# Higher order evolution inequalities with nonlinear convolution terms

Roberta Filippucci  Dipartimento di Matematica e Informatica, Universitá degli
Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy  Email:
<EMAIL_ADDRESS>Marius Ghergu  School of Mathematics and
Statistics, University College Dublin, Belfield, Dublin 4, Ireland  Institute
of Mathematics Simion Stoilow of the Romanian Academy, 21 Calea Grivitei St.,
010702 Bucharest, Romania  Corresponding author, Email<EMAIL_ADDRESS>
###### Abstract

We are concerned with the study of existence and nonexistence of weak
solutions to

$\begin{cases}&\displaystyle\frac{\partial^{k}u}{\partial
t^{k}}+(-\Delta)^{m}u\geq(K\ast|u|^{p})|u|^{q}\quad\mbox{ in
}\mathbb{R}^{N}\times\mathbb{R}_{+},\\\\[7.22743pt]
&\displaystyle\frac{\partial^{i}u}{\partial t^{i}}(x,0)=u_{i}(x)\,\,\text{ in
}\mathbb{R}^{N},\,0\leq i\leq k-1,\\\ \end{cases}$

where $N,k,m\geq 1$ are positive integers, $p,q>0$ and $u_{i}\in L^{1}_{\rm
loc}(\mathbb{R}^{N})$ for $0\leq i\leq k-1$. We assume that $K$ is a radial
positive and continuous function which decreases in a neighbourhood of
infinity. In the above problem, $K\ast|u|^{p}$ denotes the standard
convolution operation between $K(|x|)$ and $|u|^{p}$. We obtain necessary
conditions on $N,m,k,p$ and $q$ such that the above problem has solutions. Our
analysis emphasizes the role played by the sign of
$\displaystyle\frac{\partial^{k-1}u}{\partial t^{k-1}}$.

Keywords: Higher-order evolution inequalities; nonlinear convolution terms;
nonlinear capacity estimates.

2020 AMS MSC: 35G20, 35K30, 35L30, 35B45

##  1 Introduction and the main results

Let $N,k,m\geq 1$ be positive integers. In this work we are concerned with the
problem

$\begin{cases}&\displaystyle\frac{\partial^{k}u}{\partial
t^{k}}+(-\Delta)^{m}u\geq(K\ast|u|^{p})|u|^{q}\quad\mbox{ in
}\mathbb{R}^{N+1}_{+}:=\mathbb{R}^{N}\times\mathbb{R}_{+},\\\\[7.22743pt]
&\displaystyle\frac{\partial^{i}u}{\partial t^{i}}(x,0)=u_{i}(x)\,\,\text{ in
}\mathbb{R}^{N},\,0\leq i\leq k-1,\\\ \end{cases}$ (1)

where $\mathbb{R}_{+}=(0,\infty)$, $N\geq 1$, $p,q>0$ and $u_{i}\in L^{1}_{\rm
loc}(\mathbb{R}^{N})$ for $0\leq i\leq k-1$. We assume that $K\in
C(\mathbb{R}_{+};\mathbb{R}_{+})$ satisfies:

$K(|x|)\in L^{1}_{\rm loc}(\mathbb{R}^{N})$

and

$(A)\;\;$ there exists $R_{0}>1$ such that
$\displaystyle\inf_{r\in(0,R)}K(r)=K(R)$ for all $R>R_{0}$.

In particular, condition $(A)$ above implies that $K$ is decreasing on the
interval $(R_{0},\infty)$. Typical examples of potentials $K$ satisfying the
above conditions are the constant functions as well as

$K(r)=r^{-\alpha},\;\alpha\in(0,N)\quad\mbox{ or }\quad
K(r)=r^{-\alpha}\log^{\beta}(1+r),\;\alpha\in(0,N),\beta\in\mathbb{R},\beta>\alpha-N.$

The picture below illustrates some other functions $K(r)$ satisfying the
hypothesis $(A)$.

By $K\ast|u|^{p}$ we denote the standard convolution operator defined by

$(K\ast|u|^{p})(x,t)=\int\limits_{\mathbb{R}^{N}}K(|x-y|)|u(y,t)|^{p}dy\quad\mbox{
for all }(x,t)\in\mathbb{R}^{N+1}_{+}$ (2)

and we shall require the above integral to be finite for almost all
$(x,t)\in\mathbb{R}^{N+1}_{+}$.

We are interested in weak solutions of (1), that is, functions
$u\in{L}^{p}_{\rm loc}(\mathbb{R}^{N+1}_{+})$ such that

  1. (i)

$(K\ast|u|^{p})|u|^{q}\in{L}^{1}_{\rm loc}(\mathbb{R}^{N+1}_{+})$;

  2. (ii)

for any nonnegative test function $\varphi\in
C^{\infty}_{c}(\mathbb{R}^{N+1}_{+})$, we have

$\sum_{i=1}^{k}\\!(-1)^{i}\\!\\!\int\limits_{\mathbb{R}^{N}}u_{k-i}(x)\frac{\partial^{i-1}\varphi}{\partial
t^{i-1}}(x,0)dx+\int\limits_{{\mathbb{R}^{N+1}_{+}}}u\Big{[}(-1)^{k}\frac{\partial^{k}\varphi}{\partial
t^{k}}+(-\Delta)^{m}\varphi\Big{]}\,dxdt\\\
\geq\int\limits_{{\mathbb{R}^{N+1}_{+}}}(K\ast|u|^{p})|u|^{q}\varphi\,dxdt.$
(3)

Using a standard integration by parts it is easily seen that any classical
solution of (1) is also a weak solution. Let us point out that condition (i)
above implies that the convolution integral in (2) is finite for a.a.
$(x,t)\in\mathbb{R}^{N+1}_{+}$. This further yields $u\in L^{p}_{\rm
loc}(\mathbb{R}^{N+1}_{+})$. Indeed for $R>R_{0}$ and $x,y\in B_{R}$ we have
$|x-y|\leq 2R$, so that by the definition of $K$ and its monotonicity we
deduce

$\infty>(K*|u|^{p})(x,t)\geq K(2R)\int\limits_{B_{R}}|u(y,t)|^{p}dy.$ (4)

Since early 1980s many research works have been devoted to the study of the
prototype evolution inequalities

$\frac{\partial u}{\partial t}-\Delta u\geq|u|^{q}\quad\mbox{ and
}\quad\frac{\partial^{2}u}{\partial t^{2}}-\Delta u\geq|u|^{q}\quad\mbox{ in
}\mathbb{R}^{N+1}_{+}.$

To the best of our knowledge, the first study of nonexistence of solutions to
higher order hyperbolic inequalities is due to L. Véron and S.I. Pohozaev [21]
related to

$\frac{\partial^{2}u}{\partial
t^{2}}-\mathcal{L}_{m}\big{(}\phi(u)\big{)}\geq|u|^{q}\quad\mbox{ in
}\mathbb{R}^{N+1}_{+},$ (5)

where

$\mathcal{L}_{m}v=\sum_{|\alpha|=m}D^{\alpha}(a_{\alpha}(x,t)v)\quad\mbox{ for
some integer }m\geq 1$

and $\phi$ is a locally bounded real-valued function such that $|\phi(u)|\leq
c|u|^{p}$, for some $c,p>0$. It is obtained in [21] that if $q>\max\\{p,1\\}$
and one of the following conditions hold

$\mbox{ either }\quad m\geq 2N\quad\mbox{ or }\quad
m<2N\leq\frac{m(q+1)}{q-p},$ (6)

then (5) has no solutions in $\mathbb{R}^{N+1}_{+}$ satisfying

$\int_{\mathbb{R}^{N}}\frac{\partial u}{\partial t}(x,0)dx\geq 0.$ (7)

Condition (7) is essential in the study of nonexistence of solutions to (5)
and will also play an important role in the study of (1) (see Theorem 1.1
below). Further, if $m=2$, it is obtained in [21] that if (6) fails to hold
then (5) has a positive solution. Among the results for $m=2$, we quote also
[10], where a weighted nonlinearity is considered.

Another set of results that motivate our present work are due to G.G. Laptev
[16, 17] where the following problem is studied

$\begin{cases}&\displaystyle\frac{\partial^{k}u}{\partial t^{k}}-\Delta
u\geq|x|^{-\sigma}|u|^{q},u\geq 0\quad\mbox{ in
}\Omega\times\mathbb{R}_{+},\\\\[7.22743pt]
&\displaystyle\frac{\partial^{k-1}u}{\partial t^{k-1}}(x,0)\geq 0\,\,\text{ in
}\Omega.\end{cases}$ (8)

In the above, $\Omega\subset\mathbb{R}^{N}$ is either the exterior of a ball
or an unbounded cone-like domain; for other results on hyperbolic inequalities
in exterior domains see [14, 15, 18]. We observe that solutions of (8) are
also required to satisfy (7). It is obtained in [17] that if $\sigma>-2$ then
the above problem has no solutions provided that

$1<q<q_{k}^{*}:=1+\frac{2+\sigma}{N-2+2/k}.$

The above exponent $q_{k}^{*}$ coincides with the Fujita-Hayakawa critical
exponent if $k=1$, that is, $q_{1}^{*}=1+\frac{2+\sigma}{N}$ and with the Kato
critical exponent if $k=2$, that is, $q_{2}^{*}=1+\frac{2+\sigma}{N-1}$.

