$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle\int_{\frac{T}{2}}^{T}\int_{|x|\leq
t+R}(2+|D|^{p^{\prime}})\phi_{0}^{-\frac{p^{\prime}}{p}}\Psi_{0}^{p^{\prime}}dxdt$
$\displaystyle{\lesssim}$
$\displaystyle\int_{\frac{T}{2}}^{T}\int_{0}^{t+R}(1+r)^{-p^{\prime}\frac{1-d_{\infty}}{2}+1}(\ln(r+2))^{-\frac{p^{\prime}}{p}}e^{(r-t)p^{\prime}}drdt$
$\displaystyle{\lesssim}$ $\displaystyle
T^{2-p^{\prime}\frac{1-d_{\infty}}{2}}(\ln T)^{-\frac{p^{\prime}}{p}}\ .$

Plugging these estimates in (4.12), we derive that

(5.5)
$\varepsilon^{p}T^{p}{\lesssim}F_{0}F_{1}^{p-1}{\lesssim}T^{2(p-1)-p\frac{1-d_{\infty}}{2}}\times\left\\{\begin{array}[]{ll}T^{1-p^{\prime}}(\ln
T)^{-1}&p<2\\\ T^{1-p^{\prime}}\ln T&p=2\\\ T^{3-2p^{\prime}}&p>2\
.\end{array}\right.\ .$

Similarly, based on this inequality, we could extract the second set (fourth
to sixth) of lifespan estimates in (1.11) for $p\in(1,p_{S}(n+d_{\infty}))$
and $n=2$, which have certain log adjustment for the case $p<n/(n-1)=2$.

##  6\. Proof of Theorem 1.2

###  6.1. Subcritical case

In this subsection, we present the proof of Theorem 1.2 for $p<p_{S}(n)$,
under the assumption that $V=0$ and $D(x)\in C(\mathbb{R}^{n})\cap
C^{\delta}(B_{\delta})$, which is of short range, in the sense that $D\in
L^{n}\cap L^{\infty}(\mathbb{R}^{n})$. The proof follows the same lines as
that of Theorem 1.5 or Theorem 1.3.

At first, with $\phi_{0}=1$, by (4.3), we have

(6.1)
$C_{1}(f,g)\varepsilon+\int_{0}^{T}\int_{\mathbb{R}^{n}}|u|^{p}\eta_{T}^{2p^{\prime}}dxdt\leq
C\int_{T/2}^{T}\int_{B_{t+R}}(T^{-2}+|D|T^{-1})^{p^{\prime}}dxdt=F_{0}\ .$

When $p^{\prime}\geq n$, i.e., $p\leq\frac{n}{n-1}$, we know that $D\in
L^{n}\cap L^{\infty}(\mathbb{R}^{n})\subset L^{p^{\prime}}$, and so

$F_{0}=C\int_{T/2}^{T}\int_{B_{t+R}}(T^{-2}+|D|T^{-1})^{p^{\prime}}dxdt{\lesssim}T^{n+1-2p^{\prime}}+T^{1-p^{\prime}}{\lesssim}T^{1-p^{\prime}}.$

Otherwise, for $p^{\prime}<n$, i.e., $p>\frac{n}{n-1}$, by using Hölder’s
inequality, we obtain

$\int_{B_{T+R}}|D|^{p^{\prime}}dx{\lesssim}\||D|^{p^{\prime}}\|_{L^{n/p^{\prime}}}\|1\|_{L^{n/(n-p^{\prime})}(B_{T+R})}{\lesssim}T^{n-p^{\prime}}\
,$

and thus

$F_{0}{\lesssim}T^{n+1-2p^{\prime}}+T^{1-p^{\prime}}\int_{B_{T+R}}|D|^{p^{\prime}}dx{\lesssim}T^{n+1-2p^{\prime}}\
.$