The study of convolution terms in time-dependent partial differential
equations goes back to near a century ago. Indeed, the equation

$i\psi_{t}-\Delta\psi=(|x|^{\alpha-N}\ast\psi^{2})\psi\quad\mbox{in
}{\mathbb{R}}^{N},\alpha\in(0,N),N\geq 1,$ (9)

was introduced by D.H. Hartree [11, 12, 13] in 1928 for $N=3$ and $\alpha=2$
in relation to the Schrödinger equation in quantum physics. Nowadays, (9)
bears the name of Choquard equation. Stationary solutions to (9) and its
related equations have been extensively investigated from various
perspectives: ground states, isolated singularities, symmetry of solutions;
see e.g., [3, 6, 7, 8, 9, 19, 20]. In a different direction, G. Whithman [22,
Section 6] considered the nonlocal equation

$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial
x}-\int_{-\infty}^{\infty}K(x-y)\frac{\partial u}{\partial x}(y,t)dy=0$

to study general dispersion in water waves; for recent results on nonlocal
evolution equations we refer the reader to [1, 2]. In [4] it is studied a
class of quasilinear parabolic inequalities featuring nonlocal convolution
terms in the form

$\displaystyle\frac{\partial u}{\partial t}-\mathcal{L}u\geq(K\ast
u^{p})u^{q}\quad\mbox{ in }\mathbb{R}^{N+1}_{+},$

where $\mathcal{L}u$ is a quasilinear operators that contains as prototype
model the $m$-Laplacian and the generalized mean curvature operator.

In the present work we investigate the existence and nonexistence of weak
solutions to (1). The stationary solutions to (1) were discussed recently in
[6]. Our main result concerning the inequality (1) reads as follows.

######  Theorem 1.1.

Assume $N,m,k\geq 1$ and $p,q>0$.

  1. (i)

If $k\geq 1$ is a even integer and $q\geq 1$, then (1) admits some positive
solutions $u\in C^{\infty}(\mathbb{R}^{N+1}_{+})$ which verify

$u_{k-1}=\frac{\partial^{k-1}u}{\partial t^{k-1}}(\cdot,0)<0\quad\mbox{ in
}\mathbb{R}^{N}.$

  2. (ii)

If $p+q>2$ and

$\limsup_{R\to\infty}K(R)R^{\frac{2N+2m/k}{p+q}-N+2m(1-1/k)}>0,$ (10)

then (1) has no nontrivial solutions such that

$u_{k-1}\geq 0\quad\mbox{ or }\quad u_{k-1}\in
L^{1}(\mathbb{R}^{N})\quad\mbox{ and
}\;\int\limits_{\mathbb{R}^{N}}u_{k-1}(x)dx>0.$ (11)

Let us note that condition (10) and the fact that $K$ is decreasing in a
neighbourhood of infinity imply that

$\frac{2N+2m/k}{p+q}\geq N-2m\Big{(}1-\frac{1}{k}\Big{)}.$

Also, under extra assumptions on $u_{k-1}$ and strengthening (10), we can
handle the case $\int_{\mathbb{R}^{N}}u_{k-1}(x)=0$ in (11) for which the same
conclusion as in Theorem 1.1(ii) holds (see Proposition 2.3).

The proof of Theorem 1.1 relies on nonlinear capacity estimates specifically
adapted to the nonlocal setting of our problem. More precisely, we derive
integral estimates in time for the new quantity

$J(t)=\int_{\mathbb{R}^{N}}u^{\frac{p+q}{2}}(x,t)\varphi(x,t)dx\,,\,\,\,t\geq
0,$

where $\varphi$ is a specially constructed test function with compact support
(see (26)).

Theorem 1.1 shows a sharp contrast in the existence/nonexistence diagram
according to whether $\frac{\partial^{k-1}u}{\partial t^{k-1}}(x,0)$ has
constant sign (positive or negative) on $\mathbb{R}^{N}$. To better illustrate
this fact, let us discuss the case of pure powers in the potential
$K(r)=r^{-\alpha}$, $\alpha\in(0,N)$ and $k=1,2$.

Let us first consider the parabolic problem

$\begin{cases}&\frac{\partial u}{\partial
t}+(-\Delta)^{m}u\geq(|x|^{-\alpha}\ast|u|^{p})|u|^{q}\quad\mbox{ in
}\mathbb{R}^{N+1}_{+}:=\mathbb{R}^{N}\times\mathbb{R}_{+},\\\
&u(x,0)=u_{0}(x)\,\,\text{ in }\mathbb{R}^{N}.\end{cases}$ (12)

Note that since $k=1$ in (1), condition (11) is satisfied for all nonnegative
solutions of (12).

######  Corollary 1.2.

Let $N,m\geq 1$, $p,q>0$ and $\alpha\in(0,N)$.

  1. (i)

If $0<\alpha<m$ and

$2<p+q\leq\frac{2N+2m}{N+\alpha},$

then (1) has no nontrivial nonnegative solutions;

  2. (ii)

If $N>2m$ and

$\min\\{p,q\\}>\frac{N-\alpha}{N-2m}\quad\mbox{ and
}p+q>\frac{2N-\alpha}{N-2m},$

then (12) has positive solutions.

Part (i) in the above result follows from Theorem 1.1 while part (ii) follows
from [6, Theorem 1.4] where the stationary case of (12) was discussed. Let us
note that for $m=1$, the nonexistence of a nonnegative solution in Corollary
1.2(i) was already observed in [4].

We next take $K(r)=r^{-\alpha}$, $k=2$ and $m=1$ in Theorem 1.1. We thus
consider

$\begin{cases}&\displaystyle\frac{\partial^{2}u}{\partial t^{2}}-\Delta
u\geq(|x|^{\alpha}\ast|u|^{p})|u|^{q}\quad\mbox{ in
}\mathbb{R}^{N+1}_{+}:=\mathbb{R}^{N}\times\mathbb{R}_{+},\\\\[7.22743pt]
&u(x,0)=u_{0}(x)\,\,\text{ in }\mathbb{R}^{N},\\\ &\displaystyle\frac{\partial
u}{\partial t}(x,0)=u_{1}(x)\,\,\text{ in }\mathbb{R}^{N}.\end{cases}$ (13)

Our result on problem (13) is stated below.

######  Corollary 1.3.

Let $N,m\geq 1$, $p,q>0$ and $\alpha\in(0,N)$.

  1. (i)

If $q\geq 1$, then (1) admits some positive solutions $u\in
C^{2}(\mathbb{R}^{N+1}_{+})$ which verify

$\frac{\partial u}{\partial t}(\cdot,0)<0\quad\mbox{ in }\mathbb{R}^{N}.$

  2. (ii)

If $0<\alpha<\min\\{N,3/2\\}$ and

$2<p+q\leq\frac{2N+1}{N+\alpha-1},$

then (13) has no nontrivial solutions satisfying

$\frac{\partial u}{\partial t}(x,\cdot)\geq 0\mbox{ in
}\mathbb{R}^{N}\quad\mbox{ or }\quad\int\limits_{\mathbb{R}^{N}}\frac{\partial
u}{\partial t}(x,\cdot)dx>0.$ (14)

  3. (iii)

If $N>2$ and

$\min\\{p,q\\}>\frac{N-\alpha}{N-2}\quad\mbox{ and }\quad
p+q>\frac{2N-\alpha}{N-2},$ (15)

then (13) admits some positive solutions which verify
$\displaystyle\frac{\partial u}{\partial t}(x,\cdot)>0$ in $\mathbb{R}^{N}$.

The diagram of existence/nonexistence of a weak solution to (13) satisfying
(14) in the $pq$-plane is given below. The light shaded region represents the
region for which (13) admits solutions satisfying (14) while the dark shaded
region describes the pairs $(p,q)$ for which no such solutions exist.
Corollary 1.3 leaves open the isue of existence and nonexistence in the white
regions of the $(p>0,q>0)$ quadrant.

Theorem 1.1 also applies to the case where $K\equiv 1$ for which (1) reads

$\begin{cases}&\displaystyle\frac{\partial^{k}u}{\partial
t^{k}}-\Delta^{m}u\geq\Big{(}\int_{\mathbb{R}^{N}}|u(y)|^{p}dy\Big{)}|u|^{q}\quad\mbox{
in }\mathbb{R}^{N+1}_{+},\\\\[7.22743pt]
&\displaystyle\frac{\partial^{i}u}{\partial t^{i}}(x,0)=u_{i}(x)\,\,\text{ in
}\mathbb{R}^{N},\,0\leq i\leq k-1.\\\ \end{cases}$ (16)

A similar conclusion to Corollary 1.2 and Corollary 1.3 (in which we let
$\alpha=0$) hold for (16).