In conclusion, we have

(6.2)
$\int_{0}^{T}\int_{\mathbb{R}^{n}}|u|^{p}\eta_{T}^{2p^{\prime}}dxdt\lesssim\left\\{\begin{array}[]{ll}T^{1-p^{\prime}}&\mbox{if}\
1<p\leq\frac{n}{n-1},\\\ T^{n+1-2p^{\prime}}&\mbox{if}\ p\geq\frac{n}{n-1}.\\\
\end{array}\right.$

As $D\in L^{\infty}(\mathbb{R}^{n})$, there exists $\lambda_{0}>0$ such that
$|D|\leq\lambda_{0}$ and then we choose $\phi_{\lambda_{0}}$ which is ensured
by Lemma 2.4. Let
$\Psi(t,x)=\eta_{T}^{2p^{\prime}}\Phi(t,x)=\eta_{T}^{2p^{\prime}}e^{-\lambda_{0}t}\phi_{\lambda_{0}}(x)$
be the test function, as for (4.3) and (4.12), we get from (4.1) and Lemma 4.1
that

(6.3) $\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{n}}|u|^{p}\Psi
dxdt+C_{2}(f,g)\varepsilon$ $\displaystyle{\lesssim}$
$\displaystyle\int_{T/2}^{T}\int_{B_{t+R}}|u|\eta_{T}^{2(p^{\prime}-1)}\frac{2+|D|}{T}\Phi
dxdt$ $\displaystyle{\lesssim}$ $\displaystyle
T^{-1}\left(\int_{T/2}^{T}\int_{\mathbb{R}^{n}}\eta_{T}^{2p^{\prime}}|u|^{p}dxdt\right)^{\frac{1}{p}}\left(\int_{T/2}^{T}\int_{B_{t+R}}\Phi^{p^{\prime}}dxdt\right)^{\frac{1}{p^{\prime}}}$
$\displaystyle\leq$ $\displaystyle
CT^{-1+(n-\frac{n-1}{2}p^{\prime})\frac{1}{p^{\prime}}}\left(\int_{T/2}^{T}\int_{\mathbb{R}^{n}}\eta_{T}^{2p^{\prime}}|u|^{p}dxdt\right)^{\frac{1}{p}},$

which yields

(6.4)
$\displaystyle\varepsilon^{p}T^{n-\frac{n-1}{2}p}=\varepsilon^{p}T^{p-(n-\frac{n-1}{2}p^{\prime})(p-1)}\lesssim\int_{T/2}^{T}\int_{\mathbb{R}^{n}}\eta_{T}^{2p^{\prime}}|u|^{p}dxdt\
.$

Based on (6.2) and (6.4), we obtain the first and second lifespan estimates in
(1.9) in Theorem 1.2.

###  6.2. Critical case

Turning to the critical case, $p=p_{S}(n)$, the proof is parallel to that in
[18], which heavily relies on Lemma 2.5. For completeness, we present a proof
here.

Based on the family of test functions $\phi_{\lambda}$, with
$\lambda\in(0,1]$, satisfying

(6.5) $\displaystyle\Delta\phi_{\lambda}-\lambda
D(x)\phi_{\lambda}=\lambda^{2}\phi_{\lambda},~{}~{}~{}x\in\mathbb{R}^{n}\ ,$

we construct a new class of test functions, with parameters $q>0$,

$b_{q}(t,x)=\int_{0}^{1}e^{-\lambda t}\phi_{\lambda}(x)\lambda^{q-1}d\lambda\
.$

The magic of the test functions $b_{q}(t,x)$ lie on the facts that they
satisfy

(6.6) $\partial^{2}_{t}b_{q}-\Delta
b_{q}-D(x)\partial_{t}b_{q}=0,~{}~{}~{}\partial_{t}b_{q}=-b_{q+1}\ ,$

and enjoy the asymptotic behavior for $n\geq 2$ and $r\leq t+R$

(6.7) $b_{q}(t,x)\gtrsim(t+R)^{-q},\ q>0\ ,$

and

(6.8) $b_{q}(t,x)\lesssim\left\\{\begin{array}[]{ll}(t+R)^{-q}&\mbox{if}\
0<q<\frac{n-1}{2},\\\
(t+R)^{-\frac{n-1}{2}}(t+R+1-|x|)^{\frac{n-1}{2}-q}&\mbox{if}\
q>\frac{n-1}{2}.\\\ \end{array}\right.$

Based on (6.6)-(6.8), the same argument in [18] will yield a proof for the
last lifespan of Theorem 1.2.

####  6.2.1. Estimates of the test functions: (6.7) and (6.8)

In [18], the asymptotic behavior (6.8) was proved by employing the property of
the hypergeometric function, when $\beta>2$ and $n\geq 3$. In the following we
will use a relatively simpler way to show it, in the general case $\beta>1$
and $n\geq 2$, inspired by the method in [17].

We first consider the lower bound (6.7). From the definition of $b_{q}$ we
know

(6.9) $\displaystyle b_{q}(t,x)$
$\displaystyle\gtrsim\int_{\frac{1}{2(t+R)}}^{\frac{1}{t+R}}e^{-\lambda
t}\phi_{\lambda}(x)\lambda^{q-1}d\lambda$
$\displaystyle\gtrsim\int_{\frac{1}{2(t+R)}}^{\frac{1}{t+R}}e^{-\lambda(t+R)}\lambda^{q-1}d\lambda$
$\displaystyle\gtrsim(t+R)^{-q}\int_{\frac{1}{2}}^{1}e^{-\theta}\theta^{q-1}d\theta$
$\displaystyle\gtrsim(t+R)^{-q},$

where we used the fact $\phi_{\lambda}\thicksim 1$ when
$r\lambda\leq\frac{r}{t+R}\leq 1$ by (2.8).

For the upper bound (6.8), we divide the proof into two parts:
$r\leq\frac{t+R}{2}$ and $\frac{t+1}{2}\leq r\leq t+R$. For the former case,
we have

(6.10) $\displaystyle b_{q}(t,x)$
$\displaystyle\lesssim\int_{0}^{1}e^{-\frac{\lambda(t+R)}{2}}(1+\lambda
r)^{-\frac{n-1}{2}}\lambda^{q-1}d\lambda$
$\displaystyle\lesssim\int_{0}^{1}e^{-\frac{\lambda(t+R)}{2}}\lambda^{q-1}d\lambda$
$\displaystyle\lesssim(t+R)^{-q}\int_{0}^{\infty}e^{-\theta}\theta^{q-1}d\theta$
$\displaystyle\lesssim(t+R)^{-q}.$

If $\frac{t+R}{2}\leq r\leq t+R$ and $0<q<\frac{n-1}{2}$, it is clear that

(6.11) $\displaystyle b_{q}(t,x)$
$\displaystyle\lesssim\int_{0}^{1}(1+\lambda(t+R))^{-\frac{n-1}{2}}\lambda^{q-1}d\lambda$
$\displaystyle\lesssim(t+R)^{-q}\int_{0}^{\infty}(1+\theta)^{-\frac{n-1}{2}}\theta^{q-1}d\theta$
$\displaystyle\lesssim(t+R)^{-q}.$

For the remaining case: $\frac{t+R}{2}\leq r\leq t+R$ and $q>\frac{n-1}{2}$,
we see that

(6.12) $\displaystyle b_{q}(t,x)$
$\displaystyle\lesssim\int_{0}^{1}e^{-\lambda(t+R+1-r)}(\lambda(t+R))^{-\frac{n-1}{2}}\lambda^{q-1}d\lambda$
$\displaystyle\lesssim(t+R)^{-\frac{n-1}{2}}\int_{0}^{1}e^{-\lambda(t+R+1-r)}\lambda^{q-1-\frac{n-1}{2}}d\lambda$
$\displaystyle\lesssim(t+R)^{-\frac{n-1}{2}}(t+R+1-r)^{\frac{n-1}{2}-q}\int_{0}^{\infty}e^{-\theta}\theta^{q-1-\frac{n-1}{2}}d\theta$
$\displaystyle\lesssim(t+R)^{-\frac{n-1}{2}}(t+R+1-r)^{\frac{n-1}{2}-q}\ .$