We can further employ the ideas in the study of (1) to the case of systems of
type

$\begin{cases}&\displaystyle\frac{\partial^{k}u}{\partial
t^{k}}+(-\Delta)^{m}u\geq(K\ast|v|^{p})|v|^{q}\quad\mbox{ in
}\mathbb{R}^{N+1}_{+}=\mathbb{R}^{N}\times\mathbb{(}0,\infty),\\\\[7.22743pt]
&\displaystyle\frac{\partial^{k}v}{\partial
t^{k}}+(-\Delta)^{m}v\geq(L\ast|u|^{n})|u|^{s}\quad\mbox{ in
}\mathbb{R}^{N+1}_{+},\\\\[7.22743pt]
&\displaystyle\frac{\partial^{i}u}{\partial t^{i}}(x,0)=u_{i}(x)\,\,\text{ in
}\mathbb{R}^{N},\,0\leq i\leq k-1,\\\\[7.22743pt]
&\displaystyle\frac{\partial^{i}v}{\partial t^{i}}(x,0)=v_{i}(x)\,\,\text{ in
}\mathbb{R}^{N},\,0\leq i\leq k-1,\\\ \end{cases}$ (17)

where $N,m,k\geq 1$, $p,q,n,s>0$. We assume $u_{i},v_{i}\in L^{1}_{\rm
loc}(\mathbb{R}^{N})$, $0\leq i\leq k-1$ and that $K,L$ satisfy condition
$(A)$ for some $R_{0}>1$.

A pair $(u,v)\in{L}^{n}_{\rm loc}(\mathbb{R}^{N+1}_{+})\times{L}^{p}_{\rm
loc}(\mathbb{R}^{N+1}_{+})$ is a weak solution of (17) if:

  1. (i)

$(L\ast u^{n})u^{s},(K\ast v^{p})v^{q}\in{L}^{1}_{\rm
loc}(\mathbb{R}^{N+1}_{+})$;

  2. (ii)

for any nonnegative test function $\varphi\in
C^{\infty}_{c}(\mathbb{R}^{N}\times\mathbb{R}),$ we have

$\left\\{\begin{aligned}
\sum_{i=1}^{k}\\!(-1)^{i}\\!\\!\int\limits_{\mathbb{R}^{N}}u_{k-i}(x)\frac{\partial^{i-1}\varphi}{\partial
t^{i-1}}(x,0)dx+\int\limits_{{\mathbb{R}^{N+1}_{+}}}u\Big{[}(-1)^{k}\frac{\partial^{k}\varphi}{\partial
t^{k}}&+(-\Delta)^{m}\varphi\Big{]}\,dxdt\\\
&\geq\int\limits_{{\mathbb{R}^{N+1}_{+}}}(K\ast|v|^{p})|v|^{q}\varphi\,dxdt,\\\
\sum_{i=1}^{k}\\!(-1)^{i}\\!\\!\int\limits_{\mathbb{R}^{N}}v_{k-i}(x)\frac{\partial^{i-1}\varphi}{\partial
t^{i-1}}(x,0)dx+\int\limits_{{\mathbb{R}^{N+1}_{+}}}v\Big{[}(-1)^{k}\frac{\partial^{k}\varphi}{\partial
t^{k}}&+(-\Delta)^{m}\varphi\Big{]}\,dxdt\\\
&\geq\int\limits_{{\mathbb{R}^{N+1}_{+}}}(L\ast|u|^{n})|u|^{s}\varphi\,dxdt.\end{aligned}\right.$
(18)

Our main result concerning (18) is stated below.

######  Theorem 1.4.

Assume $\min\\{p+q,n+s\\}>2$. If either

$\limsup_{R\to\infty}K(R)L(R)^{\frac{1}{n+s}}R^{\frac{2N+2m/k}{(n+s)(p+q)}+\frac{N+2m}{n+s}-N+2m\big{(}1-\frac{1}{k}\big{)}}>0,$
(19)

or

$\limsup_{R\to\infty}K(R)^{\frac{1}{p+q}}L(R)R^{\frac{2N+2m/k}{(n+s)(p+q)}+\frac{N+2m}{p+q}-N+2m\big{(}1-\frac{1}{k}\big{)}}>0,$
(20)

then (18) has no nontrivial solutions such that

$u_{k-1},v_{k-1}\geq 0\quad\mbox{ or }\quad u_{k-1},v_{k-1}\in
L^{1}(\mathbb{R}^{N})\quad\mbox{ and
}\;\int_{\mathbb{R}^{N}}u_{k-1}(x)dx,\,\int_{\mathbb{R}^{N}}v_{k-1}(x)dx>0.$
(21)

The next sections contain the proofs of the above results.

##  2 Proof of Theorem 1.1

Proof of Theorem 1 (i) Let $\gamma>0$ be such that $p\gamma>N$ and define

$u(x,t)=e^{-Mt}v(x)^{-\gamma/2}\quad\mbox{ with }\quad v(x)=1+|x|^{2},$ (22)

where $M>1$ will be specified later. Since $K>0$ is continuous in
$\mathbb{R}^{+}$ and decreasing in a neighbourhood of infinity (by condition
$(A)$) it follows that

$\sup_{[1,\infty)}K=\max_{[1,\infty)}K<\infty.$

Furthermore, we have

$\displaystyle\big{(}K\ast v^{-p\gamma/2}\big{)}(x)$
$\displaystyle=\int_{\mathbb{R}^{N}}K(|z|)(1+|z-x|^{2})^{-p\gamma/2}dz$ (23)
$\displaystyle\leq\int_{B_{1}}K(|z|)dz+\big{(}\max_{[1,\infty)}K\big{)}\int_{\mathbb{R}^{N}\setminus
B_{1}}(1+|z-x|^{2})^{-p\gamma/2}dz$
$\displaystyle\leq\int_{B_{1}}K(|z|)dz+\big{(}\max_{[1,\infty)}K\big{)}\int_{\mathbb{R}^{N}}(1+|z-x|^{2})^{-p\gamma/2}dz$
$\displaystyle=\int_{B_{1}}K(|z|)dz+\big{(}\max_{[1,\infty)}K\big{)}\int_{\mathbb{R}^{N}}(1+|y|^{2})^{-p\gamma/2}dy$
$\displaystyle\leq C(K,p,\gamma),$

since $p\gamma>N$.

Further, a direct calculation shows that
$-\Delta\big{(}v^{-\gamma/2}\big{)}=c_{1}v^{-\gamma/2-1}+c_{2}v^{-\gamma/2-2}$
in $\mathbb{R}^{N}$, where $c_{1},c_{2}$ are real coefficients depending on
$N$ and $\gamma$. Hence, an induction argument yields

$(-\Delta)^{m}\bigl{(}v^{-\gamma/2}\bigr{)}=v^{-2m-\gamma/2}\sum_{j=0}^{m}c_{j}v^{j}\quad\mbox{
in }\mathbb{R}^{N},$

where $c_{j}=c_{j}(\gamma,N,m)\in\mathbb{R}$. Thus, the function $u$ given by
(22) satisfies

$\frac{\partial^{k}u}{\partial
t^{k}}+(-\Delta)^{m}u=e^{-Mt}\Big{(}(-1)^{k}M^{k}v^{2m}+\sum_{j=0}^{m}c_{j}v^{j}\Big{)}v^{-2m-\gamma/2}\quad\mbox{
in }\mathbb{R}^{N+1}_{+},$ (24)

where $c_{j}\in\mathbb{R}$ are independent of $M$. Using the fact that $k$ is
an even integer, by taking $M>1$ large, we may ensure that

$M^{k}v^{2m}+\sum_{j=0}^{m}c_{j}v^{j}\geq C(K,p,\gamma)v^{2m}\quad\mbox{ in
}\mathbb{R}^{N},$ (25)

where $C(K,p,\gamma)>0$ is the constant from (23). Now, combining (23)-(25)
and using that $u^{p}e^{Mpt}=v^{-p\gamma/2}$, we deduce

$\displaystyle\frac{\partial^{k}u}{\partial t^{k}}-\Delta^{m}u$
$\displaystyle\geq C(K,p,\gamma)e^{-Mt}v^{-\gamma/2}$ $\displaystyle\geq
C(K,p,\gamma)e^{-(p+q)Mt}v^{-\gamma/2}$ $\displaystyle\geq\big{(}K\ast
u^{p}\big{)}e^{-qMt}v^{-\gamma/2}$ $\displaystyle\geq\big{(}K\ast
u^{p}\big{)}\Big{(}e^{-Mt}v^{-\gamma/2})^{q}$ $\displaystyle\geq\big{(}K\ast
u^{p}\big{)}u^{q}\quad\mbox{ in }\mathbb{R}^{N+1}_{+},$

being $q\geq 1$, which concludes the proof of part (i). Since $k$ is even, one
has

$u_{k-1}(x)=\frac{\partial^{k-1}u}{\partial
t^{k-1}}(x,0)=(-1)^{k-1}M^{k}v(x)<0.$

(ii) Assume that $p+q>2$, (10) and (11) hold and that (1) admits a weak
solution $u$. We shall first construct a suitable test function $\varphi$ for
(3). To do so, take a standard cut off function $\varrho\in
C^{\infty}_{c}(\mathbb{R})$ such that:

  * •

$\varrho=1$ in $(0,1)$, $\varrho=0$ in $(2,\infty)$;

  * •

$0\leq\varrho\leq 1$, $\text{supp}\varrho\subseteq[0,2]$.

Now take $R>0$, $\gamma>0$ to be precised later, $\kappa\geq 1$ sufficiently
large and consider the function

$\varphi(x)=\varrho^{\kappa}\biggl{(}\frac{|x|}{R}\biggr{)}\varrho^{\kappa}\biggl{(}\frac{t}{R^{\gamma}}\biggr{)}\quad\text{in}\quad\mathbb{R}^{N+1}_{+}.$
(26)

Clearly

$\text{supp\,}\varphi\subset
B_{2R}\times[0,2R^{\gamma})\subset\mathbb{R}^{N+1}_{+},$ (27)

and

$\text{supp\,}\frac{\partial^{k}\varphi}{\partial t^{k}}\subset
B_{2R}\times[R^{\gamma},2R^{\gamma})\subset\text{supp\,}\varphi,\qquad\text{supp\,}\Delta\varphi\subset(B_{2R}\setminus
B_{R})\times[0,2R^{\gamma})\subset\text{supp\,}\varphi.$ (28)

As in [5, Lemma 3.1], we have the following estimates:

######  Proposition 2.1.

Let $\varphi$ be defined in (26). Then for $\varsigma>1$ and $\kappa\geq
2m\varsigma$ one has

$\int_{\text{\rm
supp}\,\varphi}\frac{1}{\varphi^{\varsigma-1}}\Big{|}\frac{\partial^{i}\varphi}{\partial
t^{i}}\Big{|}^{\varsigma}dxdt\leq CR^{N+\gamma-i\gamma\varsigma},\quad i\geq
1,$ (29) $\int_{\text{\rm
supp}\,\varphi}\frac{|\Delta^{m}\varphi|^{\varsigma}}{\varphi^{\varsigma-1}}dxdt\leq
CR^{N+\gamma-2m\varsigma},$ (30)

where $C$ is a positive constant changing from line to line.

Since $\frac{\partial^{i}\varphi}{\partial t^{i}}(x,0)=0$ in $\mathbb{R}^{N}$
for all $i\geq 1$, from (3) we deduce

$\displaystyle\int_{\mathbb{R}^{N}}u_{k-1}(x)\varphi(x,0)$ $\displaystyle
dx+\int_{\mathbb{R}^{N+1}_{+}}(K\ast|u|^{p})|u|^{q}\varphi\,dxdt$ (31)
$\displaystyle\leq\int_{\mathbb{R}^{N+1}_{+}}|u|\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}\,dxdt+\int_{\mathbb{R}^{N+1}_{+}}|u||\Delta^{m}\varphi|\,dxdt.$

Observe that by (11) we have $u_{k-1}\geq 0$ or $u_{k-1}\in
L^{1}(\mathbb{R}^{N})$ and $\int_{\mathbb{R}^{N}}u_{k-1}(x)dx>0$. In the
latter case, from $u_{k-1}\in L^{1}_{loc}(\mathbb{R}^{N})$, we deduce, using
(26),

$\displaystyle\lim_{R\to\infty}\int_{\mathbb{R}^{N}}u_{k-1}(x)\varphi(x,0)dx$
$\displaystyle=\lim_{R\to\infty}\int_{\mathbb{R}^{N}}u_{k-1}(x)\varrho^{\kappa}\biggl{(}\frac{|x|}{R}\biggr{)}dx=\int_{\mathbb{R}^{N}}u_{k-1}(x)dx.$
(32)

Thus, from (11) we deduce that for large $R>0$ we have

$\int_{\mathbb{R}^{N}}u_{k-1}(x)\varphi(x,0)dx\geq 0,$ (33)

case in which (31) yields

$\int_{\mathbb{R}^{N+1}_{+}}(K\ast|u|^{p})|u|^{q}\varphi\,dxdt\leq\int_{\mathbb{R}^{N+1}_{+}}|u|\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}\,dxdt+\int_{\mathbb{R}^{N+1}_{+}}|u||\Delta^{m}\varphi|\,dxdt,$
(34)

provided $R>0$ in the definition of $\varphi$ (see (26)) is large enough.

An important tool of our approach is the following result.

######  Lemma 2.2.

For almost all $t\geq 0$ we have $u(\cdot,t)\in
L^{\frac{p+q}{2}}_{loc}(\mathbb{R}^{N})$ and

$\int_{\mathbb{R}^{N}}(K\ast|u|^{p})|u(x,t)|^{q}\varphi(x,t)dx\geq
K(4R)J^{2}(t)\quad\mbox{ for all }\;R\geq R_{0},$ (35)

where

$J(t)=\int_{\mathbb{R}^{N}}|u(x,t)|^{\frac{p+q}{2}}\varphi(x,t)dx.$ (36)

###### Proof.

First note that for $x,y\in B_{2R}$ one has $|x-y|\leq 4R$, so that, thanks to
the monotonicity of $K$,

$\int_{\mathbb{R}^{N}}K(|x-y|)|u(y)|^{p}dy\geq\int_{B_{2R}}K(|x-y|)|u(y)|^{p}dy\geq
K(4R)\int_{B_{2R}}|u(y)|^{p}dy.$

Hence

$\int_{\mathbb{R}^{N}}(K\ast|u|^{p})|u(x,t)|^{q}\varphi(x,t)dx\geq
K(4R)\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}|u(y,t)|^{p}\varphi(y,t)|u(x,t)|^{q}\varphi(x,t)dxdy,$
(37)

where we have used that $\varphi\leq 1$ and that $\varphi(\cdot,t)\equiv 0$
outside of $B_{2R}$ for all $t$.

Furthermore, by Hölder’s inequality we have

$\displaystyle\biggl{(}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}|u(y,t)|^{p}\varphi(y,t)$
$\displaystyle|u(x,t)|^{q}\varphi(x,t)\,dx\,dy\biggr{)}^{2}$ $\displaystyle=$
$\displaystyle\biggl{(}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}|u(y,t)|^{p}\varphi(y,t)|u(x,t)|^{q}\varphi(x,t)\,dx\,dy\biggr{)}$
$\displaystyle\cdot\biggl{(}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}|u(x,t)|^{p}\varphi(x,t)|u(y,t)|^{q}\varphi(y,t)\,dx\,dy\biggr{)}$
$\displaystyle\geq$
$\displaystyle\biggl{(}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}|u(x,t)|^{\frac{p+q}{2}}|u(y,t)|^{\frac{p+q}{2}}\varphi(x,t)\varphi(y,t)\,dx\,dy\biggr{)}^{2}$
$\displaystyle=$
$\displaystyle\biggl{(}\int_{\mathbb{R}^{N}}|u(x,t)|^{\frac{p+q}{2}}\varphi(x,t)\,dx\biggr{)}^{4}=J(t)^{4},$

which, by (37) and part (i) in the definition of a solution, yields
$u(\cdot,t)\in L^{\frac{p+q}{2}}_{loc}(\mathbb{R}^{N})$ and (35). ∎

Inserting (35) into (34) we find

$K(4R)\int_{0}^{2R^{\gamma}}J^{2}(t)dt\leq\int_{\mathbb{R}^{N+1}_{+}}|u|\left(\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}+|\Delta^{m}\varphi|\right)\,dxdt.$ (38)

We next estimate the integral term on the right hand side of (38). Using
Hölder’s inequality, we have

$\displaystyle\int_{\mathbb{R}^{N+1}_{+}}|u|\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}dxdt$
$\displaystyle\leq\biggl{(}\int_{\text{supp}(\frac{\partial^{k}\varphi}{\partial
t^{k}})}|u|^{\frac{p+q}{2}}\varphi\,dxdt\biggr{)}^{\frac{2}{p+q}}\cdot\biggl{(}\int_{\text{supp}(\frac{\partial^{k}\varphi}{\partial
t^{k}})}\varphi^{-\frac{2}{p+q-2}}\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}^{\frac{p+q}{p+q-2}}\biggr{)}^{\frac{p+q-2}{p+q}}$ (39)
$\displaystyle\leq
C\biggl{(}\int_{\text{supp}(\frac{\partial^{k}\varphi}{\partial
t^{k}})}|u|^{\frac{p+q}{2}}\varphi\,dxdt\biggr{)}^{\frac{2}{p+q}}R^{-k\gamma+(N+\gamma)\frac{p+q-2}{p+q}},$

where in the last inequality we have used (29) with
$\varsigma=\frac{p+q}{p+q-2}$, $i=k$ and thanks to (28). Note that

$\text{supp}(\frac{\partial^{k}\varphi}{\partial t^{k}})\subset
B_{2R}\times[R^{\gamma},2R^{\gamma}].$ (40)