####  6.2.2. Proof

With $b_{q}$ and its asymptotic behavior in hand, we use
$\Psi_{T}=\eta_{T}^{2p^{\prime}}b_{q}$ as the test function, which gives us

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{n}}\eta_{T}^{2p^{\prime}}b_{q}|u|^{p}dxdt$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{n}}\left(\partial_{t}^{2}u-\Delta
u+D(x)\partial_{t}u\right)b_{q}\eta_{T}^{2p^{\prime}}dxdt$
$\displaystyle\lesssim$
$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{n}}|u|\left(|2\partial_{t}b_{q}\partial_{t}\eta_{T}^{2p^{\prime}}|+|b_{q}\partial_{t}^{2}\eta_{T}^{2p^{\prime}}|+\Big{|}D(x)b_{q}\partial_{t}\eta_{T}^{2p^{\prime}}\Big{|}\right)dxdt$
$\displaystyle\leq$
$\displaystyle(\int_{\frac{T}{2}}^{T}\int_{\mathbb{R}^{n}}|u|^{p}\Psi_{T}dxdt)^{\frac{1}{p}}\left(\int_{\frac{T}{2}}^{T}\int_{B_{t+R}}b_{q}^{-\frac{p^{\prime}}{p}}\left[(T^{-1}+D)b_{q}+b_{q+1}\right]^{p^{\prime}}T^{-p^{\prime}}dxdt\right)^{\frac{1}{p^{\prime}}}\
.$

Let $q=\frac{n-1}{2}-\frac{1}{p}$. As $p>(n+1)/(n-1)$, $p^{\prime}<(n+1)/2\leq
n$, $D\leq\mu(1+r)^{-\beta}$, we have

$\displaystyle\int_{\frac{T}{2}}^{T}\int_{B_{t+R}}(T^{-1}+D)^{p^{\prime}}T^{-p^{\prime}}b_{q}dxdt$
$\displaystyle{\lesssim}$
$\displaystyle\int_{\frac{T}{2}}^{T}\int_{B_{t+R}}(t+R)^{\frac{1}{p}-\frac{n-1}{2}}T^{-p^{\prime}}(1+r)^{-p^{\prime}}dxdt$
$\displaystyle{\lesssim}$ $\displaystyle
T^{\frac{1}{p}-\frac{n-1}{2}-2p^{\prime}+n+1}=1$

where we used (6.8) and the fact that

$\frac{1}{p}-\frac{n-1}{2}-2p^{\prime}+n+1=\frac{1}{p}+\frac{n-1}{2}-\frac{2}{p-1}=0$

for $p=p_{S}(n)$. By (6.7)-(6.8), we see that

$b_{q}\simeq(t+R)^{-q},b_{q+1}{\lesssim}(t+R)^{-\frac{n-1}{2}}(t+R+1-|x|)^{\frac{n-1}{2}-q-1}\simeq(t+R)^{-\frac{n-1}{2}}(t+R+1-|x|)^{-\frac{1}{p^{\prime}}}\
,$

and so

$\displaystyle\int_{\frac{T}{2}}^{T}\int_{B_{t+R}}b_{q+1}^{p^{\prime}}b_{q}^{-\frac{p^{\prime}}{p}}T^{-p^{\prime}}dxdt$
$\displaystyle{\lesssim}$
$\displaystyle\int_{\frac{T}{2}}^{T}\int_{B_{t+R}}(t+R)^{-\frac{n-1}{2}-\frac{1}{p(p-1)}}(t+R+1-|x|)^{-1}(t+R)^{-(\frac{n-1}{2}-q)\frac{1}{p-1}}T^{-p^{\prime}}dxdt$
$\displaystyle{\lesssim}$
$\displaystyle\int_{\frac{T}{2}}^{T}\int_{0}^{{t+R}}T^{\frac{n-1}{2}-\frac{1}{p(p-1)}-p^{\prime}}(t+R+1-r)^{-1}drdt$
$\displaystyle{\lesssim}$
$\displaystyle\int_{\frac{T}{2}}^{T}\int_{0}^{{t+R}}T^{-1}(t+R+1-r)^{-1}drdt{\lesssim}\ln
T\ .$

In conclusion, we have

(6.13)
$\int_{0}^{T}\int_{\mathbb{R}^{n}}\eta_{T}^{2p^{\prime}}b_{q}|u|^{p}dxdt{\lesssim}(\ln
T)^{1/p^{\prime}}Z(T)^{1/p}\ ,$

where

$Z(T)\triangleq\int_{T/2}^{T}\int_{\mathbb{R}^{n}}|u|^{p}b_{q}\eta_{T}^{2p^{\prime}}dxdt\leq\int_{0}^{T}\int_{\mathbb{R}^{n}}|u|^{p}b_{q}\eta_{T}^{2p^{\prime}}dxdt\triangleq
X(T)\ .$

To relate $Z$ and $X$, and recall the critical nature of the situation, let
$Z=TY^{\prime}$ and $Y(2)=0$, then

$Y(M)=\int_{2}^{M}Z(T)T^{-1}dT,$

and

$\displaystyle Y(M)=$
$\displaystyle\int_{2}^{M}\left(\int_{T/2}^{T}\int_{\mathbb{R}^{n}}b_{q}|u|^{p}(\eta_{T}(t))^{2p^{\prime}}dxdt\right)T^{-1}dT$
$\displaystyle\leq$
$\displaystyle\int_{1}^{M}\int_{\mathbb{R}^{n}}b_{q}|u|^{p}\int_{t}^{\min(M,2t)}(\eta_{T}(t))^{2p^{\prime}}T^{-1}dTdxdt$
$\displaystyle=$
$\displaystyle\int_{1}^{M}\int_{\mathbb{R}^{n}}b_{q}|u|^{p}\int^{1}_{\max(t/M,1/2)}(\eta(s))^{2p^{\prime}}s^{-1}dsdxdt$
$\displaystyle\leq$
$\displaystyle\int_{1}^{M}\int_{\mathbb{R}^{n}}b_{q}|u|^{p}(\eta(t/M))^{2p^{\prime}}\int^{1}_{1/2}s^{-1}dsdxdt$
$\displaystyle\leq$ $\displaystyle\ln
2\int_{1}^{M}\int_{\mathbb{R}^{n}}b_{q}|u|^{p}(\eta(t/M))^{2p^{\prime}}dxdt\leq\ln
2\int_{0}^{M}\int_{\mathbb{R}^{n}}|u|^{p}b_{q}\eta_{T}dxdt{\lesssim}X(M)\ ,$

where we used the assumption that $\eta$ is decreasing. Thus, recalling
(6.13), we have

(6.14) $Y(T){\lesssim}(\ln T)^{1/p^{\prime}}(TY^{\prime}(T))^{1/p},\
TY^{\prime}(T)\geq cY^{p}(\ln T)^{1-p},\forall T\in[2,T_{\varepsilon})\ .$

In addition, by (6.4) and (6.7), we have

(6.15)
$TY^{\prime}=Z(T)\triangleq\int_{T/2}^{T}\int_{\mathbb{R}^{n}}|u|^{p}b_{q}\eta_{T}^{2p^{\prime}}dxdt\gtrsim\varepsilon^{p}T^{n-\frac{n-1}{2}p-q}=\varepsilon^{p}.$

where we used the fact

$n-\frac{n-1}{2}p=q=\frac{n-1}{2}-\frac{1}{p}$

for $p=p_{S}(n)$.