By Hölder’s inequality we find

$\displaystyle\int_{\mathbb{R}^{N+1}_{+}}|u|\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}dxdt$ $\displaystyle\leq
CR^{-k\gamma+(N+\gamma)\frac{p+q-2}{p+q}}\biggl{(}\int_{R^{\gamma}}^{2R^{\gamma}}\Big{(}\int_{\mathbb{R}^{N}}|u|^{\frac{p+q}{2}}\varphi\,dx\Big{)}\,dt\biggr{)}^{\frac{2}{p+q}}$
(41) $\displaystyle\leq
CR^{-k\gamma+(N+\gamma)\frac{p+q-2}{p+q}+\frac{\gamma}{p+q}}\biggl{(}\int_{R^{\gamma}}^{2R^{\gamma}}\Big{(}\int_{\mathbb{R}^{N}}|u|^{\frac{p+q}{2}}\varphi\,dx\Big{)}^{2}\,dt\biggr{)}^{\frac{1}{p+q}}.$

Similarly, by Hölder’s inequality, (30) with $\varsigma=\frac{p+q}{p+q-2}$ and
(28), we find

$\displaystyle\int_{\mathbb{R}^{N+1}_{+}}|u||\Delta^{m}\varphi|dxdt$
$\displaystyle\leq\biggl{(}\int_{\text{supp}(\Delta^{m}\varphi)}|u|^{\frac{p+q}{2}}\varphi\,dxdt\biggr{)}^{\frac{2}{p+q}}\cdot\biggl{(}\int_{\text{supp}(\Delta^{m}\varphi)}\varphi^{-\frac{2}{p+q-2}}|\Delta^{m}\varphi|^{\frac{p+q}{p+q-2}}\biggr{)}^{\frac{p+q-2}{p+q}}$
(42) $\displaystyle\leq
C\biggl{(}\int_{\text{supp}(\Delta^{m}\varphi)}|u|^{\frac{p+q}{2}}\varphi\,dxdt\biggr{)}^{\frac{2}{p+q}}R^{-2m+(N+\gamma)\frac{p+q-2}{p+q}}.$

Since

$\text{supp}(\Delta^{m}\varphi)\subset(B_{2R}\setminus
B_{R})\times[0,2R^{\gamma}],$ (43)

a new application of Hölder’s inequality in the above estimate yields

$\displaystyle\int_{\mathbb{R}^{N+1}_{+}}|u||\Delta^{m}\varphi|dxdt$
$\displaystyle\leq
CR^{-2m+(N+\gamma)\frac{p+q-2}{p+q}}\biggl{(}\int_{0}^{2R^{\gamma}}\Big{(}\int_{B_{2R}\setminus
B_{R}}|u|^{\frac{p+q}{2}}\varphi\,dx\Big{)}dt\biggr{)}^{\frac{2}{p+q}}$ (44)
$\displaystyle\leq
CR^{-2m+(N+\gamma)\frac{p+q-2}{p+q}+\frac{\gamma}{p+q}}\biggl{(}\int_{0}^{2R^{\gamma}}\Big{(}\int_{B_{2R}\setminus
B_{R}}|u|^{\frac{p+q}{2}}\varphi\,dx\Big{)}^{2}dt\biggr{)}^{\frac{1}{p+q}}.$

Comparing the powers of $R$ in (41) and (44) we are led to choose $\gamma>0$
so that $k\gamma=2m$, that is $\gamma=2m/k$. Also,

$-2m+(N+\gamma)\frac{p+q-2}{p+q}+\frac{\gamma}{p+q}=N-2m\Big{(}1-\frac{1}{k}\Big{)}-\frac{2N+\frac{2m}{k}}{p+q}.$

Thus, (38) together with (41) and (44) yield

$\displaystyle K(4R)$ $\displaystyle\int_{0}^{2R^{\gamma}}J^{2}(t)dt\leq
CR^{N-2m\big{(}1-\frac{1}{k}\big{)}-\frac{2N+2m/k}{p+q}}\times$ (45)
$\displaystyle\times\left[\biggl{(}\int_{R^{\gamma}}^{2R^{\gamma}}\Big{(}\int_{\mathbb{R}^{N}}|u|^{\frac{p+q}{2}}\varphi\,dx\Big{)}^{2}\,dt\biggr{)}^{\frac{1}{p+q}}+\biggl{(}\int_{0}^{2R^{\gamma}}\Big{(}\int_{B_{2R}\setminus
B_{R}}|u|^{\frac{p+q}{2}}\varphi\,dx\Big{)}^{2}dt\biggr{)}^{\frac{1}{p+q}}\right].$

Using (36), the above estimate implies

$K(4R)\int_{0}^{2R^{\gamma}}\\!J^{2}(t)dt\leq
CR^{N-2m\big{(}1-\frac{1}{k}\big{)}-\frac{2N+2m/k}{p+q}}\Big{(}\int_{0}^{2R^{\gamma}}J^{2}(t)dt\Big{)}^{\frac{1}{p+q}}$
(46)

which further yields

$\biggl{(}\int_{0}^{2R^{\gamma}}J^{2}(t)dt\biggr{)}^{\frac{p+q-1}{p+q}}\leq
C\frac{1}{K(4R)R^{\frac{2N+2m/k}{p+q}-N+2m(1-1/k)}}.$ (47)

Let $\\{R_{j}\\}_{j}$ be a divergent sequence that achieves the limsup in
(10), namely

$K(4R_{j})R_{j}^{\frac{2N+2m/k}{p+q}-N+2m(1-1/k)}\to\ell>0\quad\text{as}\quad
j\to\infty.$

Passing to a subsequence we may assume $R_{j}>2R_{j-1}$ for all $j>1$.

If $\ell=\infty$, we can pass to the limit in (47), by replacing $R$ with
$R_{j}$, to raise $\int_{0}^{\infty}J^{2}(t)dt=0$ from where $J\equiv 0$ in
$\mathbb{R}^{+}$ and then, by the definition of $J$,

$\int_{B_{R}\times(0,\infty)}|u|^{(p+q)/2}dxdt=0\quad\text{for all}\quad R>0,$

namely $u\equiv 0$ in $\mathbb{R}^{N+1}_{+}$ as required.

If $\ell\in(0,\infty)$, then (47) shows that $J\in L^{2}(0,\infty)$. Using
this fact we infer that

$\int_{R_{j}^{m}}^{2R_{j}^{m}}\\!\\!\Big{(}\int_{\mathbb{R}^{N}}|u(x,t)|^{\frac{p+q}{2}}\varphi(x,t)dx\Big{)}^{2}dt\to
0\,,\;\int_{0}^{2R_{j}^{m}}\\!\\!\Big{(}\int_{B_{2R_{j}}\setminus
B_{R_{j}}}|u(x,t)|^{\frac{p+q}{2}}\varphi(x,t)dx\Big{)}^{2}dt\to 0,$ (48)

as $j\to\infty$. Indeed, the first limit in (48) follows immediately by the
fact that $J\in L^{2}(0,\infty)$. To check the second limit in (48) we observe
that

$\displaystyle\infty>\int_{0}^{\infty}\Big{(}\int_{\mathbb{R}^{N}}|u(x,t)|^{\frac{p+q}{2}}dx\Big{)}^{2}dt$
$\displaystyle\geq\int_{0}^{\infty}\Big{(}\sum_{j\geq
1}\int_{B_{2R_{j}}\setminus B_{R_{j}}}|u(x,t)|^{\frac{p+q}{2}}dx\Big{)}^{2}dt$
$\displaystyle\geq\int_{0}^{\infty}\sum_{j\geq
1}\Big{(}\int_{B_{2R_{j}}\setminus
B_{R_{j}}}|u(x,t)|^{\frac{p+q}{2}}dx\Big{)}^{2}dt$ $\displaystyle=\sum_{j\geq
1}\int_{0}^{\infty}\Big{(}\int_{B_{2R_{j}}\setminus
B_{R_{j}}}|u(x,t)|^{\frac{p+q}{2}}dx\Big{)}^{2}dt.$

The convergence of the last series in the above estimate implies the second
part of (48). Using this fact in (45) we find

$\biggl{(}\int_{0}^{2R_{j}^{\gamma}}J^{2}(t)dt\biggr{)}^{\frac{p+q-2}{p+q}}\leq
C\frac{1}{K(4R_{j})R_{j}^{\frac{2N+2m/k}{p+q}-N+2m(1-1/k)}}\,o(1)\quad\mbox{
as }j\to\infty.$

Again, by letting $R_{j}\to\infty$, we can conclude, also when
$\ell\in(0,\infty)$, that $\int_{0}^{\infty}J^{2}(t)dt=0$, namely $u\equiv 0$
in $\mathbb{R}^{N+1}_{+}$.∎

A similar argument allows us to treat the case
$\int_{\mathbb{R}^{N}}u_{k-1}(x)dx=0$. In this case we need to be more precise
on the behavior of $u_{k-1}$.

######  Proposition 2.3.

Let $\varrho\in C^{\infty}_{c}(\mathbb{R})$ satisfy $\text{\rm supp
}\varrho\subseteq[0,2]$, $0\leq\varrho\leq 1$ and $\varrho=1$ in $(0,1)$.

Assume that for some $\kappa\geq 2m$ we have

$\int_{\mathbb{R}^{N}}u_{k-1}(x)\varrho^{\kappa}\left(\frac{x}{R}\right)dx=O(R^{-\beta})\quad\mbox{
as }R\to\infty,\quad\mbox{ for some }\beta>0.$ (49)

If

$\limsup_{R\to\infty}K(R)R^{\min\\{\beta,\frac{2N+2m/k}{p+q}-N+2m(1-1/k)\\}}>0,$
(50)

then (1) has no nontrivial solutions.

###### Proof.

With the help of the above $\varrho$ we construct the test function $\varphi$
as defined in (26). Thus, (33) is no more in force and (31) changes to

$\int_{\mathbb{R}^{N+1}_{+}}(K\ast|u|^{p})|u|^{q}\varphi\,dxdt\leq
CR^{-\beta}+\int_{\mathbb{R}^{N+1}_{+}}|u|\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}\,dxdt+\int_{\mathbb{R}^{N+1}_{+}}|u||\Delta^{m}\varphi|\,dxdt.$

Consequently using (41) and (44), with $\gamma=2k/m$, and (35), then the above
inequality gives

$K(4R)\int_{0}^{2R^{\gamma}}\\!J^{2}(t)dt\leq
CR^{-\beta}+CR^{N-2m\big{(}1-\frac{1}{k}\big{)}-\frac{2N+2m/k}{p+q}}\Big{(}\int_{0}^{2R^{\gamma}}J^{2}(t)dt\Big{)}^{\frac{\color[rgb]{0,0,1}1}{p+q}}.$
(51)

Let $\sigma=\frac{2N+2m/k}{p+q}-N+2m(1-1/k)$. By Young’s inequality we have

$\displaystyle
CR^{N-2m\big{(}1-\frac{1}{k}\big{)}-\frac{2N+2m/k}{p+q}}\Big{(}\int_{0}^{2R^{\gamma}}J^{2}(t)dt\Big{)}^{\frac{\color[rgb]{0,0,1}1}{p+q}}$
$\displaystyle=CR^{-\sigma}\Big{(}\int_{0}^{2R^{\gamma}}J^{2}(t)dt\Big{)}^{\frac{\color[rgb]{0,0,1}1}{p+q}}$
$\displaystyle=CR^{-\sigma}K(4R)^{-\frac{\color[rgb]{0,0,1}1}{p+q}}\Big{(}K(4R)\int_{0}^{2R^{\gamma}}J^{2}(t)dt\Big{)}^{\frac{\color[rgb]{0,0,1}1}{p+q}}$
$\displaystyle\leq\frac{K(4R)}{2}\int_{0}^{2R^{\gamma}}J^{2}(t)dt+CR^{-\frac{\sigma(p+q)}{p+q-\color[rgb]{0,0,1}1}}K(4R)^{-\frac{\color[rgb]{0,0,1}1}{p+q-\color[rgb]{0,0,1}1}}.$

Using this last inequality into (51) we find

$\frac{K(4R)}{2}\int_{0}^{2R^{\gamma}}J^{2}(t)dt\leq
CR^{-\beta}+CR^{-\frac{\sigma(p+q)}{p+q-\color[rgb]{0,0,1}1}}K(4R)^{-\frac{\color[rgb]{0,0,1}1}{p+q-\color[rgb]{0,0,1}1}},$

that is,

$\int_{0}^{2R^{\gamma}}J^{2}(t)dt\leq\frac{C}{R^{\beta}K(4R)}+\frac{C}{\Big{(}R^{\sigma}K(4R)\Big{)}^{\frac{p+q}{p+q-\color[rgb]{0,0,1}1}}}.$

Now, in virtue of (50) we may let $R\to\infty$ in the above estimate to deduce
$J=0$ and then $u\equiv 0$. ∎

##  3 Proof of Corollary 1.3

Part (i) and (ii) in Corollary 1.3 follow directly from Theorem 1.1.

(iii) Let $p,q,\alpha$ satisfy (15) which we may write as

$(N-2)\min\\{p,q\\}>N-\alpha\quad\mbox{ and }\quad(N-2)(p+q-1)>N+2-\alpha.$

Thus, we may choose $\beta\in(0,N-2)$ such that

$\beta\min\\{p,q\\}>N-\alpha\,,\quad\beta p\neq N\quad\mbox{ and
}\quad\beta(p+q-1)>N+2-\alpha$ (52)

Set $w(x,t)=(1+t)^{-a}+1+|x|^{2}$ where $a>0$ will be precised later and
$u(x,t)=w^{-\beta/2}(x,y)$. Then

$\displaystyle-\Delta u$ $\displaystyle=\beta
Nw^{-\beta/2-1}-\beta(\beta+2)w^{-\beta/2-2}|x|^{2}$ (53)
$\displaystyle=\beta\Big{[}N-(\beta+2)\frac{|x|^{2}}{w}\Big{]}w^{-\beta/2-1}\geq\beta(N-\beta-2)w^{-(\beta+2)/2}\quad\mbox{
in }\mathbb{R}^{N+1}_{+}.$

Also,

$\displaystyle\frac{\partial^{2}u}{\partial t^{2}}$
$\displaystyle=-\frac{a(a+1)\beta}{2}(1+t)^{-a-2}w^{-(\beta+2)/2}+\frac{a^{2}\beta(\beta+2)}{4}(1+t)^{-2a-2}w^{-(\beta+2)/2-1}$
(54)
$\displaystyle=\frac{a\beta}{2}w^{-(\beta+2)/2}(1+t)^{-a-2}\biggl{[}-a-1+\frac{a}{2}(\beta+2)\frac{(1+t)^{a}}{w}\biggr{]}$
$\displaystyle\geq-\frac{a(a+1)\beta}{2}w^{-(\beta+2)/2}\quad\mbox{ in
}\mathbb{R}^{N+1}_{+}.$

Combining now (53) and (54) we have

$\frac{\partial^{2}u}{\partial t^{2}}-\Delta
u\geq\beta\Big{(}N-2-\beta-\frac{a(a+1)}{2}\Big{)}w^{-(\beta+2)/2}\quad\mbox{
in }\mathbb{R}^{N+1}_{+}.$

Thus, by letting $a>0$ small enough, the above estimate yields

$\frac{\partial^{2}u}{\partial t^{2}}-\Delta u\geq
Cw^{-(\beta+2)/2}\quad\mbox{ in }\mathbb{R}^{N+1}_{+},$ (55)

for some constant $C>0$. To proceed further, we need the following result.

######  Lemma 3.1.

Let $\alpha\in(0,N)$, $\sigma>N-\alpha$ and $1\leq\lambda\leq 2$. Then, there
exists a constant $C=C(N,\alpha,\sigma)>0$ (note that $C$ is independent of
$\lambda$) such that

$\int_{\mathbb{R}^{N}}\frac{dy}{|x-y|^{\alpha}(\lambda+|y|^{2})^{\sigma/2}}\leq
C\left\\{\begin{aligned} &(\lambda+|x|^{2})^{(N-\alpha-\sigma)/2}&&\quad\mbox{
if }\sigma<N,\\\ &(\lambda+|x|^{2})^{-\alpha/2}&&\quad\mbox{ if
}\sigma>N,\end{aligned}\right.\qquad\mbox{ for all }|x|\geq 1.$

For $\lambda=0$, similar estimates are available in [19, Lemma A.1] as well as
in [6, 7, 9].

###### Proof.

Let $|x|\geq 1$. We split the integral into

$\int_{\mathbb{R}^{N}}\frac{dy}{|x-y|^{\alpha}(\lambda+|y|^{2})^{\sigma/2}}=\left\\{\,\int\limits_{|y|\geq
2|x|}+\int\limits_{\frac{1}{2}|x|\leq|y|\leq
2|x|}+\int\limits_{|y|\leq|x|/2}\right\\}\frac{dy}{|x-y|^{\alpha}(\lambda+|y|^{2})^{\sigma/2}}.$

If $|y|\geq 2|x|$, then $|x-y|\geq|y|-|x|\geq|y|/2$ and we have

$\displaystyle\int\limits_{|y|\geq
2|x|}\frac{dy}{|x-y|^{\alpha}(\lambda+|y|^{2})^{\sigma/2}}$ $\displaystyle\leq
C\int\limits_{|y|\geq 2|x|}\frac{dy}{|y|^{\alpha+\sigma}}\leq
C|x|^{N-\alpha-\sigma}$ $\displaystyle\leq
C\Big{(}\frac{\lambda+|x|}{3}\Big{)}^{N-\alpha-\sigma}$ $\displaystyle\leq
C\big{(}\lambda+|x|\big{)}^{N-\alpha-\sigma}$ $\displaystyle\leq
C(\lambda+|x|^{2})^{(N-\alpha-\sigma)/2},$

where we have used that

$|x|\geq\frac{\lambda+|x|}{3}\quad\text{if}\quad|x|\geq
1\quad\text{and}\quad\lambda\leq 2.$ (56)

Next we have

$\displaystyle\int\limits_{\frac{1}{2}|x|\leq|y|\leq
2|x|}\frac{dy}{|x-y|^{\alpha}(\lambda+|y|^{2})^{\sigma/2}}$ $\displaystyle\leq
C(\lambda+|x|^{2})^{-\sigma/2}\int\limits_{\frac{1}{2}|x|\leq|y|\leq
2|x|}\frac{dy}{|x-y|^{\alpha}}$ $\displaystyle\leq
C(\lambda+|x|^{2})^{-\sigma/2}\int\limits_{|y-x|\leq
3|x|}\frac{dy}{|x-y|^{\alpha}}$ $\displaystyle\leq
C(\lambda+|x|^{2})^{-\sigma/2}|x|^{N-\alpha}$ $\displaystyle\leq
C(\lambda+|x|^{2})^{(N-\alpha-\sigma)/2},$

where we have used that $|y|\leq 2|x|$ implies $|y-x|\leq 3|x|$.