Equipped with (6.15) and (6.14), we could apply Lemma 3.10 in [11] to conclude
the last lifespan in (1.9). Actually, by (6.15), integration from $2$ to $T>4$
yields

(6.16) $Y(T)\geq Y(2)+c\varepsilon^{p}(\ln T-\ln 2)\gtrsim\varepsilon^{p}\ln
T,\ \forall T\in(4,T_{\varepsilon})\ .$

Similarly, for (6.14), integration from $T_{1}$ to $T_{2}>T_{1}$ gives us

$Y(T_{2})^{1-p}\leq Y(T_{1})^{1-p}-c(p-1)\int_{\ln T_{1}}^{\ln
T_{2}}\tau^{1-p}d\tau\ ,\forall 2<T_{1}<T_{2}<T_{\varepsilon}\ .$

As $Y(T)\geq 0$, letting $T_{2}\to T_{\varepsilon}$, and using (6.16), we see
that

$\int_{\ln T_{1}}^{\ln
T_{\varepsilon}}\tau^{1-p}d\tau{\lesssim}Y(T_{1})^{1-p}{\lesssim}\varepsilon^{-p(p-1)}(\ln
T_{1})^{1-p},\forall 4<T_{1}<T_{\varepsilon}\ .$

Setting $T_{1}=\sqrt{T}_{\varepsilon}$, it follows that

$\ln T_{\varepsilon}{\lesssim}\varepsilon^{-p(p-1)}\ ,$

which gives us the desired lifespan for the critical case, in (1.9).

ACKNOWLEDGMENTS

The first author is supported by NSF of Zhejiang Province(LY18A010008) and
NSFC 11771194. The second and fourth author were supported in part by NSFC
11971428.

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assumption $J_{32}=J_{13}=J_{42}=0$, formula (4.5) simplifies to

$c_{k}^{+}=-J_{14}\sum_{j=1}^{k-1}(-1)^{j}\sigma_{j}^{-}(0)\sigma_{k-j}^{+}(1),\qquad
k\in\\{1,\ldots,{n}\\}.$ (7.70)

Since $Q_{12}^{(j_{0})}(0)\neq 0$, without loss of generality we can assume
that $Q_{12}^{(j)}(0)=0$ for $j\in\\{0,\ldots,j_{0}-1\\}$. With account of
notation (4.26) we have

$q_{+}^{(j)}(0)=0,\quad j\in\\{0,\ldots,j_{0}-1\\},\qquad
q_{+}^{(j_{0})}(0)\neq 0.$ (7.71)

Combining Corollary 6.3(ii) with condition (7.71) now implies

$\sigma^{-}_{j}(0)=0,\quad
j\in\\{1,\ldots,{j_{0}}\\},\qquad\sigma^{-}_{j_{0}+1}(0)=(-1)^{j_{0}}q_{+}^{(j_{0})}(0)\neq
0.$ (7.72)

Similarly, we infer from (6.29) and condition $Q_{21}^{(j_{1})}(1)\neq 0$ that

$\sigma^{+}_{j}(1)=0,\quad
j\in\\{1,\ldots,{j_{1}}\\},\qquad\sigma^{+}_{j_{1}+1}(1)=(-1)^{j_{1}}q_{-}^{(j_{1})}(1)\neq
0.$ (7.73)

In turn, combining (7.72) with (7.73) yields

$\sigma_{j}^{-}(0)\sigma_{k^{+}-j}^{+}(1)=0,\quad
k^{+}:=j_{0}+j_{1}+2\leqslant n,\quad j\neq j_{0}+1,$ (7.74)

for $1\leqslant j\leqslant k^{+}-1$. Combining (7.70) with relations
(7.72)–(7.74) and condition $J_{14}\neq 0$, we arrive at

$c_{k^{+}}^{+}=-J_{14}(-1)^{j_{0}+1}\sigma_{j_{0}+1}^{-}(0)\sigma_{j_{1}+1}^{+}(1)\neq
0.$ (7.75)

Therefore, $c_{0}^{-}\neq 0$ and $c_{k^{+}}^{+}\neq 0$. Hence Theorem 5.1
implies the desired completeness property. ∎

######  Remark 7.14.

If $j_{0}=j_{1}=1$ in condition (7.69) $($i.e. $Q_{12}(0)Q_{21}(1)\neq 0)$,
then Proposition 2.6 guarantees the desired completeness property under much
more relaxed smoothness assumption of continuity of $Q(\cdot)$ at the
endpoints, as opposed to smoothness assumption $Q\in
W_{1}^{2}([0,1];\mathbb{C}^{2\times 2})$ imposed by Corollary 7.13. This shows
some advantage of the Birkhoff solution approach used in [17].

If $J_{32}=J_{13}=0$ and $J_{42}\neq 0$, then condition of completeness is
also very simple and explicit.

######  Corollary 7.15.

Let $Q_{12},Q_{21}\in W_{1}^{n}[0,1]$ for some $n\in\mathbb{N}$. Let the
following condition hold

$J_{32}=J_{13}=0,\qquad J_{42}\neq 0,\qquad Q_{21}^{(j_{1})}(1)\neq
0,\qquad|J_{14}|+|Q_{21}^{(j_{0})}(0)|\neq 0,$ (7.76)

for some $j_{0},j_{1}\in\\{0,1,\ldots,n-1\\}$. Then the system of root vectors
of the BVP (1.1)–(1.4) is complete and minimal in
$L^{2}([0,1];\mathbb{C}^{2})$.

###### Proof.

Since $J_{32}=J_{13}=0$, formulas (4.5)–(4.6) for coefficients $c_{k}^{\pm}$,
$k\in\\{0,1,\ldots,n\\}$, take the form

$\displaystyle c_{0}^{+}=0,\qquad
c_{k}^{+}:=J_{42}\sigma_{k}^{+}(1)-J_{14}\sum_{j=1}^{k-1}(-1)^{j}\sigma_{j}^{-}(0)\sigma_{k-j}^{+}(1),$
(7.77) $\displaystyle c_{0}^{-}=J_{14},\qquad
c_{k}^{-}:=J_{42}\sigma_{k}^{+}(0).$ (7.78)

Since $Q_{21}^{(j_{1})}(1)\neq 0$, then just as in the proof of Corollary 7.13
we infer relation (7.73). In turn, combining (7.77) with (7.73) and condition
$J_{42}\neq 0$ implies

$c_{j_{1}+1}^{+}=J_{42}\sigma_{j_{1}+1}^{+}(1)-J_{14}\sum_{j=1}^{j_{1}}(-1)^{j}\sigma_{j}^{-}(0)\cdot\underset{\text{$=0$}}{\underbrace{\sigma_{j_{1}+1-j}^{+}(1)}}=J_{42}(-1)^{j_{1}}q_{-}^{(j_{1})}(1)\neq
0.$ (7.79)

First, let $J_{14}\neq 0$. Then combining (7.79) with Theorem 5.1 implies the
desired completeness property.

Now, let $J_{14}=0$. Then condition (7.76) implies that
$Q_{21}^{(j_{0})}(0)\neq 0$. Similarly to how relation (7.72) was established
in the proof of Corollary 7.13 we derive that

$\sigma^{+}_{j}(0)=0,\quad
j\in\\{1,\ldots,{j_{0}}\\},\qquad\sigma^{+}_{j_{0}+1}(0)=(-1)^{j_{0}}q_{-}^{(j_{0})}(0)\neq
0.$ (7.80)

Since $J_{42}\neq 0$, then combining (7.80) with (7.78) implies that
$c_{j_{0}+1}^{+}\neq 0$. Theorem 5.1 completes the proof. ∎

For reader convenience we also formulate the previous results in a similar
case $J_{14}=J_{42}=0$.