Finally, if $|y|\leq|x|/2$ then $|x-y|\geq|x|-|y|\geq|x|/2$. We have

$\int\limits_{|y|\leq|x|/2}\frac{dy}{|x-y|^{\alpha}(\lambda+|y|^{2})^{\sigma/2}}\leq
C|x|^{-\alpha}\int\limits_{|y|\leq|x|/2}\frac{dy}{(\lambda+|y|^{2})^{\sigma/2}}.$
(57)

If $\sigma<N$ then

$\int\limits_{|y|\leq|x|/2}\frac{dy}{(\lambda+|y|^{2})^{\sigma/2}}\leq\int\limits_{|y|\leq|x|/2}\frac{dy}{|y|^{\sigma}}=C|x|^{N-\sigma},$

so that by (56) and being $N-\alpha-\sigma<0$

$\int\limits_{|y|\leq|x|/2}\frac{dy}{|x-y|^{\alpha}(\lambda+|y|^{2})^{\sigma/2}}\leq
C|x|^{N-\alpha-\sigma}\leq
C\Big{(}\frac{\lambda+|x|}{3}\Big{)}^{N-\alpha-\sigma}\leq
C(\lambda+|x|^{2})^{(N-\alpha-\sigma)/2}.$

If $\sigma>N$ then

$\int\limits_{|y|\leq|x|/2}\frac{dy}{(\lambda+|y|^{2})^{\sigma/2}}\leq\int\limits_{\mathbb{R}^{N}}\frac{dy}{(1+|y|^{2})^{\sigma/2}}<C<\infty,$

and from (57) and (56) one has

$\int\limits_{|y|\leq|x|/2}\frac{dy}{|x-y|^{\alpha}(\lambda+|y|^{2})^{\sigma/2}}\leq
C|x|^{-\alpha}\leq C\Big{(}\frac{\lambda+|x|}{3}\Big{)}^{-\alpha}\leq
C(\lambda+|x|^{2})^{-\alpha/2},$

which concludes our proof. ∎

Let us return to the proof of Corollary 1.3 and observe that

$\big{(}|x|^{-\alpha}*u^{p}\big{)}u^{q}\leq(1+|x|^{2})^{-\beta
q/2}\int\limits_{\mathbb{R}^{N}}\frac{dy}{|x-y|^{\alpha}(1+|y|^{2})^{\beta
p/2}}\quad\mbox{ for all }(x,t)\in\mathbb{R}^{N+1}_{+},$

since $w(x,t)\geq 1+|x|^{2}$. Since from (52) we have $\beta p>0$ and
$\alpha<N$, the above integral is finite and thus
$\big{(}|x|^{-\alpha}*u^{p}\big{)}u^{q}$ is uniformly bounded from above for
$(x,t)\in B_{1}\times[0,\infty)$. In addition, since $1\leq w\leq 3$ in
$(x,t)\in B_{1}\times[0,\infty)$, then by (55) $\frac{\partial^{2}u}{\partial
t^{2}}-\Delta u$ and $\big{(}|x|^{-\alpha}*u^{p}\big{)}u^{q}$ are positive,
continuous and uniformly bounded functions from below and from above
respectively, on $(x,t)\in B_{1}\times[0,\infty)$. We may thus find $C_{1}>0$
such that

$\frac{\partial^{2}u}{\partial t^{2}}-\Delta u\geq
C_{1}\big{(}|x|^{-\alpha}*u^{p}\big{)}u^{q}\quad\mbox{ in
}B_{1}\times[0,\infty).$ (58)

On the other hand, by Lemma 3.1 for $\lambda=1+(1+t)^{-a}$ and $\sigma=\beta
p$, where $\sigma>N-\alpha$ by (52), we have

$\big{(}|x|^{-\alpha}*u^{p}\big{)}u^{q}\leq C\left\\{\begin{aligned}
&w^{\frac{N-\alpha-\beta(p+q)}{2}}&&\quad\mbox{ if }\sigma=\beta p<N,\\\
&w^{-\frac{\beta q+\alpha}{2}}&&\quad\mbox{ if }\sigma=\beta
p>N,\end{aligned}\right.\qquad\mbox{ for all }(x,t)\in(\mathbb{R}\setminus
B_{1})\times[0,\infty).$

Using the above estimate together with (52) and (55) we find

$\frac{\partial^{2}u}{\partial t^{2}}-\Delta u\geq Cw^{-(\beta+2)/2}\geq
Cw^{\max\\{\frac{N-\alpha-\beta(p+q)}{2},-\frac{\beta q+\alpha}{2}\\}}\geq
C_{2}\big{(}|x|^{-\alpha}*u^{p}\big{)}u^{q},$ (59)

for all $(x,t)\in(\mathbb{R}\setminus B_{1})\times[0,\infty)$. Letting now
$M=\big{(}\max\\{C_{1},C_{2}\\}\big{)}^{1/(p+q)-1}$, it follows from (58) and
(59) that $U=Mu$ is a $C^{\infty}(\mathbb{R}^{N+1}_{+})$ solution of (13) such
that

$\frac{\partial U}{\partial t}(x,0)=\frac{a\beta
M}{2}\big{(}2+|x|^{2}\big{)}^{-(\beta+2)/2}>0.$

##  4 Proof of Theorem 1.4

Let $(u,v)$ be a weak solution of (17) which satisfies (21). Let $\varphi$ be
the test function given by (26) with $\gamma=2m/k$. Arguing as in (31) we have

$\left\\{\begin{aligned}
\int_{\mathbb{R}^{N}}u_{k-1}(x)\varphi(x,0)+\int_{\mathbb{R}^{N+1}_{+}}(K\ast|v|^{p})|v|^{q}\varphi\,dxdt&\leq\int_{\mathbb{R}^{N+1}_{+}}u\left(\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}+|\Delta^{m}\varphi|\right)dxdt,\\\
\int_{\mathbb{R}^{N}}v_{k-1}(x)\varphi(x,0)+\int_{\mathbb{R}^{N+1}_{+}}(L\ast|u|^{n})|u|^{s}\varphi\,dxdt&\leq\int_{\mathbb{R}^{N+1}_{+}}v\left(\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}dxdt+|\Delta^{m}\varphi|\right)dxdt.\\\ \end{aligned}\right.$
(60)

Setting

$\displaystyle I(t)$
$\displaystyle=\int_{\mathbb{R}^{N}}|u(x,t)|^{\frac{n+s}{2}}\varphi(x,t)dx,$
$\displaystyle J(t)$
$\displaystyle=\int_{\mathbb{R}^{N}}|v(x,t)|^{\frac{p+q}{2}}\varphi(x,t)dx,$

we have supp $I\subset[0,2R^{\gamma})$ and supp $J\subset[0,2R^{\gamma})$.
With the same approach as in Lemma 2.2, for $R>R_{0}$ we have

$\left\\{\begin{aligned}
\int_{\mathbb{R}^{N}}(K\ast|v|^{p})|v(x,t)|^{q}\varphi(x,t)dx\geq
K(4R)J^{2}(t),\\\
\int_{\mathbb{R}^{N}}(L\ast|u|^{n})|u(x,t)|^{s}\varphi(x,t)dx\geq
L(4R)I^{2}(t),\end{aligned}\right.\qquad\mbox{ for all }\;t\geq 0.$ (61)

Thus, (60), (32) and (21) yield

$\left\\{\begin{aligned}
K(4R)\int_{0}^{2R^{\gamma}}J^{2}(t)\,dt&\leq\int_{\mathbb{R}^{N+1}_{+}}|u|\left(\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}+|\Delta^{m}\varphi|\right)dxdt,\\\
L(4R)\int_{0}^{2R^{\gamma}}I^{2}(t)\,dt&\leq\int_{\mathbb{R}^{N+1}_{+}}|v|\left(\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}+|\Delta^{m}\varphi|\right)dxdt,\\\ \end{aligned}\right.$ (62)

for $R>0$ large. We next employ the estimates (39) and (42) with
$\gamma=2m/k$. We find

$\displaystyle\int_{\mathbb{R}^{N+1}_{+}}$
$\displaystyle|u|\left(\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}+|\Delta^{m}\varphi|\right)dxdt$ $\displaystyle\leq
CR^{-2m+\big{(}N+\frac{2m}{k}\big{)}\frac{n+s-2}{n+s}}\left[\biggl{(}\int_{\text{supp}(\rho_{tt})}|u|^{\frac{n+s}{2}}\varphi\,dxdt\biggr{)}^{\frac{2}{n+s}}+\biggl{(}\int_{\text{supp}(\Delta^{m}\rho)}|u|^{\frac{n+s}{2}}\varphi\,dxdt\biggr{)}^{\frac{2}{n+s}}\right].$

Since supp$(\frac{\partial^{k}\varphi}{\partial
t^{k}})\subset[R^{\gamma},2R^{\gamma}]\times\mathbb{R}^{N}$ and
supp$(\Delta^{m}\varphi)\subset[0,2R^{\gamma}]\times(B_{2R}\setminus B_{R})$,
applying Hölder’s inequality we derive

$\displaystyle\int\limits_{\mathbb{R}^{N+1}_{+}}|u|$
$\displaystyle\left(\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}+|\Delta^{m}\varphi|\right)dxdt\leq
CR^{N-2m\big{(}1-\frac{1}{k}\big{)}-\frac{2N+2m/k}{n+s}}\times$ (63)
$\displaystyle\times\left[\biggl{(}\int_{R^{\gamma}}^{2R^{\gamma}}\Big{(}\int_{\mathbb{R}^{N}}|u|^{\frac{n+s}{2}}\varphi\,dx\Big{)}^{2}dt\biggr{)}^{\frac{1}{n+s}}+\biggl{(}\int_{0}^{2R^{\gamma}}\Big{(}\int_{B_{2R}\setminus
B_{R}}|u|^{\frac{n+s}{2}}\varphi\,dx\Big{)}^{2}dt\biggr{)}^{\frac{1}{n+s}}\right].$

In particular,

$\displaystyle\int_{\mathbb{R}^{N+1}_{+}}$
$\displaystyle|u|\left(\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}+|\Delta^{m}\varphi|\right)dxdt$ $\displaystyle\leq
CR^{N-2m\big{(}1-\frac{1}{k}\big{)}-\frac{2N+2m/k}{n+s}}\Big{(}\int_{0}^{2R^{\gamma}}I^{2}(t)\,dt\Big{)}^{\frac{1}{n+s}}.$
(64)

Similarly

$\displaystyle\int\limits_{\mathbb{R}^{N+1}_{+}}|v|$
$\displaystyle\left(\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}+|\Delta^{m}\varphi|\right)dxdt\leq
CR^{N-2m\big{(}1-\frac{1}{k}\big{)}-\frac{2N+2m/k}{p+q}}\times$ (65)
$\displaystyle\times\left[\biggl{(}\int_{R^{\gamma}}^{2R^{\gamma}}\Big{(}\int_{\mathbb{R}^{N}}|v|^{\frac{p+q}{2}}\varphi\,dx\Big{)}^{2}dt\biggr{)}^{\frac{1}{p+q}}+\biggl{(}\int_{0}^{2R^{m}}\Big{(}\int_{B_{2R}\setminus
B_{R}}|v|^{\frac{p+q}{2}}\varphi\,dx\Big{)}^{2}dt\biggr{)}^{\frac{1}{p+q}}\right],$

and in particular,

$\displaystyle\int_{\mathbb{R}^{N+1}_{+}}$
$\displaystyle|v|\left(\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}+|\Delta^{m}\varphi|\right)dxdt$ $\displaystyle\leq
CR^{N-2m\big{(}1-\frac{1}{k}\big{)}-\frac{2N+2m/k}{p+q}}\Big{(}\int_{0}^{2R^{\gamma}}J^{2}(t)\,dt\Big{)}^{\frac{1}{p+q}}.$
(66)

From (62), (64) and (66) we derive

$\left\\{\begin{aligned} K(4R)\int_{0}^{2R^{\gamma}}J^{2}(t)\,dt&\leq
CR^{N-2m\big{(}1-\frac{1}{k}\big{)}-\frac{2N+2m/k}{n+s}}\Big{(}\int_{0}^{2R^{\gamma}}I^{2}(t)\,dt\Big{)}^{\frac{1}{n+s}},\\\
L(4R)\int_{0}^{2R^{\gamma}}I^{2}(t)\,dt&\leq
CR^{N-2m\big{(}1-\frac{1}{k}\big{)}-\frac{2N+2m/k}{p+q}}\Big{(}\int_{0}^{2R^{\gamma}}J^{2}(t)\,dt\Big{)}^{\frac{1}{p+q}}.\\\
\end{aligned}\right.$ (67)

Assume now that (19) holds and let $\\{R_{j}\\}$ be an increasing sequence of
positive real numbers such that $R_{j}>2R_{j-1}$ for all $j\geq 2$ and

$K(R_{j})L(R_{j})^{\frac{1}{n+s}}R_{j}^{\frac{2N+2m/k}{(n+s)(p+q)}+\frac{N+2m}{n+s}-N+2m\big{(}1-\frac{1}{k}\big{)}}\longrightarrow\ell\in(0,\infty]\quad\mbox{
as }j\to\infty.$ (68)

Assume first $\ell=\infty$. Then, using the second estimate of (67) into the
first we find

$\Big{(}\int_{0}^{2R_{j}^{\gamma}}J^{2}(t)\,dt\Big{)}^{1-\frac{1}{(n+s)(p+q)}}\leq\frac{C}{K(R_{j})L(R_{j})^{\frac{1}{n+s}}R_{j}^{\frac{2N+2m/k}{(n+s)(p+q)}+\frac{N+2m}{n+s}-N+2m\big{(}1-\frac{1}{k}\big{)}}}\longrightarrow
0$ (69)

as $j\to\infty$. This implies that $J(t)\equiv 0$ so that $v\equiv 0$. From
the second estimate of (67) it follows now that $I(t)\equiv 0$, hence $u\equiv
0$.

Assume now $\ell\in(0,\infty)$. From (69) it follows that $J^{2}\in
L^{1}(0,\infty)$ and arguing as in (48), this further implies

$\int_{R_{j}^{\gamma}}^{2R_{j}^{\gamma}}\Big{(}\int_{\mathbb{R}^{N}}|v(x,t)|^{\frac{p+q}{2}}\varphi(x,t)dx\Big{)}^{2}dt\to
0\,,\;\int_{0}^{2R_{j}^{\gamma}}\Big{(}\int_{B_{2R_{j}}\setminus
B_{R_{j}}}|v(x,t)|^{\frac{p+q}{2}}\varphi(x,t)dx\Big{)}^{2}dt\to 0,$ (70)

as $j\to\infty$. Using (70) into (65) (with $R_{j}$ instead of $R$) it follows
that

$\int_{\mathbb{R}^{N+1}_{+}}|v|\left(\Big{|}\frac{\partial^{k}\varphi}{\partial
t^{k}}\Big{|}+|\Delta^{m}\varphi|\right)dxdt\leq
CR^{N-2m\big{(}1-\frac{1}{k}\big{)}-\frac{2N+2m/k}{p+q}}o(1)\quad\mbox{ as
}j\to\infty.$

Using this fact in (62) along with (64) we deduce

$\left\\{\begin{aligned} K(4R)\int_{0}^{2R^{\gamma}}J^{2}(t)\,dt&\leq
CR^{N-2m\big{(}1-\frac{1}{k}\big{)}-\frac{2N+2m/k}{n+s}}\Big{(}\int_{0}^{2R^{\gamma}}I^{2}(t)\,dt\Big{)}^{\frac{1}{n+s}}\\\
L(4R)\int_{0}^{2R^{\gamma}}I^{2}(t)\,dt&\leq
CR^{N-2m\big{(}1-\frac{1}{k}\big{)}-\frac{2N+2m/k}{p+q}}o(1)\\\
\end{aligned}\right.\quad\mbox{ as }j\to\infty.$ (71)

Using the second estimate of (71) into the first one obtains

$\Big{(}\int_{0}^{2R_{j}^{\gamma}}J^{2}(t)\,dt\Big{)}^{1-\frac{1}{(n+s)(p+q)}}\leq\frac{C}{K(R_{j})L(R_{j})^{\frac{2}{n+s}}R_{j}^{\frac{2N+2m/k}{(n+s)(p+q)}+\frac{N+2m}{n+s}-N+2m\big{(}1-\frac{1}{k}\big{)}}}\,o(1)\longrightarrow
0,$

as $j\to\infty$. As before we deduce again $u\equiv v\equiv 0$.

## Acknowledgments

We thank the Referees for the valuable comments which helped us to prepared
this improved version of our results.

RF is a member of the Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta
Matematica (INdAM) and she was partly supported by Fondo Ricerca di Base di
Ateneo Esercizio 2017–19 of the University of Perugia, Italy, titled Problemi
con non linearità dipendenti dal gradiente.

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