######  Corollary 7.16.

Let $Q_{12},Q_{21}\in W_{1}^{n}[0,1]$ for some $n\in\mathbb{N}$ and let

$J_{14}=J_{42}=0,\qquad Q_{21}^{(j_{0})}(0)\neq 0,$ (7.81)

for some $j_{0}\in\\{0,1,\ldots,n-1\\}$. Let one of the following conditions
hold

$\displaystyle(i)\quad$ $\displaystyle J_{13}=0,\quad J_{32}\neq 0,\quad
Q_{12}^{(j_{1})}(1)\neq 0\quad\text{for some}\quad 0\leqslant j_{1}\leqslant
n-2-j_{0},$ (7.82) $\displaystyle(ii)\quad$ $\displaystyle J_{13}\neq
0,\quad|J_{32}|+|Q_{21}^{(j_{1})}(1)|\neq 0,\quad\text{for some}\quad
j_{1}\in\\{0,1,\ldots,n-1\\}.$ (7.83)

Then the system of root vectors of the BVP (1.1)–(1.4) is complete and minimal
in $L^{2}([0,1];\mathbb{C}^{2})$.

In the case when potential matrix function $Q(\cdot)$ is analytic, Corollary
7.15 yields the following criterion of completeness of the system of root
vectors for BVP (1.1)–(1.4) for a certain class of boundary conditions. Note
that if $b_{1}+b_{2}\neq 0$ then any degenerate boundary conditions (either
$\Delta_{0}(\cdot)\equiv 0$ or $\Delta_{0}(\cdot)$ has no zeros) can be
transformed to a form (7.84) using transformations $y_{1}\leftrightarrow
y_{2}$ and $x\rightarrow 1-x$.

######  Corollary 7.17.

Let $Q(\cdot)$ be an analytic matrix function on $[0,1]$, and let boundary
conditions (1.4) be of the form

$\alpha_{1}y_{1}(0)+\alpha_{2}y_{2}(0)+\alpha_{3}y_{1}(1)=0,\quad
y_{2}(1)=0,\quad\alpha_{2}\neq 0.$ (7.84)

Then the system of root vectors of the BVP (1.1), (7.84) is complete in
$L^{2}([0,1];\mathbb{C}^{2})$ if and only if $Q_{21}(\cdot)\not\equiv 0$.

###### Proof.

It follows from the form of boundary conditions (7.84) that $J_{32}=J_{13}=0$
and $J_{42}=-\alpha_{2}\neq 0$.

Let $Q_{21}(\cdot)\not\equiv 0$. Since $Q_{21}(\cdot)$ is analytic on $[0,1]$,
then for some $j_{0},j_{1}\in\mathbb{Z}_{+}$ we have

$Q_{21}^{(j_{1})}(1)\neq 0\quad\text{and}\quad Q_{21}^{(j_{0})}(0)\neq 0,$
(7.85)

which trivially implies condition (7.76). Hence Corollary 7.15 implies that
the system of root vectors of the BVP (1.1), (7.84) is complete in
$L^{2}([0,1];\mathbb{C}^{2})$.

If $Q_{21}(\cdot)\equiv 0$ then for any solution
$y=\operatorname{col}(y_{1},y_{2})$ of the system (1.1), (7.84) we have

$-ib_{2}^{-1}y_{2}^{\prime}(x)=\lambda y_{2}(x),\quad x\in[0,1],\qquad
y_{2}(1)=0.$ (7.86)

Clearly (7.86) is an initial value problem that has the only solution
$y_{2}(\cdot)\equiv 0$. Thus, any root vector of the system (1.1), (7.84) is
of the form $y_{n}=\operatorname{col}(y_{n,1},0)$ and hence the system of root
vectors of the BVP (1.1), (7.84) is incomplete in
$L^{2}([0,1];\mathbb{C}^{2})$ and of infinite co-dimension. The proof is now
complete. ∎

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